77un/ Edition
DIGITAL
SIGNAL
PROCESSING
Principles, Algorithms, m l Applications
J o h n G. Proakis
Dimitris G. M anolakis
Digital Signal
Processing
Principles, Algorithms, and Applications
Third E dition
John G. Proakis
Northeastern U niversity
Dimitris G. Manolakis
Boston C ollege
PRENTICE-HALL INTERNATIONAL, INC.
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© 1996 by Prentice-Hall, Inc.
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Printed in the United States of America
10
9
8
7
6
5
ISBN 0-13-3TM33fl-cl
Prentice-Hall International (U K ) Limited. L ondon
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Contents
PREFACE
xiii
1
1 INTRODUCTION
1.1
S ignals, S ystem s, and S ignal P ro cessin g 2
1.1.1
Basic Elements of a Digital Signal Processing System. 4
1.1.2 A dvantages of Digital over Analog Signal Processing, 5
1.2
C lassificatio n o f Signals 6
1.2.1
Multichannel and Multidimensional Signals. 7
1.2.2 Continuous-Time Versus Discrete-Time Signals. 8
1.2.3 Continuous-Valued Versus Discrete-Valued Signals. 10
1.2.4 Determ inistic Versus Random Signals, 11
1.3
T h e C o n c e p t o f F re q u e n c y in C o n tin u o u s -T im e an d
D isc re te -T im e S ignals 14
1.3.1
Continuous-Time Sinusoidal Signals, 14
1.3.2 Discrete-Time Sinusoidal Signals. 16
1.3.3 Harmonically Related Complex Exponentials, 19
1.4
A n a lo g -to -D ig ita l an d D ig ita l-to -A n a lo g C o n v e rs io n 21
1.4.1
Sampling of Analog Signals, 23
1.4.2 The Sampling Theorem , 29
1.4.3 Q uantization of Continuous-Am plitude Signals, 33
1.4.4 Quantization of Sinusoidal Signals, 36
1.4.5 Coding of Quantized Samples, 38
1.4.6 Digital-to-Analog Conversion, 38
1.4.7 Analysis of Digital Signals and Systems Versus Discrete-Time
Signals and Systems, 39
S u m m a ry a n d R e fe re n c e s
Problems
39
40
iii
iv
2
Contents
DISCRETE-TIME SIGNALS AND SYSTEMS
2.1
D isc rete-T im e S ignals 43
2.1.1 Some Elem entary Discrete-Time Signals, 45
2.1.2 Classification of Discrete-Time Signals, 47
2.1.3 Simple Manipulations of Discrete-Time Signals, 52
2.2
D isc re te -T im e S ystem s 56
2.2.1
Input-O utput Description of Systems, 56
2.2.2 Block Diagram Representation of Discrete-Time Systems, 59
2.2.3 Classification of Discrete-Time Systems, 62
2.2.4 Interconnection of Discrete-Tim e Systems, 70
2.3
A n alysis o f D isc re te -T im e L in e a r T im e -In v a ria n t S ystem s 72
2.3.1 Techniques for the Analysis of Linear Systems, 72
2.3.2 Resolution of a Discrete-Time Signal into Impulses, 74
2.3.3 Response of LTI Systems to A rbitrary Inputs: The Convolution
Sum, 75
2.3.4 Properties of Convolution and the Interconnection of LTI
Systems, 82
2.3.5 Causal Linear Tim e-Invariant Systems. 86
2.3.6 Stability of Linear Tim e-Invariant Systems, 87
2.3.7 Systems with Fim te-D uration and Infinite-Duration Impulse
Response. 90
2.4
D isc rete-T im e System s D e s c rib e d by D iffe re n c e E q u a tio n s 91
2.4.1
Recursive and Nonrecursive Discrete-Tim e Systems, 92
2.4.2 Linear Time-Invariant Systems Characterized by
Constant-Coefficient Difference Equations, 95
2.4.3 Solution of Linear Constant-Coefficient Difference Equations. 100
2.4.4 The Impulse Response of a Linear Tim e-Invariant Recursive
System, 108
2.5
Im p le m e n ta tio n o f D isc re te -T im e S ystem s 111
2.5.1
Structures for the Realization of Linear Tim e-Invariant
Systems, 111
2.5.2 Recursive and Nonrecursive Realizations of FIR Systems, 116
2.6
C o rre la tio n of D isc re te -T im e S ignals 118
2.6.1
Crosscorrelation and A utocorrelation Sequences, 120
2.6.2 Properties of the A utocorrelation and Crosscorrelation
Sequences, 122
2.6.3 Correlation of Periodic Sequences, 124
2.6.4 Com putation of Correlation Sequences, 130
2.6.5 Input-O utput Correlation Sequences, 131
2.7
S u m m ary a n d R e fe re n c e s
Problems
135
134
43
Contents
3
THE Z-TRANSFORM AND ITS APPLICATION TO THE ANALYSIS
OF LTI SYSTEMS
3.1
T h e r-T ra n sfo rm
151
3.1.1
The Direct ^-Transform. 152
3.1.2 The inverse : -Transform, 160
3.2
P ro p e rtie s o f th e ; -T ra n sfo rm
3.3
R a tio n a l c-T ran sfo rm s 172
3.3.1
Poles and Zeros, 172
3.3.2 Pole Location and Time-Domain Behavior for Causal Signals. 178
3.3.3 The System Function of a Linear Tim e-Invariant System. 181
3.4
In v e rs io n o f th e ^ -T ra n sfo rm 184
3.4.1
The Inverse ; -Transform by Contour Integration. 184
3.4.2 The Inverse ;-Transform by Power Series Expansion. 186
3.4.3 The Inverse c-Transform by Partial-Fraction Expansion. 188
3.4.4
Decomposition of Rational c-Transforms. 195
3.5
T h e O n e -sid e d ^ -T ra n sfo rm
197
3.5.1
Definition and Properties, 197
3.5.2
Solution of Difference Equations. 201
3.6
A n aly sis o f L in e a r T im e -In v a ria n t S ystem s in th e --D o m a in
3.6.1
Response of Systems with Rational System Functions. 203
3.6.2 Response of P ole-Z ero Systems with Nonzero Initial
Conditions. 204
3.6.3 Transient and Steady-State Responses, 206
3.6.4 Causality and Stability. 208
3.6.5 P ole-Z ero Cancellations. 210
3.6.6 M ultiple-Order Poles and Stability. 211
3.6.7 The Schur-C ohn Stability Test, 213
3.6.8 Stability of Second-Order Systems. 215
3.7
S u m m ary an d R e fe re n c e s
P ro b le m s
4
151
161
203
219
220
FREQUENCY ANALYSIS OF SIGNALS AND SYSTEMS
4.1
F re q u e n c y A n aly sis o f C o n tin u o u s-T im e Signals 230
4.1.1 The Fourier Series for Continuous-Time Periodic Signals. 232
4.1.2 Power Density Spectrum of Periodic Signals. 235
4.1.3 The Fourier Transform for Continuous-Time Aperiodic
Signals, 240
4.1.4 Energy Density Spectrum of Aperiodic Signals. 243
4.2
F re q u e n c y A n aly sis o f D isc re te -T im e Signals 247
4.2.1 The Fourier Series for Discrete-Time Periodic Signals, 247
230
Contents
V)
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
4.2.7
4.2.8
Power Density Spectrum of Periodic Signals. 250
The Fourier Transform of Discrete-Time Aperiodic Signals. 253
Convergence of the Fourier Transform. 256
Energy Density Spectrum of Aperiodic Signals, 260
Relationship of the Fourier Transform to the i-Transform , 264
The Cepstrum, 265
The Fourier Transform of Signals with Poles on the Unit
Circle, 267
4.2.9 The Sampling Theorem Revisited, 269
4.2.10 Frequency-Domain Classification of Signals: The Concept of
Bandwidth, 279
4.2.11 The Frequency Ranges of Some N atural Signals. 282
4.2.12 Physical and M athematical Dualities. 282
4.3
P ro p e rtie s of th e F o u rie r T ra n s fo rm fo r D isc re te -T im e
S ignals 286
4.3.1
Symmetry Properties of the Fourier Transform, 287
4.3.2 Fourier Transform Theorems and Properties, 294
4.4
F re q u e n c y -D o m a in C h a ra c te ristic s of L in e a r T im e -In v a ria n t
S ystem s 305
4.4.1
Response to Complex Exponential and Sinusoidal Signals: The
Frequency Response Function. 306
44.2
Steady-State and Transient Response to Sinusoidal Input
Signals. 314
4.4.3 Steady-State Response to Periodic Input Signals, 315
4.4.4 Response to Aperiodic Input Signals. 316
4.4.5 Relationships Between the System Function and the Frequency
Response Function. 319
4.4.6 Com putation of the Frequency Response Function. 321
4.4.7
Input-O utput Correlation Functions and Spectra, 325
4.4.8 Correlation Functions and Power Spectra for Random Input
Signals. 327
4.5
L in e a r T im e -In v a ria n t S ystem s as F re q u e n c y -S e le c tiv e
F ilters 330
Ideal Filter Characteristics, 331
4.5.1
4.5.2 Lowpass, Highpass, and Bandpass Filters, 333
4,5.3 Digital Resonators, 340
4.5.4
Notch Filters, 343
4.5.5
Comb Filters. 345
4.5.6 All-Pass Filters. 350
4.5.7
Digital Sinusoidal Oscillators, 352
4.6
In v e rse S y stem s an d D e c o n v o lu tio n 355
4.6.1
Invertibility of Linear Tim e-Invariant Systems, 356
4.6.2 Minimum-Phase. Maximum-Phase, and Mixed-Phase Systems. 359
4.6.3 System Identification and Deconvolution, 363
4.6.4 Hom om orphic Deconvolution. 365
vii
Contents
4.7
S u m m ary a n d R e fe re n c e s
P ro b le m s
5
368
THE DISCRETE FOURIER TRANSFORM: ITS PROPERTIES AND
APPLICATIONS
5.1
F re q u e n c y D o m a in Sam pling: T h e D isc re te F o u rie r
T ra n s fo rm 394
5.1.1
Frequency-Dom ain Sampling and Reconstruction of
Discrete-Time Signals. 394
5.1.2 The Discrete Fourier Transform (DFT). 399
5.1.3 The D FT as a Linear Transform ation. 403
5.1.4 Relationship of the DFT to O ther Transforms, 407
5.2
P ro p e rtie s o f th e D F T 409
5.2.1
Periodicity. Linearity, and Symmetry Properties, 410
5.2.2
Multiplication of Two DFTs and Circular Convolution. 415
5.2.3
Additional DFT Properties, 421
5.3
L in e a r F ilte rin g M e th o d s B ased on th e D F T
5.3.1
Use of the DFT in Linear Filtering. 426
5.3.2 Filtering of Long Data Sequences. 430
425
5.4
F re q u e n c y A n aly sis o f S ignals U sing th e D F T
433
5.5
S u m m ary an d R e fe re n c e s
P ro b le m s
6
367
394
440
440
EFFICIENT COMPUTATION OF THE DFT: FAST FOURIER
TRANSFORM ALGORITHMS
448
6.1
E fficien t C o m p u ta tio n of th e D F T : F F T A lg o rith m s 448
6.1.1
Direct Com putation of the DFT, 449
6.1.2 D ivide-and-Conquer Approach to Com putation of the DFT. 450
6.1.3 Radix-2 FFT Algorithms. 456
6.1.4 Radix-4 FFT Algorithms. 465
6.1.5
Split-Radix FFT Algorithms, 470
6.1.6 Im plem entation of FFT Algorithms. 473
6.2
A p p lic a tio n s o f F F T A lg o rith m s 475
6.2.1
Efficient Com putation of the D FT of Two Real Sequences. 475
6.2.2 Efficient Com putation of the D FT of a Z N -Point Real
Sequence, 476
6.2.3 Use of the FFT Algorithm in Linear Filtering and Correlation, 477
6.3
A L in e a r F ilte rin g A p p ro a c h to C o m p u ta tio n o f th e D F T
6.3.1 The Goertzel Algorithm, 480
6.3.2 The Chirp-z Transform Algorithm, 482
479
viii
Contents
6.4
Q u a n tiz a tio n E ffects in the C o m p u ta tio n o f th e D F T 486
6.4.1
Quantization Errors in the Direct Com putation of the DFT. 487
6.4.2 Quantization Errors in FFT Algorithms. 489
6.5
S u m m ary an d R e fe re n c e s
P ro b le m s
493
494
500
7 IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
7.1
S tru c tu res fo r th e R e a liz a tio n o f D isc re te -T im e S ystem s
7.2
S tru c tu res fo r F IR System s 502
7.2.1
Direcl-Form Structure, 503
7.2.2
Cascade-Form Structures. 504
7.2.3
Frequency-Sampling S tructures1. 506
7.2.4 Lattice Structure. 511
500
S tru c tu re s for IIR S ystem s 519
7.3.1
Direct-Form Structures. 519
7.3.2 Signal Flow Graphs and Transposed Structures. 521
7.3.3 Cascade-Form Structures, 526
7.3.4 Parallel-Form Structures. 529
7.3.5
Lattice and Lattice-Ladder Structures for IIR Systems, 531
S tate-S p a ce System A n aly sis a n d S tru c tu re s 539
7.4.1
State-Space Descriptions of Systems Characterized by Difference
Equations. 540
7.4.2 Solution of the State-Space Equations. 543
7.4.3 Relationships Between Input-O utput and State-Space
Descriptions, 545
7.4.4 State-Space Analysis in the z-Domain, 550
7.4.5 Additional State-Space Structures. 554
R e p re s e n ta tio n of N u m b e rs 556
7.5.1 Fixed-Point Representation of Numbers. 557
7.5.2 Binary Floating-Point R epresentation of Numbers. 561
7.5.3 E rrors Resulting from R ounding and Truncation. 564
Q u a n tiz a tio n of F ilte r C o e fficien ts 569
7.6.1
Analysis of Sensitivity to Quantization of Filter Coefficients. 569
7.6.2 Q uantization of Coefficients in FIR Filters. 578
7.7
R o u n d -O ff E ffects in D igital F ilte rs 582
7.7.1 Limit-Cycle Oscillations in Recursive Systems. 583
7.7.2 Scaling to Prevent Overflow, 588
7.7.3 Statistical Characterization of Q uantization Effects in Fixed-Point
Realizations of Digital Filters. 590
7.8
S u m m ary a n d R e fe re n c e s
P ro b le m s
600
598
Contents
ix
8
614
DESIGN OF DIGITAL FILTERS
8.1
G e n e ra l C o n s id e ra tio n s 614
8.1.1
Causality and Its Implications. 615
8.1.2 Characteristics of Practical Frequency-Selective Filters. 619
8.2
D e sig n o f F IR F ilters 620
8.2.1
Symmetric and Antisym m eiric FIR Filters, 620
8.2.2 Design of Linear-Phase FIR Filters Using Windows, 623
8.2.3 Design of Linear-Phase FIR Filters by the Frequency-Sampling
M ethod, 630
8.2.4 Design of Optimum Equiripple Linear-Phase FIR Filters, 637
8.2.5 Design of FIR Differentiators, 652
8.2.6 Design of Hilbert Transformers, 657
8.2.7 Comparison of Design M ethods for Linear-Phase FIR Filters, 662
8.3
D esig n o f I I R F ilters F ro m A n a lo g F iiters 666
8.3.1
IIR Filter Design by Approxim ation of Derivatives. 667
8.3.2 IIR Filter Design by Impulse Invariance. 671
8.3.3 IIR Filter Design by the Bilinear Transform ation, 676
8.3.4 The M atched-; Transform ation, 681
8.3.5
Characteristics of Commonly Used Analog Filters. 681
8.3.6 Some Examples of Digital Filter Designs Based on the Bilinear
Transform ation. 692
8.4
F re q u e n c y T ra n s fo rm a tio n s 692
8.4.1
Frequency Transform ations in the Analog Dom ain, 693
8.4.2 Frequency Transform ations in the Digital Dom ain. 698
8.5
D esig n o f D ig ital F ilters B a sed on L e a st-S q u a re s M e th o d
8.5.1
Pade Approxim ation Method, 701
8.5.2 Least-Squares Design Methods, 706
8.5.3 FIR Least-Squares Inverse (W iener) Filters, 711
8.5.4 Design of IIR Filters in the Frequency Dom ain, 719
8.6
S u m m ary an d R e fe re n c e s
P ro b le m s
9
701
724
726
SAMPLING AND RECONSTRUCTION OF SIGNALS
9.1
S am p lin g o f B a n d p a ss S ignals 738
9.1.1
R epresentation of Bandpass Signals, 738
9.1.2 Sampling of Bandpass Signals, 742
9.1.3 Discrete-Time Processing of Continuous-Time Signals, 746
9.2
A n a lo g -to -D ig ita l C o n v e rsio n 748
9.2.1 Sample-and-Hold. 748
9.2.2 Quantization and Coding, 750
9.2.3 Analysis of Q uantization Errors, 753
9.2.4 Oversampling A /D Converters, 756
738
Contents
X
9.3
D ig ita l-to -A n a lo g C o n v e rsio n 763
9.3.1
Sample and Hold, 765
9.3.2 First-Order Hold. 768
9.3.3 Linear Interpolation with Delay, 771
9.3.4
Oversampling D/A Converters, 774
9.4
S u m m ary an d R e fe re n c e s
P ro b le m s
774
775
10 MULTIRATE DIGITAL SIGNAL PROCESSING
782
10.1
In tro d u c tio n
10.2
D e c im a tio n by a F a c to r D
784
10.3
In te rp o la tio n by a F a c to r /
787
10.4
S am p lin g R a te C o n v e rsio n by a R a tio n a l F a c to r I ID
10.5
F iite r D esig n an d Im p le m e n ta tio n for S a m p lin g -R ate
C o n v e rsio n 792
10.5.1 Direct-Form FIR Filter Structures, 793
10.5.2 Polyphase Filter Structures, 794
10.5.3 Time-Variant Filter Structures. 800
10.6
M u ltistag e Im p le m e n ta tio n o f S a m p lin g -R a te C o n v e rs io n
10.7
S a m p lin g -R a te C o n v e rsio n o f B a n d p a ss S ignals 810
10.7.1 Decim ation and Interpolation by Frequency Conversion, 812
10.7.2 M odulation-Free Method for Decimation and Interpolation. 814
10.8
S a m p lin g -R a te C o n v e rsio n by an A rb itra ry F a c to r 815
10.8.1 First-O rder Approxim ation, 816
10.8.2 Second-Order Approximation (Linear Interpolation). 819
10.9
A p p lic a tio n s o f M u ltira te Signal P ro c essin g 821
10.9.1 Design of Phase Shifters. 821
10.9.2 Interfacing of Digital Systems with Different Sampling Rates, 823
10.9.3 Im plem entation of Narrowband Lowpass Filters, 824
10.9.4 Im plem entation of Digital Filter Banks. 825
10.9.5 Subband Coding of Speech Signals, 831
10.9.6 Q uadrature M irror Filters. 833
10.9.7 Transmultiplexers. 841
10.9.8 Oversampling A/D and D /A Conversion, 843
10.10
S u m m ary an d R e fe re n c e s
P ro b le m s
846
783
844
790
806
Contents
xi
11 LINEAR PREDICTION AND OPTIMUM LINEAR FILTERS
852
11.1
In n o v a tio n s R e p re s e n ta tio n o f a S ta tio n a ry R a n d o m
P ro c e ss 852
11.1.1 Rational Power Spectra. 854
11.1.2 Relationships Between the Filter Param eters and the
Autocorrelation Sequence, 855
11.2
F o rw a rd an d B a ck w ard L in e a r P re d ictio n 857
11.2.1 Forw ard Linear Prediction, 857
11.2.2 Backward Linear Prediction, 860
11.2.3 The Optimum Reflection Coefficients for the Lattice Forward and
Backward Predictors, 863
11.2.4 Relationship of an A R Process to Linear Prediction. 864
11.3
S o lu tio n o f th e N o rm al E q u a tio n s 864
11.3.1 The Levinson-Durbin Algorithm. 865
11.3.2 The Schiir Algorithm. 868
11.4
P ro p e rtie s o f th e L in e a r P re d ic tio n -E rro r F ilte rs
11.5
A R L attice an d A R M A L a ttic e -L a d d e r F ilters 876
11.5.1 AR LaLtice Structure. 877
11.5.2 A RM A Processes and Lattice-Ladder Filters. 878
11.6
W ie n e r F ilters fo r F ilterin g a n d P re d ictio n 880
11.6.1 FIR W iener Filter, 881
11.6.2 Orthogonality Principle in Linear M ean-Square Estimation, 884
11.6.3 IIR W iener Filter. 885
11.6.4 Noncausal Wiener Filter. 889
11.7
S u m m ary an d R e fe re n c e s
P ro b le m s
873
890
892
12 POWER SPECTRUM ESTIMATION
12.1
E stim a tio n o f S p e c tra from F in ite -D u ra tio n O b s e rv a tio n s o f
Signals 896
12.1.1 Com putation of the Energy Density Spectrum. 897
12.1.2 Estim ation of the Autocorrelation and Power Spectrum of
Random Signals: The Periodogram. 902
12.1.3 The Use of the DFT in Power Spectrum Estim ation, 906
12.2
N o n p a ra m e tric M e th o d s fo r P o w er S p ectru m E s tim a tio n 908
12.2.1 The B artlett Method: Averaging Periodograms, 910
12.2.2 The Welch Method: Averaging Modified Periodogram s, 911
12.2.3 The Blackman and Tukey Method: Smoothing the
Periodogram, 913
12.2.4 Perform ance Characteristics of N onparam etric Power Spectrum
Estim ators, 916
896
xii
Contents
12.2.5 Com putational Requirem ents of Nonparam etric Power Spectrum
Estimates, 919
12.3
P a ra m e tric M e th o d s fo r P o w er S p e c tru m E stim a tio n 920
12.3.1 Relationships Between the A utocorrelation and the Model
Param eters, 923
12.3.2 The Y ule-W alker M ethod for the A R Model Param eters, 925
12.3.3 The Burg M ethod for the A R Model Param eters, 925
12.3.4 Unconstrained Least-Squares M ethod for the A R Model
Param eters, 929
12.3.5 Sequential Estim ation M ethods for the A R Model Param eters, 930
12.3.6 Selection of A R Model O rder, 931
12.3.7 MA Model for Power Spectrum Estim ation, 933
12.3.8 A R M A Model for Power Spectrum Estim ation, 934
12.3.9 Some Experim ental Results, 936
12.4
M in im u m V a rian ce S p ectral E stim a tio n
12.5
E ig e n an aly sis A lg o rith m s fo r S p e c tru m E stim a tio n 946
12.5.1 Pisarenko Harm onic Decom position M ethod, 948
12.5.2 Eigen-decomposition of the A utocorrelation Matrix for Sinusoids
in White Noise, 950
12.5.3 MUSIC Algorithm. 952
12.5.4 ESPR IT Algorithm, 953
12.5.5 O rder Selection Criteria. 955
12.5.6 Experim ental Results, 956
12.6
S u m m ary an d R e fe re n c e s
P ro b le m s
942
959
960
,
A RANDOM SIGNALS CORRELATION FUNCTIONS, AND POWER
SPECTRA
A1
B RANDOM NUMBER GENERATORS
B1
C TABLES OF TRANSITION COEFFICIENTS FOR THE DESIGN OF
LINEAR-PHASE FIR FILTERS
C1
D LIST OF MATLAB FUNCTIONS
D1
REFERENCES AND BIBLIOGRAPHY
R1
INDEX
11
Lj_ Preface
T h is b o o k w as d e v e lo p e d b ased on o u r te ach in g o f u n d e rg ra d u a te and g ra d u ­
a te level co u rse s in d ig ital signal p ro cessin g o v er th e p a s t several y ears. In this
b o o k w e p re se n t th e fu n d a m e n ta ls o f d isc re te -tim e signals, system s, and m o d e rn
d ig ital p ro cessin g a lg o rith m s an d a p p lic a tio n s fo r stu d e n ts in electrical e n g in e e r­
ing. c o m p u te r en g in eerin g , a n d c o m p u te r science. T h e b o o k is su itab le fo r e ith e r
a o n e -se m e s te r o r a tw o -se m e ste r u n d e rg ra d u a te level c o u rse in d isc re te system s
a n d dig ital signal p ro cessin g . It is also in te n d e d fo r use in a o n e -se m e s te r first-year
g ra d u a te -le v e l co u rse in digital signal processing.
It is a ssu m ed th a t th e s tu d e n t in electrical and c o m p u te r e n g in e e rin g has h ad
u n d e rg ra d u a te c o u rses in a d v an ce d calculus (in clu d in g o rd in a ry d iffe re n tia l e q u a ­
tio n s). an d lin ear sy stem s fo r c o n tin u o u s-tim e signals, including an in tro d u c tio n
to th e L ap lace tran sfo rm . A lth o u g h the F o u rie r se ries a n d F o u rie r tra n sfo rm s of
p e rio d ic an d a p e rio d ic signals a re d escrib ed in C h a p te r 4, we ex p ect th a t m any
s tu d e n ts m ay have h ad th is m a te ria l in a p rio r course.
A b ala n c e d co v erag e is p ro v id e d of b o th th e o ry an d p ra c tic a l ap p licatio n s.
A larg e n u m b e r o f w ell d esigned p ro b le m s a re p ro v id e d to h e lp th e s tu d e n t in
m a ste rin g th e su b ject m a tte r. A so lu tio n s m a n u a l is av ailab le fo r th e b en efit o f
th e in stru c to r an d can be o b ta in e d fro m th e p u b lish er.
T h e th ird e d itio n o f th e b o o k covers basically th e sa m e m a te ria l as th e se c­
o n d e d itio n , b u t is o rg an ized d ifferen tly . T h e m a jo r d ifferen ce is in th e o rd e r in
w hich th e D F T a n d F F T alg o rith m s are co v ered . B a sed o n su g g estio n s m a d e by
se v era l rev iew ers, w e n o w in tro d u c e th e D F T a n d d esc rib e its efficient c o m p u ta ­
tio n im m e d ia te ly fo llo w ing o u r tr e a tm e n t of F o u rie r analysis. T his re o rg a n iz a tio n
h as also allo w ed us to elim in a te re p e titio n o f so m e to p ics c o n cern in g th e D F T and
its ap p licatio n s.
In C h a p te r 1 w e d escrib e th e o p e ra tio n s in v o lv ed in th e an alo g -to -d ig ital
c o n v ersio n o f an alo g signals. T h e p ro cess o f sa m p lin g a sin u so id is d escrib ed in
so m e d e ta il an d th e p ro b le m o f aliasing is ex p lain ed . Signal q u a n tiz a tio n an d
d ig ita l-to -a n a lo g co n v ersio n a re also d escrib ed in g e n e ra l term s, b u t th e analysis
is p re s e n te d in su b s e q u e n t c h a p te rs.
C h a p te r 2 is d e v o te d e n tire ly to th e c h a ra c te riz a tio n a n d analysis o f lin e a r
tim e -in v a ria n t (sh ift-in v arian t) d isc re te -tim e system s a n d d isc re te -tim e signals in
th e tim e d o m a in . T h e co n v o lu tio n sum is d e riv e d a n d system s a re categ o rized
a c co rd in g to th e d u ra tio n of th e ir im p u lse re sp o n s e as a fin ite -d u ra tio n im p u lse
xiii
xiv
Preface
re sp o n se (F IR ) an d as an in fin ite -d u ra tio n im pulse re sp o n se ( II R ) . L in e a r tim ein v a ria n t sy stem s c h a ra c te riz e d by d ifferen ce e q u a tio n s are p r e s e n te d an d th e so ­
lu tio n o f d ifferen ce e q u a tio n s w ith initial c o n d itio n s is o b ta in e d . T h e c h a p te r
co n clu d es w ith a tre a tm e n t o f d isc re te -tim e c o rre la tio n .
T h e z -tra n sfo rm is in tro d u c e d in C h a p te r 3. B o th th e b ila te ra l an d th e
u n ila te ra l z -tra n sfo rm s are p re se n te d , a n d m e th o d s fo r d e te rm in in g th e in v erse
z -tra n sfo rm are d esc rib e d . U se o f the z -tra n s fo rm in the analysis o f lin ear tim ein v a ria n t sy stem s is illu stra te d , an d im p o rta n t p ro p e rtie s o f system s, su c h as c a u s a l­
ity a n d stab ility , a re re la te d to z-d o m ain ch aracteristics.
C h a p te r 4 tr e a ts th e analysis o f signals and sy stem s in th e fre q u e n c y d o m ain .
F o u rie r se ries an d th e F o u rie r tra n sfo rm a re p re s e n te d fo r b o th co n tin u o u s-tim e
an d d isc rete-tim e signals. L in e a r tim e -in v a ria n t (L T I) d isc rete sy stem s are c h a r­
a c terized in th e fre q u e n c y d o m a in by th e ir freq u e n c y resp o n se fu n c tio n an d th e ir
re sp o n se to p e rio d ic an d a p e rio d ic signals is d e te rm in e d . A n u m b e r of im p o rta n t
ty p es o f d isc re te -tim e system s are d esc rib e d , in clu d in g re s o n a to rs , n o tc h filters,
co m b filters, all-p ass filters, a n d o scillato rs. T h e desig n of a n u m b e r of sim ple
F IR a n d IIR filters is also co n sid ered . In a d d itio n , th e stu d e n t is in tro d u c e d to
th e co n c e p ts o f m in im u m -p h a se , m ix ed -p h ase, an d m a x im u m -p h a se system s an d
to th e p ro b le m o f d e c o n v o lu tio n .
T h e D F T . its p ro p e rtie s an d its a p p licatio n s, a re th e topics c o v e re d in C h a p ­
te r 5. T w o m e th o d s a re d e sc rib e d fo r using th e D F T to p e rfo rm lin e a r filtering.
T h e use o f th e D F T to p e rfo rm fre q u e n c y analysis o f signals is also d escrib ed .
C h a p te r 6 co v ers th e efficient c o m p u ta tio n o f th e D F T . In c lu d e d in this c h a p ­
te r are d e sc rip tio n s o f radix-2, ra d ix -4, a n d sp lit-ra d ix fast F o u rie r tra n sfo rm (F F T )
alg o rith m s, a n d a p p lic a tio n s o f th e F F T a lg o rith m s to th e c o m p u ta tio n o f c o n v o ­
lu tio n a n d c o rre la tio n . T h e G o e rtz e l alg o rith m a n d the ch irp -z tra n sfo rm are
in tro d u c e d as tw o m e th o d s fo r c o m p u tin g th e D F T using lin e a r filtering.
C h a p te r 7 tre a ts th e re a liz a tio n o f I I R an d F IR system s. T h is tre a tm e n t
in clu d es d irect-fo rm , cascad e, p a ra lle l, lattice, a n d la ttic e -la d d e r re a liz a tio n s. T h e
c h a p te r in clu d es a tr e a tm e n t o f sta te -sp a c e analysis an d s tru c tu re s fo r d isc rete-tim e
system s, an d ex am in es q u a n tiz a tio n effects in a d igital im p le m e n ta tio n o f F IR and
I IR system s.
T e c h n iq u e s fo r d esign o f digital F IR a n d IIR filters are p r e s e n te d in C h a p ­
te r 8. T h e d esign te c h n iq u e s in clu d e b o th d irect design m e th o d s in d isc re te tim e
an d m e th o d s involv in g th e co n v ersio n o f an a lo g filters in to digital filters by v ario u s
tra n sfo rm a tio n s. A lso tre a te d in this c h a p te r is th e d esig n o f F I R a n d IIR filters
by le a st-sq u a re s m e th o d s.
C h a p te r 9 fo cu ses o n th e sam pling o f c o n tin u o u s-tim e sig n a ls a n d th e r e ­
c o n s tru c tio n o f such signals fro m th e ir sam ples. In th is c h a p te r, w e d eriv e th e
sam p lin g th e o re m fo r b a n d p a ss co n tin u o u s-tim e -sig n a ls an d th e n co v e r th e A /D
an d D /A co n v ersio n te c h n iq u e s, including o v e rsam p lin g A /D a n d D /A co n v erters.
C h a p te r 10 p ro v id e s an in d e p th tre a tm e n t o f sa m p lin g -ra te c o n v ersio n and
its a p p lic a tio n s to m u ltira le d ig ital signal p ro cessin g . In a d d itio n to d escrib in g d e c ­
im atio n a n d in te rp o la tio n by in te g e r facto rs, we p re s e n t a m e th o d o f sa m p lin g -rate
Preface
xv
co n v e rsio n by an a rb itra ry facto r. S ev eral a p p licatio n s to m u ltira te signal p ro c e ss­
ing a re p re s e n te d , in clu d in g th e im p le m e n ta tio n o f d igital filters, su b b a n d cod in g
o f sp e ech sig n als, tra n sm u ltip le x in g , an d o v ersam p lin g A /D a n d D /A c o n v e rte rs.
L in e a r p re d ic tio n an d o p tim u m lin e a r (W ien er) filters a re tr e a te d in C h a p ­
te r 11. A lso in clu d ed in this c h a p te r are d escrip tio n s o f th e L e v in s o n -D u rb in
alg o rith m a n d Schiir a lg o rith m fo r solving th e n o rm a l e q u a tio n s , as w ell as th e
A R la ttic e a n d A R M A la ttic e -la d d e r filters.
P o w e r sp e c tru m e stim a tio n is th e m ain to p ic of C h a p te r 12. O u r co v erag e
in clu d es a d e s c rip tio n o f n o n p a ra m e tric an d m o d el-b ased (p a ra m e tric ) m e th o d s.
A lso d e s c rib e d a re e ig e n -d e c o m p o sitio n -b a se d m e th o d s, in clu d in g M U S IC an d
E S P R IT .
A t N o r th e a s te r n U n iv ersity , w e h av e u se d th e first six c h a p te rs o f this b o o k
fo r a o n e -se m e s te r (ju n io r level) c o u rse in d isc rete sy stem s a n d d ig ital signal p r o ­
cessing.
A o n e -s e m e s te r se n io r level c o u rse fo r stu d e n ts w h o h av e h a d p rio r e x p o su re
to d isc rete sy stem s can u se th e m a te ria l in C h a p te rs 1 th ro u g h 4 for a q u ick rev iew
a n d th e n p ro c e e d to co v er C h a p te r 5 th ro u g h 8.
In a first-v ear g ra d u a te level c o u rse in digital signal p ro cessin g , th e first five
c h a p te rs p ro v id e th e s tu d e n t w ith a goo d rev iew of d isc re te -tim e system s. T h e
in stru c to r can m o v e q u ick ly th ro u g h m o st o f th is m aterial a n d th e n co v e r C h a p te rs
6 th ro u g h 9, fo llo w ed by e ith e r C h a p te rs 10 and 11 o r by C h a p te rs 11 an d 12.
W e h a v e in c lu d e d m an y ex am p les th ro u g h o u t th e b o o k an d a p p ro x im a te ly
500 h o m e w o rk p ro b le m s. M an y o f th e h o m ew o rk p ro b le m s can b e so lv ed n u m e r ­
ically on a c o m p u te r, using a so ftw are p ack ag e such as M A T L A B © . T h e se p r o b ­
lem s a re id e n tifie d by an asterisk . A p p e n d ix D co n tain s a list o f M A T L A B fu n c­
tio n s th a t th e s tu d e n t can use in solving th e se p ro b lem s. T h e in s tru c to r m ay also
w ish to c o n s id e r th e u se o f a s u p p le m e n ta ry b o o k th a t c o n ta in s c o m p u te r b ased
exercises, su c h as th e b o o k s Digilal Signal Processing Us ing M A T L A B (P.W .S.
K e n t, 1996) by V. K. In g le a n d J. G . P ro a k is a n d C o m p u te r- B a s e d Exercises f o r
S ignal P ro cessing Using M A T L A B (P re n tic e H all, 1994) by C. S. B u rru s e t al.
T h e a u th o rs a re in d e b te d to th e ir m an y facu lty c o lleag u es w ho h av e p ro v id e d
v alu ab le su g g e stio n s th ro u g h review s o f the first an d se co n d ed itio n s o f this b o o k .
T h e se in clu d e D rs. W . E . A le x a n d e r, Y. B re sle r, J. D e lle r, V. Ingle, C. K eller,
H . L e v -A ri, L. M e ra k o s , W. M ik h a e l, P. M o n ticcio lo , C. N ikias, M . S ch etzen ,
H . T ru ssell, S. W ilso n , a n d M. Z o lto w sk i. W e a re also in d e b te d to D r. R , P ric e fo r
re c o m m e n d in g th e in clu sion o f sp lit-ra d ix F F T alg o rith m s a n d re la te d su g g estio n s.
F in ally , w e w ish to ac k n o w le d g e th e su g g e stio n s an d c o m m e n ts o f m an y fo rm e r
g ra d u a te s tu d e n ts , a n d especially th o se by A . L. K ok, J. L in an d S. S rin id h i w ho
assisted in th e p r e p a r a tio n o f several illu stra tio n s an d th e so lu tio n s m an u al.
J o h n G . P ro a k is
D im itris G , M a n o lak is
Introduction
D ig ital signal p ro cessin g is an are a o f science a n d e n g in e e rin g th a t h a s d ev e lo p e d
rap id ly o v e r th e p ast 30 y ears. T his rap id d e v e lo p m e n t is a resu lt o f th e signif­
ican t ad v an ce s in digital c o m p u te r tech n o lo g y an d in te g ra te d -c irc u it fab rica tio n .
T h e digital c o m p u te rs an d asso ciated digital h ard w are of th re e d e c a d e s ago w ere
relativ ely larg e an d ex p en siv e and, as a co n seq u en ce, th e ir use w as lim ited to
g e n e ra l-p u rp o s e n o n -re a l-tim e (o ff-line) scientific c o m p u ta tio n s an d business a p ­
p licatio n s. T h e ra p id d ev e lo p m e n ts in in te g ra te d -c irc u it te c h n o lo g y , sta rtin g with
m ed iu m -scale in te g ra tio n (M S I) an d p ro g ressin g to large-scale in te g ra tio n (L S I),
a n d now , v ery -larg e-scale in te g ra tio n (V L S I) of e le c tro n ic circuits has sp u rre d
th e d e v e lo p m e n t o f p o w erfu l, sm a ller, faster, an d c h e a p e r digital c o m p u te rs an d
sp e cial-p u rp o se d igital h a rd w a re . T h e se in ex p en siv e an d re lativ ely fast digital c ir­
cuits h av e m a d e it p o ssib le to co n stru c t highly so p h istic a te d digital system s cap ab le
o f p e rfo rm in g co m p lex digital signal p ro cessin g fu n ctio n s a n d tasks, w hich are u su ­
ally to o difficult a n d /o r to o expensive to be p e rfo rm e d by an a lo g circuitry or a n alo g
signal p ro cessin g system s. H e n c e m an y of th e signal p ro cessin g task s th a t w ere
c o n v en tio n ally p e rfo rm e d by an alo g m e a n s a re realized to d a y by less ex p en siv e
an d o fte n m o re re lia b le digital h a rd w a re .
W e do n o t w ish to im ply th a t digital signal p ro cessin g is th e p ro p e r so lu ­
tio n fo r all signal p ro cessin g p ro b lem s. In d e e d , fo r m a n y signals w ith e x tre m e ly
w ide b a n d w id th s, real-tim e p ro cessin g is a re q u ire m e n t. F o r such signals, a n a ­
log o r, p e rh a p s, o p tical signal p ro cessin g is th e only p o ssib le so lu tio n . H o w ev er,
w h ere dig ital circuits are av ailab le an d h av e sufficient sp e e d to p e rfo rm th e signal
p ro cessin g , th ey a re usually p re fe ra b le .
N o t only d o d igital circuits yield c h e a p e r an d m o re re lia b le system s fo r signal
p ro cessin g , th e y h av e o th e r a d v an tag es as w ell. In p a rtic u la r, digital pro cessin g
h a rd w a re allow s p ro g ra m m a b le o p e ra tio n s. T h ro u g h so ftw are, on e can m o re easily
m o d ify th e sig n al p ro cessin g fu n ctio n s to b e p e rfo rm e d by th e h a rd w a re . T h u s
dig ital h a rd w a re a n d a s so ciated so ftw are p ro v id e a g re a te r d eg re e o f flexibility in
sy stem d esign. A lso , th e re is o ften a h ig h e r o rd e r of p re c isio n ach iev ab le w ith
d ig ital h a rd w a re an d so ftw are c o m p a re d w ith an alo g circu its a n d an alo g signal
p ro cessin g system s. F o r all th e se re a so n s, th e re h as b e e n an explosive grow th in
d ig ital signal p ro cessin g th e o ry a n d a p p licatio n s o v e r th e p ast th re e decades.
2
Introduction
Chap. 1
In this b o o k o u r o b jectiv e is to p re se n t an in tro d u c tio n o f th e basic analysis
to ols an d te c h n iq u e s fo r d igital p ro cessin g o f signals. W e b eg in by in tro d u c in g
so m e o f th e n ecessa ry term in o lo g y an d by d escrib in g th e im p o rta n t o p e ra tio n s
asso ciated w ith th e p ro cess of c o n v ertin g an an alo g signal to d ig ital fo rm su itab le
fo r d igital p ro cessin g . A s we shall se e, digital p ro cessin g o f a n a lo g signals has
som e d raw b ack s. F irst, an d fo re m o st, c o n v ersio n o f an a n a lo g signal to digital
fo rm , acco m p lish ed by sa m p lin g th e signal an d q u a n tiz in g th e sa m p le s, resu lts in a
d isto rtio n th a t p re v e n ts us fro m re c o n stru c tin g th e o rig in a l a n a lo g signal fro m the
q u a n tiz e d sam p les. C o n tro l o f th e a m o u n t o f th is d isto rtio n is ach ie v e d by p ro p e r
choice o f th e sam p lin g ra te a n d th e p recisio n in th e q u a n tiz a tio n p ro cess. S eco n d ,
th e re a re finite p re c isio n effects th a t m u st be c o n s id e re d in th e d igital pro cessin g
o f th e q u a n tiz e d sam p les. W hile th e se im p o rta n t issues are c o n s id e re d in som e
d etail in this b o o k , th e em p h asis is on th e analysis a n d d esig n o f digital signal
p ro cessin g sy stem s a n d c o m p u ta tio n a l te ch n iq u es.
1.1 SIGNALS, SYSTEMS, AND SIGNAL PROCESSING
A signal is d efin ed as any physical q u a n tity th a t varies w ith tim e, sp ace, o r any
o th e r in d e p e n d e n t v ariab le o r variables. M a th em atic ally , we d e sc rib e a signal as
a fu n ctio n o f o n e o r m o re in d e p e n d e n t variab les. F o r e x am p le, th e fu n ctio n s
* i( r ) = 5/
(1.1.1)
S2(t) = 20 r
d escrib e tw o signals, o n e th a t varies lin early w ith the in d e p e n d e n t v ariab le t (tim e)
an d a seco n d th a t v aries q u a d ra tic a lly w ith t. A s a n o th e r ex a m p le , co n sid e r the
fu n ctio n
v) = 3x + 2 x y + 1 0 y 2
(1.1.2)
T his fu n ctio n d escrib es a signal o f tw o in d e p e n d e n t v a riab les x a n d y th a t could
r e p re s e n t th e tw o sp a tia l c o o rd in a te s in a p lan e.
T h e signals d e sc rib e d by (1.1.1) an d (1.1.2) b e lo n g to a class o f signals th a t
are p recisely d efin ed by specifying th e fu n c tio n a l d e p e n d e n c e on th e in d e p e n d e n t
v ariab le. H o w ev er, th e re are cases w h ere such a fu n c tio n a l re la tio n sh ip is u n k n o w n
o r to o highly c o m p licated to be o f any p ractical use.
F o r ex am p le, a sp e ech signal (see Fig. 1.1) c a n n o t be d e s c rib e d fu n ctio n ally
by ex p ressio n s such as (1.1.1). In g e n eral, a se g m e n t o f sp e ech m ay be re p re se n te d
to a high d eg re e o f accu racy as a sum of se v era l sin u so id s o f d iffe re n t am p litu d e s
a n d freq u e n cies, th a t is, as
N
A j ( t ) s i n [ 2 ; r f } ( r ) f + #,■(/)]
(1.1.3)
i=i
w h ere {/!,(/)}, {F ,(r)j, a n d {t9,(r)} a re th e se ts of (p o ssib ly tim e -v a ry in g ) a m p litu d es,
freq u e n cies, an d p h a se s, resp ectiv ely , o f th e sinusoids. In fact, o n e w ay to in te rp re t
th e in fo rm a tio n c o n te n t o r m essag e co n v ey ed by an y sh o rt tim e se g m e n t o f th e
Sec. 1.1
—
#
S
^
#
i j ^
3
Signals, Systems, and Signal Processing
I
Th
... ‘ ^
•
ft
A
N
D
---------- ,|* y y y > v y y m w
■'r r m
■'
w m
' W W W ’ ......................
1
Figure 1.1
Example of a speech signal.
sp e e c h signal is to m e a s u re the a m p litu d es, freq u e n cies, a n d p h a se s c o n ta in e d in
th e sh o rt tim e se g m e n t o f the signal.
A n o th e r ex am p le o f a n a tu ra l signal is an e le c tro c a rd io g ra m (E C G ). Such a
signal p ro v id e s a d o c to r w ith in fo rm a tio n a b o u t th e co n d itio n o f the p a tie n t's h e a rt.
S im ilarly, an e le c tro e n c e p h a lo g ra m (E E G ) signal p ro v id es in fo rm a tio n a b o u t th e
activ ity o f th e b rain .
S p eech , e le c tro c a rd io g ra m , a n d e le c tro e n c e p h a lo g ra m signals a re ex am p les
o f in fo rm a tio n -b e a rin g signals th a t evolve as fu n ctio n s o f a single in d e p e n d e n t
v ariab le, n am elv , tim e. A n ex am p le o f a signal th at is a fu n ctio n o f tw o in d e ­
p e n d e n t v ariab les is an im age signal. T h e in d e p e n d e n t v ariab les in th is case are
th e sp atial c o o rd in a te s. T h e se a re b u t a few ex am p les o f th e co u n tless n u m b e r of
n a tu ra l signals e n c o u n te re d in practice.
A s so c ia te d w ith n a tu ra l signals are the m ean s by w hich such signals are g e n ­
e ra te d . F o r ex am p le, sp e ech signals are g e n e ra te d by fo rcin g air th ro u g h th e vocal
co rd s. Im ag es a re o b ta in e d by ex p o sin g a p h o to g ra p h ic film to a scene o r an o b ­
ject. T h u s signal g e n e ra tio n is usually asso ciated w ith a sy stem th a t re sp o n d s to a
stim u lu s o r fo rce. In a sp e ech signal, th e system consists o f th e vocal cords a n d
th e vocal tra c t, also called th e vocal cavity. T h e stim ulus in c o m b in a tio n w ith th e
sy stem is called a signal source. T h u s w e have sp eech so u rces, im ag es so u rces, an d
v ario u s o th e r ty p es o f signal sources.
A sy stem m ay also be defin ed as a physical device th a t p e rfo rm s an o p e r a ­
tio n on a signal. F o r e x am p le, a filter u sed to red u c e th e n o ise an d in te rfe re n c e
co rru p tin g a d e s ire d in fo rm a tio n -b e a rin g signal is called a system . In this case th e
filter p e rfo rm s so m e o p e ra tio n (s ) on th e signal, w hich h as th e effect o f red u cin g
(filterin g ) th e n o ise a n d in te rfe re n c e from th e d e sire d in fo rm a tio n -b e a rin g signal.
W h en w e pass a signal th ro u g h a system , as in filterin g , w e say th a t we h av e
p ro c e sse d th e signal. In this case th e p ro cessin g of th e signal involves filtering th e
n o ise an d in te rfe re n c e fro m th e d e s ire d signal. In g e n e ra l, th e system is c h a ra c ­
te riz e d by th e ty p e o f o p e ra tio n th a t it p e rfo rm s on th e signal. F o r ex am p le, if
th e o p e ra tio n is lin ear, th e system is called linear. If th e o p e ra tio n o n th e signal
is n o n lin e a r, th e system is said to be n o n lin e a r, a n d so fo rth . S uch o p e ra tio n s a re
u su a lly re fe rre d to as signal processing.
4
Introduction
Chap. 1
F o r o u r p u rp o se s, it is c o n v en ien t to b r o a d e n th e d efin itio n o f a system to
include n o t o n ly physical devices, b u t also so ftw are re a liz a tio n s o f o p e ra tio n s on
a signal. In d igital p ro cessin g o f signals on a digital c o m p u te r, th e o p e ra tio n s p e r­
fo rm e d on a signal co n sist of a n u m b e r of m a th e m a tic a l o p e ra tio n s as specified by
a so ftw are p ro g ram . In this case, th e p ro g ra m r e p re s e n ts an im p le m e n ta tio n o f the
system in software. T h u s we h ave a system th a t is re a liz e d on a d igital c o m p u te r
by m ean s o f a se q u en ce o f m a th e m a tic a l o p e ra tio n s; th a t is, w e h av e a digital
signal p ro cessin g system realized in so ftw are. F o r e x am p le, a d ig ital c o m p u te r can
be p ro g ra m m e d to p e rfo rm digital filtering. A lte rn a tiv e ly , th e d igital processing
o n th e signal m ay be p e rfo rm e d by digital h ard w a re (logic circu its) co nfigured to
p e rfo rm th e d e sire d specified o p e ra tio n s. In such a re a liz a tio n , w e h av e a physical
d ev ice th a t p e rfo rm s th e specified o p e ra tio n s. In a b r o a d e r se n se, a digital system
can be im p le m e n te d as a c o m b in a tio n o f digital h a rd w a re an d so ftw are, each of
w hich p e rfo rm s its ow n set of specified o p e ra tio n s.
T h is b o o k d eals w ith th e p ro cessin g o f signals by digital m e a n s, e ith e r in so ft­
w are o r in h a rd w a re . Since m an y of the signals e n c o u n te re d in p ra c tic e are analog,
w e will also co n sid er th e p ro b lem of c o n v ertin g an a n a lo g signal in to a digital sig­
n al fo r pro cessin g . T h u s we will be d ealin g p rim a rily w ith d ig ital system s. T he
o p e ra tio n s p e rfo rm e d by such a system can u su ally be specified m ath em atically .
T h e m e th o d o r set o f ru les for im p le m e n tin g th e sy stem by a p ro g ra m th a t p e r ­
fo rm s th e c o rre sp o n d in g m a th e m a tic a l o p e ra tio n s is called an algorithm. U sually,
th e re are m an y w ays o r alg o rith m s by w hich a system can be im p le m e n te d , e ith e r
in so ftw are o r in h a rd w a re , to p e rfo rm th e d e sire d o p e ra tio n s a n d c o m p u tatio n s.
In p ra c tic e , we h av e an in te re st in devising a lg o rith m s th a t are c o m p u ta tio n a lly
efficient, fast, an d easily im p lem en ted . T h u s a m a jo r to p ic in o u r stu d y o f digi­
tal signal p ro cessin g is th e discussion o f efficient a lg o rith m s fo r p e rfo rm in g such
o p e ra tio n s as filterin g , c o rre la tio n , an d sp e c tra l analysis.
1.1.1 Basic Elements of a Digital Signal Processing
System
M o st o f th e signals e n c o u n te re d in science an d e n g in e e rin g a re a n a lo g in n a tu re.
T h a t is. th e signals a re fu n ctio n s of a c o n tin u o u s v a ria b le , such as tim e o r space,
an d u su ally ta k e o n v alues in a co n tin u o u s ran g e. S uch signals m ay be p ro cessed
directly by a p p ro p ria te an alo g system s (such as filters o r fre q u e n c y an aly zers) or
fre q u e n c y m u ltip lie rs for th e p u rp o se of ch an g in g th e ir c h a ra c te ristic s o r ex tractin g
so m e d esired in fo rm a tio n . In such a case w e say th a t th e signal h as b e e n p ro cessed
d irectly in its an alo g fo rm , as illu strated in Fig. 1.2. B o th th e in p u t signal a n d the
o u tp u t signal a re in an a lo g form .
Analog
input
signal
Analog
signal
processor
Analog
output
signal
Figure 1.2
A nalog signal processing.
Sec. 1.1
Signals, Systems, and Signal Processing
5
Analog
output
signal
Analog
input
signal
Digital
input
signal
Figure 1.3
Digital
output
signal
Block diagram of a digital signal processing system.
D ig ital signal p ro cessin g p ro v id e s an a lte rn a tiv e m e th o d fo r p ro cessin g th e
a n a lo g signal, as illu stra te d in Fig. 1.3. T o p e rfo rm th e p ro cessin g digitally, th e re
is a n e e d fo r an in te rfa c e b e tw e e n th e an a lo g signal a n d th e digital p ro cesso r.
T h is in te rfa c e is called an analog-to-digital ( A / D ) converter. T h e o u tp u t of th e
A /D c o n v e rte r is a d ig ital signal th a t is a p p ro p ria te as an in p u t to th e d igital
p ro cesso r.
T h e dig ital signal p ro c e ss o r m ay be a larg e p ro g ra m m a b le digital c o m p u te r
o r a sm all m ic ro p ro c e s so r p ro g ra m m e d to p e rfo rm th e d e s ire d o p e ra tio n s on th e
in p u t signal. It m ay also be a h a rd w ire d digital p ro c e ss o r co n fig u red to p e rfo rm
a specified se t o f o p e ra tio n s on th e in p u t signal. P ro g ra m m a b le m ach in es p r o ­
v id e th e flexibility to ch an g e th e signal p ro cessin g o p e ra tio n s th ro u g h a ch an g e
in th e so ftw are, w h e re a s h a rd w ire d m ach in es a re difficult to reco n fig u re. C o n s e ­
q u e n tly , p ro g ra m m a b le signal p ro c e ss o rs a re in very c o m m o n use. O n th e o th e r
h an d , w h en signal p ro cessin g o p e ra tio n s are w ell d efin ed , a h a rd w ire d im p le m e n ­
ta tio n o f th e o p e ra tio n s can be o p tim ized , re su ltin g in a c h e a p e r signal p ro c e sso r
a n d , u su ally , o n e th a t ru n s fa ste r th a n its p ro g ra m m a b le c o u n te rp a rt. In a p p li­
c atio n s w h e re th e d ig ital o u tp u t fro m th e d igital signal p ro c e sso r is to be given
to th e u se r in an alo g form , such as in sp e ech co m m u n icatio n s, w e m ust p r o ­
vid e a n o th e r in te rfa c e fro m th e digital d o m a in to th e a n a lo g d o m ain . S uch an
in te rfa c e is called a digital-to-analog ( D / A ) converter. T h u s th e signal is p r o ­
v id ed to th e u se r in an a lo g form , as illu stra te d in th e b lo ck d iag ram o f Fig. 1.3.
H o w e v e r, th e re a re o th e r p ractical a p p lic a tio n s involving signal analysis, w h ere
th e d e s ire d in fo rm a tio n is co n v ey ed in digital form a n d n o D /A c o n v e rte r is
re q u ire d . F o r ex am p le, in th e d igital p ro cessin g o f r a d a r signals, th e in fo rm a ­
tio n e x tra c te d fro m th e ra d a r signal, such as th e p o sitio n o f th e aircra ft a n d its
sp e ed , m ay sim ply b e p rin te d on p a p e r. T h e re is n o n e e d fo r a D /A c o n v e rte r in
th is case.
1.1.2 Advantages of Digital over Analog Signal
Processing
T h e re a re m an y re a so n s w hy d ig ital signal p ro cessin g o f an an alo g signal m ay be
p re fe ra b le to p ro cessin g th e signal directly in th e an a lo g d o m ain , as m e n tio n e d
briefly e a rlie r. F irst, a digital p ro g ra m m a b le sy stem allow s flexibility in r e c o n ­
figuring th e digital signal p ro cessin g o p e ra tio n s sim ply by ch anging th e p ro g ra m .
6
Introduction
Chap. 1
R e c o n fig u ra tio n o f an an a lo g system usually im plies a re d e sig n o f th e h a rd w a re
follow ed by te stin g a n d v erification to see th a t it o p e ra te s p ro p e rly .
A ccu racy c o n s id e ra tio n s also p lay an im p o rta n t role in d e te rm in in g th e fo rm
o f th e signal p ro cesso r. T o le ra n c e s in an alo g c ircu it c o m p o n e n ts m a k e it e x tre m e ly
difficult fo r th e system d esig n er to co n tro l th e accu racy o f an an a lo g signal p r o ­
cessing system . O n th e o th e r h an d , a digital system p ro v id e s m uch b e tte r c o n tro l
o f accu racy re q u ire m e n ts . Such re q u ire m e n ts , in tu rn , re s u lt in specifying th e a c ­
cu racy r e q u ire m e n ts in th e A /D c o n v e rte r a n d th e d igital sig n a l p ro c e sso r, in te rm s
o f w ord le n g th , flo atin g -p o in t v ersu s fix ed -p o in t arith m e tic , a n d sim ilar facto rs.
D ig ita l signals are easily sto re d o n m a g n e tic m ed ia (ta p e o r disk) w ith o u t d e ­
te rio ra tio n o r loss o f signal fidelity b e y o n d th a t in tro d u c e d in th e A /D co n v ersio n .
A s a c o n se q u e n c e , th e signals b e c o m e tra n s p o rta b le an d can b e p ro cessed off-line
in a re m o te la b o ra to ry . T h e digital signal p ro cessin g m e th o d also allow s for th e im ­
p le m e n ta tio n o f m o re so p h istic a te d signal p ro cessin g alg o rith m s. It is usually very
difficult to p e rfo rm p recise m a th e m a tic a l o p e ra tio n s on signals in a n a lo g fo rm b u t
th ese sam e o p e ra tio n s can b e ro u tin e ly im p le m e n te d on a d ig ital c o m p u te r using
so ftw are.
In so m e cases a d igital im p le m e n ta tio n of th e signal p ro cessin g system is
c h e a p e r th a n its an a lo g c o u n te rp a rt. T h e lo w er cost m ay be d u e to th e fact th a t
th e dig ital h a rd w a re is c h e a p e r, o r p e rh a p s it is a re su lt o f th e flexibility fo r m o d ­
ifications p ro v id e d by th e digital im p le m e n ta tio n .
A s a c o n se q u e n c e o f th ese ad v a n ta g e s, d igital signal p ro c e ssin g has b e e n
a p p lied in p ractical sy stem s co v erin g a b ro a d ra n g e of d iscip lin es. W e cite, fo r ex ­
am p le, th e a p p licatio n o f d igital signal p ro cessin g te c h n iq u e s in sp e ech p ro cessin g
an d signal tran sm issio n o n te le p h o n e ch an n els, in im age p ro c e ssin g an d tra n sm is­
sio n , in seism o lo g y an d geophysics, in oil e x p lo ra tio n , in th e d e te c tio n of n u c le a r
ex p lo sio n s, in th e p ro cessin g of signals receiv ed fro m o u te r sp a ce, an d in a vast
v ariety o f o th e r a p p licatio n s. S om e o f th e se a p p lic a tio n s a re cited in su b s e q u e n t
ch ap ters.
A s a lre a d y in d icated , h o w ev er, digital im p le m e n ta tio n has its lim itatio n s.
O n e p ractical lim ita tio n is th e sp e ed o f o p e ra tio n o f A /D c o n v e rte rs a n d digital
signal p ro cesso rs. W e shall see th a t signals hav in g e x tre m e ly w id e b a n d w id th s re ­
q u ire fa st-sam p lin g -rate A /D c o n v e rte rs an d fast d igital signal p ro cesso rs. H e n c e
th e re a re an alo g signals w ith larg e b a n d w id th s fo r w hich a digital p ro cessin g a p ­
p ro a c h is b ey o n d th e s ta te of th e a rt o f digital h a rd w a re .
1.2 CLASSIFICATION OF SIGNALS
T h e m e th o d s we use in p ro cessin g a signal o r in an aly zin g th e re s p o n s e o f a system
to a sig n al d e p e n d h eavily on th e ch a ra c te ristic a ttr ib u te s o f th e specific signal.
T h e re a re te c h n iq u e s th a t ap p ly only to specific fam ilies o f signals. C o n seq u en tly ,
an y in v estig atio n in signal p ro cessin g sh o u ld sta rt w ith a classification o f th e signals
in v o lv ed in th e specific ap p licatio n .
Sec. 1.2
Classification of Signals
7
1.2.1 Multichannel and Multidimensional Signals
A s e x p lain ed in S ectio n 1.1, a signal is d escrib ed by a fu n c tio n o f o n e o r m o re
in d e p e n d e n t v ariab les. T h e v alue of th e fu n ctio n (i.e., th e d e p e n d e n t v ariab le) can
be a re a l-v a lu e d sc alar q u a n tity , a co m p lex -v alu ed q u a n tity , o r p e rh a p s a v ecto r.
F o r e x am p le, th e signal
si( r ) = A sin37rr
is a re a l-v a lu e d signal. H o w e v e r, th e signal
s2(f) = A e ji7Tt = A cos 37t t
j'A sin 3:r r
is co m p lex v alu ed .
In so m e a p p lic a tio n s, signals a re g e n e ra te d by m u ltip le so u rces or m u ltip le
sen so rs. Such signals, in tu rn , can be re p re s e n te d in v e c to r fo rm . F ig u re 1.4 show s
th e th re e c o m p o n e n ts of a v e c to r signal th a t re p re se n ts th e g ro u n d a c c e le ra tio n
d u e to an e a r th q u a k e . T h is a c c e le ra tio n is the re su lt of th re e basic ty p es of elastic
w aves. T h e p rim a ry (P ) w aves an d th e se co n d a ry (S) w aves p ro p a g a te w'ithin th e
b o d y o f rock a n d a re lo n g itu d in al a n d tra n sv e rsa l, resp ec tiv ely . T h e th ird ty p e
o f elastic w ave is called th e su rface w ave, b e c a u se it p ro p a g a te s n e a r th e g ro u n d
su rface. If $*(/). k = 1. 2. 3. d e n o te s th e electrical signal from th e £ th se n so r as a
fu n ctio n o f tim e, th e se t of p = 3 signals can be re p re se n te d by a v e c to r S?(f )< w h ere
r si (O '
S;,(r) =
Si(t)
-Sl(t) J
W e re fe r to such a v e c to r o f signals as a m u ltich a n n el signal. In e le c tro c a rd io g ra ­
p hy. for ex am p le, 3 -lead an d 12-lead e le c tro c a rd io g ra m s (E C G ) are o ften used in
p ractice, w hich resu lt in 3 -ch an n el a n d 12-channel signals.
L e t us n o w tu rn o u r a tte n tio n to th e in d e p e n d e n t v a ria b le (s). If the signal is
a fu n ctio n o f a single in d e p e n d e n t v ariab le, th e signal is called a o ne-d im en sio n a l
signal. O n th e o th e r h a n d , a signal is called M -d i m e n s i o n a l if its v alu e is a fu n ctio n
of M in d e p e n d e n t v ariab les.
T h e p ic tu re sh o w n in Fig. 1.5 is an ex am p le of a tw o -d im e n sio n al signal, since
th e in ten sity o r b rig h tn e ss I ( x . y) a t each p o in t is a fu n ctio n of tw o in d e p e n d e n t
v ariab les. O n th e o th e r h a n d , a b la c k -a n d -w h ite telev isio n p ic tu re m ay be r e p ­
r e se n te d as I ( x . y . t ) since th e b rig h tn e ss is a fu n ctio n of tim e. H e n c e th e T V
p ic tu re m ay b e tr e a te d as a th re e -d im e n s io n a l signal. In c o n tra st, a co lo r T V p ic ­
tu re m ay b e d e sc rib e d by th re e in te n sity fu n ctio n s of th e fo rm Ir (x, y. ?), Is (x. y. t ),
a n d I i , ( x . y , t ) , c o rre sp o n d in g to th e b rig h tn e ss of the th re e p rin cip al colors (red .
g re e n , b lu e) as fu n ctio n s o f tim e. H e n c e th e co lo r T V p ic tu re is a th re e -c h a n n e l,
th re e -d im e n s io n a l signal, w hich can b e re p re s e n te d by th e v e c to r
-/,(* ,> ■ . O '
I U , y. t) —
. l b(x, v ,r ) _
In this b o o k we d e a l m ainly w ith sin g le-ch an n el, o n e -d im e n sio n a l real- or
co m p lex -v alu ed signals a n d w e re fe r to th e m sim ply as signals. In m a th e m a tic a l
Introduction
Chap. 1
Up
/ % East
7jJL______
South
bouth
1
I____ i
]____ I.
f S waves
_4 P waves
1____ 1____ I
t Surface waves
r1 -2
I
I
i
i
i_______ I ,
I____ _ J _______I_______ I----------- 1-----------1----------- 1-----------1----------- 1-----------1
0
2
4
6
8
12
10
14
16
18
20
22
24
26
28
30
Time (seconds)
(b)
Figure 1.4 Three components of ground acceleration measured a few kilometers
from the epicenter of an earthquake. (From Earthquakes, by B. A . Bold. © 1988
by W. H. Freeman and Company. Reprinted with permission of the publisher.)
te rm s th ese signals are d escrib ed by a fu n ctio n o f a single in d e p e n d e n t v ariable.
A lth o u g h th e in d e p e n d e n t variab le n e e d n o t be tim e, it is c o m m o n p ractice to use
t as th e in d e p e n d e n t v ariab le. In m an y cases th e signal p ro c e ssin g o p e ra tio n s and
a lg o rith m s d e v e lo p e d in this tex t for o n e -d im e n sio n a l, sin g le-ch an n el signals can
b e e x te n d e d to m u ltic h a n n e l an d m u ltid im e n sio n a l signals.
1.2.2 Continuous-Time Versus Discrete-Time Signals
Signals can b e fu rth e r classified in to fo u r d iffe re n t c a te g o rie s d e p e n d in g on the
ch a ra c te ristic s o f th e tim e (in d e p e n d e n t) v a ria b le an d th e v alu es th ey tak e.
Con tin u o u s -tim e signals o r a nalog signals a re d e fin ed for ev e ry value o f tim e an d
Sec. 1.2
9
Classification of Signals
Figure 1.5
Example of a two-dimensional signal.
th ey ta k e on v alu es in the co n tin u o u s in terv al (a . b ). w h e re a can be —oc a n d b
can be oc. M a th em atic ally , th ese signals can be d e sc rib e d by fu n ctio n s o f a c o n ­
tin u o u s v ariab le. T h e sp eech w av efo rm in Fig. 1.1 an d th e signals x i(r) = c o s 7i t ,
x j { t ) = e ^ 1' 1, —oc < t < oq are ex am p les o f an alo g signals. Discrete-time signals
a re d efin ed o n ly at c e rta in specific v alu es o f tim e. T h e se tim e in sta n ts n eed n o t be
e q u id ista n t, b u t in p ractice th ey are usually ta k e n a t e q u a lly sp a ced in terv als fo r
c o m p u ta tio n a l c o n v en ien ce an d m a th e m a tic a l tra c ta b ility . T h e signal x(t„) =
n = 0, ± 1 , ± 2 , . . . p ro v id es an ex am p le o f a d isc re te -tim e signal. If we use th e
in d ex n o f th e d isc rete-tim e in sta n ts as th e in d e p e n d e n t v ariab le, th e signal v alu e
b eco m es a fu n ctio n o f an in te g e r v ariab le (i.e., a se q u e n c e of n u m b e rs). T h u s a
d isc re te -tim e signal can be re p re s e n te d m a th e m a tic a lly by a se q u e n c e of real o r
c o m p lex n u m b ers. T o em p h asize the d isc rete-tim e n a tu r e o f a signal, w e sh all
d e n o te such a signal as x{n) in ste a d o f x ( t ) . If th e tim e in stan ts t„ are e q u ally
sp a ced (i.e., t„ = n T ), th e n o ta tio n x ( n T ) is also used. F o r ex am p le, th e se q u e n c e
x(n)
if n > 0
o th erw ise
( 1 .2 . 1 )
is a d isc re te -tim e signal, w hich is r e p re s e n te d g rap h ically as in Fig. 1.6.
In ap p licatio n s, d isc rete-tim e signals m ay arise in tw o ways:
1. B y se lectin g v alu es o f an an alo g signal a t d isc re te -tim e in stan ts. T his p ro c e ss
is called s am plin g an d is discussed in m o re d etail in S ectio n 1.4. A ll m e a s u r­
ing in stru m e n ts th a t ta k e m e a s u re m e n ts at a re g u la r in te rv a l o f tim e p ro v id e
d isc rete-tim e signals. F o r ex am p le, th e signal x ( n ) in Fig. 1.6 can be o b ta in e d
10
Introduction
Chap. 1
x{n)
I I T
Figure 1.6 Graphical representation of the discrete time signal x[n) = 0.8" for
n > 0 and x(n) = 0 for n < 0.
bv sa m p lin g th e an a lo g signal x ( t ) — 0 .8 ', t > 0 an d x ( t ) = 0. t < 0 once
ev ery seco n d .
2. By accu m u latin g a v a ria b le o v e r a p e rio d o f tim e. F o r e x a m p le , c o u n tin g th e
n u m b e r o f cars using a given s tre e t every h o u r, o r re c o rd in g th e v alu e of gold
ev ery day, resu lts in d isc re te -tim e signals. F ig u re 1.7 show s a g rap h o f the
W o lfer su n sp o t n u m b e rs. E a c h sam ple o f this d isc re te -tim e signal p ro v id es
th e n u m b e r o f su n s p o ts o b se rv e d d u rin g an in te rv a l o f 1 y e a r.
1.2.3 Continuous-Valued Versus Discrete-Valued Signals
T h e v alu es o f a c o n tin u o u s-tim e or d isc re te -tim e signal can be c o n tin u o u s or d is­
crete. If a signal ta k e s on all po ssib le values on a finite or an infinite ran g e, it
Year
Figure 1.7
W olfer annual sunspot num bers (1770-1869).
Sec. 1.2
Classification of Signals
11
is said to b e c o n tin u o u s-v a lu e d signal. A lte rn a tiv e ly , if th e signal ta k e s on v alu es
fro m a finite se t o f p o ssib le values, it is said to be a d isc re te -v a lu e d signal. U su ally ,
th e s e v alu es a re e q u id ista n t a n d h en ce can be e x p ressed as an in te g e r m u ltip le of
th e d istan c e b e tw e e n tw o successive values. A d isc re te -tim e signal hav in g a set of
d isc rete v alu es is called a digital signal. F ig u re 1,8 show s a d igital signal th a t ta k e s
o n o n e o f fo u r p o ssib le values.
In o r d e r fo r a signal to be p ro c e sse d digitally, it m u st be d isc rete in tim e
a n d its v alu es m u st b e d isc re te (i.e., it m ust b e a digital sig n al). If th e signal to
b e p ro c e sse d is in an a lo g fo rm , it is c o n v e rte d to a digital signal by sam pling th e
an alo g signal at d isc rete in stan ts in tim e, o b ta in in g a d isc re te -tim e signal, and th e n
by q ua n tizin g its v alu es to a set o f d isc re te v alu es, as d e sc rib e d la te r in th e c h a p te r.
T h e p ro cess o f co n v e rtin g a c o n tin u o u s-v a lu e d signal in to a d isc re te -v a lu e d signal,
called quan tizatio n, is basically an a p p ro x im a tio n p ro cess. It m ay be acco m p lish ed
sim ply bv ro u n d in g o r tru n c a tio n . F o r ex am p le, if th e allo w ab le signal v alu es
in th e d ig ital signal are in teg ers, say 0 th ro u g h 15, the c o n tin u o u s-v a lu e signal is
q u a n tiz e d in to th ese in te g e r values. T h u s th e signal v alue 8.58 will be a p p ro x im a te d
by th e v alu e 8 if th e q u a n tiz a tio n p ro cess is p e rfo rm e d by tru n c a tio n o r by 9 if
th e q u a n tiz a tio n p ro c e ss is p e rfo rm e d by ro u n d in g to th e n e a re s t in teg er. A n
ex p la n a tio n o f th e a n a lo g -to -d ig ita l co n v ersio n p ro cess is given la te r in th e c h a p te r.
Figure 1.8
Digital signal with four different amplitude values.
1.2.4 Deterministic Versus Random Signals
T h e m a th e m a tic a l an aly sis an d p ro cessin g o f signals re q u ire s th e availability o f a
m a th e m a tic a l d e sc rip tio n fo r th e signal itself. T h is m a th e m a tic a l d e scrip tio n , o fte n
re fe rre d to as th e signal m o d e l, lead s to a n o th e r im p o rta n t classification of signals.
A n y signal th a t can b e u n iq u ely d esc rib e d by an explicit m a th e m a tic a l ex p ressio n ,
a tab le o f d a ta , o r a w ell-defined ru le is called deterministic. T his te rm is used to
em p h asize th e fact th a t all p ast, p re se n t, an d fu tu re values o f th e signal are kno w n
precisely , w ith o u t an y u n c e rta in ty .
In m a n y p ractical ap p lic a tio n s, h o w ev er, th e re are sig n a ls th a t e ith e r c a n n o t
b e d esc rib e d to an y re a so n a b le d e g re e o f accu racy by explicit m a th e m a tic a l fo r­
m u las, o r such a d e sc rip tio n is to o c o m p licated to b e of any p ractical use. T h e lack
12
Introduction
Chap. 1
o f such a re la tio n sh ip im plies th a t such signals ev o lv e in tim e in an u n p re d ic ta b le
m a n n e r. W e re fe r to th ese signals as ra n d o m . T h e o u tp u t o f a n oise g e n e ra to r,
th e seism ic signal o f Fig. 1.4, an d th e sp e ech signal in Fig. 1.1 are ex am p les of
ra n d o m signals.
F ig u re 1.9 show s tw o signals o b ta in e d fro m th e sa m e n o ise g e n e ra to r an d
th e ir asso ciated h isto g ram s. A lth o u g h th e tw o signals do n o t re s e m b le each o th e r
visually, th e ir h isto g ra m s re v e a l som e sim ilarities. T his p ro v id e s m o tiv a tio n fo r
—3>—4 1
---------------------------------------------------------- 1
0
200
400
600
-------■*------------------- —
800
1000
----- ------ ---------------------------------------
1200
1400
1600
(a)
400 f ----------------- ------------------- — - — -------------------------- ------------------- ------------------- 1
1
350 r
i
300 -
-j
250 <200 -
150 100 -
I
(b)
Figure 1.9
tograms.
Two random signals from the same signal generator and their his­
Sec. 1.2
Classification of Signals
13
4
4
0
—----- '
200
------------ '-------------------------------------------■---------------—--- --------1
400
600
800
1000
1200
1400
1600
<c)
Figure 1.9
C ontinued
th e an aly sis a n d d e sc rip tio n of ra n d o m signals using statistical te c h n iq u e s in stea d
o f ex p licit fo rm u las. T h e m ath e m a tic a l fra m e w o rk fo r th e th e o re tic a l analysis of
ra n d o m sig n als is p ro v id e d by th e th e o ry of p ro b a b ility a n d sto c h astic processes.
S o m e b asic e le m e n ts o f this a p p ro a c h , a d a p te d to th e n e e d s o f this bo o k , are
p re s e n te d in A p p e n d ix A .
It sh o u ld b e em p h a siz e d a t th is p o in t th a t th e classification o f a real-world
sig n al as d e te rm in is tic o r ra n d o m is n o t alw ays clear. S o m etim es, b o th a p p ro a c h e s
le a d to m e a n in g fu l resu lts th a t p ro v id e m o re insight in to signal b eh av io r. A t o th e r
Introduction
14
Chap. 1
tim es, th e w ro n g classification m ay le a d to e rro n e o u s resu lts, since som e m a th e ­
m atical to o ls m ay ap p ly o n ly to d e te rm in istic signals w hile o th e rs m ay apply only
to ra n d o m signals. T h is will b e c o m e c le a re r as w e ex am in e specific m a th e m a tic a l
tools.
1.3 THE CONCEPT OF FREQUENCY IN CONTINUOUS-TIME AND
DISCRETE-TIME SIGNALS
T h e co n cep t o f fre q u e n c y is fam iliar to stu d e n ts in e n g in e e rin g a n d th e sciences.
T h is co n cep t is basic in. fo r ex am p le, th e design of a rad io re c e iv e r, a high-fidelity
sy stem , o r a sp e c tra l filter fo r co lo r p h o to g ra p h y . F ro m physics w e k n o w th a t
freq u e n cy is closely re la te d to a specific ty p e o f p e rio d ic m o tio n called h arm o n ic
o scillatio n , w hich is d e s c rib e d by sin u so id al fu n ctio n s. T h e c o n c e p t o f freq u e n cy
is d irectly re la te d to th e c o n c e p t o f tim e. A ctu ally , it has th e d im en sio n of inverse
tim e. T h u s w e sh o u ld e x p ect th a t th e n a tu re of tim e (c o n tin u o u s o r d isc rete) w ould
affect th e n a tu re o f th e fre q u e n c y accordingly.
1.3.1 Continuous-Time Sinusoidal Signals
A sim ple h a rm o n ic o sc illatio n is m a th e m a tic a lly d escrib ed
c o n tin u o u s-tim e sin u so id al signal:
x a(t) = A cos(Q t + 0). —oc < t < oc
by th e follow ing
(1.3.1)
show n in Fig. 1.10. T h e su b sc rip t a u se d w ith x { t ) d e n o te s an an a lo g signal. T his
signal is co m p letely c h a ra c te riz e d by th re e p a ra m e te rs: A is th e a m p litu d e of the
sin u so id . ft is th e fre q u e n c y in ra d ia n s p e r se co n d (rad /s), a n d 6 is th e p h a se in
rad ian s. In ste a d o f ft, we o fte n use th e fre q u e n c y F in cycles p e r seco n d o r h e rtz
(H z), w h ere
Q — In F
(1.3.2)
In term s o f F. (1.3.1) can be w ritte n as
x a(t) = A cos(2n F t + 6 ), —oo < t < oc
(1.3.3)
W e will use b o th fo rm s, (1.3.1) an d (1.3.3), in re p re s e n tin g sin u so id al signals.
x j t ) = A cos(2nFt + 8)
Figure 1.10 Example of an analog
sinusoidal signal.
Sec. 1.3
Frequency Concepts in Continuous-Discrete-Tim e Signals
15
T h e an a lo g sin u so id al signal in (1.3.3) is c h a ra c te riz e d by th e follow ing p r o p ­
erties:
A L F o r ev ery fixed v alu e o f th e fre q u e n c y F, x a(r) is p erio d ic. In d e e d , it can
easily b e sh o w n , using e le m e n ta ry trig o n o m e try , th a t
x a(.t + Tp ) = A„(r)
w h e re Tp = 1 / F is th e fu n d a m e n ta l p e rio d o f the sin u so id al signal.
A 2 . C o n tin u o u s -tim e sin u so id al signals w ith d istin ct (d iffe re n t) fre q u e n c ie s a re
th e m se lv e s d istin ct.
A 3 . In c re a sin g th e fre q u e n c y F resu lts in an in crease in th e ra te o f o sc illatio n
o f th e signal, in th e sense th a t m o re p e rio d s are included in a given tim e
in terv al.
W e o b se rv e th a t fo r F = 0. th e value Tp — oc is co n sisten t w ith the fu n ­
d a m e n ta l re la tio n F = 1 / T r . D u e to co n tin u ity o f th e tim e v a ria b le r, w e can
in crease th e fre q u e n c y F, w ith o u t lim it, w ith a c o rre sp o n d in g increase in th e ra te
o f oscillatio n .
T h e re la tio n sh ip s we h ave d e sc rib e d fo r sin u so id al signals carry o v er to th e
class o f co m p lex e x p o n e n tia l signals
xa {t) - A e JlSi,+{"
(1.3.4)
T h is can easily b e seen by e x p ressin g th e se signals in te rm s o f sinusoids using th e
E u le r id en tity
e±j4: = cos <p i j sin <p
(1.3.5)
B y d efin itio n , freq u e n c y is an in h e re n tly p o sitiv e physical q u an tity . T his
is o b v io u s if we in te r p re t fre q u e n c y as the n u m b e r o f cycles p e r u n it tim e in a
p e rio d ic signal. H o w e v e r, in m any cases, only fo r m a th e m a tic a l co n v en ien ce , w e
n e e d to in tro d u c e n e g a tiv e freq u e n cies. T o see this w e recall th a t th e sin u so id al
signal (1.3.1) m ay be e x p ressed as
xa (t) = A c o s (^ r + 6 ) = j
eJ(Q,+f>) + ~ e - J(a+9)
(1.3.6)
w hich follow s fro m (1.3.5). N o te th a t a sin u so id al signal can be o b ta in e d by ad d in g
tw o e q u a l-a m p litu d e c o m p lex -co n ju g ate e x p o n e n tia l signals, so m e tim e s called p h aso rs, illu stra te d in Fig. 1.11. A s t i m e p ro g re sse s th e p h a s o rs ro ta te in o p p o site
d ire c tio n s w ith a n g u la r fre q u e n c ie s ±£2 ra d ia n s p e r se co n d . Since a positive f r e ­
q u e n c y c o rre sp o n d s to co u n te rc lo c k w ise u n ifo rm a n g u la r m o tio n , a negative f r e ­
q u e n c y sim ply c o rre sp o n d s to clockw ise a n g u la r m o tio n .
F o r m a th e m a tic a l c o n v en ien ce , w e use b o th n e g a tiv e a n d po sitiv e freq u e n cies
th ro u g h o u t th is b o o k . H e n c e th e fre q u e n c y ra n g e fo r a n a lo g sinusoids is —oo <
F < oo.
16
Introduction
Chap. 1
Re
Figure 1.11 Representation of a cosine
function by a pair of complex-conjugate
exponentials (phasors).
1.3.2 Discrete-Time Sinusoidal Signals
A d isc rete-tim e sin u so id al signal m ay be e x p ressed as
x ( n ) — A cos (ton + 8), —oo < n < oc
(1.3.7)
w h ere n is an in te g e r v ariab le, called th e sam p le n u m b e r. A is th e am plitu de o f the
sin u so id , co is th e fr e q u e n c y in rad ian s p e r sa m p le , and 8 is th e p h a s e in radians.
If in ste a d o f a> we use the freq u e n cy v ariab le / defin ed by
a; = 2 n f
(1.3.8)
x( n ) — A cos(2n f n + 8). —oc < n < oc
(1.3.9)
th e re la tio n (1.3.7) b eco m es
T h e freq u e n cy / h as d im en sio n s o f cycles p e r sa m p le . In S ectio n 1.4. w here
we co n sid e r th e sam p lin g o f an alo g sinusoids, w e re la te th e fre q u e n c y v ariab le
/ o f a d isc re te -tim e sin u so id to th e fre q u e n c y F in cycles p e r se co n d fo r the
an a lo g sin u so id . F o r th e m o m e n t we co n sid er th e d isc re te -tim e sin u so id in (1.3.7)
in d e p e n d e n tly of th e co n tin u o u s-tim e sinusoid given in (1.3.1). F ig u re 1.12 show s
a sin u so id w ith fre q u e n c y co — n /6 ra d ia n s p e r sa m p le ( f ~ ~ cycles p e r sam ple)
a n d p h ase 8 — n / 3 .
x(n) - A cos (urn + 8)
Figure 1.12 Example of a discrete-time
sinusoidal signal (w — 7t / 6 and b — 7r/3).
Sec. 1.3
Frequency Concepts in Continuous-Discrete-Tim e Signals
17
In c o n tra s t to co n tin u o u s-tim e sinusoids, th e d isc re te -tim e sin u so id s are c h a r­
a c te riz e d by th e fo llow ing p ro p e rtie s:
B l . A discrete-time sinu so id is p er iod ic o n ly i f its fr e q u e n c y f is a rational n u m b e r .
B y d efin itio n , a d isc rete-tim e signal x( n ) is p erio d ic w ith p e rio d N ( N > 0) if
a n d only if
x ( n + N ) = x(n )
fo r all n
(1.3.10)
T h e sm a llest v alu e o f N fo r w hich (1.3.10) is tru e is called th e f u n d a m e n ta l p eriod .
T h e p r o o f o f th e p erio d icity p ro p e rty is sim ple. F o r a sin u so id w ith fre q u e n c y
/o to b e p e rio d ic , we sh o u ld have
cos[27t /o( A7 + n) + 8} — co s(2 ,t/o « + 6)
T h is re la tio n is tru e if an d only if th e re exists an in te g e r k such th a t
2 n f ) N = 2kn
o r, eq u iv alen tly .
/o = 4
N
(1.3.11)
A cc o rd in g to (1.3.11). a d isc rete-tim e sin u so id al signal is p e rio d ic only if its f re ­
q u e n c y /o can be ex p re sse d as th e ra tio o f tw o in teg ers (i.e.. / () is ra tio n a l).
T o d e te rm in e th e fu n d a m e n ta l p e rio d N o f a p e rio d ic sin u so id , w e e x p ress its
fre q u e n c y /o as in (1.3.11) an d cancel com m on facto rs so th a t k a n d N are relativ ely
p rim e . T h e n th e fu n d a m e n ta l p e rio d o f the sin u so id is e q u a l to N. O b serv e th a t a
sm all ch an g e in fre q u e n c y can resu lt in a large change in th e p erio d . F o r ex am p le,
n o te th a t f \ = 3 1 /6 0 im plies th at N\ = 60, w h ereas f i = 3 0 /6 0 resu lts in Nz = 2.
B 2. Disc rete-tim e sinu so ids whose fre q u en cies are separa ted by an integer m ultiple
o f 2 n are identical.
T o p ro v e this assertio n , let us c o n sid er th e sin u so id cos(£oo« + 0). It easily
fo llow s th a t
cos[(wo + 2 n )n + B\ = cos(wo/i + 2 n n + 9) — co%{a)Qn + 9)
(1.3.12)
A s a re su lt, all sin u so id al se q u en ces
x k(n) — A cos(a>tn + 8).
k = 0 ,1 ,2 ,...
(1.3.13)
w h ere
Wk = cl>c + 2k n ,
—7r < o>o < n
are in distinguishable (i.e., identical). O n th e o th e r h a n d , th e se q u e n c e s o f any tw o
sin u so id s w ith fre q u e n c ie s in th e ra n g e - n < a> < n or —^ < f < ~ a re distinct.
C o n s e q u e n tly , d isc rete-tim e sinusoidal signals w ith fre q u e n c ie s M < n o r | / | < \
Introduction
18
Chap. 1
are u n iq u e. A n y se q u e n c e re su ltin g fro m a sin u so id w ith a fre q u e n c y M > n , or
| / | > j , is id en tical to a se q u e n c e o b ta in e d fro m a sin u so id al signal w ith fre q u e n c y
\co\ < n . B e c a u se o f th is sim ilarity, we call th e sin u so id h av in g th e freq u e n cy M >
tt an alias o f a c o rre sp o n d in g sinusoid w ith fre q u e n c y jwj < n . T h u s w e re g a rd
freq u e n cies in th e ra n g e —tt < a> < tt, o r —1 < / < 1 as u n iq u e a n d all freq u e n cies
|o>[ > t t , o r | / | > ~, as aliases. T h e r e a d e r sh o u ld n o tice th e d iffe re n c e b e tw e e n
d isc rete-tim e sin u so id s an d co n tin u o u s-tim e sinusoids, w h e re th e la tte r re su lt in
d istin ct signals fo r £2 o r F in th e e n tir e ran g e —oc < £2 < oc o r —oc < F < oc.
B 3. The highest rate o f oscillation in a discrete-time sin u s o i d is attained whe n
to — 7i (or cu = —tt) or, eq uivalently, f — \ (or f = —\ ) .
T o illu stra te th is p ro p e rty , let u s in v estig ate th e c h a ra c te ristic s of th e sin u ­
so id a l signal se q u e n c e
x ( n ) = cos c^on
w h en th e fre q u e n c y v aries fro m 0 to tt. T o sim plify th e a rg u m e n t, we ta k e values
o f (l>
o = 0, 7t / 8, tt/4 , jt/2 , n c o rre sp o n d in g to / = 0,
5,
w hich resu lt in
p erio d ic se q u e n c e s h av in g p e rio d s N = oc, 16, 8, 4, 2. as d e p ic te d in Fig. 1.13. W e
n o te th a t th e p e rio d o f th e sinusoid d e c re a se s as th e fre q u e n c y in creases. In fact,
we can see th a t th e ra te o f o scillatio n in crease s as th e fre q u e n c y increases.
,, xin)
Figure 1.13
Signal x ( n ) = c o s cu^n for various values of the frequency cdq.
Sec. 1.3
Frequency Concepts in Continuous-Discrete-Time Signals
19
T o see w h at h a p p e n s for tt < ioq < 2tt . we co n sid e r th e sinusoids w ith
fre q u e n c ie s a>\ = a>(> a n d 0J2 — 2 n — ojq. N o te th at as co\ varies from tt to 2n . a>z
v aries fro m ir to 0. it can be easily se en th at
= A cos co} n — A cos won
X2 (n) = A cos uhn — A cos(27r — coo)n
(1.3.14)
= A c o s (—coqii) — x \ (h)
H e n ce ur± is an alias o f w\. If we h ad u sed a sine fu n ctio n in stea d o f a cosine fu n c ­
tio n , th e resu lt w ou ld basically be th e sam e, ex cep t fo r a 180' p h a se d ifferen ce
b etw een th e sin u so id s A](«) and xi ( n ) . In any case, as we increase th e re lativ e
freq u e n cy coo o f a d isc re te -tim e sin u so id from tt to 27r. its ra te of o sc illatio n d e ­
creases. F o r coo = 2 tt th e re su lt is a c o n sta n t signal, as in th e case fo r oju = 0.
O b v io u sly , fo r co{) = tt (o r f = k) w e h ave th e hig h est ra te o f oscillation.
A s fo r th e case o f co n tin u o u s-tim e signals, n eg ativ e fre q u e n c ie s can be in ­
tro d u c e d as w ell for d isc rete-tim e signals. F o r this p u rp o se w e use th e id en tity
A
A
x (n ) = Acos(con + 0) = — (>Jiwn+0) + —
(1.3.15)
Since d isc re te -tim e sinusoidal signals w ith fre q u e n c ie s th a t are se p a ra te d by
an in teg er m u ltip le o f 27r are iden tical, it follow s th a t th e fre q u e n c ie s in any in terv al
co] < a> < co\ + 2 tt c o n stitu te all the existing d isc rete-tim e sinusoids o r com plex
e x p o n en tials. H en ce th e fre q u e n c y ran g e fo r d isc rete-tim e sinusoids is finite with
d u ra tio n 2 n . U sually, we choose th e ran g e 0 < co < 2n o r —tt < co < tt ({) < f < 1.
—1 < / < | ) , w hich we call th e f u n d a m e n t a l range.
1.3.3 Harmonically Related Complex Exponentials
S in u so id al signals a n d com plex e x p o n e n tia ls play a m a jo r role in th e analysis o f
signals an d system s. In so m e cases we deal w ith sets of h arm on ic a lly related c o m ­
plex e x p o n e n tia ls (o r sin u so id s). T h e se are sets of p e rio d ic co m p lex e x p o n e n tia ls
w ith fu n d a m e n ta l fre q u e n c ie s th a t are m u ltip le s o f a single positive freq u e n cy .
A lth o u g h we confine o u r discussion to com plex e x p o n e n tia ls, th e sam e p r o p e r ­
ties clearly hold fo r sin u so id al signals. W e co n sid e r h a rm o n ically re la te d co m p lex
e x p o n e n tia ls in b o th co n tin u o u s tim e and d isc rete tim e.
Continuous-time exponentials.
T h e basic signals for co n tin u o u s-tim e ,
h arm o n ically re la te d e x p o n e n tia ls are
sk(t) = ejkno' = e ll7TkFn'
jt = 0 . ± l . ± 2 . . . .
(1.3.16)
W e n o te th a t for ea c h value o f k, s^U) is p erio d ic w ith fu n d a m e n ta l p e rio d
1 /(kFo) = Tp/ k o r fu n d a m e n ta l freq u e n cy kFo. Since a signal th a t is p e rio d ic
w ith p e rio d Tp / k is also p e rio d ic w ith p erio d k ( T p/ k ) = Tp fo r any positive in te g e r
k, w e see th a t all o f th e s*(r) h av e a co m m o n p e rio d of Tp, F u rth e rm o re , acco rd in g
20
Introduction
Chap. 1
to S ectio n 1.3.1, Fo is allo w ed to ta k e any v alu e an d all m e m b e rs of th e set are
d istin ct, in th e se n se th a t if k\ ^ k2, th en 5*1 (7) ^
F ro m th e basic signals in (1.3.16) we can c o n s tru c t a lin e a r c o m b in a tio n of
h arm o n ically re la te d co m p lex ex p o n e n tia ls o f th e form
cc
SC
=
x ° ( ' ) =
Ckelkiltit
k — — oc
( 1 -3 -1 7 )
k ~ — oc
w h ere ck, k = 0, ± 1 , ± 2 . . . . are a rb itra ry co m p lex co n sta n ts. T h e signal x a(t)
is p erio d ic w ith fu n d a m e n ta l p erio d Tp = l / f o , a n d its r e p re s e n ta tio n in term s
o f (1.3.17) is called th e F ourier series ex p an sio n fo r x a (t). T h e co m p lex -v alu ed
co n sta n ts are th e F o u rie r se rie s coefficients a n d th e signal sk (r) is called th e fcth
h a rm o n ic o f x (l(t).
Discrete-time exponentials. Since a d isc re te -tim e co m p le x e x p o n e n tia l is
p erio d ic if its relativ e fre q u e n c y is a ra tio n a l n u m b e r, w e ch o o se f Q — 1/A' an d we
d efine th e sets o f h arm o n ically re la te d co m p lex e x p o n e n tia ls by
k = 0. ± 1 . ± 2 , . . .
sk (n) = ej2* kf" \
(1.3.18)
In c o n tra st to th e c o n tin u o u s-tim e case, we n o te th at
sk+Nln) = eJ7* n'k+N,/N = e ^ s k (n) = sk (n)
T his m ean s th a t, c o n siste n t w ith (1.3.10), th e re are only N d istin ct p e rio d ic com plex
e x p o n en tials in th e se t d e sc rib e d by (1.3.18). F u rth e rm o re , all m e m b e rs of the set
h av e a co m m o n p e rio d o f N sam ples. C learly, w e can ch o o se a n y co n secu tiv e A'
co m p lex e x p o n e n tia ls, say from k = no to k — no 4- N — 1 to fo rm a h arm o n ically
re la te d set w ith fu n d a m e n ta l freq u e n cy /(, = 1 / N . M o st o fte n , fo r co n v en ien ce ,
we ch o o se th e set th a t c o rre sp o n d s to no = 0 , th a t is, th e set
* = 0 . 1 . 2 .........N - 1
sk(n) = ejl n k n / s .
(1.3.19)
A s in th e case o f c o n tin u o u s-tim e signals, it is o b v io u s th a t th e lin e a r co m ­
b in atio n
N- 1
\-l
x ( n ) = £ c * s * ( n ) = Y L ckei2nkn' N
k= 0
Jt= 0
resu lts in a p e rio d ic signal w ith fu n d a m e n ta l p e rio d N . A s w e shall see later,
this is th e F o u rie r series re p re s e n ta tio n for a p e rio d ic d isc re te -tim e se q u e n c e w ith
F o u rie r co efficien ts {q}. T h e se q u e n c e sk (n) is called th e /tth h a rm o n ic o f x(n ).
Example 1.3.1
Stored in the memory of a digital signal processor is one cycle of the sinusoidal signal
. ( 2nn
x ( n ) = sin I
+ 6
where 6 — 2 n q / N , where q and N are integers.
Sec. 1.4
Analog-to-Digital and Digital-to-Analog Conversion
21
(a) Determ ine how this table of values can be used to obtain values of harmonically
related sinusoids having the same phase.
(b) D eterm ine how this table can be used to obtain sinusoids of the same frequency
but different phase.
Solution
(a) Let
denote the sinusoidal signal sequence
( 2yrnk
xk(n) = sin I --------- !
V N
This is a sinusoid with frequency f k = k / N. which is harmonically related to
x{n). But xk{n) may be expressed as
xk{n) = sin
2 tt ( k n )
——
= x(kn)
Thus we observe that ,vt (0) = .v(0). **(1) = x ( k ) . x k (2) = x ( 2 k ) . and so on.
Hence the sinusoidal sequence
can be obtained from the table of values
of x ( n ) by taking every k th value of * ( « ) . beginning with .v(0). In this m anner we
can generate the values of all harmonically related sinusoids with frequencies
fk = k / N for k = 0. 1....... N - 1.
(b) We can control the phase 8 of the sinusoid with frequency j\ — k / N by taking
the first value of the sequence from memory location q — 9 N/ 2 tt. where q is
an integer. Thus the initial phase 6 controls the starling location in the table
and we wrap around the table each time the index (kn) exceeds N.
1.4 ANALOG-TO-DIGITAL AND DIGITAL-TO-ANALOG CONVERSION
M o st sig n als o f p ractical in terest, such as sp e ech , bio lo g ical signals, seism ic signals,
ra d a r signals, so n a r signals, and v ario u s co m m u n ic a tio n s signals such as au d io a n d
v id eo signals, are an alo g . T o p ro cess an a lo g signals by digital m e a n s , it is first
n ecessa ry to c o n v e rt th e m into digital form , th a t is, to c o n v e rt th e m to a se q u e n c e
o f n u m b e rs h av in g finite precision. T his p ro c e d u re is called analog-to-digital ( A / D )
co n v e r sio n , an d th e c o rre sp o n d in g devices a re called A / D converters ( A D C s ) .
C o n c e p tu a lly , w e view A /D co n v ersio n as a th re e -s te p p ro cess. T his p ro cess
is illu stra te d in Fig. 1.14.
1. S am p lin g . T his is th e c o n v ersio n o f a c o n tin u o u s-tim e signal in to a d is c re te ­
tim e signal o b ta in e d by tak in g “ sa m p le s’" o f th e c o n tin u o u s-tim e signal at
d isc re te -tim e in stan ts. T h u s, if x a(t) is th e in p u t to th e sa m p le r, th e o u tp u t
is x a ( n T ) = x ( n ), w h ere T is called th e s a m p lin g interval.
2. Q ua ntiza tio n . T h is is th e co n v ersio n o f a d isc re te -tim e c o n tin u o u s-v a lu e d
signal in to a d isc rete-tim e, d isc rete-v alu ed (d ig ital) signal. T h e v alu e o f ea c h
Introduction
22
Chap. 1
A/D converter
01011...
"7
Analog
signal
Discrete-time
signal
Quantized
signal
Digital
signal
Figure 1.14 Basic parts of an analog-to-digital (A /D ) converter.
signal sam p le is re p re s e n te d by a v alu e se lected from a finite set o f p o ssi­
b le values. T h e d ifferen ce b e tw e e n th e u n q u a n tiz e d sa m p le x ( n ) an d the
q u a n tiz e d o u tp u t x q(n) is called th e q u a n tiz a tio n erro r.
3. Coding. In th e co d in g p ro cess, each d isc rete v alu e x q{n) is re p re s e n te d by a
6 -b it b in ary se q u en ce.
A lth o u g h w e m o d e l th e A /D c o n v e rte r as a sa m p le r follow ed by a q u a n tiz e r
an d co d er, in p ractice th e A /D co n v ersio n is p e rfo rm e d by a single device th at
ta k e s x a(t) an d p ro d u c e s a b in a ry -c o d e d n u m b e r. T h e o p e ra tio n s of sa m p lin g an d
q u a n tiz a tio n can be p e rfo rm e d in e ith e r o rd e r b u t. in p ractice, sa m p lin g is alw ays
p e rfo rm e d b e fo re q u a n tiz a tio n .
In m an y cases o f p ractical in te re st (e.g., sp e ech p ro cessin g ) it is d esirab le
to co n v ert th e p ro cessed digital signals in to an a lo g form . (O b v io u sly , we can n o t
listen to th e se q u e n c e of sa m p le s re p re se n tin g a sp e ech signal o r see th e n u m ­
b ers co rre sp o n d in g to a T V signal.) T h e p ro cess o f co n v e rtin g a digital signal
in to an an alo g signal is kno w n as digital-to-analog ( D / A ) co nversion. A ll D /A
c o n v e rte rs “c o n n ect th e d o ts ’" in a digital signal by p e rfo rm in g so m e kind of in te r­
p o la tio n , w hose accu racy d e p e n d s on th e q u ality of th e D /A c o n v ersio n process.
F ig u re 1.15 illu strates a sim ple fo rm o f D /A c o n v ersio n , called a z e ro -o rd e r hold
o r a sta ircase a p p ro x im a tio n . O th e r a p p ro x im a tio n s a re p o ssib le, such as lin early
co n n ectin g a p a ir o f successive sa m p le s (lin e a r in te rp o la tio n ), fittin g a q u a d ra tic
th ro u g h th re e successive sa m p le s (q u a d ra tic in te rp o la tio n ), an d so on. Is th e re an
o p tim u m (ideal) in te rp o la to r? F o r signals hav in g a limited f r e q u e n c y co ntent (finite
b an d w id th ), th e sa m p lin g th e o re m in tro d u c e d in th e follow ing se c tio n specifies the
o p tim u m form o f in te rp o la tio n .
S am p lin g a n d q u a n tiz a tio n are tr e a te d in th is sectio n . In p a rtic u la r, we
d e m o n s tra te th a t sa m p lin g d o e s n o t re su lt in a loss of in fo rm a tio n , n o r d o es it
in tro d u c e d isto rtio n in th e signal if th e signal b a n d w id th is finite. In p rin cip le , th e
an alo g signal can b e re c o n stru c te d from th e sam ples, p ro v id e d th a t th e sam p lin g
ra te is sufficiently high to avoid th e p ro b le m co m m o n ly called aliasing. O n th e
o th e r h an d , q u a n tiz a tio n is a n o n in v e rtib le o r irre v e rsib le p ro c e ss th a t resu lts in
signal d isto rtio n . W e shall sh o w th a t th e a m o u n t o f d isto rtio n is d e p e n d e n t on
Sec. 1.4
Analog-to-Digital and Digital-to-Analog Conversion
Original
Signal
23
Staircase
Approximation
/
/
/
V
*o
/
/
/
-/
I
I
o
67Time
Figure 1.15
Zero-ordcr hold digital-to-analog (D /A ) conversion.
th e accu racy , as m e a s u re d by th e n u m b e r of bits, in th e A /D c o n v ersio n process.
T h e facto rs affe ctin g the choice of th e d esired accu racy of th e A /D c o n v e rte r are
cost an d sam p lin g ra te . In g en eral, th e cost in crease s w ith an in crease in accuracy
a n d /o r sa m p lin g rate.
1.4.1 Sampling of Analog Signals
T h e re are m an y ways to sa m p le an an a lo g signal. W e lim it o u r discussion to
p er iod ic o r u n i f o r m s a m p li n g , w hich is th e ty p e of sa m p lin g used m o st o ften in
p ractice. T h is is d escrib ed by the re la tio n
x(n) = xa (nT).
—o c < n < c c
(1.4.1)
w h ere x ( n ) is th e d isc re te -tim e signal o b ta in e d by “ ta k in g sa m p le s” o f th e an alo g
signal x aU) ev ery T se co n d s. T his p ro c e d u re is illu stra te d in Fig. 1.16. T h e tim e
in te rv a l T b etw e e n successive sa m p les is called th e sa m p li n g p er io d o r sa m p le
interval an d its recip ro c al 1 / 7 — Fs is called the s a m p lin g rate (sam p les p e r second)
o r th e sa m p lin g f r e q u e n c y (h ertz).
P e rio d ic sa m p lin g estab lish es a re la tio n sh ip b e tw e e n th e tim e v ariab les t an d
n o f c o n tin u o u s-tim e an d d isc re te -tim e signals, resp ec tiv ely . In d e e d , th e s e v a ri­
ab les a re lin early re la te d th ro u g h th e sa m p lin g p e rio d T or, e q u iv alen tly , th ro u g h
th e sa m p lin g ra te Fs — l / 7 \ as
(1.4.2)
A s a c o n s e q u e n c e o f (1.4.2), th e re exists a re la tio n sh ip b e tw e e n th e freq u e n cy
v a ria b le F (o r Q) fo r an a lo g signals a n d the freq u e n c y v a ria b le / (o r co) for
d isc re te -tim e signals. T o estab lish th is re la tio n sh ip , co n sid e r an an alo g sinusoidal
signal o f th e fo rm
x B{t) = A c o s ( 2 t t F t + 8 )
(1.4.3)
24
Introduction
*(n) = xa(nT)
Analog
signal
Fs = 1IT
Chap. 1
Discrete-time
signal
Sampler
Figure 1.16
Periodic sampling of an analog signal.
w hich, w h en sa m p le d p erio d ically at a ra te Fs — 1 / 7 sa m p le s p e r se co n d , yields
x a( n T ) == x{ n) = A cos(27 t F n T -f 9)
/2 n n F
\
= A cos ( —
+ 9 \
(1-4.4)
If we c o m p are (1.4.4) w ith (1.3.9). w e n o te th a t th e fre q u e n c y v ariab les F
an d / a re lin early re la te d as
Is
or, eq u iv alen tly , as
co = Q T
(1.4.5)
Fs=sampling frequency
F=frequency of analoa
f=frequency of digital signal
(1.4.6)
=relative or normalized frequency
T h e re la tio n in (1.4.5) ju stifies th e n am e relative o r n o r m a li z e d f r e q u e n c y , w hich is
so m e tim e s u sed to d esc rib e th e fre q u e n c y v a ria b le / . A s (1.4.5) im p lies, w e can use
/ to d e te rm in e th e fre q u e n c y F in h e rtz only if th e sa m p lin g fre q u e n c y Fs is k now n.
W e recall fro m S ectio n 1.3.1 th a t th e ran g e o f th e fre q u e n c y v a ria b le F o r £2
fo r co n tin u o u s-tim e sin u so id s are
—oc < F < oo
—oc < £2 < 00
( I .4 .7 )
H o w e v e r, th e situ a tio n is d iffe re n t fo r d isc re te -tim e sinusoids. F ro m S ectio n 1.3.2
w e recall th a t
2
c / < \
2
(1.4.8)
—n < co < n
B y su b stitu tin g fro m (1.4.5) a n d (1.4.6) in to (1.4.8), w e find th a t th e freq u e n cy
o f th e co n tin u o u s-tim e sin u so id w h e n sa m p le d a t a ra te Fs = 1 / 7 m u st fall in
Sec. 1.4
25
Analog-to-Digitai and Digital-to-Analog Conversion
th e ra n a e
(1.4.9)
or, e q u iv alen tly .
T — n F, .< -Q < _Ti F,
-----------=
=
—
T
(1.4.10)
T h e se re la tio n s a re su m m a riz e d in T ab le 1.1.
RELATIO NS AM ONG FR E Q U E N C Y VARIABLES
Discrete-time sienals
Continuous-time sienals
co = 2,t f
radians
cycles
sample
sample
F
u ..
\
u> = nT ,f=F /F,
/
/
'
‘
\
1
Q =
radians
sec
1A
5
IA
TA B LE 1.1
- /
q = to/T.F~f- Fs
\
\
.................. - ..... ..
—tt/T < Q < ,T /r
-f-2,/2 5 F < /-;«/-
- o c < fi < oc
—oc < f-' < oc
F ro m th ese re la tio n s w e o b se rv e th at the fu n d a m e n ta l d ifferen ce b etw een
c o n tin u o u s-tim e an d d isc rete-tim e signals is in th e ir ran g e of v alu es o f th e fre ­
q u en cy v a riab les F an d / , o r Q and w. P erio d ic sa m p lin g o f a c o n tin u o u s-tim e
signal im p lies a m a p p in g of th e infinite freq u e n cy ran g e fo r th e v ariab le F (o r £2)
in to a finite fre q u e n c y ran g e for the v ariab le / (o r a>). Since th e highest freq u e n cy
in a d isc re te -tim e signal is co — tt o r / = •*, it follow s th a t, w ith a sa m p lin g rate
Fs, th e c o rre sp o n d in g h ig h est values o f F and £2 are
H
_J_
Y ~ 2T
(1.4.11)
^max — ft Fs —
T h e re fo re , sa m p lin g in tro d u c e s an am b ig u ity , since th e h ig h est fre q u e n c y in a
co n tin u o u s-tim e signal th a t can be u n iq u ely d istin g u ish ed w h en such a signal is
sa m p le d at a ra te Fs = l / T is Fm&li — F J 2 , o r Qmax = n F s. T o see w h at h a p p e n s
to fre q u e n c ie s a b o v e F J 2, let us co n sid er the follow ing ex am p le.
Example 1.4.1
The implications of these frequency relations can be fully appreciated by considering
the two analog sinusoidal signals
xi (!) — cos 2ji(l0)t
xi(t) — cos2;r(50)f
(1.4.12)
26
Introduction
Chap. 1
which are sampled at a rate Fs = 40 Hz. The corresponding discrete-time signals or
sequences are
(1.4.13)
However. cos5,t«/2 = cos(2^n + 7rn/2) — co s7rn /2 . Hence
= *i(n)- Thus the
sinusoidal signals are identical and, consequently, indistinguishable. If we are given
the sampled values generated by cos(7r/'2)n, there is some ambiguity as to whether
these sampled values correspond to x\{i) or xz(D- Since x2(r) yields exactly the same
values as
when the two are sampled at F, — 40 samples per second, we say that
the frequency F2 — 50 Hz is an alias of the frequency F\ = 10 Hz at the sampling
rate of 40 samples per second.
It is im portant to note that F2 is not the only alias of F]. In fact at the sampling
rate of 40 samples per second, the frequency F3 = 90 Hz is also an alias of F], as is
the frequency F4 = 130 Hz, and so on. All of the sinusoids cos2tt(Fj -f 40k)i. k = 1.
2. 3. 4 . . . . sampled at 40 samples per second, yield identical values. Consequently,
they are all aliases of F\ = 10 Hz.
In g en eral, th e sam pling o f a c o n tin u o u s-tim e sin u so id al signal
x a(t) = A cos(27rF{)/ + 8)
(1 .4 .1 4 )
w ith a sam pling rate F v = 1 / T resu lts in a d isc re te -tim e signal
x ( n ) = A cos(27r/ 0« -f 6)
(1.4.15)
w h ere /o = F()/F , is th e re la tiv e fre q u e n c y o f th e sin u so id . If w e assum e th a t
- F s /2 < Fo < F J 2 . th e fre q u e n c y f 0 o f x ( n ) is in th e ran g e —^ < /o < L w hich is
th e freq u e n cy ran g e fo r d isc re te -tim e signals. In this case, the re la tio n sh ip b etw een
Fo an d f {) is o n e -to -o n e , a n d h en ce it is p o ssib le to identify (o r re c o n stru c t) th e
a n alo g signal xa (t) from th e sa m p les x ( n) .
O n th e o th e r h a n d , if th e sinusoids
x a {t) = A c o s (27z F kt + 6)
(1.4.16)
w h ere
Fk = F0 + k F s .
k = ± l.± 2 .
(1.4.17)
are sam p led a t a ra te F,, it is c le a r th a t th e fre q u e n c y F* is o u tsid e th e fu n d a m e n ta l
fre q u e n c y ran g e —Fs / 2 < F < F J 2 . C o n se q u e n tly , th e sa m p le d signal is
= A c o s (27zn F o / F s + 6 + 2 n k n )
— A c o s ( 2 n f o n + 6)
Sec. 1.4
Analog-to-Digital and Digital-to-Analog Conversion
27
w hich is id en tical to th e d isc re te -tim e signal in (1.4.15) o b ta in e d by sam p lin g .
(1.4.14). T h u s an infinite n u m b e r of c o n tin u o u s-tim e sin u so id s is re p re se n te d by
sa m p lin g th e sa m e d isc re te -tim e signal (i.e.. by th e sam e se t o f sam p le s). C o n ­
se q u e n tly , if w e a re given th e se q u en ce
an am b ig u ity exists as to w hich
c o n tin u o u s-tim e signal x a(t) th e se v alu es re p re se n t. E q u iv a le n tly , we can say th a t
th e fre q u e n c ie s Fk — F o + k F s, —oo < k < oo (k in te g e r) a re in d istin g u ish a b le fro m
th e fre q u e n c y Fo a fte r sa m p lin g an d h e n c e th ey are aliases o f Fo. T he re la tio n sh ip
b e tw e e n th e fre q u e n c y v a riab les o f th e c o n tin u o u s-tim e a n d d isc rete-tim e signals
is illu stra te d in Fig. 1.17.
A n ex a m p le o f aliasing is illu stra te d in Fig. 1.18. w h e re tw o sinusoids w ith
fre q u e n c ie s F 0 = | H z an d Fj = —| H z yield id en tical sa m p le s w hen a sam p lin g
ra te o f Fs — 1 H z is used. F ro m (1.4.17) it easily follow s th a t fo r k = - 1 , Fo =
F, + Fs = ( —| + 1) H z = i H z.
Figure 1.17 Relationship between the continuous-time and discrete-time fre­
quency variables in the case of periodic sampling.
Figure 1.18
Illustration of aliasing.
28
Introduction
Chap. 1
Since Fsf2. w hich c o rre sp o n d s to w = tt, is th e hig h est fre q u e n c y th a t can be
r e p re s e n te d u n iq u ely w ith a sam p lin g ra te Fs , it is a sim ple m a tte r to d e te rm in e
th e m ap p in g o f any (alias) fre q u e n c y above Fs/2 (co — tt) in to th e e q u iv a le n t
freq u e n cy b elo w Fs /2. W e can use F J 2 or a) — t t as the p iv o ta l p o in t a n d reflect
or “ fo ld ” th e alias freq u e n c y to the ran g e 0 < w < tt. Since th e p o in t o f reflectio n
is Fsj 2 (co = t t ) , th e fre q u e n c y F J 2 (cu = re) is called th e f o l d i n g frequency.
Example 1.4.2
Consider the analog signal
Xait) = 3 cos IOOtt/
(a) Determ ine the minimum sampling rate required to avoid aliasing.
(b) Suppose that the signal is sampled at the rate Fs = 200 Hz.
What is the
discrete-time signal obtained after sampling?
(c) Suppose that the signal is sampled at the rate Fs ~ 75 Hz. W hat is the discretetime signal obtained after sampling?
(d) What is the frequency 0 < F < FJ2 of a sinusoid that yields samples identical
to those obtained in part (c)?
Solution
(a) The frequency of the analog signal is F — 50 Hz. Hence the minimum sampling
rate required to avoid aliasing is Ff = 100 Hz.
(b) If the signal is sampled at Fs = 200 Hz. the discrete-time signal is
lOOtf
TT
x{n) = j cos — — n — j cos —n
200
2
(c) If the signal is sampled at F, = 75 Hz. the discrete-time signal is
100?r
A tt
x(n) — 3 cos —j ^ —n — 3 cos — n
— 3 cos —~n
3
(d) For the sampling rate of Fs ~ 75 Hz. we have
F = f F, = 7 5 /
The frequency of the sinusoid in part (c) is / — |. Hence
F = 25 Hz
Clearly, the sinusoidal signal
ya(i) — 3 cos I n Ft
— 3 cos 507ti
sampled at Fs — 75 samples/s yields identical samples. H ence F — 50 Hz is an
alias of F = 25 Hz for the sampling rate Fs = 75 Hz.
Sec. 1.4
Analog-to-Digital and Digital-to-Analog Conversion
29
1.4.2 The Sampling Theorem
G iv en an y an a lo g signal, how sh o u ld w e select th e sa m p lin g p e rio d T or, eq u iv ­
alen tly , th e sa m p lin g ra te FJ. T o an sw er this q u e stio n , w e m u st h ave som e in­
fo rm a tio n a b o u t th e c h aracteristics of th e signal to be sa m p le d . In p a rtic u la r, we
m u st h av e so m e g e n e ra l in fo rm a tio n c o n cern in g th e fr e q u e n c y con tent o f th e sig­
nal. S uch in fo rm a tio n is g en erally av ailab le to us. F o r e x am p le, w e k n o w g en erally
th a t th e m a jo r fre q u e n c y co m p o n e n ts o f a sp eech signal fall b elo w 3000 H z. O n
th e o th e r h a n d , telev isio n signals, in g e n eral, c o n ta in im p o rta n t fre q u e n c y co m ­
p o n e n ts u p to 5 M H z. T h e in fo rm a tio n c o n te n t of such signals is c o n ta in e d in
th e a m p litu d e s, fre q u e n c ie s, an d p h ases o f the v ario u s fre q u e n c y co m p o n e n ts, b ut
d e ta ile d k n o w le d g e o f th e c h aracteristics of such signals is n o t a v a ilab le to us p rio r
to o b ta in in g th e signals. In fact, the p u rp o se of p ro cessin g the signals is usually to
e x tra c t th is d e ta ile d in fo rm a tio n . H o w ev er, if w e k n o w th e m ax im u m freq u e n cy
c o n te n t o f th e g e n e ra l class o f signals (e.g.. th e class of sp e ech signals, the class
o f v id eo signals, etc.). w e can specify th e sam pling ra te n ecessa ry to co n v ert the
a n alo g signals to dig ital signals.
L et us su p p o se th a t any an alo g signal can be r e p re s e n te d as a sum o f sin u so id s
o f d iffe re n t a m p litu d e s, freq u e n cies, a n d p h ases, th a t is.
N
(1.4.18)
w h ere N d e n o te s th e n u m b e r o f freq u e n cy c o m p o n e n ts. A ll signals, such as speech
an d v id eo , len d th em se lv e s to such a re p re s e n ta tio n o v er an y sh o rt tim e segm ent.
T h e a m p litu d e s, freq u e n cies, a n d p h ases usually ch an g e slow ly w ith tim e from one
tim e se g m en t to a n o th e r. H o w e v e r, su p p o se th a t th e fre q u e n c ie s do n o t exceed
som e k n o w n fre q u e n c y , say Fmax. F o r ex am p le, F max = 3000 H z fo r th e class
o f sp e ech signals a n d Fmax = 5 M H z fo r telev isio n signals. Since th e m ax im u m
freq u e n cy m ay v ary slightly fro m d iffe re n t re a liz a tio n s am o n g signals of any given
class (e.g., it m ay vary slightly from s p e a k e r to sp e a k e r), w e m ay wish to e n su re
th a t Fmax d o e s n o t ex ceed som e p re d e te rm in e d v alue by passin g th e an a lo g signal
th ro u g h a filter th a t se v ere ly a tte n u a te s freq u e n cy c o m p o n e n ts ab o v e Fmax. T hus
we a re c e rta in th a t no signal in the class co n tain s fre q u e n c y c o m p o n e n ts (having
significant a m p litu d e o r p o w e r) above Fmax. In p ra c tic e , such filtering is com m only
u sed p rio r to sam p lin g .
F ro m o u r k n o w led g e o f Fmax, w e can se lect th e a p p ro p ria te sam pling rate.
W e k n o w th a t th e h ig h est freq u e n cy in an an alo g signal th a t can be u n a m b ig u ­
ously re c o n s tru c te d w h en th e signal is sa m p le d a t a ra te F, = 1 / T is F J 7. A ny
fre q u e n c y a b o v e Fsf 2 o r b elo w - F J 2 resu lts in sa m p le s th a t a re id en tical w ith a
c o rre sp o n d in g fre q u e n c y in th e ra n g e — F J 2 < F < Fs/2. T o avoid th e am b ig u ities
re su ltin g fro m aliasin g , we m u st se lect th e sa m p lin g ra te to be sufficiently high.
T h a t is, w e m u st select F J 2 to be g re a te r th an Fmax. T h u s to avoid th e p ro b le m
o f aliasin g , Fs is se le c te d so th a t
Fs > 2 Fmax
(1.4.19)
30
Introduction
Chap. 1
w h ere Fmax is th e larg est fre q u e n c y c o m p o n e n t in the a n a lo g signal. W ith the
sa m p lin g ra te se le c te d in this m a n n e r, any fre q u e n c y c o m p o n e n t, say |F ;| < Fmax,
in th e an alo g signal is m a p p e d in to a d isc re te -tim e sinusoid w ith a freq u e n cy
1
F
2
Fs ~ 2
1
(1.4.20)
o r, eq u iv alen tly ,
— tt < a)j = 2n f < 7r
(1.4.21)
S ince, | / | = \ o r \co\ = n is th e h ig h est (u n iq u e ) freq u e n c y in a d isc re te -tim e signal,
th e choice o f sam p lin g ra te acco rd in g to (1.4.19) avoids th e p ro b le m of aliasing.
In o th e r w o rd s, th e co n d itio n Fs > 2 Fmax e n s u re s th a t all th e sin u so id al c o m p o ­
n e n ts in th e a n a lo g signal are m a p p e d in to c o rre sp o n d in g d isc re te -tim e freq u e n cy
c o m p o n e n ts w ith fre q u e n c ie s in th e fu n d a m e n ta l in terv al. T h u s all the freq u e n cy
c o m p o n e n ts o f th e a n alo g signal a re re p re s e n te d in sa m p le d fo rm w ith o u t am b i­
guity, a n d h en ce th e an alo g signal can be re c o n stru c te d w ith o u t d isto rtio n from
th e sa m p le v alu es u sing an “ a p p r o p ria te ” in te rp o la tio n (d ig ita l-to -a n a lo g c o n v e r­
sio n ) m e th o d . T h e ‘■ ap p ro p riate” o r ideal in te rp o la tio n fo rm u la is specified by the
s a m p ling theorem.
Sam pling T h eorem .
If the h ig h est fre q u e n c y c o n ta in e d in an an alo g signal
x a (t) is Fmax = B a n d th e signal is sa m p le d at a ra te F, > 2 F max = 2 B. th e n A.u(r)
can b e exactly re c o v e re d fro m its sa m p le values using th e in te rp o la tio n fu n ctio n
s i n 2 jr B /
8(t) =
2n B t
_ ,
(
}
T h u s jcfl(f) m ay be e x p ressed as
* . ( £ ) * ( ' - £ )
(1-4.23)
w h ere x a(n / F s ) = x a( n T ) = Jt(rc) a re th e sa m p les o f x a(t).
W h en th e sa m p lin g of x a(t) is p e rfo rm e d at the m in im u m sam p lin g ra te
Fs = 2 B , th e re c o n stru c tio n fo rm u la in (1.4.23) b eco m es
^
=
/ n \ sin2nB (t — n flB )
, ,
( zb)
(1'4 2 4 )
,
T h e sa m p lin g r a te F ^ = 2 B = 2 Fmax is called th e N y q u is t rate. F ig u re 1.19 illus­
tra te s th e ideal D /A c o n v ersio n p ro cess using th e in te rp o la tio n fu n c tio n in (1.4.22).
A s can b e o b se rv e d from e ith e r (1.4.23) o r (1.4.24), th e re c o n stru c tio n o f x a(t)
fro m th e se q u e n c e x ( n ) is a c o m p lic a te d p ro cess, involving a w e ig h te d sum o f the
in te rp o la tio n fu n ctio n g (t) an d its tim e -sh ifte d v ersio n s g ( t —n T ) fo r —oo < n < oo,
w h ere th e w eig h tin g facto rs a re th e sa m p le s x ( n ) . B e c a u se o f th e co m p lex ity an d
th e infinite n u m b e r o f sa m p les re q u ire d in (1.4.23) o r (1.4.24), th e s e re c o n stru c tio n
Sec. 1.4
Analog-to-Digital and Digital-to-Analog Conversion
31
sample of ,v„(n
(/[ —^ /
{ri — l )l
fll
\’t -r l 11
Figure 1.19 Ideal D /A conversion
(interpolation).
fo rm u ias are p rim a rily o f th e o re tic a l in te re st. P ra ctical in te rp o la tio n m e th o d s are
given in C h a p te r 9.
Example 1.4.3
Consider the analog signal
xu(r) = 3cos50;rz
10sin300;n —cos 100tt?
What is the Nvquist rate for this signal?
Solution
The frequencies present in the signal above are
F = 25 Hz.
F: = 150 Hz.
F, = 50 Hz
Thus Fnm = 150 Hz and according to (1.4.19),
F > 2 Fmax = 300 Hz
The Nvquist rale is FA = 2 Fm;„. Hence
Fs = 300 Hz
Discussion It should be observed that the signal component 10sin300;r/. sampled at
the Nvquist raie FA- = 300, results in the samples 10 sin 7r/j. which are identically zero.
In other words, we are sampling the analog sinusoid at its zero-crossing points, and
hence we miss this signal component completely. This situation would not occur if the
sinusoid is offset in phase by some am ount 8. In such a case we have lOsinGOOin -ffl)
sampled at the Nvquist rate FA- = 300 samples per second, which yields the samples
-c o s t t h sin f>)
10 sin(7rn+ ^) = 10(sin n n cos & -+
= lOsin 6 cos nn
Thus if 6 ^ 0 or tt, the samples of the sinusoid taken at the Nvquist rate are not all
zero. However, we still cannot obtain the correct amplitude from the samples when
the phase 9 is unknown. A simple rem edy that avoids this potentially troublesome
situation is to sample the analog signal at a rate higher than the Nvquist rate.
Example 1.4.4
Consider the analog signal
*a(t) = 3cos2000irf + 5 sin6000;rr + lOcos 12.000;?;
Introduction
Chap. 1
(a) What is the Nvquist rate for this signal?
(b) Assume now that we sample this signal using a sampling rate Fs = 5000
samples/s. What is the discrete-time signal obtained after sampling?
(c) What is the analog signal y„(r) we can reconstruct from the samples if we use
ideal interpolation?
— = 2.5 kHz
2
and this is the maximum frequency that can be represented uniquely by the
sampled signal. By making use of (1.4.2) we obtain
= 3 cos 2jt( j )h + 5 sin 2n- (^ )« + 10 cos 2n(^)n
= 3 c o s2 ;r(|)n + 5 sin 2 7 r(l — §)« 4- 10cos2,t(1 4- ^)u
= 3cos2jr({)n 4- 5sin27r(—1)« 4- 1 0 c o s 2 ^ (|)«
Finally, we obtain
x(n) = 13 cos2^({)/i - 5sin27r( = )fl
The same result can be obtained using Fig. 1.17. Indeed, since F, = 5 kHz.
the folding frequency is FJ2 = 2.5 kHz. This is the maximum frequency that
can be represented uniquely by the sampled signal. From (1.4.17) we have
Fti = Fk — kFs. Thus Fo can be obtained by subtracting from Fk an integer
m ultiple of Fs such that —F t/2 < F0 < F J 2. The frequency F: is less than Fsf2
and thus it is not affected by aliasing. However, the other two frequencies are
above the folding frequency and they will be changed by the aliasing effect.
Indeed.
F'2 = Fj ~ Fs = - 2 kHz
f;
= Fj — Fs = 1 kHz
From (1.4.5) it follows that /] =
with the result above.
h —
and h = ^ which are in agreem ent
Sec. 1.4
A nalog-toD igital and Digital-to-Analog Conversion
33
(c) Since only the frequency components at 1 kHz and 2 kHz are present in the
sampled signal, the analog signal we can recover is
x„(t) =
13 cos 2000^r - 5sin400()-Tf
which is obviously different from the original signal x„U). This distortion of the
original analog signal was caused by the aliasing effect, due to the low sampling
rate used.
A lth o u g h aliasin g is a pitfall to be av o id ed , th e re are tw o useful p ractical
a p p licatio n s b ased on th e ex p lo itatio n of the aliasing effect. T h e se a p p licatio n s
are th e stro b o sc o p e an d the sam pling oscilloscope. B o th in stru m e n ts are d esigned
to o p e ra te as aliasin g devices in o rd e r to re p re se n t high fre q u e n c ie s as low f re ­
q u en cies.
T o e la b o ra te , co n sid er a signal w ith h ig h -freq u en cy c o m p o n e n ts confined to
a given fre q u e n c y b an d B\ < F < B2. w h ere Bz — B\ = B is d efined as the
b a n d w id th o f th e signal. W e assum e th a t B < < B\ < B 2. T h is co n d itio n m ean s
th a t th e fre q u e n c y c o m p o n e n ts in the signal are m uch larg er th an th e b an d w id th
B of th e signal. Such signals are usually called p a ssb a n d or n a rro w b a n d signals.
N ow . if this signal is sa m p le d at a rate Fs > 2B. b u t F^ << B\. th e n all th e f re ­
q u en cy c o m p o n e n ts c o n ta in e d in the signal will be aliases of fre q u e n c ie s in the
ran g e 0 < F < F J 2 . C o n seq u en tly , if we o b se rv e the freq u e n cy c o n te n t of the
signal in th e fu n d a m e n ta l range 0 < F < F J 2 . we k now precisely the freq u e n cy
co n te n t o f th e an a lo g signal since we k now the fre q u e n c y b an d B\ < F < B2 u n d e r
c o n sid e ra tio n . C o n se q u e n tly , if the signal is a n a rro w b a n d (p a ss b a n d ) signal, we
can re c o n stru c t th e o riginal signal from the sam p le s, p ro v id e d th a t the signal is
sa m p le d at a ra te Fs > 2 B. w h ere B is th e b an d w id th . T h is s ta te m e n t c o n stitu te s
a n o th e r fo rm o f th e sam pling th e o re m , w hich we call the p a s s b a n d f o r m in o rd e r
to d istin g u ish it fro m th e p rev io u s form o f the sa m p lin g th e o re m , w hich ap p lies in
g en eral to all ty p es of signals. T he la tte r is so m e tim es called th e bas e ban d f or m.
T h e p a s s b a n d f o r m o f th e sam pling th e o re m is d escrib ed in d e ta il in S ectio n 9.1.2.
1.4.3 Quantization of Continuous-Amplitude Signals
A s w e h av e se en , a dig ital signal is a se q u en ce of n u m b e rs (sa m p le s) in w hich each
n u m b e r is re p re s e n te d by a finite n u m b e r of digits (finite p recisio n ).
T h e p ro c e ss o f co n v ertin g a d isc rete-tim e c o n tin u o u s-a m p litu d e signal in to a
dig ital signal by ex p ressin g each sa m p le value as a finite (in ste a d of an infinite)
n u m b e r o f d igits, is called quant izati on. T he e rro r in tro d u c e d in re p re se n tin g th e
c o n tin u o u s-v a lu e d signal by a finite set o f d isc rete v alu e levels is called quant i zat i on
error o r quant i zat i on noise.
W e d e n o te th e q u a n tiz e r o p e ra tio n o n th e sa m p le s x ( n ) as Q[x{n)] an d let
x q{n) d e n o te th e se q u en ce o f q u a n tiz e d sam ples a t th e o u tp u t o f th e q u a n tiz e r.
H e n ce
Xq(n)
= Q[x(n)]
34
Introduction
Chap. 1
T h e n th e q u a n tiz a tio n e r ro r is a se q u e n c e eq (n) d efin ed as th e d iffe re n c e b etw e e n
th e q u a n tiz e d v alu e a n d th e actu al sa m p le value. T h u s
eq (n) = x q {n) - x ( n )
W e illu strate th e q u a n tiz a tio n p ro cess w ith a n ex am p le.
d isc rete-tim e signal
(1.4.25)
L e t us co n sid e r the
o b ta in e d by sa m p lin g th e an a lo g e x p o n e n tia l signal x a( t) = 0 .9 ', t > 0 w ith a
sam p lin g freq u e n cy f , = 1 H z (see Fig. 1.20(a)). O b se rv a tio n o f T a b le 1.2, w hich
show s th e v alu es o f th e first 10 sa m p les o f x ( n ) , rev eals th a t th e d e sc rip tio n o f the
sam p le v alu e x{n) re q u ire s n significant digits. It is o b v io u s th a t th is signal can n o t
(a)
Figure 1.20
Illustration of quantization.
Sec. 1.4
35
Analog-to-Digital and Digital-to-Analog Conversion
TA B LE 1.2
NU M ER IC A L ILLU S TR A TIO N O F Q U A N TIZA TIO N W ITH ONE
S IG N IF IC A N T D IG IT USING T R U N C A T IO N O R R O UN DING
x,(n )
(Truncation)
(Rounding)
1
0.9
1.0
0.9
1.0
0.9
3
4
5
6
7
0.81
0.729
0.6561
0.59049
0.531441
0.4782969
0.8
0.7
0.8
0.7
0.7
8
9
0.43046721
0.387420489
0.4
x ( n)
n
0
1
2
D iscrete-tim e signal
.voi)
0.6
0.5
0.5
0.4
0.3
0.6
0.5
0.5
0.4
0.4
(Rounding)
0.0
0.0
0.01
- 0.029
-
0.0439
0.00951
- 0.031441
-
0.0217031
0.03046721
0.012579511
be p ro cessed by u sing a c a lcu lato r o r a digital c o m p u te r since only the first few
sam p les can be sto re d an d m a n ip u la te d . F o r ex am p le, m ost c alcu lato rs process
n u m b e rs w ith only eig h t significant digits.
H o w e v e r, let us assum e th a t w e w an t to use only o n e significant digit. To
elim in ate th e excess digits, w e can e ith e r sim ply d iscard th em (tru n ca tion ) o r dis­
card th e m bv ro u n d in g th e resu ltin g n u m b e r (ro un din g). T h e resu ltin g q u an tized
signals x q (n) a re show n in T ab le 1.2. W e discuss only q u a n tiz a tio n by ro u n d in g ,
alth o u g h it is ju st as easy to tr e a t tru n c a tio n . T h e ro u n d in g p ro c e ss is g raphically
illu stra te d in Fig. 1.20b. T h e values allo w ed in th e digital signal are called the
quan tizatio n levels, w h ereas the d istan c e A b etw een tw o successive q u a n tiz a tio n
levels is called th e q uantization step siz e o r resolution. T h e ro u n d in g q u a n tiz e r
assigns each sa m p le of x ( n ) to th e n e a re s t q u a n tiz a tio n level. In c o n tra st, a q u a n ­
tizer th a t p e rfo rm s tru n c a tio n w ould have assigned each sa m p le of jc(/z) to the
q u a n tiz a tio n level b elo w it. T h e q u a n tiz a tio n e r ro r eq (n) in ro u n d in g is lim ited to
th e ra n g e of —A /2 to A /2 , th a t is,
A
A
- y <<?,(«)<•f
(1A26)
In o th e r w o rd s, th e in sta n ta n e o u s q u a n tiz a tio n e r ro r c a n n o t exceed half of the
q u a n tiz a tio n ste p (see T a b le 1.2).
If jcmjn an d j:max r e p re s e n t th e m in im u m an d m ax im u m v alu e of x (n ) a n d L
is th e n u m b e r o f q u a n tiz a tio n levels, th en
A = Xmax ~ ^
L - 1
(1.4.27)
W e d efin e th e d y n a m i c range of th e signal as jrmax — -*min- 1° o u r ex am p le we
h av e Jtmax = 1, Jtmjn = 0, a n d L — 11, w hich leads to A = 0.1. N o te th a t if the
d y n am ic ran g e is fixed, in creasin g th e n u m b e r o f q u a n tiz a tio n levels, L resu lts in a
d e c re a se o f th e q u a n tiz a tio n ste p size. T h u s th e q u a n tiz a tio n e r ro r d e c re a s e s and
th e accu racy o f th e q u a n tiz e r in crease s. In p ra c tic e w e can re d u c e th e q u a n tiz a tio n
36
Introduction
Chap. 1
error to an insignificant am ount by choosing a sufficient num ber o f quantization
levels.
T h eoretically, quantization o f analog signals always resu lts in a loss o f in­
form ation. T his is a result o f the am biguity introduced by quantization. Indeed,
quantization is an irreversible or noninvertible process (i.e., a m any-to-on e m ap­
ping) since all sam ples in a distance A /2 about a certain quantization level are
assigned the sam e value. This am biguity m akes the exact quantitative analysis of
quantization extrem ely difficult. This subject is discussed further in C hapter 9,
where w e use statistical analysis.
1.4.4 Quantization of Sinusoidal Signals
Figure 1.21 illustrates the sam pling and quantization o f an an alog sinusoidal signal
x a (/) = A cos
using a rectangular grid. H orizontal lines w ithin the range of the
quantizer indicate the allow ed levels o f quantization. V ertical lines indicate the
sam pling tim es. Thus, from the original analog signal x a{t) w e obtain a discrete­
tim e signal x ( n ) = x a {nT) by sam pling and a discrete-tim e, discrete-am plitude
signal x q ( nT) after quantization. In practice, the staircase sign al xv (r) can be
obtained by using a zero-order hold. This analysis is useful b ecau se sinusoids are
used as test signals in A /D converters.
If the sam pling rate Fs satisfies the sam pling theorem , quantization is the only
error in the A /D conversion process. Thus w e can evalu ate the quantization error
Time
Figure L21
Sampling and quantization of a sinusoidal signal.
Sec. 1.4
Analog-to-Digital and Digital-to-Analog Conversion
37
bv q u a n tiz in g th e an alo g signal x„(t) in stead of th e d isc re te -tim e signal a (« ) =
x a( nT) . In sp e c tio n o f Fig. 1.21 in d icates th at the signal x u {t) is alm o st lin ear
b e tw e e n q u a n tiz a tio n levels (see Fig. 1.22). T he c o rre sp o n d in g q u a n tiz a tio n e rro r
eq (t) — x u(t) — x q(t) is show n in Fig. 1.22. In Fig. 1.22. r d e n o te s th e tim e th at
x a {t) stay s w ithin th e q u a n tiz a tio n levels. T he m e a n -sq u a re e r ro r p o w e r Pq is
Pq = 2 t /
e«{!)cJl = 7 j
(1.4.28)
Since eq (i) = ( A / 2 r ) t . - t < t < t . we have
If th e q u a n tiz e r has b bits of accuracy an d the q u a n tiz e r co v ers the e n tire range
2A . th e q u a n tiz a tio n step is A = 2 A / 2 h. H en ce
A 2/ 3
P., =
(1.4.30)
T h e a v erag e p o w e r o f the signal xu(D is
1 f 1'
,
A2
P, = — / (A cos Qi,i r d i = ~
TP Jo
2
(1.4.31)
T h e q u ality o f th e o u tp u t o f the A /D c o n v e rte r is usually m e a s u re d by th e signalto- quanti zati on noise ratio ( S Q N R ) . w hich pro v id es th e ratio o f th e signal p o w er
to th e no ise po w er:
S Q N R = — = - • 22b
P
">
' i/
E x p re s se d in d ecib els (d B ), th e S Q N R is
S Q N R (d B ) = 101og1(l S Q N R = 1.76 + 6.026
(1.4.32)
T his im p lies th a t th e S Q N R in crease s a p p ro x im ately 6 dB fo r ev ery bit a d d e d to
th e w o rd le n g th , th a t is. fo r each d o u b lin g of th e q u a n tiz a tio n levels.
A lth o u g h fo rm u la (1.4.32) was d eriv ed fo r sin u so id al signals, w e shall see in
C h a p te r 9 th a t a sim ilar resu lt holds fo r every signal w hose d y n am ic ran g e sp a n s the
ran g e o f th e q u a n tiz e r. T h is re la tio n sh ip is ex trem ely im p o rta n t b e c a u se it d ictates
Figure 1.22
T he quantization error eq (t) — x a (t) - x q (t).
38
Introduction
Chap. 1
th e n u m b e r of bits re q u ire d by a specific a p p lic a tio n to assu re a given signal-ton o ise ratio . F o r ex am p le, m o st co m p a c t disc p lay ers use a sa m p lin g freq u e n cy
o f 44.1 k H z an d 16-bit sa m p le re so lu tio n , w hich im plies a S Q N R of m o re th an
96 dB .
1.4.5 Coding of Quantized Samples
T h e co d in g p ro cess in an A /D c o n v e rte r assigns a u n iq u e b in a ry n u m b e r to each
q u a n tiz a tio n level. If we h av e L levels w e n e e d at least L d iffe re n t binary' n u m b ers.
W ith a w ord len g th o f b bits w e can c re a te 2b d iffe re n t b in ary n u m b e rs. H e n c e we
h av e 2h > L. o r e q u iv alen tly , b > log 2 L. T h u s th e n u m b e r of bits re q u ire d in the
co d e r is th e sm allest in te g e r g re a te r th a n o r eq u al to log 2 L. In o u r ex am p le it can
easily be seen th a t we n eed a c o d e r w ith b = 4 bits. C o m m ercially av ailab le A /D
c o n v e rte rs m ay be o b ta in e d w ith finite p recisio n of b — 16 o r less. G e n e ra lly , the
h ig h er th e sam p lin g sp e ed a n d th e fin er th e q u a n tiz a tio n , th e m o re ex p en siv e the
d evice becom es.
1.4.6 Digital-to-Analog Conversion
Am plifude
T o co n v ert a d igital signal in to an an a lo g signal we can use a d ig ital-to -an alo g
(D /A ) co n v erter. A s sta te d p rev io u sly , th e task o f a D /A c o n v e rte r is to in te rp o la te
b e tw e e n sam ples.
T h e sam p lin g th e o re m specifies th e o p tim u m in te rp o la tio n fo r a bandlim ited signal. H o w ev er, this ty p e o f in te rp o la tio n is to o c o m p lic a te d an d . h en ce
im p ractical, as in d icated p rev io u sly . F ro m a p ractical v iew p o in t, the sim p lest D /A
co n v e rte r is th e z e ro -o rd e r h o ld show n in Fig. 1.15. w hich sim p ly holds c o n sta n t
th e v alu e o f o n e sam p le u n til th e n ex t o n e is receiv ed . A d d itio n a l im p ro v e m e n t
can b e o b ta in e d by u sing lin e a r in te rp o la tio n as show n in Fig. 1.23 to c o n n e c t
successive sa m p les w ith stra ig h t-lin e se g m en ts. T h e z e ro -o rd e r hold an d lin ear
in te rp o la to r are an aly zed in S ectio n 9.3. B e tte r in te rp o la tio n can be ach iev ed by
u sing m o re so p h isticated h ig h e r-o rd e r in te rp o la tio n tech n iq u es.
In g en eral, su b o p tim u m in te rp o la tio n te c h n iq u e s re su lt in p assin g fre q u e n c ie s
ab o v e th e fo ld in g freq u e n cy . S uch fre q u e n c y c o m p o n e n ts are u n d e s ira b le a n d are
u su ally rem o v ed by p assin g th e o u tp u t o f th e in te rp o la to r th ro u g h a p ro p e r an alo g
Sec. 1.5
Summ ary and References
39
filter, w hich is called a postfilter or sm o o th in g filter. T h u s D /A c o n v ersio n usually
involves a su b o p tim u m in te rp o la to r follow ed by a p ostfilter. D /A c o n v e rte rs are
tre a te d in m o re d e ta il in S ectio n 9.3.
1.4.7 Analysis of Digital Signals and Systems Versus
Discrete-Time Signals and Systems
W e h av e seen th a t a d igital signal is defin ed as a fu n ctio n o f an in te g e r in d e p e n d e n t
v ariab le an d its v alu es are ta k e n from a finite set of po ssib le values. T he usefulness
of such signals is a c o n s eq u en ce of th e possibilities o ffe re d by digital co m p u ters.
C o m p u te rs o p e ra te on n u m b ers, w hich are re p re s e n te d by a strin g of 0 's an d l's .
T h e len g th of th is strin g (w o r d length) is fixed an d finite an d usually is 8. 12. 16. or
32 bits. T h e effe cts o f finite w ord len g th in c o m p u ta tio n s cause co m p licatio n s in
th e an aly sis of d ig ital signal p ro cessin g system s. T o avoid th ese co m p licatio n s, we
neg lect th e q u a n tiz e d n a tu re of digital signals an d system s in m uch o f o u r analysis
an d c o n sid e r th em as d isc rete-tim e signals an d system s.
In C h a p te rs 6 . 7. and 9 we in v estig ate th e c o n se q u e n c e s o f using a finite w ord
len g th . T h is is an im p o rta n t topic, since m any digital signal p ro cessin g p ro b lem s are
solved w ith sm all c o m p u te rs o r m icro p ro cesso rs th at em p lo y fix ed -p o in t arith m etic.
C o n se q u e n tly , o n e m ust look carefully at the p ro b le m of fin ite-p recisio n arith m e tic
an d a c c o u n t for it in th e design of so ftw are an d h a rd w a re th at p e rfo rm s the d esired
signal p ro cessin g tasks.
1.5 SUMMARY AND REFERENCES
In th is in tro d u c to ry c h a p te r w e have a tte m p te d to p ro v id e the m o tiv a tio n for digital
signal p ro cessin g as an a lte rn a tiv e to a n a lo g signal pro cessin g . W e p re se n te d the
basic e le m e n ts o f a digital signal p ro cessin g system an d d efin ed th e o p e ra tio n s
n e e d e d to c o n v e rt an an alo g signal in to a digital signal re a d y fo r processing. O f
p a rtic u la r im p o rta n c e is the sam pling th e o re m , w hich w as in tro d u c e d by N vquist
(1928) an d la te r p o p u la riz e d in the classic p a p e r by S h a n n o n (1949). T h e sam pling
th e o re m as d e sc rib e d in S ection 1.4.2 is d eriv ed in C h a p te r 4. S in u so id al signals
w ere in tro d u c e d p rim a rily fo r the p u rp o se of illu stra tin g th e aliasin g p h e n o m e n o n
an d fo r th e s u b s e q u e n t d e v e lo p m e n t o f th e sa m p lin g th e o re m .
Q u a n tiz a tio n effects th a t are in h e re n t in the A /D co n v e rsio n of a signal w ere
also in tro d u c e d in th is c h a p te r. Signal q u a n tiz a tio n is b est tre a te d in statistical
term s, as d esc rib e d in C h a p te rs 6 , 7. an d 9.
F in ally , th e to p ic o f signal re c o n stru c tio n , o r D /A co n v e rsio n , w as d escrib ed
briefly. Signal re c o n stru c tio n b ased on sta ircase o r lin e a r in te rp o la tio n m eth o d s is
tre a te d in S ectio n 9.3.
T h e re a re n u m e ro u s p ractical a p p lic a tio n s of d igital signal processing. T he
b o o k e d ite d by O p p e n h e im (1978) tre a ts a p p lic a tio n s to sp e ech p rocessing, im age
p ro cessin g , ra d a r signal pro cessin g , so n a r signal p ro cessin g , a n d g eophysical signal
p ro cessin g .
40
Introduction
Chap. 1
PROBLEMS
LI Classify the following signals according to whether they are (1) one- or multi­
dimensional; (2) single or multichannel, (3) continuous time or discrete time, and
(4) analog or digital (in amplitude). Give a brief explanation.
(a) Closing prices of utility stocks on the New York Stock Exchange.
(b) A color movie.
(c) Position of the steering wheel of a car in motion relative to car’s reference frame.
(d) Position of the steering wheel of a car in motion relative to ground reference
frame.
(e) Weight and height measurem ents of a child taken every month.
1.2 D eterm ine which of the following sinusoids are periodic and com pute their funda­
mental period.
30n \
( 62m
(a) cosO.OIjt/i
(b) c°s n ——- I
(c) cos 3™
(d) sin3«
(e) sin
105 /
'
'
V 10
1 3 Determ ine whether or not each of the following signals is periodic. In case a signal
is periodic, specify its fundamental period.
(a) xu(r) — 3cos(5r + 7r/6)
(b)
= 3 cos(5n + ;r/6)
(c) j:(h) = 2 e x p [j(n /6 - 7i)]
(d) x(n) = cos(«/8) cos(?rn/8)
(e) x(n) = cos(7rn/2) — sin(7rn/8) + 3cos(jrn/4 + 7t / 3)
1.4 (a) Show that the fundamental period Nr of the signals
,s>(«) = ei2nkr,IN.
* = 0 .1 .2 ,...
is given by Np = N / C C D ( k . N ), where GCD is the greatest common divisor of k
and N.
(b) What is the fundam ental period of this set for N =71
(c) What is it for N = 16?
1.5 Consider the following analog sinusoidal signal:
xa(t) — 3 sin(1007rr)
(a) Sketch the signal xa{t) for 0 < t < 30 ms.
(b) The signal xa(t) is sampled with a sampling rate Fs = 300 samples/s. Determ ine
the frequency of the discrete-time signal x{n) = xa( nT), T = 1 /F„. and show that
it is periodic.
(c) Compute the sample values in one period of .x(n). Sketch _*<n) on the same
diagram with x„(t). W hat is the period of the discrete-time signal in milliseconds?
(d) Can you find a sampling rate Fs such that the signal x(n) reaches its peak value
of 3? W hat is the minimum Fs suitable for this task?
L6 A continuous-time sinusoid xa(t) with fundamental period Tp = 1/F0 is sampled at a
rate F, = 1/ T to produce a discrete-time sinusoid x(n) = x„(,nT).
(») Show that x(n) is periodic if T / T p = k / N (i.e., T/ T p is a rational num ber).
(b) If x(n) is periodic, what is its fundam ental period Tp in seconds?
Chap. 1
41
Problem s
(c) Explain the statement: ,r(n) is periodic if its fundamental period Tr . in seconds,
is equal to an integer number of periods of .v„u).
1.7 An analog signal contains frequencies up to 10 kHz.
(a) W hat range of sampling frequencies allows exact reconstruction of this signal
from its samples?
(b ) Suppose that we sample this signal with a sampling frequency F, = 8 kHz. Ex­
amine what happens to the frequency F| = 5 kHz.
(c) Repeat part (b) for a frequency F> = 9 kHz.
1.8 An analog electrocardiogram (ECG) signal contains useful frequencies up to 100 Hz.
(a) W hat is the Nvquist rate for this signal?
(b ) Suppose that we sample this signal at a rate of 250 samples/s. What is the highest
frequency that can be represented uniquely at this sampling rate?
1.9 An analog signal a „ u ) = sin(480;rr) + 3sin(720:rr) is sampled 600 times per second.
(a) D eterm ine the Nvquist sampling rate for x a{t).
(b ) D eterm ine the folding frequency.
(c) What are the frequencies, in radians, in the resulting discrete time signal t (/j )?
(d) If
is passed through an ideal D/A converter, what is the reconstructed signal
v„(n?
1.10 A digital communication link carries binary-coded words representing samples of an
input signal
(t) — 3 cos 600.7 r
2 cos 1800jt /
The link is operated at 10.000 bits/s and each input sample is quantized into 1024
different voltage levels.
(a) W hat is the sampling frequency and the folding frequency?
( b ) W hat is the Nvquist rate for the signal .*„(;)?
(c) What are the frequencies in the resulting discrete-time signal x(n)7
(d) W hat is the resolution A?
1.11 Consider the simple signal processing system shown in Fig. P I.11. The sampling
periods of the A/D and D /A converters are T = 5 ms and T' = 1 ms. respectively.
Determ ine the output v„U) of the system, if the input is
a:„(n = 3 cos 100;rf -+- 2 sin 250;rt
(t in seconds)
The postfilter removes any frequency component above F J 2.
Figure P l . l l
1.12 (a) Derive the expression for the discrete-time signal .r(n) in Example 1.4.2 using the
periodicity properties of sinusoidal functions.
(b) W hat is the analog signal we can obtain from x(n) if in the reconstruction process
we assume that Fs = 10 kHz?
42
Introduction
Chap. 1
1.13 The discrete-time signal x(n) = 6.35cos(jr/10)n is quantized with a resolution (a) A =
0.1 or (b) A = 0.02. How many bits are required in the A/D converter in each case?
1.14 D eterm ine the bit rate and the resolution in the sampling of a seismic signal with
dynamic range of 1 volt if the sampling rate is Fs = 20 samples/s and we use an S-bit
A/D converter? W hat is the maximum frequency that can be present in the resulting
digital seismic signal?
1.15* Sampling o f sinusoidal signals: aliasing Consider the following continuous-time si­
nusoidal signal
XoO) = sin 2jr/r0f,
—oc < t < oo
Since xa(t) is described m athematically, its sampled version can be described by values
every T seconds. The sampled signal is described by the formula
where Fs = l / T is the sampling frequency.
(a) Plot the signal j:(n), 0 < n < 99 for F, = 5 kHz and Fu = 0.5, 2, 3, and 4.5 kHz.
Explain the similarities and differences am ong the various plots.
(b) Suppose that F0 = 2 kHz and Fs = 50 kHz.
(1) Plot the signal x(n). W hat is the frequency / (l of the signal je(n)?
(2) Plot the signal v(n) created by taking the even-numbered samples of x(n).
Is this a sinusoidal signal? Why? If so, what is its frequency?
1.16* Quantization error in A /D conversion o f a sinuoidal signal Let xq(n) be the signal
obtained by quantizing the signal x(n) = sin27r/on. The quantization error power PQ
is defined by
The “quality” of the quantized signal can be measured by the signal-to-quantization
noise ratio (SQ NR) defined by
SQ N R = 10 log,0
“a
where Px is the power of the unquantized signal x (n).
(«) For /o = 1/50 and N = 200, write a program to quantize the signal Jt(n), using
truncation, to 64, 128, and 256 quantization levels. In each case plot the signals
x (n), Xq(n), and e(n) and com pute the corresponding SQNR.
(b) Repeat part (a) by using rounding instead of truncation.
(c) Comment on the results obtained in parts (a) and (b).
(d) Compare the experimentally m easured SQNR with the theoretical SQNR pre­
dicted by formula (1.4.32) and com m ent on the differences and similarities.
2
Discrete-Time Signals and
Systems
In C h a p te r 1 w e in tro d u c e d th e re a d e r to a n u m b e r o f im p o rta n t ty p es o f signals
an d d escrib ed th e sa m p lin g p ro cess bv w hich an a n a lo g signal is c o n v e rte d to a
d isc rete-tim e signal. In a d d itio n , we p re s e n te d in so m e d etail th e c h a racteristics
o f d isc re te -tim e sin u so id al signals. T h e sin u so id is an im p o rta n t e le m e n ta ry signal
th a t se rv es as a b asic b u ild in g block in m o re co m p lex signals. H o w e v e r, th e re are
o th e r e le m e n ta ry signals th a t are im p o rta n t in o u r tr e a tm e n t o f signal processing.
T h ese d isc re te -tim e signals a re in tro d u c e d in this c h a p te r a n d are used as basis
fu n ctio n s o r b u ild in g b locks 1o d escrib e m o re com plex signals.
T h e m a jo r e m p h asis in this c h a p te r is th e c h a ra c te riz a tio n o f d isc rete-tim e
sy stem s in g en era! a n d the class o f lin ear tim e -in v a ria n t (L T I) system s in p articu lar.
A n u m b e r o f im p o rta n t tim e-d o m ain p ro p e rtie s o f L T I sy stem s a re d efined and
d e v e lo p e d , an d an im p o rta n t fo rm u la, called th e c o n v o lu tio n fo rm u la, is d eriv ed
w hich allow s us to d e te rm in e th e o u tp u t of an L T I system to any given a rb itra ry
in p u t signal. In a d d itio n to th e c o n v o lu tio n fo rm u la , d iffe re n c e e q u a tio n s are in ­
tro d u c e d as an a lte rn a tiv e m e th o d fo r describ in g th e in p u t- o u tp u t re la tio n sh ip of
an L T I sy stem , an d in a d d itio n , recursive an d n o n re c u rsiv e re a liz a tio n s of LTI
sy stem s are tre a te d .
O u r m o tiv a tio n fo r th e em p h asis on th e stu d y o f L T I sy stem s is tw ofold. F irst,
th e re is a larg e co llectio n o f m a th e m a tic a l te c h n iq u e s th a t can be ap p lied to the
an aly sis o f L T I system s. S eco n d , m an y p ractical system s a re e ith e r L T I system s
o r can b e a p p ro x im a te d by L T I system s. B ecau se of its im p o rta n c e in digital
signal p ro cessin g a p p licatio n s an d its close re se m b la n c e to th e co n v o lu tio n form ula,
we also in tro d u c e th e c o rre la tio n b e tw e e n tw o signals. T h e a u to c o rre la tio n and
c ro ssc o rre la tio n o f signals a re defined an d th e ir p ro p e rtie s a re p re se n te d .
2.1 DISCRETE-TIME SIGNALS
A s w e d iscu ssed in C h a p te r 1, a d isc re te -tim e signal x{n) is a fu n ctio n o f an in d e ­
p e n d e n t v a ria b le th a t is an in teg er. It is g rap h ically re p re s e n te d as in Fig. 2.1. It
is im p o rta n t to n o te th a t a d isc re te -tim e signal is n o t defined at in sta n ts b etw een
43
44
Discrete-Time Signals and Systems
Figure 2-1
Chap. 2
Graphical representation of a discrete-time signal.
tw o successive sam p les. A lso , it is in c o rre c t to th in k th a t .v(n) is e q u a l to z e ro if n
is n o t an in teg er. S im ply, th e signal x ( n ) is n o t defin ed fo r n o n in te g e r v alu es o f n.
In th e se q u el w e will assu m e th a t a d isc re te -tim e signal is d efin ed fo r every
in te g e r value n fo r —oo < n < oc. By tra d itio n , w e re fe r to x( n) as th e “n th sa m p le ”
o f th e signal ev en if th e signal x( n) is in h e re n tly d isc re te tim e (i.e., n ot o b ta in e d
by sam p lin g an a n a lo g signal). If, in d e e d , x ( n) w as o b ta in e d fro m sa m p lin g an
a n alo g signal x a( t ), th e n .i(n ) = x a( nT) , w h ere T is th e sa m p lin g p e rio d (i.e., th e
tim e b etw e e n successive sam p les).
B e sid es th e g rap h ical re p re s e n ta tio n of a d isc re te -tim e signal o r se q u e n c e as
illu strated in Fig. 2.1. th e re a re som e a lte rn a tiv e re p re s e n ta tio n s th at are o ften
m o re co n v e n ie n t to use. T h e se are:
1. F u n ctio n al re p re s e n ta tio n , such as
x(n) —
f 1,
fo r n = 1, 3
[ 0,
else w h e re
I4, fo r n = 2
(2 . 1.1)
2. T a b u la r r e p re s e n ta tio n , such as
n
•••
-2
-1
0
1
2
3
4
5
x ( n)
■■■
0
0
0
1
4
1
0
0
3. S eq u en ce re p re s e n ta tio n
A n in fin ite -d u ra tio n signal o r se q u e n c e w ith th e tim e o rig in (n = 0) in d ic a te d
by th e sy m b o l | is re p re s e n te d as
*<n) = { . . . 0 . 0 . 1 . 4 , 1 . 0 , 0 , . . . }
T
(2.1.2)
A se q u e n c e j:(n ), w hich is z e ro fo r n < 0, can be re p re s e n te d as
jc(«) =
{ 0 ,1 .4 .1 .0 .0 ....}
T
(2.1.3)
T h e tim e o rig in fo r a se q u e n c e x ( n ) , w hich is z e ro fo r n < 0, is u n d e rs to o d to be
th e first (le ftm o st) p o in t in th e seq u en ce.
Sec. 2.1
45
Discrete-Time Signals
A fin ite -d u ra tio n se q u e n c e can be re p re se n te d as
T
x i n) = {3. - 1 . - 2 . 5 .0 .4 . -1 }
(2.1.4)
w h ereas a fin ite -d u ra tio n se q u e n c e th a t satisfies the c o n d itio n x(/i) ~ 0 for n < 0
can be r e p re s e n te d as
T
jc(n) = { 0 .1 .4 . 1)
(2.1.5)
T h e signal in (2.1.4) co n sists of seven sa m p le s or p o in ts (in tim e), so it is called or
id en tified as a se v e n -p o in t se q u e n c e . S im ilarly, the se q u e n c e given by (2.1.5) is a
f o u r-p o in t se q u e n c e .
2.1.1 Some Elementary Discrete-Time Signals
In o u r stu d y o f d isc re te -tim e signals an d system s th e re a re a n u m b e r o f b asic signals
th at a p p e a r o ften a n d play an im p o rta n t role. T h ese signals a re d efin ed below .
1. T h e unit s a m p l e se quence is d e n o te d as <5(n) an d is defin ed as
<5( / i )
=
0.
for n - 0
for n ^ 0
( 2 . 1. 6 )
In w o rd s, th e u n it sa m p le se q u e n c e is a signal th at is zero e v erv w h ere, ex cep t
a t n — 0 w h e re its value is unity. T his signal is so m e tim e s re fe rre d to as a
unit impul se. In c o n tra st to the an alo g signal 8(t). w hich is also called a
u n it im p u lse an d is d efin ed to be ze ro ev ery w h ere ex cep t / = 0. an d has unit
a re a , th e u n it sa m p le se q u e n c e is m uch less m ath e m a tic a lly c o m p licated . T he
g rap h ical re p re s e n ta tio n o f <5(n ) is show n in Fig. 2.2.
2. T h e unil step signal is d e n o te d a s w (n ) an d is defin ed as
u(n) =
1.
0.
fo r n > 0
fo r n < 0
(2.1.7)
F ig u re 2.3 illu stra te s th e u nit ste p signal.
3. T h e uni t r a m p signal is d e n o te d as u r (n) an d is d efin ed as
u r (n) -
fo r n > 0
fo r n < 0
( 2 . 1.
T h is signal is illu stra te d in Fig. 2.4.
fi(n)
Figure 2.2 G raphical rep resen tatio n of
the unit sample signal.
46
Discrete-Tim e Signals and Systems
Chap. 2
u(n)
T
0
12
3 4 5 6 7
n
Figure 2 3 G raphical rep resen tatio n of
the unit step signal.
n
Figure 2,4 G raphical rep resen tatio n of
the unit ram p signal.
ur(n)
T
4. T h e exponent i al signal is a se q u e n c e o f th e fo rm
jr(n) = a ”
fo r all n
(2.1.9)
If th e p a r a m e te r a is real, th e n jr(n) is a real signal. F ig u re 2.5 illu stra te s x ( n)
fo r v ario u s v alu es o f th e p a r a m e te r a.
W h en th e p a ra m e te r a is co m p lex v a lu e d , it can b e e x p re ss e d as
a s rejf1
w h e re r an d 6 a re n o w th e p a ra m e te rs. H e n c e w e c a n e x p ress x ( n ) as
x ( n ) = r nej0n
(2 . 1. 10 )
= r n (cos On -j- y s in # n )
TiinilU
Figure 2.5
G raphical representation of exponential signals.
Sec. 2.1
47
Discrete-Time Signals
Since x (/j ) is now co m plex valu ed , it can be re p re s e n te d g rap h ically by p lo ttin g
th e real p a rt
x K(n) = r" cos6#fi
(2.1.11)
as a fu n ctio n of n. an d se p a ra te ly p lo ttin g th e im ag in ary p a rt
xi (n) = r ’1sin 6n
(2.1.12)
as a fu n ctio n o f n. F ig u re 2.6 illu strates th e g ra p h s o f x R(n) a n d x / ( n ) for r — 0.9
an d 6 = tt/1 0 . W e o b se rv e th a t th e signals x R(?i) a n d x / ( n ) a re a d a m p e d (decaying
e x p o n e n tia l) co sin e fu n ctio n an d a d a m p e d sine fu n c tio n . T h e angle v ariab le 6
is sim ply th e freq u e n c y of th e sinusoid, p rev io u sly d e n o te d by th e (n o rm alized )
fre q u e n c y v ariab le w. C learly, if r — 1. th e d a m p in g d is a p p e a rs an d x K(n). x/ ( n).
an d A'(n) h av e a fixed am p litu d e, w hich is unity.
A lte rn a tiv e ly , th e signal .*(/?) given by (2.1.10) can be re p re s e n te d g raphically
by th e a m p litu d e fu n ctio n
|.i(h )| = A{n) = r ”
(2.1.13)
_ v (/;) = <f>{n) = Bn
(2.1.14)
an d th e p h ase fu n ctio n
F ig u re 2.7 illu stra te s -4(/?) an d 0 (/i) fo r r — 0.9 a n d B = ,t/1 0 . W e o b se rv e th at
th e p h ase fu n ctio n is lin ear w ith n. H o w ev er, th e p h ase is d efin e d only o v e r the
in terv al —n < B < t t o r. eq u iv alen tly , o v er th e in terv al 0 < 6 < 2 t t . C o n seq u en tly ,
by co n v e n tio n 4>(n) is p lo tte d o v er th e finite in terv al —n < B < t t o t Q < $ < 2tt.
In o th e r w o rd s, w e su b tra c t m u ltip lies o f I n fro m <p(n) b e fo re p lo ttin g . In one
case. <p(n) is c o n s tra in e d to th e range —n < 0 < n a n d in th e o th e r case <p(n) is
c o n stra in e d to th e ran g e 0 < fi < 2n. T h e su b tra c tio n o f m u ltip le s o f 2n from <p(n)
is e q u iv a le n t to in te rp re tin g th e fu n ctio n 4>(n) as 4>{n), m o d u lo 2n. T he g rap h for
<p{n). m o d u lo 2n . is show n in Fig. 2.7b.
2.1.2 Classification of Discrete-Time Signals
T h e m a th e m a tic a l m e th o d s e m p lo y ed in th e analysis of d isc re te -tim e signals and
system s d e p e n d on th e c h aracteristics o f th e signals. In this se ctio n we classify
d isc rete-tim e signals acco rd in g to a n u m b e r of d iffe re n t c h aracteristics.
Energy signals and power signals.
T h e en e rg y £ of a signal x( n ) is
d efin ed as
OC
E=
|jc(n)|2
(2.1.15)
n=-oc
W e h av e u se d th e m a g n itu d e -sq u a re d v alu es o f jr(n), so th a t o u r d efin itio n applies
to c o m p le x -v a lu e d signals as w ell as re a l-v a lu e d signals. T h e e n e rg y of a signal can
be fin ite o r in fin ite. If E is finite (i.e., 0 < E < oo), th e n x ( n ) is called an energy
Figure 2.6
signal.
G raph of the real and im aginary com ponents of a com plex-valued exponential
Sec. 2.1
49
Discrete-Time Signals
-1 0
1 2
3
4
5
b
11
~ S
9
(a* Gra ph of A {n I = r" . r = 0.9
( b t G r a p h ol
^
n . m o d u l o 2ir p l o n e d in t h e r a n g e I—t t , it I
Figure 2.7 G r a p h o f a m p l i tu d e and p h a s e function of a co m p l e x -v a l u e d e x p o n e n ­
tial sicnal: ( a ) gr ap h of A i m = r " . 4 = 0 . 9 : ( b ) g rap h of c/>in) = I . t / I O v j . m o d u lo
2ji p lo tt e d in the ran ge i - j i . t t J .
signal. S o m etim es w e add a su b scrip t .v to £ an d w rite £ , to em p h asize th a t £ , is
the en e rg y o f th e signal x( n).
M anv signals th a t possess infinite en erg y , h av e a finite av e ra g e p o w er. T he
av erag e p o w e r o f a d isc re te -tim e signal x( n) is defin ed as
P =
1
lim
■**— 2 N -j- 1 n - —
-,\
(2 .1.1 6)
If we define th e signal en e rg y o f x(?i) o v er th e finite in terv al —N < n < N as
(2.1.17)
th e n w e can e x p ress th e signal energy £ as
£ =
lim £/v'
,\ —oc
(2.1.18)
an d th e a v erag e p o w e r o f th e signal x ( n ) as
1
En
P = lim
n -+x 2 N + 1
(2.1.19)
50
Discrete-Tim e Signals and Systems
Chap. 2
C learly , if E is finite. P = 0. O n th e o th e r h a n d , if E is infinite, th e av erag e
p o w e r P m ay be e ith e r finite o r infinite. If P is finite (an d n o n z e ro ), th e signal is
called a p o w e r signal. T h e follow ing e x am p le illu stra te s such a signal.
Example 2.1.1
Determ ine the power and energy of the unit step sequence. The average power of
the unit step signal is
=
N + 1
lim --------------
A1-** 2 N
1
1 + l/N
1
lim ------------- --- -
a—9c 2 4- l / N
2
Consequently, the unit step sequence is a power signal. Its energy is infinite.
Sim ilarly, it can b e show n th a t th e co m p lex e x p o n e n tia l se q u e n c e x(n') =
A e JUJan h as av erag e p o w e r A 2, so it is a p o w e r signal. O n th e o th e r h an d , th e unit
ra m p se q u en ce is n e ith e r a p o w e r signal n o r an en e rg y signal.
Periodic signals and aperiodic signals. A s d efin e d o n S ection 1.3, a
signal x( n ) is p erio d ic w ith p e rio d N ( N > 0) if a n d only if
x( n + N ) = x( n ) fo r all n
(2 . 1.20 )
T h e sm allest v alu e o f N fo r w hich (2.1.20) h o ld s is called th e (fu n d a m e n ta l) p erio d .
If th e re is no v alu e o f N th a t satisfies (2.1.20), th e signal is called nonper i odi c or
aperiodic.
W e h av e a lread y o b se rv e d th a t th e sin u so id al signal o f th e form
x ( n) = A sin 2n f o n
( 2 . 1 .21 )
is p e rio d ic w h en /J, is a ra tio n a l n u m b e r, th a t is, if /o can be e x p re sse d as
(2 .1.22 )
w h ere k an d N a re integers.
T h e e n erg y o f a p erio d ic signal x ( n ) o v e r a single p e rio d , say. o v er th e in terv al
0 5 n < N - 1, is fin ite if x («) ta k e s on finite v alu es o v e r th e p e rio d . H o w ev er, the
e n e rg y o f th e p e rio d ic signal fo r —oc < n < oo is infinite. O n th e o th e r h an d , the
a v e ra g e p o w e r o f th e p erio d ic signal is finite an d it is e q u a l to th e av erag e p o w er
o v e r a single p e rio d . T h u s if x («) is a p e rio d ic signal w ith fu n d a m e n ta l p e rio d N
an d ta k e s o n fin ite v alues, its p o w e r is g iven by
(2.1.23)
n=0
C o n s e q u e n tly , p erio d ic signals are p o w e r signals.
Sec. 2.1
Discrete-Time Signals
51
Symmetric (even) and antisymmetric (odd) signals.
A real v a lu ed sig­
n al x ( n ) is called sy m m etric (ev en ) if
j r ( - n ) = j («)
(2.1.24)
O n th e o th e r h a n d , a signal x ( n ) is called a n tisy m m etric (o d d ) if
.v( - h ) = - x ( n )
(2.1.25)
W e n o te th a t if .v(/7) is odd, th e n x(0) = 0. E x am p le s o f signals w ith ev en an d odd
sy m m etry are illu stra te d in Fig. 2.8.
W e w ish to illu strate th a t any a rb itra ry signal can be e x p re sse d as th e sum of
tw o signal c o m p o n e n ts , o n e o f w hich is even an d th e o th e r o d d . T h e ev en signal
co m p o n e n t is fo rm e d by ad d in g x(/i) to x ( —n) and div id in g by 2. th a t is.
= j[.v{7!) + x ( - » ) ]
(2.1.26)
.v(n]
<'
4►
T
!I
j T
] !
•
l M ! ....................
*
ll
-4-3-2-I
0
12
3 4
i,
(a)
.r(n)
•
- 5 - 4 - 3 - 2 -1
.ill'
12
3 4 5
<>
(b)
Figure 2.8
Exam ple of even (a) and odd (b) signals.
rt
52
Discrete-Time Signals and Systems
Chap. 2
Clearly, x e(n) satisfies the sym m etry con d ition (2.1.24). Sim ilarly, w e form an odd
signal com p onent x„(n) according to the relation
x„(n) = j[.x(n) - * ( - / i ) ]
(2.1.27)
A gain , it is clear that x 0(n) satisfies (2.1.25); hence it is in d eed odd. N ow , if we
add the tw o signal com p onents, defined by (2.1.26) and (2.1.27), w e ob tain ;c(n),
that is,
*(n ) = x e(n) + x n{n)
(2.1.28)
Thus any arbitrary signal can be exp ressed as in (2.1.28).
2.1.3 Simple Manipulations of Discrete-Time Signals
In this section w e consider som e sim ple m odifications or m anipulations involving
the in d ep en dent variable and the signal am plitude (depend en t variable).
Transformation of the independent variable (time). A signal x ( n ) may
be shifted in tim e by replacing the in d ep en dent variable n by n — k, w here k is an
integer. If A: is a positive integer, the tim e shift results in a delay of the signal by
k units o f tim e. If k is a negative integer, the tim e shift results in an advance of
the signal by \k\ units in time.
Example 2.L2
A signal x ( n ) is graphically illustrated in Fig. 2.9a. Show a graphical representation
of the signals x ( n — 3) and x ( n -I- 2).
Solution The signal x (/i —3) is obtained by delaying ;t(n) by three units in time. The
result is illustrated in Fig. 2.9b. On the other hand, the signal x(n + 2 ) is obtained by
advancing x ( n ) by two units in time. The result is illustrated in Fig. 2.9c. Note that
delay corresponds to shifting a signal to the right, whereas advance implies shifting
the signal to the left on the time axis.
If the signal x ( n ) is stored on m agnetic tape or on a disk or, perhaps, in the
m em ory o f a com puter, it is a relatively sim ple operation to m odify the base by
introducing a delay or an advance. O n the other hand, if the signal is not stored but
is b ein g generated by som e physical p h en om en on in real tim e, it is not p ossible
to advance the signal in tim e, since such an op eration in volves signal sam ples
that have not yet b een generated. W hereas it is alw ays possib le to insert a delay
into signal sam ples that have already b een generated, it is physically im possible
to view the future signal sam ples. C on sequ en tly, in real-tim e signal processing
applications, the operation o f advancing the tim e base o f the signal is physically
unrealizable.
A n o th er useful m odification o f the tim e base is to replace th e in d ep en dent
variable n by —n. T h e result o f this op eration is a f o l d i n g or a reflection o f the
signal about the tim e origin n = 0.
Sec. 2.1
53
Discrete-Time Signals
xin)
4 i
■j— 1—
I —4 —3 —2 — 1 0
1 2
3
4
x i n - 3)
il
T -l
0
1 2
3 4
S 6
(b)
xin +2)
- 6 - 5 - 4 -3 - 2 - 1
0
Figure 2.9 Graphical representation of
a signal, and its delayed and advanced
versions.
1
Example 2.1.3
Show the graphical representation of the signal x { - n ) and x i - n + 2). where x ( n ) is
the signal illustrated in Fig. 2.10a.
The new signal yin) = x ( —n) is shown in Fig. 2.10b. Note that y(0) = *(0).
v ( l ) = x( — 1). y(2) = _t( —2). and so on. Also, y (—1) = jc(1 ) , v(—2) = x(2), and so on.
Solution
Therefore, yin) is simply xin) reflected or folded about the lime origin n = 0. The
signal yin) — x ( —n ■
+
• 2) is simply x ( —n) delayed by two units in time. The resulting
signal is illustrated in Fig. 2.10c. A simple way to verify that the result in Fig. 2.10c
is correct is to compute samples, such as y(0) = *(2), y (l) = jc(1 >, v(2) = ;t(0),
v(—1) = jr(3). and so on.
It is im p o rta n t to n o te th a t th e o p e ra tio n s o f folding an d tim e d elay in g (o r
ad v an cin g ) a signal a re n o t co m m u ta tiv e . If we d e n o te th e tim e-d e la y o p e ra tio n
by T D a n d th e fo ld in g o p e ra tio n by F D . we can w rite
T D i[jc(n )l = jc(n - k)
k > 0
(2.1.29)
FD [jc(/j)] = x ( —n)
N ow
T D A(FD[.r(n)]l = T D *[.t(-n)] = x ( - n + k)
(2.1.30)
54
Discrete-Time Signals and Systems
Chap. 2
y(n) = x(-n + 2)
-2 -1
in
0 1 2 3 4 5
(c)
Figure 2.10 Graphical illustration of
the folding and shifting operations.
w hereas
F D {T D i[j:(n )]} = FD[j:(/i — &)] = x ( —n — k )
(2.1.31)
N o te that becau se the signs o f n and k in x { n —k) and Jt(-n + ik ) are different, the re­
sult is a shift o f the signals x ( n ) and x ( —n) to the right by k sam p les, corresponding
to a tim e delay.
A third m odification o f the in d ep en dent variable in volves replacing n by fin,
w here /x is an integer. W e refer to this tim e-b ase m odification as time scaling or
dow nsam pling.
Example 2.L4
Show the graphical representation of the signal y(n) = x(2n), where x(n) is the signal
illustrated in Fig. 2.11a.
Solution We note that the signal y(n) is obtained from x(n) by taking every other
sample from jc(«), starting with x(0). Thus y(0) = x(0), y{l) = x(2), y(2) = jc(4),
and y ( - l ) = x(~ 2 ), y ( -2 ) = jc(—4), and so on. In other words, we have skipped
Sec. 2.1
55
Discrete-Time Signals
v(n) = .v(2n)
I
-4
I
! -2
0 I 2 3
(b)
Figure 2.11
Graphical illustration ot down-samplinc operation.
the odd-num bered samples in *(«) and retained the even-num bered samples. The
resulting signal is illustrated in Fig. 2.11b.
If th e signal ,\ («) w as o riginally o b ta in e d by sa m p lin g an a n a lo g signal x a(t),
th e n jc(«) = Xa(nT), w h ere T is the sa m p lin g in terv al. Nowr. v(n) = x ( 2n) =
x a(2Tn). H e n c e th e tim e-scalin g o p e ra tio n d escrib ed in E x a m p le 2.1.4 is e q u iv a le n t
to ch an g in g th e sam p lin g ra te from 1 /T to 1/27". th a t is, to d e c re a s in g the ra te by
a facto r o f 2. T h is is a d o w n s a mp l i n g o p e ra tio n .
Addition, multiplication, and scaling of sequences. A m p litu d e m o d ifi­
catio n s in clu d e addition, multiplication, an d scaling o f d isc re te -tim e signals.
A m p l i t u d e scaling o f a signal by a c o n sta n t A is acco m p lish ed by m u ltiplying
th e v alu e o f ev ery signal sa m p le by A. C o n se q u e n tly , w e o b ta in
v(n) = Ax ( n )
— oc < fi < oc
T h e s u m o f tw o signals xj ( n) a n d xz i n) is a signal _v(n), w h o se v alu e at any
in stan t is equal to th e sum o f th e values o f th e se tw o signals a t th a t in sta n t, th a t is.
y(/t) = jq (n) + X2 <n)
— oc < n < oc
T h e p r o d u c t o f tw o signals is sim ilarly defin ed o n a sa m p le -to -sa m p le basis as
v(n) = X](n)X 2 (n)
— oo < n < oo
56
Discrete-Time Signals and Systems
Chap. 2
2.2 DISCRETE-TIME SYSTEMS
In m a n y ap p lic a tio n s o f d ig ital signal p ro cessin g w e w ish to desig n a d ev ice or
an alg o rith m th a t p e rfo rm s so m e p re sc rib e d o p e ra tio n o n a d isc re te -tim e signal.
S uch a d evice o r a lg o rith m is called a d isc re te -tim e system . M o re specifically, a
discrete-time s y s t em is a d ev ice o r a lg o rith m th a t o p e ra te s on a d isc re te -tim e signal,
called th e i nput o r excitation, acco rd in g to so m e w ell-defined ru le , to p ro d u c e a n ­
o th e r d isc rete-tim e signal called th e out p u t o r response of th e system . In g en eral,
we view a system as an o p e r a tio n o r a se t o f o p e ra tio n s p e rfo rm e d on th e in p u t
sig n al x( n) to p ro d u c e th e o u tp u t signal _v(n). W e say th a t th e in p u t signal x( n) is
t r ans f ormed by th e sy stem in to a signal >■(«), an d ex p ress th e g e n e ra l re la tio n sh ip
b e tw e e n jc («) a n d y ( n ) as
y( n) = T[x{n) }
(2 .2 .1)
w h ere th e sym bol T d e n o te s th e tra n s fo rm a tio n (also called an o p e r a to r) , o r p r o ­
cessing p e rfo rm e d by th e sy stem on ;c(n) to p ro d u c e y(n). T h e m a th e m a tic a l
re la tio n sh ip in (2.2.1) is d e p ic te d g rap h ically in Fig. 2.12.
T h e re are v ario u s w ays to d escrib e th e c h a ra c te ristic s of th e system an d th e
o p e ra tio n it p e rfo rm s on x( n ) to p ro d u c e y(n ). In this c h a p te r w e shall b e c o n ­
c e rn e d w ith th e tim e -d o m a in c h a ra c te riz a tio n o f system s. W e shall begin w ith
an in p u t-o u tp u t d e sc rip tio n o f th e system . T h e in p u t-o u tp u t d e sc rip tio n focuses
on th e b e h a v io r at th e te rm in a ls of th e system a n d ignores th e d e ta ile d in te rn a l
co n stru ctio n o r re a liz a tio n o f th e system . L a te r, in S ection 7.5. w e in tro d u c e th e
sta te -sp a c e d e sc rip tio n o f a system . In this d e sc rip tio n w e d e v e lo p m a th e m a ti­
cal e q u a tio n s th a t n o t o nly d esc rib e th e in p u t- o u tp u t b e h a v io r o f th e sy stem b u t
specify its in te rn a l b e h a v io r a n d stru c tu re .
2.2.1 Input-Output Description of Systems
T h e in p u t-o u tp u t d e sc rip tio n o f a d isc re te -tim e sy stem consists o f a m a th e m a tic a l
ex p ressio n o r a ru le, w hich explicitly d efin es th e re la tio n b e tw e e n th e in p u t an d
o u tp u t signals ( i n p u t - o u t p u t relationship). T h e ex act in te rn a l s tru c tu re o f th e sys­
tem is e ith e r u n k n o w n o r ig n o red . T h u s th e o n ly w ay to in te ra c t w ith th e sy stem is
by u sin g its in p u t an d o u tp u t te rm in a ls (i.e., th e system is a ssu m ed to be a “black
b o x ” to th e u se r). T o reflec t th is p h ilo so p h y , w e u se th e g ra p h ic a l re p re se n ta -
x (n J
Input signal
or excitation
Figure 2.12
Discrete-time
System
Output signal
or response
Block diagram representation of a discrete-tim e system.
Sec. 2.2
57
Discrete-Time Systems
tio n d e p ic te d in Fig. 2.12, an d th e g e n e ra l in p u t-o u tp u t re la tio n sh ip in (2.2.1) or,
a lte rn a tiv e ly , th e n o ta tio n
Jt(n)
y(n)
(2.2.2)
w hich sim ply m e a n s th a t v(n) is th e re sp o n se of the system T to th e e x c ita tio n
x{n). T h e fo llo w in g e x am p les illu stra te se v era l d iffe re n t system s.
Example 2.2.1
D eterm ine the response of the following sytems to the input signal
x(n) = , A
u,
(a)
(b)
(c)
(d)
-3 < n < 3
otherwise
y(n) = x{n)
v(«) = x in — i)
y(n) = x i n 4- i)
y i n ) = j[A-(n + 1) + x ( n ) + x i n - D]
(e) y ( n) = m a x { x( n + 1), x ( n ) . x ( n — 1)1
(0 y ( n ) = Z L . x x ( k ) = x ( n ) + x ( n — 1) + x{n — 2) -t
Solution
(2.2.3)
First, we determ ine explicitly the sample values of the input signal
xin) = ( ....0 .3 ,2 .1 .0 .1 .2 , 3 ,0 ,...)
T
Next, we determ ine the output of each system using its input-output relationship.
(a) In this case the output is exactly the same as the input signal. Such a system is
known as the identity system.
(b) This system simply delays the input by one sample. Thus its output is given by
x{n) = { ...,0 ,3 .2 .1 ,0 ,1 .2 .3 ,0 ,..,)
t
(c) In this case the system “advances” the input one sample into the future. For
example, the value of the output at time n = 0 is y(0) = *(1). The response of
this system to the given input is
x(n) = { ...,0 ,3 . 2 .1 .0 ,1 ,2 . 3 ,0 ....}
t
(d) The output of this system at any time is the mean value of the present, the
im m ediate past, and the immediate future samples. For example, the output at
time n = 0 is
y(0) =
+ x(0) + jr(l)] = |[1 + 0 + 1] = |
R epeating this com putation for every value of n, we obtain the output signal
>■(«) = {• ...0 ,1 , f , 2 , l j . l . 2 , § , 1 .0 ,...)
t
58
Discrete-Time Signals and Systems
Chap. 2
(e> This system selects as its output at time n the maximum value of the three input
samples x(n - l.l, .v(n). and ,r(n + 1). Thus the response of this system to the
input signal .\{n) is
v(n) = {0.3. 3. 3. 2 .1 .2 . 3, 3, 3 . 0 . . . . )
t
(f) This system is basically an accumulator that computes the running sum of all
the past input values up to present time. The response of this system to the
given input is
v(n) = {.. ..0 .3 . 5. 6. 6, 7, 9. 1 2 .0 ....}
T
W e o b se rv e th a t fo r several of th e sy stem s c o n sid e re d in E x a m p le 2.2.1 the
o u tp u t at tim e n — no d e p e n d s n ot only on th e v alu e of the in p u t at n = n (, [i.e.,
jc(«o)]- b u t also on th e values o f the in p u t a p p lie d to th e system b e fo re an d after
n = n (). C o n sid er, fo r in stan ce, th e a c c u m u la to r in th e ex a m p le . W e see th at the
o u tp u t at tim e n = ?i() d e p e n d s n ot only on th e in p u t a t tim e n = no. b u t also on
x ( n ) a t tim es n = no — 1. no - 2, and so on. By a sim ple a lg e b ra ic m a n ip u latio n
th e in p u t-o u tp u t re la tio n o f th e a c c u m u la to r can b e w ritte n as
= y(/i - 1) + x(ti)
w hich justifies th e te rm accumul at or. In d e e d , th e system c o m p u te s th e c u rre n t
v alu e o f th e o u tp u t by a d d in g (a ccu m u latin g ) th e c u rre n t v alu e o f th e in p u t to th e
p rev io u s o u tp u t value.
T h e re are so m e in te re stin g co n clu sio n s th a t can be d raw n by tak in g a close
lo o k in to this a p p a re n tly sim ple system . S u p p o se th a t we are given th e in p u t signal
x(rt ) fo r n > no. a n d we wish to d e te rm in e th e o u tp u t v(/i) o f th is system fo r n > no.
F o r n = no. no + 1........ (2.2.4) gives
v (n n) = v(«o - 1) -f x (/i0)
_v(no + l l
=
v ( « o ) + x ( n o + 1)
an d so on. N o te th a t we h ave a p ro b le m in c o m p u tin g y ( n a), since it d e p e n d s on
y(«o - 1). H o w ev er,
y(t io - 1) =
^
x(k)
k — — ’X .
th a t is. y(no - 1) “sum m arizes*’ th e effect on th e system from all the in p u ts w hich
h ad b e e n ap p lied to th e system b efo re tim e no- T h u s th e re sp o n s e of th e system
fo r n > no to th e in p u t x(/7j th a t is a p p lie d a t tim e no is th e c o m b in e d resu lt of this
in p u t an d all in p u ts th a t h ad b e e n a p p lie d p re v io u sly to th e sy stem . C o n seq u en tly .
y(/i), n > no is n o t u n iq u ely d e te rm in e d by th e in p u t x ( n ) fo r n > no.
Sec. 2.2
59
Discrete-Time Systems
T he additional inform ation required to d eterm ine y ( n) for n > no is the initial
condi t i on y(no - 1). T his value sum m arizes the effect o f all p reviou s inputs to the.
system . Thus the initial con d ition y(«o - 1) togeth er with the input seq u en ce x ( n )
for n > no uniquely d eterm ine the output sequ en ce y(n ) for n > n0.
If the accum ulator had no excitation prior to n 0, the initial con d ition is y(no —
1) = 0. In such a case w e say that the system is initially relaxed. S ince y(no —1) = 0,
the output seq u en ce y(n) d ep en d s only on the input seq u en ce x ( n ) for n > n0.
It is custom ary to assum e that every system is relaxed at n = —oo. In this
case, if an input x ( n ) is applied at n = —co, the corresponding output y( n) is solely
and uni qu e l y determ ined by the given input.
Example 2.2.2
T he accum ulator described by (2.2.3) is excited by the sequence x(n) = nu(n). D e­
term ine its output under the condition that:
(a) It is initially relaxed [i.e., v ( - l ) = 0].
<b) Initially, y (— 1 )= 1.
Solution
The output of the system is defined as
tl
-]
y(n) = ^
x(k) =
x(k) +
*=-oc
*=-oc
r
i= < l
x(k)
= y (-l) +
k=o
But
n(n -f 1)
(a) If the system is initially relaxed, v(—1) = 0 and hence
n(n + l)
v (n)
=
-------- 2 --------
"
-
0
(b) On the other hand, if the initial condition is y ( - l ) = 1, then
n(n -I-1)
n2 + n + 2
v(n) = 1 + ---- -— - = -------------
n > 0
2.2.2 Block Diagram Representation of Discrete-Time
Systems
It is useful at this point to introduce a block diagram representation o f d iscrete­
tim e system s. For this purpose w e n eed to define som e basic building blocks that
can b e intercon n ected to form com p lex system s.
An adder. F igure 2.13 illustrates a system (adder) that perform s the addi­
tion o f tw o signal seq u en ces to form another (th e sum ) seq u en ce, w hich w e d en ote
Discrete-Time Signals and Systems
60
Chap. 2
x|(n )
y(n) = i,( n ) + x2(n)
Figure 2.13 Graphical representation
of an adder.
as y (n ). N o te th a t it is n o t n ecessa ry to sto re e ith e r o n e o f th e se q u e n c e s in o rd e r
to p e rfo rm th e a d d itio n . In o th e r w ords, th e a d d itio n o p e ra tio n is memor yl ess .
A constant multiplier. T his o p e ra tio n is d e p ic te d by Fig. 2.14, an d sim ply
re p re s e n ts ap p ly in g a scale fa c to r on th e in p u t x ( n) . N o te th a t th is o p e ra tio n is
also m em oryless.
a
------------------------ -
v(n) = o i( n )
»■
Figure 2.14 Graphical representation
of a constant multiplier.
A signal multiplier. F ig u re 2.15 illu stra te s th e m u ltip lic a tio n o f tw o sig­
nal se q u en ces to fo rm a n o th e r (th e p ro d u c t) se q u e n c e , d e n o te d in th e figure as
y (n ). A s in th e p re c e d in g tw o cases, w e can view th e m u ltip lic a tio n o p e ra tio n as
m em o ry less.
A|(n)
v(n) = jT|(n)Ai(n)
---- -0 ^ —
Figure 2.1S Graphical representation
of a signal multiplier.
x2( n )
A unit delay element. T h e u n it d elay is a sp e cial sy stem th a t sim ply d elay s
th e signal passing th ro u g h it by one sam p le. F ig u re 2.16 illu stra te s such a system .
If th e in p u t signal is x ( n) , th e o u tp u t is x( n — 1). In fact, the sa m p le x{n — 1) is
sto re d in m em o ry at tim e n — 1 a n d it is recalled fro m m e m o ry a t tim e n to form
v ( n ) = x( n - 1)
T h u s th is basic b u ild in g b lock re q u ire s m em o ry . T h e use o f th e sym bol ; _1 to
d e n o te th e u n it o f d eiay will b eco m e a p p a r e n t w h e n w e discuss th e z -tra n sfo rm in
C h a p te r 3.
x(n)
------------------ »-
y (n ) = jr ( n— 1)
_____
Figure 2,16 Graphical representation
of the unit delay element.
A unit advance element. In c o n tra st to th e u n it d e la y , a u nit ad v an ce
m o v es th e in p u t x ( n ) a h e a d by o n e sa m p le in tim e to yield x ( n + 1). F ig u re 2.17
illu stra te s th is o p e ra tio n , w ith th e o p e r a to r ; b ein g used to d e n o te th e u n it advance.
Sec. 2.2
Discrete-Time Systems
61
y( n ) = x( n + I )
x( n)
Figure 2.17 Graphical representation
of the unit advance element.
W e o b se rv e th a t an y such ad v an ce is physically im possible in real tim e, since, in
fact, it in v o lv es lo o k in g in to th e fu tu re o f th e signal. O n th e o th e r h an d , if we store
th e signal in th e m em o ry o f th e c o m p u te r, w e can recall any sa m p le a t any tim e.
In such a n o n re a l-tim e ap p lic a tio n , it is p o ssible to advance th e signal jr(?r) in tim e.
Example 2.2.3
Using basic building blocks introduced above, sketch the block diagram representa­
tion of the discrete-time system described by the input-output relation.
v(n) = 3.v(« - 1) + | x(n) + \x(n - 1)
where x(n) is the input and y(n) is the output of the system.
Solution According to (2.2.5), the output v(n) is obtained by multiplying the input
x(n) by 0.5, multiplying the previous input jr ( n - l) by 0.5. adding the two products, and
then adding the previous output v(n —1) multiplied by j. Figure 2.18a illustrates this
block diagram realization of the system. A simple rearrangem ent of (2.2.5). namely.
v (« ) =
5 .v(n
-
1 ) +
5 [jc(k) +
x(n -
1 )|
( 2 . 2 .6 )
leads to the block diagram realization shown in Fig. 2.18b. Note that if we treat "the
system” from the “viewpoint” of an input-output or an external description, we are
not concerned about how the system is realized. On the other hand, if we adopt an
Black box
0.5
-i
x( n )
(a)
Black box
-i
x( n)
Figure 2.18 Block diagram realizations of the system y(n) = 0.25y(n — 1) +
0.5 x{n) + 0.5j(n — 1).
Discrete-Time Signals and Systems
62
Chap. 2
internal description of the system, we know exactly how the system building blocks
are configured. In terms of such a realization, we can see that a system is relaxed at
time n = no if the outputs of all the delays existing in the system are zero at n = n{)
(i.e., all memory is filled with zeros).
2.2.3 Classification of Discrete-Time Systems
In th e analysis as w ell as in th e design o f system s, it is d e s ira b le to classify the
sy stem s acco rd in g to th e g e n e ra l p ro p e rtie s th a t th e y satisfy. In fact, th e m a th e ­
m atical te c h n iq u e s th a t w e d e v e lo p in th is an d in s u b s e q u e n t c h a p te rs fo r analyzing
an d d esig n in g d isc rete-tim e system s d e p e n d h eav ily on th e g e n e ra l ch aracteristics
of th e system s th a t are b ein g c o n sid e re d . F o r th is re a so n it is n ecessa ry for us
to d ev elo p a n u m b e r o f p ro p e rtie s o r c a te g o rie s th a t can be u se d to d escrib e th e
g e n e ra l ch aracteristics o f system s.
W e stress th e p o in t th a t fo r a sy stem to po ssess a given p ro p e rty , th e p ro p e rty
m u st h o ld fo r ev ery p o ssible in p u t signal to th e system . If a p ro p e rty holds for
so m e in p u t signals b u t n ot fo r o th e rs, th e system d o e s n ot p o ssess th a t p ro p erty .
T h u s a c o u n te re x a m p le is sufficient to p ro v e th a t a system d o e s n ot possess a
p ro p e rty . H o w ev er, to p ro v e th a t th e system has so m e p ro p e rty , we m ust prove
th a t th is p ro p e rty h o ld s fo r every po ssib le in p u t signal.
Static versus dynamic systems. A d isc re te -tim e sy stem is called static
o r m em o ry le ss if its o u tp u t at any in sta n t n d e p e n d s at m o st o n the in p u t sam ple
at th e sam e tim e, b u t n o t o n p a st o r fu tu re sa m p le s o f th e in p u t. In any o th e r case,
th e system is said to b e d y n a mi c o r to h av e m em o ry . If th e o u tp u t o f a system at
tim e n is co m p letely d e te rm in e d by th e in p u t sa m p le s in th e in te rv a l fro m n - N
to n ( N > 0), th e system is said to h av e m e m o r y o f d u ra tio n N . U N — 0. th e
sy stem is static. H 0 < N < oo, th e sy stem is said to have f ini te m e m o r y , w h ereas
if N = oo, th e system is said to have infinite m e m o r y .
T h e system s d escrib ed by th e follow ing in p u t- o u tp u t e q u a tio n s
y(n) = a x {n)
(2.2.7)
y ( n ) = nx ( n ) + b x 3(n)
(2.2.8)
a re b o th static o r m em o ry less. N o te th a t th e re is n o n e e d to s to re any o f th e past
in p u ts o r o u tp u ts in o rd e r to c o m p u te th e p re se n t o u tp u t. O n th e o th e r h an d , th e
sy stem s d escrib ed by th e follow ing in p u t-o u tp u t re la tio n s
y( n) = x ( n ) + 3 x ( n — 1)
(2.2.9)
y(n) = J ^ x ( n - k )
k=0
(2.2.10)
X
y( n) = J 2 x ( n - k )
Jt=0
(2.2.11)
are dynamic systems or systems with memory. The systems described by (2.2.9)
Sec. 2.2
63
Discrete-Time Systems
an d (2.2.10) h av e fin ite m em o ry , w h ereas the sy stem d e sc rib e d by (2.2.11) has
infinite m em o ry .
W e o b se rv e th a t sta tic o r m em oryless system s are d e sc rib e d in g e n e ra l by
in p u t-o u tp u t e q u a tio n s o f th e form
y(n) = T [ x ( n ) , n]
(2.2.12)
a n d th e y do n o t in clu d e d elay e le m e n ts (m em o ry ).
Time-invariant versus time-variant systems. W e can su b d iv id e th e g en ­
eral class o f sy stem s in to th e tw o b ro a d c a teg o ries, tim e -in v a ria n t system s and
tim e -v a ria n t sy stem s. A system is called tim e -in v a ria n t if its in p u t- o u tp u t c h a ra c ­
teristics d o n o t c h a n g e w ith tim e. T o e la b o ra te , su p p o se th a t w e h av e a system T
in a re la x e d s ta te w hich, w h en ex cited by an in p u t signal x ( n) , p ro d u c e s an o u tp u t
signal y(n). T h u s w e w rite
y( n) = T [ x { n) )
(2.2.13)
N o w su p p o se th a t th e sam e in p u t signal is d e la y e d by k u n its o f tim e to yield
x (n - &), an d ag ain a p p lied to th e sam e system . If th e c h a ra c te ristic s of th e system
d o n o t ch an g e w ith tim e, th e o u tp u t o f th e relax ed system will b e y(« —k). T h at is,
th e o u tp u t will b e th e sa m e as th e resp o n se to x ( n) . ex cep t th a t it will be d elay ed
by th e sam e k u n its in tim e th a t the in p u t w as d elay ed . T his lead s us to define a
tim e -in v a ria n t o r sh ift-in v a ria n t system as follow s.
Definition.
A relax ed system T is time i nvariant o r shift i nvariant if and
o n ly if
x( n)
y(n)
im p lies th a t
x {n — k) —
y( n — k)
(2.2.14)
fo r ev ery in p u t signal x (n ) a n d every tim e shift k.
T o d e te rm in e if an y given system is tim e in v a ria n t, w e n e e d to p e rfo rm the
te st specified b y th e p re c e d in g definition. B asically, we ex cite th e system w ith an
a rb itra ry in p u t se q u e n c e x ( n) , w hich p ro d u c e s an o u tp u t d e n o te d as y ( n) . N ext
w e d elay th e in p u t se q u e n c e by sam e a m o u n t k an d re c o m p u te th e o u tp u t. In
g e n eral, w e can w rite th e o u tp u t as
y(«, k) = T [ x ( n — <:)]
N o w if th is o u tp u t y{n, k) = y{n — k), for all p o ssib le v alu es o f k, th e system is
tim e in v a ria n t. O n th e o th e r h a n d , if th e o u tp u t y( n, k ) ^ y ( n — k), ev en fo r o n e
v alu e o f k, th e sy stem is tim e v arian t.
Discrete-Time Signals and Systems
64
Chap. 2
VI 71) = X( 77I- V<71 - ] I
xin )
“ D ifferentiator"
- B
x(/t)
“Time" multiplier
v( n ) = xl - n )
“ Folder"
v(n J = .u n ) cos oi„ii
Figure 2.19 Examples of a
lime-invariant (a) and some time-variant
systems (h)-(d).
Example 2.2.4
Determ ine if the systems shown in Fig. 2.19 are time invariant or time variant.
Solution
(a) This system is described by the input^output equations
y(7i) = T\ xin)} = x(n\ —x(n - 1)
(2.2.15)
Nov, if the input is delayed by k units in time and applied to the system, it is
clear from the block diagram that the output will be
y i n . k) = x i n - k) — x i n — k — 1)
(2.2.16)
On the other hand, from (2.2.14) we note that if we delay y (n) by k units in
time, we obtain
yin — k) = x(n — k) — xin — k — 1)
(2.2.17)
Since the right-hand sides of (2.2.16) and (2.2.17) are identical, it follows that
v(n. k) = yin - k). Therefore, the system is time invariant.
Sec. 2.2
Discrete-Time Systems
65
(b) The input-output equation for this system is
y(n) = T[x(n)] = nx(n)
(2.2.18)
The response of this system to x(n - Jt) is
y(n, k) = nx(n - k)
(2.2.19)
Now if we delay ;y(n) in (2.2.18) by k units in time, we obtain
y(n - k) = (n — k)x(n — k)
(2 .2.20)
= nx(n — k) - kx(n - k)
This system is time variant, since y(n, k) ^ y(n - k).
(c) This system is described by the input-output relation
>’(«) = T[x(n)\ = x ( - n )
(2.2.21)
The response of this system to jr(n - k) is
;y(n, A;) = T[x(n - *)] = x ( - n - k)
(2.2.22)
Now, if we delay the output ;y(n), as given by (2.2.21), by k units in time, the
result will be
y(n - k) = x ( - n + k)
(2.2.23)
Since y(n, k) ^ y{n - it), the system is time variant.
(d) The input-output equation for this system is
y(n) = j:(n) costDon
(2.2.24)
The response of this system to x(n - k) is
y(n, k) = x(n - k) cos o\)n
(2.2.25)
If the expression in (2.2.24) is delayed by k units and the result is compared to
(2.2.25), it is evident that the system is time variant.
Linear versus nonlinear systems. The general class o f system s can also
be subdivided into linear system s and nonlinear system s. A linear system is one
that satisfies the su pe rp ositio n princ iple. Sim ply stated, the principle o f su p erp osi­
tion requires that the resp onse o f the system to a w eighted sum o f signals b e equal
to the corresponding w eighted sum of the responses (outp u ts) of the system to each
of the individual input signals. H en ce w e have the follow ing definition o f linearity.
Definition.
A r e la x e d T s y s te m is lin e a r if a n d only if
T [ a i x i ( n ) + azx2{n)] = a\ T [ x \ ( n ) ] + a i T [ x 2{n)]
(2.2.26)
for any arbitrary input seq u en ces x\ ( n) and x 2(n), and any arbitrary constants aj
and 0 2 Figure 2.20 gives a pictorial illustration of the superposition principle.
66
Discrete-Time Signals and Systems
Chap. 2
if and only if v(n) = v'(n).
T h e su p e rp o sitio n p rin cip le e m b o d ie d in th e re la tio n (2.2.26) can be s e p a ­
r a te d in to tw o p arts. F irst, su p p o se th a t a2 = 0. T h e n (2.2.26) re d u c e s to
T{a\ X\ ( n) ] = a\ T [ x \ { n ) } = a\ vi(n)
(2.2.27)
w h ere
vi (fl) = T [ x x(n)}
T h e re la tio n (2.2.27) d e m o n s tra te s the mul ti pli cat i ve o r scaling p r o p e r t y of a lin ear
system . T h a t is, if th e re sp o n se o f th e system to th e in p u t x i ( n ) is vi(n ), the
re sp o n se to a\X](n) is sim ply a i j ’i(n ). T h u s any scaling of th e in p u t resu lts in an
id en tical scaling o f th e c o rre sp o n d in g o u tp u t.
S eco n d , su p p o se th a t ai = a2 = 1 in (2.2.26). T h e n
T [ x \ ( n ) + x 2 (n)] = T [ x \ { n ) ] + T [ x \ ( n ) }
(2.2.28)
= yi ( n) + yz(n)
T his re la tio n d e m o n s tra te s th e additivity pr ope r t y o f a lin e a r sy stem . T h e ad ditivity
an d m u ltip lic ativ e p ro p e rtie s c o n stitu te th e su p e rp o s itio n p rin c ip le as it ap p lies to
lin ear system s.
T h e lin earity co n d itio n em b o d ie d in (2.2.26) can b e e x te n d e d a rb itra rily to
any w eig h ted lin e a r c o m b in a tio n o f signals by in d u ctio n . In g e n e ra l, we h av e
M- 1
x( n) = ^ 2 GkXk(n)
M- 1
y ( n ) = ^ akyk (n)
k=l
(2.2.29)
i= l
w h ere
^ ( n ) = T [ x k(n)}
k = 1, 2, , . . , M — 1
(2.2.30)
Sec. 2.2
67
Discrete-Time Systems
W e o b se rv e fro m (2.2.27) th a t if a i — 0, th e n y (n ) = 0. In o th e r w ords, a r e ­
lax ed , lin e a r sy stem w ith z e ro in p u t p ro d u c e s a z e ro o u tp u t. If a system p ro d u c e s
a n o n z e ro o u tp u t w ith a zero in p u t, th e system m ay be e ith e r n o n re la x e d o r n o n ­
lin ear. If a re la x e d sy stem d o e s n o t satisfy th e su p e rp o s itio n p rin cip le as given by
th e d efin itio n ab o v e, it is called nonlinear.
Exam ple 2.2^
D eterm ine if the systems described by the following input-output equations are linear
or nonlinear.
(a) y(n) = ttx(n)
(b) y(n) = *(n2)
<d) y( n) = Ax{n) + B
(c) v(n) = v2(n)
(e) y(n) = ex[n]
Solution
(a) For two input sequences jti(n) and
the corresponding outputs are
Vi(n) = n.ifi(n)
(2.2.31)
y2{n) = nx2(n)
A iinear combination of the two input sequences results in the output
Vj(«) = T[a\Xy (n) + oijMh)] =
(«) +
(/i)]
(2.2.32)
= ainxi (n) + a2nx2(n)
On the other hand, a linear combination of the two outputs in (2.2.31) results
in the output
a\ V] (n) + a2y 2(n) = ainx\(n) + a2n,x2(n)
(2.2.33)
Since the right-hand sides of (2.2.32) and (2.2.33) are identical, the system is
iinear.
(b) As in part (a), we find the response of the system to two separate input signals
*i(n) and x 2(n). The result is
v,(n) = X\(n2)
(2.2.341
y2(rt) = X2(n2)
The output of the system to a linear combination of Xi(n) and *;(»?) is
y3(n) = T\a\X\ («) + a 2x 2(n)] = a hx,(n2) + a2x2(n2)
(2.2.35)
Finally, a linear combination of the two outputs in (2.2.36) yields
O] \>i(n) + c 2V2(n) = a i JCi (n2) + <i2X2(n2)
(2.2.36)
By comparing (2.2.35) with (2.2.36). we conclude that the system is linear.
(c) The output of the system is the square of the input. (Electronic devices that
have such an input-output characteristic and are called square-law devices.)
From our previous discussion it is clear that such a system is memoryless. We
now illustrate that this system is nonlinear.
68
Discrete-Time Signals and Systems
Chap. 2
The responses of the system to two separate input signals are
v,(«) = .vf(n)
'
y2(n) = x;(n)
(2.2.37)
The response of the system to a linear combination of these two input signals is
>'3<n ) = T[a 1*1 (n) + a2x2(n)}
= [fli*i(n) + a2x2(n)]2
(2.2.38)
= cfA 'fln) -I- 2a-la 2x i ( n ) x 2(n) + a 2x 2 { n )
On the other hand, if the system is linear, it would produce a linear combination
of the two outputs in (2.2.37). namely,
ai_Vi(n) + fl2.V2(n) =
(rt) + a2x2(n)
(2.2.39)
Since the actual output of the system, as given by (2.2.38). is not equal to
(2.2.39), the system is nonlinear.
(d) Assuming that the system is excited by x\(n) and x2in) separately, we obtain
the corresponding outputs
V](n) = AX](rt) + B
(2.2.40)
y2(n) = A x 2(n) + B
A linear combination of X\(n) and x 2{n) produces the output
V i(« ) =
T[u^X ]
(/i) + a 2x 2(n>]
= A[a,x,(/i) + a 2x 2(n)} + B
(2.2.41)
= A a \ X \ ( n ) -I- a2A x 2(n) + B
On the other hand, if the system were linear, its output to the linear com bina­
tion of Ji(n) and x 2 (n) would be a linear combination of vj i n) and y2(n). that is.
ai yi (n ) + a 2y 2(n) = a ] A x ] { n ) + a \ B
a 2A x 2{ n ) + a 2 B
(2.2.42)
Clearly. (2.2.41) and (2.2.42) are different and hence the system fails to satisfy
the linearity test.
The reason that this system fails to satisfy the linearity test is not that the
system is nonlinear (in fact, the system is described by a linear equation) but
the presence of the constant B. Consequently, the output depends on both the
input excitation and on the param eter B ^ 0. Hence, for B ^ 0. the system is
not relaxed. If we set B = 0, the system is now relaxed and the linearity test is
satisfied.
(e) Note that the system described by the input-output equation
y(n) = e,1',)
(2.2.43)
is relaxed. If x(n) = 0, we find that y(n) = 1. This is an indication that the
system is nonlinear. This, in fact, is the conclusion reached when the linearity
test, is applied.
Causal versus noncausal systems.
d isc re te -tim e system s.
W e b eg in w ith th e d efin itio n o f causal
Sec. 2.2
69
Discrete-Time Systems
D e fin itio n ,
a system is said to b e causal if th e o u tp u t o f th e system at any
tim e n [i.e., v(n)] d e p e n d s oniy on p re s e n t an d p ast in p u ts [i.e., x { n ), x(tt - 1),
x(rt — 2 ) , . . . ] , b u t d o e s n o t d e p e n d on fu tu re in p u ts [i.e., x( n + 1), x ( n + 2 ) , . . . ] . In
m a th e m a tic a l te rm s, th e o u tp u t o f a cau sal sy stem satisfies an e q u a tio n o f th e form
v(n) = F[x{n), x ( n - 1), x ( n - 2 ) , . . . ]
(2.2.44)
w h ere /'[■] is so m e a rb itra ry function.
If a system d o e s n o t satisfy this d efin itio n , it is called noncausal . S uch a
sy stem has an o u tp u t th a t d e p e n d s n o t oniy on p re s e n t a n d p ast in p u ts b u t also
o n fu tu re in p u ts.
It is a p p a re n t th a t in real-tim e signal p ro cessin g ap p licatio n s w e c a n n o t o b ­
serv e fu tu re v alu es o f th e signal, and h e n c e a n o n cau sal system is physically u n re a l­
izab le (i.e., it c a n n o t b e im p le m e n te d ). O n th e o th e r h an d , if th e signal is re c o rd e d
so th a t th e p ro cessin g is d o n e off-line (n o n re a l tim e ), it is p o ssible to im p lem en t
a n o n cau sal sy stem , since all v alu es o f th e signal are av ailab le a t th e tim e o f p r o ­
cessing. T h is is o fte n th e case in th e p ro cessin g o f g eophysical signals an d im ages.
Example 2.2.6
D eterm ine if the systems described by the following input-output equations are causal
or noncausal.
(a) y(n) = x(n) - x(n - 1)
(b) y(n) =
(d) y(n') = x(n) + 3jr(n + 4)
(e) y(n) = x( n2)
x(k)
(c) y(n) = ax(n)
(t) y(n) = x(2n)
(g) }'(n) = x ( -n )
Solution The systems described in parts (a), (b), and (c) are clearly causal, since the
output depends only on the present and past inputs. On the other hand, the systems
in parts (d). (e), and (f) are clearly noncausal, since the output depends on future
values of the input. The system in (g) is also noncausal, as we note by selecting, for
example, n = - 1 , which yields v(—1) = * 0 ) Thus the output at n = - 1 depends on
the input at n = 1, which is two units of time into the future.
Stable versus unstable systems. S tab ility is an im p o rta n t p ro p e rty th a t
m u st b e c o n s id e re d in an y p ractical ap p lic a tio n o f a system . U n s ta b le system s
u su ally ex h ib it e rra tic an d ex tre m e b e h a v io r an d cause overflow in an y p ractical
im p le m e n ta tio n . H e re , w e define m a th e m a tic a lly w h at w e m e a n by a sta b le system ,
a n d la te r, in S ectio n 2.3.6, w e ex p lo re th e im plicatio n s o f this definition fo r lin ear,
tim e -in v a ria n t system s.
Definition. A n a rb itra ry re la x e d system is said to be b o u n d e d in p u t-b o u n d e d
o u tp u t (B IB O ) sta b le if a n d only if ev ery b o u n d e d in p u t p ro d u ces a b o u n d e d
o u tp u t.
T h e c o n d itio n s th a t th e in p u t se q u e n c e x{n) a n d th e o u tp u t se q u e n c e y ( n) are
b o u n d e d is tra n s la te d m a th e m a tic a lly to m e a n th a t th e re exist som e finite n u m b ers,
Discrete-Time Signals and Systems
70
Chap. 2
say M x an d M v. such th at
j.v(ri)! < M K < oc
< M x < dc
(2.2.45)
fo r all n. If. fo r so m e b o u n d e d in p u t se q u e n c e ,v(»), the o u tp u t is u n b o u n d e d
(in fin ite), th e sy stem is classified as u n sta b le .
Exam ple 2.2.7
Consider the nonlinear system described by the input-output equation
V(/I ) =
— 1)
,V(/i )
As an input sequence we select the hounded signal
xin ) = C&(n )
where C is a constant. We also assume that y(—1) = 0. Then the output sequence is
y(0) = C,
y (l) = C \
y(2) = Cd............
y(n) = C : "
Clearly, the output is unbounded when ] < ICl < oc. Therefore, the system is BIBO
unstable, since a bounded input sequence has resulted in an unbounded output.
2.2.4 Interconnection ot Discrete-Time Systems
D isc rete-tim e sy stem s can be in te rc o n n e c te d to form larg er sy stem s. T h e re are
tw o b asic ways in w hich system s can be in te rc o n n e c te d : in c a scad e (series) o r in
p arallel. T h ese in te rc o n n e c tio n s are illu strated in Fig. 2.21. N o te th at th e tw o
in te rc o n n e c te d sy stem s are d ifferen t.
In th e cascad e in te rc o n n e c tio n the o u tp u t of th e first system is
yiO?) = 7j[;r(/j)]
xin)
(2.2.46)
y(n)
:
r
T\
T,
7,
(a)
v | (n )
(b)
Figure 2.21 Cascade (a) and parallel
(b) interconnections of systems.
Sec. 2.2
Discrete-Time Systems
71
an d th e o u tp u t o f th e second system is
v(n) = T2[\\(n)]
(2.2.47)
=
r 2{7 i[* (n )]}
W e o b se rv e th a t sy stem s 7"j a n d T2 can be co m b in ed o r c o n s o lid a te d in to a single
o v e ra ll sy stem
% = T271
(2.2.48)
C o n s e q u e n tly , w e can ex p ress th e o u tp u t o f th e co m b in ed sy stem as
y( n) = Tc[x(n)]
In g e n e ra l, th e o rd e r in w hich th e o p e ra tio n s T\ a n d T2 a re p e rfo rm e d is
im p o rta n t. T h a t is,
T27I # T,T2
fo r a rb itra ry system s. H o w ev er, if th e system s 7j a n d T2 a re iin e a r a n d tim e
in v a ria n t, th e n (a) % is tim e in v arian t an d (b ) T2T\ = T \ T2, th a t is, th e o r d e r in
w hich th e sy stem s p ro cess th e signal is n o t im p o rta n t. 7^71 a n d T \ T 2 yield id en tical
o u tp u t se q u e n c e s.
T h e p ro o f o f (a) follow s. T h e p ro o f o f (b ) is given in S e c tio n 2.3.4. T o p ro v e
tim e in v a ria n c e , su p p o se th a t Tj and T2 a re tim e in v arian t; th e n
x( n — k )
vi(/i - k)
an d
Vi(n - k) — '■+ y( n - k)
Thus
x{n — k) Tf
y( n — k )
an d th e re fo re , Tc is tim e in v arian t.
In th e p a ra lle l in te rc o n n e c tio n , th e o u tp u t of th e sy stem T\ is ^ ( n ) an d the
o u tp u t o f th e sy stem T2 is y2(n). H e n c e th e o u tp u t o f th e p a ra lle l in te rc o n n e c tio n is
v3(n) = .V] ( n) + >>2(n)
= Ti[x{n)\ + T2[x(n)\
= (T\ + T2)[x(n)}
= Tp[x(n)\
w h e re Tp = T\ + T2.
In g e n e ra l, w e can u se p a rallel an d cascade in te rc o n n e c tio n o f sy stem s to
c o n s tru c t la rg e r, m o re com plex system s. C o n v e rsely , w e can ta k e a la rg e r system
a n d b re a k it d o w n in to sm a ller su b sy stem s fo r p u rp o se s o f an aly sis a n d im p le­
m e n ta tio n . W e sh all u se th e s e n o tio n s la te r, in th e design a n d im p le m e n ta tio n of
d ig ital filters.
Discrete-Time Signals and Systems
72
Chap. 2
2.3 ANALYSIS OF DISCRETE-TIME LINEAR TIME-INVARIANT
SYSTEMS
In S ectio n 2.2 we classified system s in a c c o rd a n c e w ith a n u m b e r of c h aracteristic
p ro p e rtie s o r categ o ries, nam ely: lin e a rity , cau sality , stab ility , a n d tim e in variance.
H av in g d o n e so. we now tu rn o u r a tte n tio n to th e analysis o f th e im p o rta n t class
o f lin ear, tim e -in v a ria n t (L T I) system s. In p a rtic u la r, we shall d e m o n s tra te th a t
such system s are c h a ra c te riz e d in th e tim e d o m ain sim ply bv th e ir resp o n se to a
u n it sam p le se q u en ce. W e shall also d e m o n s tra te th a t any a rb itra ry in p u t signal
can b e d eco m p o sed an d r e p re s e n te d as a w eig h ted sum of u n it sa m p le seq u en ces.
A s a c o n s eq u en ce o f th e lin e a rity a n d tim e-in v arian ce p ro p e rtie s of the system ,
th e resp o n se o f th e system to any a rb itra ry in p u t signal can be e x p ressed in term s
of th e u n it sam p le re sp o n se of th e system . T h e g en eral form o f the ex p ressio n
th a t re la te s th e u n it sam ple re sp o n se o f the system an d th e a rb itra ry in p u t signal
to th e o u tp u t signal, called th e c o n v o lu tio n sum o r th e c o n v o lu tio n fo rm u la, is also
d eriv ed . T h u s we are ab le to d e te rm in e the o u tp u t o f any lin e a r, tim e-in v arian t
system to any a rb itra ry in p u t signal.
2.3.1 Techniques for the Analysis of Linear Systems
T h e re are tw o basic m e th o d s for an aly zin g th e b e h a v io r o r re sp o n s e of a lin ear
system to a given in p u t signal. O n e m e th o d is b a se d on th e d ire c t so lu tio n o f the
in p u t-o u tp u t e q u a tio n for th e system , w hich, in g e n e ra l, has th e form
v(ji) - F [v ( n - 1), v (?7 — 2 ) ........ y(n - N) , x ( n ) . x ( n — 1).......... x ( n - M)]
w h ere F[-] d e n o te s so m e fu n ctio n o f th e q u a n titie s in b ra c k e ts. Specifically, fo r
an L T I system , w e shall see la te r th a t th e g e n e ra l form o f th e in p u t-o u tp u t re la ­
tio n sh ip is
N
M
(2.3.1)
w h ere
an d {b*,.} are c o n sta n t p a ra m e te rs th a t specify th e sy stem an d a re in ­
d e p e n d e n t o f x( n) a n d y( n) . T h e in p u t-o u tp u t re la tio n sh ip in (2.3.1) is called
a d ifferen ce e q u a tio n a n d re p re se n ts o n e w ay to c h a ra c te riz e th e b e h a v io r of a
d isc re te -tim e L T I system . T h e so lu tio n o f (2.3.1) is th e su b je ct o f S ection 2.4.
T h e seco n d m e th o d fo r analyzing th e b e h a v io r o f a lin e a r sy stem to a given
in p u t signal is first to d e c o m p o se o r reso lv e th e in p u t signal in to a sum o f e le ­
m e n ta ry signals. T h e e le m e n ta ry signals a re se le c te d so th a t th e re sp o n se o f the
system to each signal c o m p o n e n t is easily d e te rm in e d . T h e n , u sin g th e lin earity
p ro p e rty o f th e sy stem , th e re sp o n se s o f th e sy stem to th e e le m e n ta ry signals are
ad d e d to o b ta in th e to ta l re s p o n s e of th e sy stem to th e given in p u t signal. T h is
seco n d m e th o d is th e o n e d e sc rib e d in th is se ctio n .
Sec. 2.3
Analysis o f Discrete-Time Linear Tim e-Invariant Systems
73
T o e la b o ra te , su p p o se th a t th e in p u t signal x (n ) is reso lv ed into a w eig h ted
sum o f elem en tary ' signal c o m p o n e n ts {*/. («)} so th a t
w h ere th e {ct} a re th e set of a m p litu d e s (w eighting coefficients) in the d e c o m ­
p o sitio n o f th e signal x( n) . N ow su p p o se th a t th e re sp o n se o f th e system to the
e le m e n ta ry signal c o m p o n e n t x k{n) is y k(n). T hus.
y*(n) = T [j.t (n)]
(2.3.3)
assu m in g th a t th e sy stem is re la x e d an d th a t th e re sp o n se to ckx k(n) is cky k(n). as
a c o n s e q u e n c e o f th e scaling p ro p e rty o f th e lin e a r system .
F in ally , th e to ta l re sp o n se to th e in p u t x( n ) is
(2.3.4)
In (2.3.4) we u se d th e ad d itiv ity p ro p e rty o f th e lin e a r system .
A lth o u g h to a larg e e x te n t, th e ch o ice o f th e e le m e n ta ry signals a p p e a rs to
b e a rb itra ry , o u r se lectio n is heavily d e p e n d e n t on th e class of input signals that
we w ish to co n sid er. If w e place n o re stric tio n on th e c h aracteristics of th e input
signals, its re so lu tio n in to a w eig h ted sum of u n it sam ple (im p u lse) se q u en ces
p ro v e s to b e m a th e m a tic a lly c o n v e n ie n t an d c o m p letely g en eral. O n th e o th e r
h a n d , if w e re stric t o u r a tte n tio n to a subclass o f in p u t signals, th e re m ay be
a n o th e r set o f e le m e n ta ry signals th a t is m o re c o n v e n ie n t m a th e m a tic a lly in the
d e te rm in a tio n o f th e o u tp u t. F o r ex am p le, if th e in p u t signal x( n) is p erio d ic
w ith p e rio d N , w e have a lre a d y o b se rv e d in S ectio n 1.3.5 th a t a m a th em atically
c o n v e n ie n t set o f elem en tary 7 signals is th e set o f e x p o n en tials
x k (n) = eJluin
k = 0. l , . . . , i V - l
(2.3.5)
w h e re th e fre q u e n c ie s {cok } a re h arm o n ic a lly re la te d , th a t is.
k = 0. 1.........N - 1
(2.3.6)
T h e fre q u e n c y 2 n / N is called th e fu n d a m e n ta l fre q u e n c y , an d all h ig h er-freq u en cy
c o m p o n e n ts a re m u ltip le s o f th e fu n d a m e n ta l fre q u e n c y c o m p o n e n t. T h is subclass
o f in p u t sig n als is c o n s id e re d in m o re d e ta il later.
F o r th e re so lu tio n o f th e in p u t signal in to a w eig h ted su m o f u n it sam ple
se q u en ces, w e m u st first d e te rm in e th e re sp o n se o f th e system to a u n it sa m ­
p le se q u e n c e a n d th e n use th e scaling a n d m u ltip lic ativ e p ro p e rtie s of th e lin ear
74
Discrete-Time Signals and Systems
Chap. 2
sy stem to d e te rm in e th e fo rm u la fo r th e o u tp u t given any a rb itra ry input, T his
d e v e lo p m e n t is d escrib ed in d etail as follow s.
2.3.2 Resolution of a Discrete-Time Signal into Impulses
S u p p o se w e h av e an a rb itra ry signal x ( n ) th a t we w ish to reso lv e in to a sum of unit
sa m p le seq u en ces. T o utilize th e n o ta tio n e sta b lish e d in th e p re c e d in g se ctio n , we
select th e e le m e n ta ry signals x k (n) to be
(2.3.7)
x k(n) = 8{n - k)
w h ere k re p re se n ts th e d elay o f th e u n it sam p le se q u e n c e . T o h a n d le an a rb itra ry
signal x ( n ) th a t m ay h ave n o n z e ro v alu es o v er an infinite d u ra tio n , th e set of unit
im p u lses m u st also b e infinite, to en co m p ass th e infinite n u m b e r of delays.
N o w su p p o se th a t we m u ltip ly th e tw o se q u e n c e s x( n) a n d <5(n - k). Since
8{n — k) is z e ro e v ery w h ere ex cep t a t n = k , w h e re its v alu e is u nity, the result
o f th is m u ltip lic atio n is a n o th e r se q u e n c e th a t is z e ro e v ery w h ere e x c e p t at n — k ,
w h ere its v alu e is x ( k) , as illu stra te d in Fig. 2,22. T h u s
x( n) 8( n — k) = x ( k) 8( n — k)
(2.3.8)
Jt(n)
T TT 111 i ’ l l
i
[ -2-10113
(a)
i , ‘ 1 I , Tt I
i
i 1
JC(Jc)
6(/i-Jt)
(b)
*(*) 6(n —k )
k
0
Figure 2.22
n
M ultiplication of a signal x i n ) with a shifted unit sam ple sequence.
Sec. 2.3
Analysis of Discrete-Time Linear Time-Invariant Systems
75
is a se q u e n c e th a t is z e ro e v e ry w h e re ex cep t at n = k , w h e re its v a lu e is x( k ) . If we
w ere to re p e a t th e m u ltip lic a tio n of x ( n ) w ith <5(a? — m ), w h ere m is a n o th e r d elay
(im =6 k), th e re su lt will b e a se q u en ce th a t is z e ro e v e ry w h e re e x cep t at n = m,
w h ere its v alu e is x ( m ) . H e n c e
x( n ) 5 ( n — m) = x ( m) 8( n — m)
(2.3.9)
In o th e r w o rd s, each m u ltip lic a tio n o f th e signal x( n ) by a u n it im p u lse at som e
d elay k, [i.e., <5(n — it)], in essen ce picks o u t th e single v alu e x ( k ) o f th e signal jc(n)
at th e d e la y w h e re th e u n it im pulse is n o n z e ro . C o n s e q u e n tly , if w e re p e a t this
m u ltip lic a tio n o v e r all p o ssib le delays, - o o < k < oo, a n d su m all th e p ro d u c t
se q u e n c e s, th e re su lt will be a se q u e n c e e q u a l to th e se q u e n c e x ( n ) , th a t is,
PCx( n) = ^
x ( k) 8( n — k)
k=—oc
(2.3.10)
W e e m p h a s iz e th a t th e rig h t-h an d side of (2.3.10) is th e su m m a tio n of an
infinite n u m b e r o f u n it sa m p le se q u en ces w h ere th e u n it sa m p le se q u e n c e 6(n - k)
h as an a m p litu d e value o f x( k ) . T h u s th e rig h t-h a n d sid e o f (2.3.10) gives th e
re so lu tio n o f o r d e c o m p o s itio n o f any a rb itra ry signal jc(n) in to a w e ig h te d (scaled)
sum o f sh ifted u n it sam p le seq u en ces.
Exam ple 2.3.1
Consider the special case of a finite-duration sequence given as
jc(«) =
(2, 4, 0,3)
T
Resolve the sequence x(n) into a sum of weighted impulse sequences.
Solution Since the sequence x(n) is nonzero for the time instants n = —1, 0. 2, we
need three impulses at delays k = —1. 0, 2, Following (2.3.10) we find that
x<n) = 2(5(n + 1) + 4<5(n) + 3<5(n —2)
2.3.3 Response of LTI Systems to Arbitrary Inputs: The
Convolution Sum
H av in g re so lv e d an a rb itra ry in p u t signal x ( n ) in to a w eig h te d su m o f im pulses,
w e a re no w re a d y to d e te rm in e th e re sp o n se of an y re la x e d lin e a r sy stem to any
in p u t signal. F irs t, w e d e n o te th e re sp o n se v(n, k) o f th e sy stem to th e in p u t unit
sa m p le se q u e n c e a t n = it by th e special sym bol h(n, k), —oo < k < oo. T h a t is,
y( n, k) = h(n. k ) = T[ S( n — £)]
(2.3.11)
In (2.3.11) w e n o te th a t n is th e tim e index a n d k is a p a r a m e te r sh ow ing th e
lo c a tio n o f th e in p u t im p u lse. If th e im pulse at th e in p u t is sc aled by an a m o u n t
ct = jc(it), th e re sp o n s e of th e system is th e c o rre sp o n d in g ly sc aled o u tp u t, th a t is,
Ckh(n, k) = x(k)h(n, k)
(2.3.12)
Discrete-Time Signals and Systems
76
Chap. 2
F in ally , if th e in p u t is th e a rb itra ry signal x(/t) th a t is e x p re ss e d as a sum of
w eig h ted im p u lses, th a t is.
(2.3.13)
th e n th e resp o n se o f the system to x(/i) is the c o rre sp o n d in g sum o f w eig h ted
o u tp u ts, th a t is,
y(rt) = 7"[.r(/;)] = T
=
^
^
x( k ) S( n — k)
x ( k ) T [ 5 (m - k)]
(2.3.14)
ii= —oc
C learly , (2.3.14) follow s from th e su p e rp o sitio n p ro p e rty of lin e a r system s, and is
k n o w n as th e superposit i on s u mma t i o n .
W e n o te th a t (2.3.14) is an e x p ressio n for th e resp o n se o f a lin e a r system to
any a rb itra ry in p u t se q u e n c e x( n) . T his ex p ressio n is a fu n ctio n of b o th .v(») and
th e resp o n ses h(n. k) of the system to th e unit im pulses Sin — k) fo r —oc < k < oc.
In d eriv in g (2.3.14) w e used th e lin earity p ro p e rty o f th e system but not its tim ein v arian ce p ro p e rty . T h u s th e ex p ressio n in (2.3.14) ap p lies to any relax ed lin ear
(tim e -v a ria n t) system .
If. in a d d itio n , th e system is tim e in v a ria n t, th e fo rm u la in (2.3.14) sim plifies
c o n sid erab ly . In fact, if the resp o n se o f th e L T I system to th e u n it sa m p le seq u en ce
<5(rc) is d e n o te d as h(n). th a t is.
h(n) = T [ b( n ) \
(2.3.15)
th en by th e tim e-in v arian ce p ro p e rty , th e resp o n se o f the system to the delay ed
u n it sa m p le se q u e n c e <5(n - k) is
h(n — k) = T [ S( n — A')]
(2,3.16)
C o n seq u en tly , th e fo rm u la in (2.3.14) re d u c e s to
(2.3.17)
k=-oc
N ow we o b serv e th a t th e relax ed L T I system is co m p letely c h a ra c te riz e d by a
single fu n ctio n h(n), n am ely , its resp o n se to th e u n it sam p le se q u e n c e
In
c o n tra st, th e g en eral c h a ra c te riz a tio n of th e o u tp u t o f a tim e -v a ria n t, lin e a r sys­
tem re q u ire s an in fin ite n u m b e r o f u n it sa m p le re sp o n s e fu n ctio n s, h{n, k), o n e for
ea c h p o ssib le d elay .
T h e fo rm u la in (2.3.17) th a t gives th e re sp o n se y( n) of th e L T I system as a
fu n c tio n o f th e in p u t signal x ( n ) a n d th e u n it sa m p le (im p u lse) re sp o n se h(n) is
called a convol ut i on s um. W e say th a t th e in p u t jt(n ) is c o n v o lv ed w ith th e im p u lse
Sec. 2.3
Analysis of Discrete-Time Linear Tim e-Invariant Systems
77
response h(n) to yield the output y in ). W e shall now explain the procedure for
com puting the resp onse y (n ). both m athem atically and graphically, given the input
x ( n ) and the im pulse response h(n) o f the system .
Suppose that we wish to com pute the output of the system at som e time
instant, say n = n 0. A ccordin g to (2.3.17), the resp onse at n = no is given as
OC
y (n 0) =
^ 2 x ( k ) h ( n 0 - k)
(2.3.18)
Jc=-oc
O ur first observation is that the index in the sum m ation is k , and h en ce both the
input signal x ( k ) and the im pulse resp onse h(no - k) are fun ction s o f k. Second,
w e ob serve that the sequ en ces x ( k ) and h(nQ — k) are m ultiplied togeth er to form
a product seq u en ce. T h e output >(«o) is sim ply the sum over all valu es o f the
product sequ en ce. T he seq u en ce h ( n 0 — k) is ob tain ed from h ( k ) by, first, folding
h (k) about k = 0 (the tim e origin), w hich results in the seq u en ce h ( —k). The
folded seq u en ce is then shifted by no to yield h(no — k). T o sum m arize, the process
o f com p utin g the con volu tion b etw een x ( k ) and h (k) in volves the follow in g four
steps.
1. Folding. F old h(k) about k = 0 to obtain h ( ~ k ) .
2. Shifting, Shift h ( —k) by n 0 to the right (left) if n o is p ositive (n egative), to
obtain h(no — £).
3. Multiplication. M ultiply
v„ Jk) = x ( k ) h ( n 0 - k).
by h (no — k) to obtain the product sequ en ce
4. S u m m a t i o n . Sum all the values o f the product seq u en ce vnt)( k ) to obtain the
value o f the output at tim e n = n 0.
W e note that this p rocedure results in the resp onse o f the system at a sin ­
gle tim e instant, say n = n 0. In gen eral, we are interested in evaluating the
response o f the system over all tim e instants - o o < n < oo. C onsequently,
steps 2 through 4 in the sum m ary m ust be rep eated , for all p ossible tim e shifts
—oo < n < oo.
In order to gain a b etter understanding o f the procedure for evaluating the
convolution sum , w e shall dem onstrate the p rocess graphically. T he graphs will
aid us in explaining the four steps in volved in the com p utation o f the convolution
sum.
Example 2.3.2
The impulse response of a linear time-invariant system is
/!(«) = [1 .2 ,1 ,-1 }
(2.3.19)
T
Determine the response of the system to the input signal
x(n) = {1,2.3,1}
t
(2.3.20)
Discrete-Time Signais and Systems
Chap. 2
Solution We shall com pute the convolution according to the formula (2.3.17). but
we shall use graphs of the sequences to aid us in the com putation. In Fig. 2.23a we
illustrate the input signal sequence x(k) and the impulse response h{k) of the system,
using k as the time index in order to be consistent with (2.3.17),
The first step in the com putation of the convolution sum is to fold h(k). The
folded sequence h(~k) is illustrated in Fig. 2.23b. Now we can compute the output
at n = 0. according to (2.3.17), which is
v(0) =
(2.3.21)
*=-cx
Since the shift n = 0, we use h( —k) directly without shifting it. The product sequence
= x(k)h(-k)
(2.3.22)
h(k)
x(k
3
4i
T
1
* -i
-1 0 ! j
10
h(-k)
•
1 2 3
L'n(k 1
2
.
. -2
T ’t . . .
-1 0 1 2
.
(b)
Shift
vAk)
Product
,,.
L’ |{t)
Product
sequence
h(\-k)
111
To
Br
T
7
(c)
T
.
2
~3 T !
j-2 -1 0 1
1■
k
0 12
(d)
Figure 2.23
G raphical com putation of convolution.
*
Sec. 2.3
Analysis of Discrete-Time Linear Time-Invariant Systems
79
is also shown in Fig. 2.23b. Finally, the sum of all the terms in the product sequence
yields
■(°) = £
vott) = 4
We continue the com putation by evaluating the response of the system at n = 1.
According to (2.3.17),
(2.3.23)
The sequence h(\ —k) is simply the folded sequence h( —k) shifted to the right by one
unit in time. This sequence is illustrated in Fig. 2.23c. The product sequence
V] (k} = x(k)h{l — k)
(2.3.24)
is also illustrated in Fig. 2.23c. Finally, the sum of all the values in the product
sequence yields
y(l) = £
ui(*) = 8
In a similar m anner, we obtain y(2) by shifting h ( - k ) two units to the right,
forming the product sequence ih(A) = x(k)h(2 — k) and then summing all the terms
in the product sequence obtaining y(2) = 8. By shifting h(—k) farther to the right,
multiplying the corresponding sequence, and summing over all the values of the re­
sulting product sequences, we obtain v(3) = 3. v(4) = - 2 , y(5) = - 1 . For u > 5, we
find that v(n) = 0 because the product sequences contain all zeros. Thus we have
obtained the response y(n) for n > 0.
Next we wish to evaluate v(n) for n < 0. We begin with n =
Then
(2.3.25)
Now the sequence h (—1 —k) is simply the folded sequence h ( —k ) shifted one time
unit to the left. The resulting sequence is illustrated in Fig. 2.23d. The corresponding
product sequence is also shown in Fig. 2.23d. Finally, summing over the values of the
product sequence, we obtain
V (-1) = 1
From observation of the graphs of Fig. 2.23, it is clear that any further shifts of
h (—1 - k) to the left always results in an all-zero product sequence, and hence
y(n) = 0
for n 5 —2
Now we have the entire response of the system for —oc < n < oc. which we
summarize below as
y(n) =
0 ,0,1, 4. 8, 8. 3, - 2 , - 1 , 0 . 0 . . . .)
t
(2.3.26)
80
Discrete-Time Signals and Systems
Chap. 2
In E x am p le 2.3.2 w e illu stra te d th e c o m p u ta tio n o f th e co n v o lu tio n sum .
using g rap h s o f th e se q u en ces to aid us in visualizing th e ste p s in volved in th e
c o m p u ta tio n p ro c e d u re .
B e fo re w o rk in g o u t a n o th e r ex a m p le , w e w ish to show th a t th e co n v o lu ­
tio n o p e ra tio n is c o m m u ta tiv e in th e se n se th a t it is irre le v a n t w hich of th e tw o
se q u e n c e s is fo ld ed a n d shifted. In d e e d , if w e b eg in w ith (2.3.17) a n d m ak e a
ch an g e in th e v ariab le o f th e su m m a tio n , fro m k to m , by defin in g a new index
m — n — k, th e n k = n — m an d (2.3.17) b e co m es
CC
y(n) =
^2
m—
—
x( n — m ) h { m )
(2.3.27)
Since m is a d u m m y in dex, w e m ay sim ply re p la c e m by k so th a t
y{n) =
x(n-k)h(k)
(2.3.28)
T h e ex p ressio n in (2.3.28) involves leav in g th e im p u lse re sp o n s e h( k) u n a lte re d ,
w hile th e in p u t se q u en ce is fo ld ed a n d sh ifted . A lth o u g h th e o u tp u t v(n) in (2.3.28)
is id en tical to (2.3.17), th e p ro d u c t se q u e n c e s in th e tw o fo rm s o f th e co n v o lu tio n
fo rm u la are not id en tical. In fact, if w e define th e tw o p ro d u c t se q u e n c e s as
v„(k) = x ( k ) h ( n — k)
w n(k) = x( n — k) h(k)
it can b e easily show n th at
un(£) = w n (n — k)
an d th e re fo re ,
v(n) =
CC
CC
^
^
k——oc
~ k)
oc
since b o th se q u en ces co n tain th e sam e sa m p le v alu es in a d iffe re n t a rra n g e m e n t.
Example 2.3.3
Determ ine the output y(n) of a relaxed linear tim e-invariant system with impulse
response
h(ri) = a"u(n), \a\ < 1
when the input is a unit step sequence, that is,
x (n) = u(n)
Solution In this case both /j(n) and jc(n) are infinite-duration sequences. We use
the form of the convolution formula given by (2.3.28) in which x (k) is folded. The
Sec. 2.3
81
Analysts of Discrete-Time Linear Tim e-Invariant Systems
h{k\
x(k)
TT i f
1
1
2
3
4
*
(b)
I'oCA')
x(- k)
l
* 1
-3
-2 -1
-1
0
1
-
A
(c)
A(1 - k )
>
i'i(i-)
a
1
- 1 0
v( 2—k)
k
1
1':(!>
II
- 2 - 1 0 1 2 3 4 5
Figure 2.24
k
Graphical computation of convolution in Example 2.3.3.
sequences h(k), x(k). and x{—k) are shown in Fig. 2.24. The product sequences vo(k).
v\(k), and v2(k) corresponding to x ( —k)h(k), x (l —k)h(k), and x(2 - k)h(k) are illus­
trated in Fig. 2.24c, d. and e. respectively. Thus we obtain the outputs
v(0) = 1
y(l) = 1 + a
y( 2) = 1 + a + a 1
Discrete-Time Signals and Systems
82
Chap. 2
Clearly, for n > 0, the output is
y(n) = 1 + a 4- a2 + ■• ■+ a”
1 _ an+1
(2.3,29)
=
1-a
On the other hand, for n < 0, the product sequences consist of all zeros. Hence
v(n) = 0
n < 0
A graph of the output y(n) is illustrated in Fig. 2.24f for the case 0 < a < 1.
Note the exponential rise in the output as a function of n. Since |a| < 1, the final
value of the output as n approaches infinity is
v(oo) = lim v(n) = ------n-*oc '
1 —a
(2.3.30)
T o su m m arize, th e co n v o lu tio n fo rm u la p ro v id e s us w ith a m ean s fo r co m ­
p u tin g th e re sp o n s e o f a re lax ed , lin ear tim e -in v a ria n t system to an y a rb itra ry in p u t
signal x( n). It ta k e s o n e o f tw o e q u iv a le n t form s, e ith e r (2.3.17) o r (2,3.28), w h ere
jt(n ) is th e in p u t sig n al to th e system , h (n ) is th e im p u lse re s p o n s e of th e system ,
an d y (n ) is th e out p u t o f th e system in re sp o n s e to th e in p u t signal x (n ). T he
e v a lu a tio n o f th e co n v o lu tio n fo rm u la involves fo u r o p e ra tio n s , n am ely: f ol di ng
e ith e r th e im p u lse re sp o n s e as specified by (2.3.17) o r th e in p u t se q u e n c e as sp ec­
ified by (2.3.28) to yield e ith e r h ( —k) o r x { —k). resp ec tiv ely , shifting th e folded
se q u e n c e by n u n its in tim e to yield e ith e r h{n — k ) o r x { n — k ). mul t i pl yi ng the
tw o se q u e n c e s to yield th e p ro d u c t se q u en ce, e ith e r x { k) h{ n — k) o r x ( n - k ) h ( k ) ,
a n d finally s u m m i n g all th e v alu es in th e p ro d u c t se q u e n c e to y ield th e o u tp u t v (n )
o f th e sy stem a t tim e n. T h e folding o p e ra tio n is d o n e only o n ce. H o w ev er, th e
o th e r th re e o p e ra tio n s a re re p e a te d fo r all p o ssib le shifts —oc < n < oo in o rd e r
to o b ta in y (n ) fo r —oo < n < oc.
2.3.4 Properties of Convolution and the Interconnection
of LTI Systems
In th is se ctio n w e in v estig ate so m e im p o rta n t p ro p e rtie s o f co n v o lu tio n an d in ­
te r p re t th e se p r o p e rtie s in te rm s o f in te rc o n n e c tin g lin ear tim e -in v a ria n t system s.
W e sh o u ld stress th a t th ese p ro p e rtie s h o ld fo r e v e ry in p u t signal.
It is c o n v e n ie n t to sim plify th e n o ta tio n by using an a s te ris k to d e n o te the
c o n v o lu tio n o p e ra tio n . T h u s
OC
y( n) = x{n) * h(n) = ^
x ( k ) h ( n — k)
(2.3.31)
Jt = - O C
In th is n o ta tio n th e se q u e n c e follow ing th e aste risk [i.e., th e im p u lse re sp o n se /i(«)]
is fo ld e d an d sh ifted . T h e in p u t to th e sy stem is ;c(n). O n th e o th e r h a n d , we also
sh o w ed th a t
OC
>>(n) = h{n) * x( n) = ^
k=-oc
h ( k ) x ( n - k)
(2.3.32)
Sec. 2.3
83
Analysis of Discrete-Time Linear Time-Invariant Systems
hin)
h(n)
Figure 2.25
<
=
>
v(n )
xin)
Interpretation of the commutative property of convolution.
In th is fo rm o f th e co n v o lu tio n fo rm u la, it is the in p u t signal th a t is fo ld ed . A lte r ­
n ativ ely . we m ay in te r p re t this fo rm o f the c o n v o lu tio n fo rm u la as re su ltin g from
an in te rc h a n g e o f th e ro les o f j:(n) an d h(n). In o th e r w ords, w e m ay re g a rd x( n)
as th e im p u lse re sp o n se o f th e system an d h( n) as the e x c ita tio n o r in p u t signal.
F ig u re 2.25 illu stra te s th is in te rp re ta tio n .
W e can view co n v o lu tio n m o re ab stractly as a m a th e m a tic a l o p e ra tio n b e ­
tw een tw o signal se q u e n c e s, say x( n ) a n d h(n), th a t satisfies a n u m b e r o f p ro p e rtie s .
T h e p ro p e rty e m b o d ie d in (2.3.31) an d (2.3.32) is called th e c o m m u ta tiv e law'.
Commutative law
jf(n) * h{n) = h(n) * x( n)
(2.3.33)
V iew ed m a th e m a tic a lly , the c o n v o lu tio n o p e ra tio n also satisfies th e asso cia­
tive law , w hich can be sta te d as follow s.
Associative law
[-v(/i) * /?[(«)] * h 2(/i) — x( n) * [/*!(«) *
(2.3.34)
F ro m a physical p o in t o f view, we can in te rp re t x (n) as the in p u t signal to
a lin e a r tim e -in v a ria n t system w ith im pulse re sp o n se /j|(/i). T h e o u tp u t o f this
sy stem , d e n o te d as v i(n ), b eco m es the in p u t to a seco n d lin e a r tim e -in v a ria n t
sy stem w ith im p u lse re sp o n se hzin). T h en th e o u tp u t is
y( n) — V] (n) * h 2(n)
= [j:(fi) * h\(n)] * h2(n)
w hich is p recisely th e left-h an d side of (2.3.34). T h u s th e le ft-h a n d side o f (2.3.34)
c o rre sp o n d s to h av in g tw o lin ear tim e -in v a ria n t system s in cascad e. N ow th e righth an d side o f (2.3.34) in d icates th a t th e in p u t x ( n ) is a p p lied to an e q u iv a le n t system
h av in g an im p u lse re sp o n se , say h(n), w hich is e q u a l to the co n v o lu tio n o f th e tw o
im p u lse resp o n se s. T h a t is,
h(n) = h](n) * h?(n)
an d
y(n) = x( n ) * h(n)
F u rth e rm o re , sin ce th e co n v o lu tio n o p e ra tio n satisfies th e c o m m u ta tiv e p ro p e rty ,
o n e can in te rc h a n g e th e o r d e r o f th e tw o system s w ith re sp o n s e s h \ ( n ) a n d hzi n)
w ith o u t a lte rin g th e o v erall in p u t-o u tp u t re la tio n sh ip . F ig u re 2.26 g rap h ically il­
lu stra te s th e asso ciativ e p ro p e rty .
84
Discrete-Time Signals and Systems
xin)
jr(n)
y(«)
h(n) =
Chap. 2
y{/i)
h\(n) * h2(n)
(a)
x(n)
h,(n)
y</i)
v(n)
h2(n)
h t(n)
(b)
Figure Z 2 6
Implications of the associative (a) and the associative and commuta­
tive (b) properties of convolution.
Example 2.3.4
D eterm ine the impulse response for the cascade of two iinear time-invariant systems
having impulse responses
h \ («) = ( j W " )
and
Solution To determ ine the overall impulse response of the two systems in cascade,
we simply convolve h}{n) with h2(n). Hence
where h2(n) is folded and shifted. We define the product sequence
v„(k) = h\(k)h2(n - k)
= (£)*(*)"-*
which is nonzero for k > 0 and n - k > 0 or n > k > 0. On the o th er hand, for n < 0,
we have v„(k) = 0 for all k, and hence
h{n) = 0, n < 0
For n > k > 0. the sum of the values of the product sequence v„(k) over all k yields
hin) =
tscO
= Hr
= (i)"(2"+l - 1 )
Sec. 2.3
Analysis of Discrete-Time Linear Tim e-Invariant Systems
85
T h e g e n e ra liz a tio n o f the associative law to m o re th a n tw o system s in cascade
follow s easily fro m th e d iscu ssio n given ab o v e. T h u s if w e h av e L lin e a r tim ein v arian t sy stem s in cascade w ith im p u lse re sp o n se s h\ ( u) . h ^ i n ) ........ h L (n). th ere
is an e q u iv a le n t lin e a r tim e -in v a ria n t sy stem hav in g an im p u lse re sp o n se th a t is
e q u a l to th e (L — l)-fo id co n v o lu tio n of th e im p u lse resp o n se s. T h a t is.
h( n) = h \ { n ) * hzi n) * ■■■* h L (n)
(2.3.35)
T h e c o m m u ta tiv e law im p lies th a t th e o rd e r in w hich the c o n v o lu tio n s a re p e r ­
fo rm e d is im m a te ria l. C o n v e rsely , an y lin e a r tim e -in v a ria n t system can be d e c o m ­
p o se d in to a cascad e in te rc o n n e c tio n o f subsystem s. A m e th o d for accom plishing
th e d e c o m p o s itio n will be d escrib ed later.
A th ird p ro p e rty th a t is satisfied by th e c o n v o lu tio n o p e ra tio n is the d istrib u ­
tive law, w hich m ay be sta te d as follow s.
Distributive law
xin) *
4-
= .xin) * h\ {n) 4- x i n ) * hzi n)
(2.3.36)
I n te rp r e te d physically, this law im plies th a t if we h ave tw o lin ear tim ein v a ria n t sy stem s w ith im p u lse re sp o n se s h \ i n ) a n d /?;(») ex cited by the sam e
in p u t signal .r(/;), th e sum of the tw o resp o n ses is identical to the resp o n se of an
o v erall system w ith im pulse resp o n se
h i n ) = )i\ in) 4- //:(/;)
T h u s th e o v erall system is view ed as a p arallel co m b in a tio n of the tw o linear
tim e -in v a ria n t sy stem s as illu stra te d in Fig. 2.27,
T h e g e n e ra liz a tio n o f (2.3.36) to m o re th an tw o lin e a r tim e-in v arian t sys­
tem s in p a rallel follow's easily by m a th e m a tic a l in d u ctio n . T h u s th e in te rc o n n e c ­
tio n of L lin ear tim e -in v a ria n t system s in p a ra lle l w ith im p u lse resp o n ses h\ i n) .
h z i n ) .........h L{n) a n d ex cited by the sa m e in p u t x i n ) is e q u iv a le n t to o n e overall
system w ith im p u lse re sp o n se
L
h(n) — ^ hj i n)
(=i
(2.3.37)
C o n v e rsely , an y lin ear tim e -in v a ria n t system can be d e c o m p o se d into a p arallel
in te rc o n n e c tio n o f su b sy stem s.
Figure 2.27 Interpretation of the distributive property of convolution: two LTI
systems connected in parallel can be replaced by a single system with h(n) =
(n) + k2{n).
Discrete-Time Signals and Systems
86
Chap. 2
2.3.5 Causal Linear Time-Invariant Systems
In S ectio n 2.2.3 w e d efin ed a causal system as o n e w hose o u tp u t at tim e n d e p en d s
o n ly on p re s e n t an d p ast in p u ts b u t d o es n o t d e p e n d on f u tu re in p u ts. In o th e r
w o rd s, th e o u tp u t o f the. sy stem at so m e tim e in s ta n t n, say n = no, d e p e n d s only
on v alu es o f jc(«) fo r n < n 0In th e case o f a lin e a r tim e -in v a ria n t system , cau sality can b e tra n sla te d
to a c o n d itio n o n th e im p u lse resp o n se. T o d e te rm in e this re la tio n sh ip , le t us
co n sid e r a lin ear tim e -in v a ria n t system having an o u tp u t a t tim e n = no given by
th e c o n v o lu tio n fo rm u la
OC
v (« o ) =
^2
h ( k ) x (n o ~ k )
k —- o c
S u p p o se th a t w e su b d iv id e th e sum in to tw o sets o f term s, o n e se t involving p re se n t
a n d p a st v alu es o f th e in p u t [i.e.. x{n) for n < n 0] a n d o n e se t involving fu tu re
valu es o f th e in p u t [i.e.,
n > no]. T h u s we o b ta in
OC
-I
y (n u) — ^ h ( k ) x ( n o - k) +
i=0
h( k) x( nu - k)
^
k ——oc
= [/ ?(0)x (n ()) + h( \ ) x( r tu -
1) + h ( 2 ) x ( n 0 - 2 ) + ■■•]
+ [/?( —1 )x(«(] + 1) + h ( - 2 ) x ( n o + 2) + ■■ ■]
W e o b se rv e th a t th e term s in th e first sum involve jr(no), x( no — 1).........w hich are
th e p re s e n t a n d p ast values of th e in p u t signal. O n th e o th e r h a n d , th e te rm s in
th e se co n d sum in volve th e in p u t signal c o m p o n e n ts ;c(no + l) , x { n o -f 2 ) .........N ow ,
if th e o u tp u t a t tim e n = n 0 is to d e p e n d only on th e p re s e n t a n d p a st in p u ts, th en ,
clearly , th e im p u lse re sp o n se o f th e system m u st satisfy th e c o n d itio n
ft(n) = 0
n < 0
(2.3.38)
Since h{n) is th e re sp o n se o f th e re lax ed lin e a r tim e -in v a ria n t sy stem to a u nit
im p u lse a p p lied a t n = 0, it follow s th a t h(n) = 0 fo r n < 0 is b o th a necessary
an d a sufficient c o n d itio n fo r causality. H e n c e an L T I syst em is causal i f a n d onl y
i f its i mpul se respons e is z e r o f o r negative values o f n.
Since fo r a cau sal sy stem , h(n) = 0 fo r n < 0, th e lim its on th e su m m a tio n of
th e co n v o lu tio n fo rm u la m ay be m odified to reflect th is re stric tio n . T h u s w e h ave
th e tw o e q u iv a le n t form s
OC
y ( n ) = ^ h ( k ) x ( n - k)
Jt=0
n
= ^
x{k) h{n — k)
k=~oc
(2.3.39)
(2.3.40)
A s in d icated p rev io u sly , cau sality is re q u ire d in an y re a l-tim e signal p ro c e ss­
ing a p p licatio n , since at a n y given tim e n w e have n o access to f u tu re v alu es o f th e
Sec. 2.3
Analysis of Discrete-Time Linear Time-Invariant Systems
87
in p u t signal. O n ly th e p re s e n t a n d p ast v alu es o f the in p u t signal are av ailab le in
c o m p u tin g th e p re s e n t o u tp u t.
It is so m e tim e s c o n v e n ie n t to call a se q u en ce th a t is z e ro fo r n < 0, a causa!
s e q u e n c e , an d o n e th a t is n o n z e ro fo r n < 0 a n d n > 0. a n on c aus al sequence. T his
te rm in o lo g y m e a n s th a t su c h a se q u e n c e could be th e u n it sa m p le re sp o n se of a
causal o r a n o n c a u sa l system , resp ectiv ely .
If th e in p u t to a causal lin e a r tim e -in v a ria n t system is a causal se q u e n c e [i.e.,
if jr(n) = 0 fo r n < 0]. th e lim its on th e co n v o lu tio n fo rm u la a re fu rth e r restricted .
In th is case th e tw o e q u iv a le n t form s o f th e c o n v o lu tio n fo rm u la b eco m e
n
y ( n ) = ^ h( k ) x ( n — k)
(2.3.41)
*=o
n
=
-k)
(2.3.42)
*■=<)
W e o b se rv e th a t in th is case, th e lim its on the su m m a tio n s fo r the tw o a lte rn a tiv e
fo rm s are id en tical, a n d th e u p p e r lim it is grow ing w ith tim e. C learly , th e resp o n se
o f a cau sal sy stem to a causal in p u t se q u e n c e is causal, since y( n) — 0 fo r n < 0.
Example 2.3.5
Determ ine the unit step response of the linear time-invariant system with impulse
response
h{n) = a " u( n )
\a\ < 1
Solution Since the input signal is a unit step, which is a causal signal, and the system
is also causal, we can use one of the special forms of the convolution formula, either
(2.3.41) or (2.3.42). Since x(n) = 1 for n > 0. (2.3.41) is simpler to use Because of the
simplicity of this problem, one can skip the steps involved with sketching the folded
and shifted sequences. Instead, we use direct substitution of the signals sequences in
(2.3.41) and obtain
y(n) = y ~ v
*=(I
1 - a"*1
1 -ci
and y(n) = 0 for n < 0. We note that this result is identical to that obtained in Ex­
ample 2.3.3. In this simple case, however, we computed the convolution algebraically
without resorting to the detailed procedure outlined previously.
2.3.6 Stability of Linear Time-Invariant Systems
A s in d ic a te d p rev io u sly , sta b ility is an im p o rta n t p ro p e rty th a t m u st be co n sid e re d
in an y p ra c tic a l im p le m e n ta tio n o f a system . W e d efin ed an a rb itra ry relax ed
system as B IB O sta b le if a n d only if its o u tp u t se q u e n c e y («) is b o u n d e d fo r every
b o u n d e d in p u t x ( n) .
88
Discrete-Time Signals and Systems
Chap. 2
If x ( n ) is b o u n d e d , th e re exists a c o n s ta n t M x such th a t
l* (n )l < M x < 0 0
S im ilarly, if th e o u tp u t is b o u n d e d , th e re exists a c o n s ta n t M y such th a t
| ^ ( n ) | < My < OO
fo r all n.
N ow , given such a b o u n d e d in p u t se q u e n c e x ( n ) to a lin e a r tim e -in v a ria n t
sy stem , le t us in v e stig a te th e im p licatio n s of th e d efin itio n o f sta b ility o n th e c h a r­
acteristics o f th e system . T o w a rd th is en d , w e w o rk again w ith th e c o n v o lu tio n
fo rm u la
OO
y( n) = £
h{k) x{n - k )
k=—oc
If w e ta k e th e a b so lu te value of b o th sides o f th is e q u a tio n , w e o b tain
h( k ) x ( n — k)
Y
Lv(«)| =
* = -O C
N ow , th e ab so lu te v alu e of th e sum o f term s is alw ays less th a n o r e q u a l to the
sum o f th e a b so lu te v alu es o f th e term s. H e n c e
OC
\y(n)\ < Y
IM*)II*(/1 - fc)l
i = —oc
If th e in p u t is b o u n d e d , th e re exists a finite n u m b e r M x such th a t |x(n)[ < M t , By
su b s titu tin g th is u p p e r b o u n d fo r x( n ) in th e e q u a tio n ab o v e, w e o b ta in
OC'
|y (n )| < M X Y
W * )l
k = -o c
F ro m th is ex p ressio n w e o b se rv e th a t th e o u tp u t is b o u n d e d if th e im p u lse resp o n se
o f th e sy stem satisfies th e co n d itio n
OO
Sh =
Y
k = -o c
IAWI < 00
<2-3-43)
T h a t is, a linear time-invari ant s y s t em is stable i f its i mpu l s e respons e is absolutely
s u m m a b l e . T h is c o n d itio n is n o t only sufficient b u t it is also n e c e ssa ry to e n su re th e
sta b ility o f th e system . In d e e d , w e shall show th a t if 5* = 00, th e r e is a b o u n d e d
in p u t fo r w hich th e o u tp u t is n o t b o u n d e d . W e c h o o se th e b o u n d e d in p u t
■
\h*(-n)\
0,
h{n) ? 0
h(n) = 0
w h e re h*(n) is th e co m p lex c o n ju g a te o f h(n). I t is sufficient to show th a t th e re is
o n e v alu e o f n fo r w h ich y( n) is u n b o u n d e d . F o r n = 0 w e h av e
,< 0 ,=
£ ;
*= “ 00^
k = —(x. 1 v /l
Thus, if Si, = 00, a bounded input produces an unbounded output since y(0) = 00.
Sec. 2.3
Analysis of Discrete-Time Linear Tim e-Invariant Systems
89
T h e c o n d itio n in (2.3.43) im plies th a t th e im p u lse re s p o n s e h(n) goes to zero
as n a p p ro a c h e s infinity. A s a co n se q u e n c e , th e o u tp u t o f th e system g o es to zero
as n a p p ro a c h e s infinity if th e in p u t is set to z e ro b ey o n d n > n 0. T o p ro v e this,
su p p o se th a t |j ( h ) | < M x fo r n < no an d x ( n ) = 0 fo r n > no- T h e n , at n = no + N,
th e system o u tp u t is
(no + AO =
^
h ( k ) x ( n 0 + N - k) + ^
*-=-oc
h { k ) x ( n 0 + N - k)
B u t th e first su m is z e ro since * («) = 0 fo r n > no- F o r th e rem ain in g p a rt, we
ta k e th e a b so lu te v alu e o f th e o u tp u t, w hich is
j oc
y («0 + AO| =
I
sc
h( k)x(no + N — £)j <
< Mx
|/i(/:)||jf(no + N — £)|
W *)!
k=N
N ow , as N a p p ro a c h e s infinity.
an d hen ce
lim ]\ {/i() + AO! = 0
T h is re su lt im p lies th a t any ex citatio n at th e in p u t to th e system , w hich is of a finite
d u ra tio n , p ro d u c e s an o u tp u t th a t is “tra n s ie n t” in n a tu re ; th a t is, its am p litu d e
d ecay s w ith tim e an d dies o u t e v en tu ally , w h en th e system is stab le.
Example 2.3.6
Determ ine the range of values of the param eter a for which the linear time-invariant
system with impulse response
hin) = a"u(n)
is stable.
Solution First, we note that the system is causal. Consequently, the lower index on
the summation in (2.3.43) begins with k = 0. Hence
Clearly, this geom etric series converges to
provided that \a\ < 1 . Otherwise, it diverges. Therefore, the system is stable if |a| < 1.
Otherwise, it is unstable. In effect, h(.n) must decay exponentially toward zero as n
approaches infinity for the system to be stable.
90
Discrete-Time Signals and Systems
Chap. 2
Example 23.7
D eterm ine the range of values of a and b for which the linear time-invariant system
with impulse response
...
( a", n > 0
h(-n) = [ iw.f , n < n0
is stable.
Solution This system is noncasual. The condition on stability given by (2.3.43) yields
OC
t :
-1
OC
i*(«)f=
n= -o c
ia i" +
fi=s(J
y i
|fe,n
n= -o c
From Example 2.3.6 we have already determ ined that the first sum converges for
|a| < 1. The second sum can be manipulated as follows:
= p( l + p + p 2 + ■■■) =
I - p
where p = \f\b\ must be less than unity for the geometric series to converge. Conse­
quently, the system is stable if both \a\ < 1 and |fc| > 1 are satisfied,
2.3.7 Systems with Finite-Duration and Infinite-Duration
Impulse Response
U p to this p oin t w e have characterized a linear tim e-invariant system in term s of
its im pulse resp onse h(n). It is also con ven ien t, how ever, to subdivide the class
o f linear tim e-invariant system s in to tw o types, those that h ave a finite-duration
im pulse resp onse (F IR ) and those that have an infinite-duration im pulse response
(IIR ). Thus an F IR system has an im pulse resp onse that is zero ou tsid e o f som e
finite tim e interval. W ithout loss o f generality, w e focus our atten tion on causal
FIR system s, so that
h (n) = 0
n < 0 and n > M
The con volu tion form ula for such a system reduces to
u-1
? ( n ) = H h ^ x(^n ~ k )
t=0
A useful interpretation o f this exp ression is ob tain ed by ob serving that th e output
at any tim e n is sim ply a w eigh ted iinear com b ination o f the in p ut signal sam ples
x (n ), x { n - 1 ) , , x ( n - M + 1). In other words, the system sim ply w eigh ts, by
the values o f the im pulse resp onse h(k), k = 0, 1,
— 1, the m ost recent
M signal sam ples and sum s the resulting M products. In e ffe ct, th e system acts
as a w in d o w that view s on ly the m ost recent M input signal sam p les in form ing
the output. It n eg lects or sim ply “forgets” all prior input sam p les [i.e., x ( n — M ),
Sec. 2.4
Discrete-Time Systems Described by Difference Equations
91
x ( n — M — 1). . . .] . T h u s we say th a t an F IR svstem has a finite m e m o ry o f len g th -M
sam p les.
In c o n tra st, an I I R lin e a r tim e-in v arian t system has an in fin ite -d u ra tio n im ­
pu lse resp o n se. Its o u tp u t, b ased on th e c o n v o lu tio n fo rm u la, is
v(n) = ^ >2h(k) x( n - k )
w h ere cau sality h as b e e n assu m ed , a lth o u g h this a ssu m p tio n is n o t n ecessary. N ow .
th e system o u tp u t is a w eig h te d [by the im p u lse re sp o n se *(/:}] lin e a r co m b in a tio n
o f th e in p u t sig n al sa m p le s .r(n), x{n - 1), x( n — 2 ) ........ S ince this w eig h te d sum
in v olves th e p re s e n t an d all th e p ast in p u t sam ples, w e say th a t th e sy stem has an
infinite m em o ry .
W e in v e stig a te th e c h aracteristics of F IR an d IIR sy stem s in m o re d etail in
su b s e q u e n t c h a p te rs.
2.4 DISCRETE-TIME SYSTEMS DESCRIBED BY DIFFERENCE
EQUATIONS
U p to this p o in t w e h ave tre a te d linear and tim e-in v arian t sy stem s th a t are c h a r­
a c terized by th e ir u n it sa m p le resp o n se h(n). In tu rn . h(n) allow s us to d e te rm in e
th e o u tp u t v{n) o f th e system for any given in p u t se q u e n c e jc(/i) by m e a n s o f the
c o n v o lu tio n su m m a tio n .
(2.4.1)
In g e n e ra l, th e n , we h av e show n th a t any linear tim e -in v a ria n t system is c h a r­
a c terized by th e in p u t- o u tp u t rela tio n sh ip in (2.4.1). M o re o v e r, th e co n v o lu tio n
su m m a tio n fo rm u la in (2.4.1) suggests a m ean s fo r th e re a liz a tio n o f th e system .
In th e case o f F IR sy stem s, such a realizatio n in v o lv es a d d itio n s, m u ltip lic atio n s,
a n d a finite n u m b e r o f m e m o ry locations. C o n se q u e n tly , an F IR system is read ily
im p le m e n te d d irectly , as im p lied by the c o n v o lu tio n su m m atio n .
If th e sy stem is IIR , h o w ev er, its practical im p le m e n ta tio n as im p lied by
c o n v o lu tio n is clearly im p o ssib le, since it re q u ire s an infinite n u m b e r o f m e m ­
o ry lo catio n s, m u ltip lic a tio n s, an d ad d itio n s. A q u e stio n th a t n a tu ra lly arises,
th e n , is w h e th e r o r n o t it is possible to realize IIR sy stem s o th e r th a n in the
form su g g e sted by th e c o n v o lu tio n su m m atio n . F o rtu n a te ly , th e an sw er is yes.
th e re is a p ra c tic a l an d c o m p u ta tio n a lly efficient m e a n s fo r im p le m e n tin g a
fam ily o f IIR sy stem s, as will b e d e m o n s tra te d in th is se ctio n . W ith in th e g en ­
eral class o f IIR sy stem s, this fam ily of d isc re te -tim e sy stem s is m o re c o n ­
v en ien tly d e s c rib e d by d ifferen ce eq u atio n s. T h is fam ily o r subclass of IIR
sy stem s is v ery u sefu l in a v ariety o f p ractical ap p licatio n s, in clu d in g th e im p le ­
m e n ta tio n o f d ig ita l filters, a n d the m o d elin g o f physical p h e n o m e n a a n d physical
system s.
92
Discrete-Time Signals and Systems
Chap. 2
2.4.1 Recursive and Nonrecursive Discrete-Time Systems
A s in d icated ab o v e, th e co n v o lu tio n su m m a tio n fo rm u la e x p re sse s th e o u tp u t of
th e lin e a r tim e -in v a ria n t sy stem explicitly a n d only in te rm s o f th e in p u t signal.
H o w e v e r, th is n e e d n o t b e th e case, as is sh o w n h e re . T h e re a re m an y system s
w h ere it is e ith e r n ecessa ry o r d e sirab le to ex p re ss th e o u tp u t o f th e system n o t
on ly in term s o f th e p re s e n t a n d p a st v alu es o f th e in p u t, b u t also in te rm s o f the
a lre a d y av ailab le p a st o u tp u t values. T h e follow ing p ro b le m illu stra te s th is point.
S u p p o se th a t w e w ish to c o m p u te th e cumul at i ve average o f a signal x ( n ) in
th e in te rv a l 0 < k < n, d efin ed as
1
"
v(n) = ------- « = 0 . 1, . . .
n + 1 ~—f.
(2.4.2)
A s im p lied b y (2.4.2), th e c o m p u ta tio n o f _v(n) re q u ire s th e sto ra g e o f all th e in p u t
sa m p le s x (k ) fo r 0 < k < n. Since n is in creasin g , o u r m em o ry r e q u ire m e n ts grow
lin early w ith tim e.
O u r in tu itio n suggests, h o w ev er, th a t y( n) can be c o m p u te d m o re efficiently
by utilizin g th e p re v io u s o u tp u t v alue y( n — 1). In d e e d , by a sim p le alg eb raic
re a rra n g e m e n t o f (2.4.2), we o b ta in
it—l
(« + l)y (n ) =
x(k) + x(n)
= ny( n - 1) + x {n)
a n d h en ce
y( n) =
1
-x(n)
■y(n - 1) +
n + 1'
n + 1'
(2.4.3)
N ow , th e cu m u lativ e a v erag e v(n) can b e c o m p u te d recu rsiv ely b y m u ltip ly in g th e
p re v io u s o u tp u t v a lu e y( n - 1) by n/ ( n 4-1 ), m u ltip ly in g th e p r e s e n t in p u t x ( n ) by
1 / (n + 1), an d a d d in g th e tw o p ro d u cts. T h u s th e c o m p u ta tio n o f y (n ) by m ean s
o f (2.4.3) re q u ire s tw o m u ltip lic atio n s, o n e a d d itio n , a n d o n e m e m o ry lo catio n , as
illu stra te d in Fig. 2.28. T his is an ex a m p le of a recursive s yst em. In g e n e ra l, a
sy stem w hose o u tp u t v(n) a t tim e n d e p e n d s o n a n y n u m b e r o f p a s t o u tp u t values
y( n - 1), y ( n - 2 ) , . . . is called a recu rsiv e system .
tin)
Figure 2.28
Realization of a recursive cum ulative averaging system.
Sec. 2.4
Discrete-Time Systems Described by Difference Equations
93
T o d e te rm in e th e c o m p u ta tio n of the recu rsiv e sy stem in (2.4.3) in m ore
d etail, su p p o se th a t we begin th e p ro cess w ith n = 0 a n d p ro c e e d fo rw a rd in tim e.
T h u s, acco rd in g to (2.4.3). we o b ta in
y(0) -
jc(0)
v (l) = 5.v(0) + U ( l )
y(2) = § y ( l ) + i* ( 2 )
an d so on. If o n e grow s fatig u ed w ith this c o m p u ta tio n an d w ishes to pass the
p ro b le m to so m e o n e else at som e tim e, say n = no. th e only in fo rm a tio n th a t one
n e e d s to p ro v id e his o r h e r su ccesso r is the p a st value y(«o - 1) a n d the new' input
sa m p le s jr(n), j (/j + 1 )........ T h u s the successor b eg in s w ith
an d p ro c e e d s fo rw ard in tim e until som e tim e, say n = n j. w h en he or she b e­
co m es fatig u ed an d passes the c o m p u ta tio n a l b u rd e n to so m e o n e else w ith the
in fo rm a tio n o n th e value _v(/?i — 1). an d so on.
T h e p o in t wc wish to m ak e in th is discussion is th a t if o n e w ishes to co m p u te
th e re sp o n se (in this case, the cu m u lativ e av e ra g e ) of the sy stem (2,4.3) to an input
signal x( n ) a p p lied at n — tiu. we n eed th e value y (n (, - 1) an d th e input sam ples
x ( n ) for /? > /in. T h e term y(>i() — 1) is called th e initial condi t i on for the system in
(2.4.3) an d c o n ta in s all the in fo rm a tio n n e e d e d to d e te rm in e th e re sp o n se o f the
sy stem for n > « () to th e in p u t signal x (n ). in d e p e n d e n t o f w h at has o ccu rred in
th e p ast.
T h e fo llow ing ex am p le illu strates th e use o f a (n o n lin e a r) recu rsiv e system
to co m p u te th e sq u a re ro o t of a n u m b e r.
Exam ple 2.4.1
Square-Root Algorithm
Many computers and calculators compute the square root of a positive number A.
using the iterative algorithm
where j_i is an initial guess (estimate) of \/~A. As the iteration converges we have
= J~A.
Consider now the recursive system
sn ~ s„_j. T hen it easily follows that
(2.4.4)
which is realized as in Fig, 2.29. If we excite this system with a step of amplitude
A [i.e.. x(n) = A u ( n )] and use as an initial condition y(—1) an estimate of
the
response v(«) of the system will tend toward
as n increases. Note that in contrast
to the system (2.4.3), we do not need to specify exactly the initial condition. A rough
estim ate is sufficient for the proper perform ance of the system. For example, if we
94
Discrete-Time Signals and Systems
Chap. 2
-0---- -0
4n)
v(n - I)
Figure Z 2 9
Realization of the square-root system.
let A = 2 and y ( - l ) = 1, we obtain j(0 ) =
>-(1) = 1,4166667, v(2) = 1,4142157.
Similarly, for y ( —1) = 1.5, we have v(0) = 1,416667, y (l) = 1.4142157. Compare
these values with the %/2, which is approximately 1.4142136.
W e h av e no w in tro d u c e d tw o sim ple recu rsiv e system s, w h ere th e o u tp u t vf/i)
d e p e n d s o n th e p re v io u s o u tp u t v alu e y( n — 1) a n d th e c u rre n t in p u t ;r(n). B o th
system s a re causal. In g e n e ra l, w e can fo rm u la te m o re co m p lex cau sal recu rsiv e
system s, in w hich th e o u tp u t y ( n ) is a fu n ctio n o f se v era l p ast o u tp u t v alu es an d
p re se n t a n d p ast in p u ts. T h e system sh o u ld h av e a finite n u m b e r o f d elay s o r,
eq u iv alen tly , sh o u ld re q u ire a finite n u m b e r o f sto ra g e lo catio n s to be p ractically
im p le m e n te d . T h u s th e o u tp u t o f a cau sal a n d p ractically re a liz a b le recursive
system can be e x p ressed in g e n e ra l as
y( n) = F[y( n — 1), y( n — 2 ) , . . . , y( n — N ) , x ( n) , x ( n — 1 ) , . . . , x ( n — M ) \
(2.4.5)
w h ere F [ ] d e n o te s so m e fu n ctio n of its a rg u m e n ts. T h is is a re c u rsiv e e q u a tio n
specifying a p ro c e d u re fo r co m p u tin g th e system o u tp u t in term s o f p rev io u s v alu es
o f th e o u tp u t a n d p re s e n t an d p ast inputs.
In c o n tra st, if y{n) d e p e n d s only o n th e p re s e n t an d p a st in p u ts, th e n
y (n ) = F[ x ( n ) , x ( n — 1)........ x(rt — M) ]
(2.4.6)
S uch a sy stem is called nonrecursi ve. W e h a s te n to ad d th a t th e c a u s a l F IR system s
d escrib ed in S ectio n 2.3.7 in te rm s o f th e co n v o lu tio n sum fo rm u la h av e th e fo rm
o f (2.4.6). In d e e d , th e c o n v o lu tio n su m m a tio n fo r a cau sal F IR sy stem is
M
y ( n) =
- *)
i=0
= h( 0 ) x ( n ) -)- h ( \ ) x ( n — 1) + • - • + h ( M ) x ( n — M )
— F [j:(n ), x( n — 1), . . . , x ( n — M )]
w h ere th e fu n ctio n F[-] is sim ply a lin e a r w eig h ted su m o f p re s e n t a n d p ast in p u ts
a n d th e im p u lse re sp o n s e values h( n), 0 < n < M , c o n s titu te th e w eig h tin g c o e f­
ficients. C o n se q u e n tly , th e cau sal lin e a r tim e -in v a ria n t F IR sy stem s d e s c rib e d by
th e c o n v o lu tio n fo rm u la in S ectio n 2.3.7, a re n o n re c u rsiv e . T h e b a sic d ifferen ces
b e tw e e n n o n re c u rsiv e a n d re c u rsiv e system s a re illu stra te d in Fig. 2.30. A sim ple
in sp e ctio n o f th is figure re v e a ls th a t th e fu n d a m e n ta l d iffe re n c e b e tw e e n th e s e tw o
Sec. 2.4
x(n)
Discrete-Time Systems Described by Difference Equations
95
V(H )
F[Mn). xin - 1),
1
x{n)
1
^
|
|
........ ti n - .Wl]
K n ) ........... v(/i -
y{n)
M)\
]
1--------------------------------- 1
(b)
Figure 2.30 Basic form for a causal
and realizable (a) nonrecursive and
(b) recursive system.
system s is the fe e d b ack loop in th e recu rsiv e system , w hich feed s b ack th e o u tp u t
o f the system in to th e in p u t. T h is feed b ack loop co n tain s a d e la y ele m e n t. T he
p resen c e o f this d elay is crucial for the realizab ility of the system , since the ab sen ce
of this d e la y w ould force the system to c o m p u te yi n) in term s o f v(n). w hich is
n o t po ssib le fo r d isc re te -tim e system s.
T h e p re se n c e o f the fe ed b ack loop o r, eq u iv alen tly , the recu rsiv e n a tu re of
(2.4.5) c re a te s a n o th e r im p o rta n t d ifferen ce b etw een recursive an d n o n re c u rsiv e
system s. F o r e x am p le, su p p o se th a t we wish to c o m p u te th e o u tp u t y(«o) of a
system w h en it is ex cited by an in p u t ap p lied at tim e n = 0. If th e system is
recu rsiv e, to co m p u te y (« 0). we first n e e d to c o m p u te all the p re v io u s v alu es y(0).
y ( l ) ........ y(«o - 1)- In c o n tra st, if the system is n o n recu rsiv e. we can c o m p u te the
o u tp u t y (n 0) im m e d ia te ly w ith o u t having y(no - 1), y(«o — 2 )........ In co n clu sio n ,
th e o u tp u t o f a recu rsiv e system sh o u ld be c o m p u te d in o rd e r [i.e., v(0), y ( l) ,
y ( 2 ) . . . w h e re a s for a n o n re c u rsiv e system , th e o u tp u t can b e c o m p u te d in any
o rd e r [i.e., y(200). y (15). y{3). y(300). etc.]. T his fe a tu re is d e sira b le in so m e
p ractical a p p licatio n s.
2.4.2 Linear Time-Invariant Systems Characterized by
Constant-Coefficient Difference Equations
In S ectio n 2.3 w e tre a te d lin e a r tim e -in v a ria n t system s a n d c h a ra c te riz e d th em
in te rm s o f th e ir im p u lse resp o n ses. In this su b sectio n w e focus o u r a tte n tio n
o n a fam ily o f iin e a r tim e -in v a ria n t system s d escrib ed by an in p u t- o u tp u t r e la ­
tio n called a d iffe re n c e e q u a tio n w ith c o n s ta n t c o e ffic ie n ts . S ystem s d e sc rib e d
by c o n s tan t-co efficien t lin e a r d ifferen ce e q u a tio n s are a subclass o f the recu rsiv e
a n d n o n re c u rsiv e sy stem s in tro d u c e d in th e p re ced in g su b sectio n . T o b rin g o u t
th e im p o rta n t id eas, w e b eg in by tre a tin g a sim ple re c u rsiv e sy stem d e sc rib e d by
a first-o rd e r d iffe re n c e e q u a tio n .
96
Discrete-Time Signals and Systems
Chap. 2
<*>
Figure 131 Block diagram realization
of a simple recursive system.
a
S u p p o se th a t w e h a v e a recu rsiv e system w ith an in p u t- o u tp u t e q u a tio n
y( n) = ay( n - 1) + x ( n )
(2.4.7)
w h ere a is a c o n sta n t. F ig u re 2.31 show s a block d ia g ra m re a liz a tio n o f th e system .
In co m p a rin g th is sy stem w ith th e cu m u la tiv e a v erag in g sy stem d e sc rib e d by the
in p u t-o u tp u t e q u a tio n (2.4.3), w e o b se rv e th a t th e system in (2.4.7) h as a c o n sta n t
co effic ie n t (in d e p e n d e n t o f tim e ), w h e re a s th e sy stem d e sc rib e d in (2.4.3) h as tim ev a ria n t coefficients. A s w e will show , (2.4.7) is an in p u t- o u tp u t e q u a tio n fo r a
lin e a r tim e -in v a ria n t sy stem , w h e re a s (2.4.3) d escrib es a lin e a r tim e -v a ria n t system .
N ow , su p p o se th a t w e ap p ly an in p u t signal x ( n ) to th e sy stem fo r n > 0.
W e m a k e n o assu m p tio n s a b o u t th e in p u t sig n a l fo r n < 0, b u t w e d o assum e
th e ex iste n ce o f th e in itial c o n d itio n v ( —1). S ince (2.4.7) d e s c rib e s th e system
o u tp u t im plicitly, w e m u st solve this e q u a tio n to o b ta in an ex p licit ex p ressio n for
th e sy stem o u tp u t. S u p p o se th a t w e co m p u te successive valu es o f y(n) fo r n > 0,
b eg in n in g w ith y(0), T h u s
y (0) = o j ( - l ) + x(0)
v (l) = ay (0 ) + jc(3 ) = a 2y ( - 1) + o j:( 0 ) + *(1)
y(2) = o j ( l ) + x ( 2 ) = o 3y (—1) + a 2jr(0) + a ^ ( l ) + x ( 2 )
y( n) = a y( n - 1) + x( n)
= e " +1y ( - l ) + a" x( 0) + fl"’ 1Jt(l) + • ■■+ a x ( n - 1) + x( n)
o r, m o re co m p actly ,
n > 0
(2.4.8)
T h e re sp o n se y ( n ) o f th e system as given by th e rig h t-h a n d sid e o f (2.4.8)
co n sists o f tw o p arts. T h e first p a rt, w hich c o n ta in s th e te rm y ( —1), is a re su lt of
th e in itial c o n d itio n y ( —1) o f th e system . T h e se co n d p a r t is th e re sp o n se o f the
sy stem to th e in p u t sig n al x( n) .
I f th e sy stem is initially re lax ed a t tim e n = 0, th e n its m e m o ry (i.e., the
o u tp u t o f th e d e la y ) sh o u ld b e zero . H e n c e y ( —1) = 0. T h u s a rec u rsiv e system is
re la x e d if it sta rts w ith z e ro in itial c o n d itio n s. B e c a u se th e m e m o ry o f th e system
d escrib es, in so m e sen se, its “ sta te ,” w e say th a t th e system is a t z e ro s ta te an d
its c o rre sp o n d in g o u tp u t is called th e zero-state respons e o r f o r c e d respons e, an d
Sec. 2.4
Discrete-Time Systems Described by Difference Equations
97
is d e n o te d by yzs(n). O b v iously, the z e ro -sta te re sp o n se o r fo rc e d resp o n se o f the
system (2.4.7) is given by
V/jittf) — Y ^ a kx( n — k)
*=()
n > 0
(2.4.9)
It is in te re stin g to n o te th at (2.4.9) is a co n v o lu tio n su m m a tio n involving the
in p u t signal co n v o lv ed w ith the im p u lse re sp o n se
h(n) = a':u ( n )
(2.4.10)
W e also o b se rv e th a t the sy stem d e sc rib e d by th e first-o rd e r d ifferen ce e q u atio n
in (2.4.7) is cau sal. A s a resu lt, th e low er lim it on th e c o n v o lu tio n su m m atio n in
(2.4.9) is k = 0. F u rth e rm o re , th e co n d itio n v (—1) = 0 im plies th a t the in p u t signal
can be a ssu m ed cau sal an d h en ce th e u p p e r lim it on th e co n v o lu tio n su m m atio n
in (2.4.9) is n. since x( n - k) = 0 for k > n. In effect, we have o b ta in e d th e resull
th a t th e re lax ed recu rsiv e system d esc rib e d by th e firs t-o rd e r d ifferen ce e q u atio n
in (2.4.7), is a lin e a r tim e-in v arian t IIR svstem w ith im p u lse resp o n se given bv
(2.4.10).
N ow . su p p o se th at th e system d e sc rib e d by (2.4.7) is in itially n o n re la x e d [i.e..
y ( —1) ^ 0] an d th e in p u t x(/i) = 0 for all //. T h e n the o u tp u t of th e system with
zero in p u t is called the zero- i nput response or nat ural r espons e an d is d e n o te d by
yZj(/j). F ro m (2.4.7). w ith ,v(») = 0 for —oc < n < oc . we o b ta in
\Y,OM = « ',+ l y ( — 1)
n
>
(2.4.11)
0
W e o b se rv e th a t a recu rsiv e system w ith n o n z e ro initial c o n d itio n is n o n relax ed
in th e se n se th a t it can p ro d u c e an o u tp u t w ith o u t b ein g ex cited . N o te th at the
z e ro -in p u t re sp o n se is d u e to th e m e m o ry of th e system .
T o su m m a riz e , th e z e ro -in p u t re sp o n se is o b ta in e d by se ttin g the in p u t signal
to z e ro , m ak in g it in d e p e n d e n t of the in p u t. It d e p e n d s only on th e n a tu re of the
system a n d th e in itial co n d itio n . T h u s th e z e ro -in p u t re sp o n s e is a ch a ra c te ristic of
th e sy stem itself, a n d it is also know n as th e nat ural o r f ree r esponse of th e svstem .
O n th e o th e r h a n d , th e z e ro -sta te resp o n se d e p e n d s on th e n a tu re of th e system
an d th e in p u t signal. Since th is o u tp u t is a re sp o n se fo rce d u p o n it by th e input
signal, it is u su ally called th e f o r c e d response of th e system . In g en eral, th e total
re sp o n se o f th e system can b e ex p ressed as v(n) = y Zi ( n ) + .Vzs( n).
T h e sy stem d escrib ed by th e first-o rd e r d iffe re n c e e q u a tio n in (2.4.7) is the
sim p lest p o ssib le recu rsiv e system in th e g en eral class o f re c u rsiv e system s d e ­
sc rib ed by lin e a r co n stan t-co efficien t d iffe re n c e e q u a tio n s . T h e g e n eral form for
such an e q u a tio n is
S
M
v(/i) = — ^ ai,y{n - k) -j- ^ bkx ( n - k)
i-=l
k=0
(2.4,12)
or, e q u iv alen tly ,
Ar
M
' Y a ky{n - k) = ' Y ^ b kx( n - k)
Jc=0
*=0
a0 m 1
(2.4.13)
98
Discrete-Time Signals and Systems
Chap. 2
T h e in te g e r N is called th e order o f the d ifferen ce e q u a tio n o r th e o r d e r of the
system . T h e n eg ativ e sign on th e rig h t-h a n d side o f (2.4.12) is in tro d u c e d as a
m a tte r o f co n v en ien ce to allow us to ex p ress th e d ifferen ce e q u a tio n in (2.4.13)
w ith o u t any n eg ativ e signs.
E q u a tio n (2.4.12) ex p re sse s th e o u tp u t of the system at tim e rt d irectly as
a w eig h ted sum o f p ast o u tp u ts \ {n - 1), y( n - 2 ).........y(/i - N ) as well as past
an d p re se n t in p u t signals sam p les. W e o b se rv e th a t in o rd e r to d e te rm in e y( n)
fo r n > 0, we n e e d th e in p u t x ( n ) for all n > 0, an d th e initial c o n d itio n s y ( —1),
y ( —2), — y ( —N ) . In o th e r w o rd s, th e initial c o n d itio n s su m m a riz e all th a t we
n eed to k n o w a b o u t th e p a s t h isto ry o f th e re sp o n se o f th e sy stem to c o m p u te
th e p re se n t an d fu tu re o u tp u ts. T h e g e n e ra l so lu tio n o f th e /V -order c o n stan tco efficien t d ifferen ce e q u a tio n is c o n sid e re d in th e follow ing su b se c tio n .
A t this p o in t we r e s ta te th e p ro p e rtie s of lin earity , tim e in v a ria n c e , an d
stab ility in th e c o n te x t o f recu rsiv e system s d e sc rib e d by lin ear c o n s ta n t-c o e ffic ie n t
d ifferen ce e q u a tio n s. A s we h av e o b se rv ed , a recu rsiv e system m a y b e relax ed o r
n o n re la x e d , d e p e n d in g o n th e initial co n d itio n s. H e n c e th e d e fin itio n s o f th ese
p ro p e rtie s m u st ta k e in to a c c o u n t th e p re se n c e o f the initial co n d itio n s.
W e begin w ith th e d e fin itio n o f lin earity . A system is lin e a r if it satisfies th e
follow ing th re e re q u ire m e n ts:
1. T h e to tal re sp o n se is e q u a l to th e sum o f th e z e ro -in p u t a n d z e ro -sta te r e ­
sp o n ses [i.e.. y ( n) = y 7.\(n) + yzs(n)].
2. T h e p rin cip le o f su p e rp o s itio n ap p lies to th e z e ro -sta te re s p o n s e (zero-state
linear).
3. T h e p rin cip le o f su p e rp o s itio n ap p lies to the z e ro -in p u t re s p o n s e (zero- i nput
linear).
A system th a t d o es n o t satisfy all three se p a ra te r e q u ire m e n ts is by d efin itio n
n o n lin ear. O b v io u sly , fo r a re lax ed system , yZj(n) = 0, an d th u s re q u ire m e n t 2,
w hich is th e d efin itio n o f lin e a rity given in S ection 2.2.4, is sufficient.
W e illu strate th e a p p lic a tio n o f th ese re q u ire m e n ts by a sim p le ex am p le.
Example 2.4.2
D eterm ine if the recursive system defined by the difference equation
v(n) = av(n —1)4- x(n)
is linear.
Solution
as
By combining (2.4.9) and (2.4.11), we obtain (2.4.8), which can be expressed
y(n) = yzi(n) + y*(n)
Thus the first requirem ent for linearity is satisfied.
Sec. 2.4
Discrete-Time Systems Described by Difference Equations
99
To check for the second requirem ent, let us assume that ,t(n) =
Then (2.4.9) gives
+
*={}
= ('iy ^ ’(n) + C2V^'(n)
Hence y„(/i> satisfies the principle of superposition, and thus the system is zero-state
linear.
Now let us assume that y(—1) = q vj(—1) 4- f 2y;( —1). From (2.4.11) we obtain
1) 4- C;V;(—1)]
=
vi (—1)4-
v; ( —1)
= Ci v'i '(n) + r ;y ’^(/i)
Hence the system is zero-input linear.
Since the system satisfies all three conditions for linearity, it is linear.
A lth o u g h it is so m e w h at ted io u s, the p ro c e d u re used in E x am p le 2.4,2 to
d e m o n s tra te lin e a rity for th e system d escrib ed by th e first-o rd e r d ifferen ce e q u a ­
tio n , ca rrie s o v er d irectly to th e g en eral recu rsiv e system s d e sc rib e d by th e c o n stan tcoefficient d ifferen ce e q u a tio n given in (2.4.13).
H en ce , a recu rsiv e system
d escrib ed by th e lin ear d iffe re n c e e q u a tio n in (2.4.13) also satisfies all th re e c o n ­
d itio n s in th e d efin itio n o f lin earity , a n d th e re fo re it is linear.
T h e n ex t q u e s tio n th a t arises is w h e th e r o r n o t the cau sal lin ear system
d e sc rib e d by th e lin e a r c o n stan t-co efficien t differen ce e q u a tio n in (2.4.13) is tim e
in v arian t. T h is is fairly easy, w h en d e alin g w ith system s d e sc rib e d by explicit
i n p u t-o u tp u t m a th e m a tic a l re la tio n sh ip s. C learly, th e system d e sc rib e d by (2.4.13)
is tim e in v a ria n t b ecau se th e coefficients ak an d bk are c o n stan ts. O n th e o th e r
h a n d , if o n e o r m o re o f th e s e coefficients d e p e n d s on tim e, th e system is tim e
v a ria n t, since its p ro p e rtie s ch an g e as a fu n ctio n of tim e. T h u s w e co n clu d e th at
the recursive sy s t em descri bed b y a linear constant-coefficient difference equat i on is
linear a n d t ime invariant.
T h e final issu e is th e sta b ility of th e recursive system d e sc rib e d by th e lin ear,
c o n stan t-co efficien t d iffe re n c e e q u a tio n in (2.4.13). In S ection 2.3.6 w e in tro d u c e d
th e c o n c e p t o f b o u n d e d in p u t-b o u n d e d o u tp u t (B IB O ) sta b ility fo r re la x e d sys­
tem s. F o r n o n re la x e d system s th a t m ay b e n o n lin e a r, B IB O sta b ility sh o u ld be
view ed w ith so m e care. H o w e v e r, in th e case o f a lin e a r tim e -in v a ria n t recursive
system d e sc rib e d by th e lin e a r co n stan t-co efficien t d iffe re n c e e q u a tio n in (2.4.13),
it suffices to s ta te th a t such a system is B IB O sta b le if a n d only if fo r every
b o u n d e d in p u t a n d ev ery b o u n d e d initial c o n d itio n , th e to ta l system re sp o n se is
bounded.
100
Discrete-Time Signals and Systems
Chap. 2
Example 2.43
Determ ine if the linear tim e-invariant recursive system described by the difference
equation given in (2.4.7) is stable.
Solution
i*(n)l <
Let us assume that the input signal x(n) is bounded in amplitude, that is,
< oc for all n > 0. From (2.4.8) we have
j v{n J | < |a',+l_y(—1)| + ^
akx(n - k) ,
n>0
If n is finite, the bound M v is finite and the output is bounded independently of the
value of a. However, as n -*• oo, the bound My remains finite only if |aj < 1 because
|a |B -»■ 0 as n -*• oc. Then M y = Ms j(\ - |o|).
Thus the system is stable only if \a\ < 1.
For the sim ple first-order system in E xam ple 2.4,3, w e w ere able to express
the con d ition for B IB O stability in term s o f the system param eter a. nam ely \a\ < 1.
W e should stress, how ever, that this task b ecom es m ore difficult for higher-order
system s. F ortunately, as we shall see in su bsequent chapters, other sim ple and
m ore efficient tech n iqu es exist for investigating the stability o f recursive system s.
2.4.3 Solution of Linear Constant-Coefficient Difference
Equations
G iven a linear con stan t-coefficien t d ifferen ce eq u ation as the in p u t-o u tp u t rela­
tionship describing a linear tim e-invariant system , our objective in this subsection
is to determ ine an explicit exp ression for the output y{n). T h e m eth od that is
d ev elo p ed is term ed the direct m e t h o d . A n alternative m eth od based on the ztransform is described in Chapter 3. For reasons that will b eco m e apparent later,
the z-transform approach is called the indirect m e th o d .
Basically, the goal is to determ ine the output y(n ), n > 0, o f the system given
a specific input x ( n ), n > 0, and a set o f initial con d ition s. T h e direct solution
m ethod assum es that the total solu tion is the sum o f tw o parts:
y ( n ) = y h(n) + }'r (n)
T he part y/,(n) is know n as the h o m o g e n e o u s or c o m p l e m e n ta r y solu tion , w hereas
yp(n) is called the particular solution.
The homogeneous solution of a difference equation. W e begin the
problem o f solving the linear con stan t-coefficien t d ifferen ce eq u ation given by
Sec. 2.4
Discrete-Time Systems Described by Difference Equations
101
(2.4.13) bv assu m in g th a t th e in p u t x (n ) = 0. T h u s w e will first o b ta in th e so lu tio n
to th e h o m o g e n e o u s di ff erence equati on
\
J 2 a ky ( n - k ) = 0
*=(>
(2.4.14)
T h e p ro c e d u re fo r solving a lin e a r c o n s tan t-co efficien t d ifferen ce e q u a tio n
d irectly is very sim ilar to th e p ro c e d u re fo r solving a iin e a r c o n stan t-co efficien t
d iffe re n tia l e q u a tio n . B asically, we assum e th a t th e so lu tio n is in th e fo rm o f an
e x p o n e n tia l, th a t is.
yi,{n) = a"
(2.4.15)
w h ere th e su b scrip t h o n \ {n) is used to d e n o te th e so lu tio n to th e h o m o g e n e o u s
d ifferen ce e q u a tio n . If w e su b stitu te this assu m ed so lu tio n in (2.4.14), w e o b tain
th e p o ly n o m ial e q u a tio n
=0
fc=U
or
k" A (k* + ci)k^ ! + a2k^ * + • • - + q n —\ k + a w ) = 0
(2.4.16)
T h e p o ly n o m ial in p a re n th e se s is called th e characteristic p o l y n o m i a l o f the
system . In g e n e ra l, it h as N roots, w hich we d e n o te as Xi. k 2.........k ^ . T h e ro o ts
can b e real o r co m p lex v alued. In p ra c tic e the co efficien ts a \ , a 2........ are usually
re a l. C o m p le x -v a lu e d ro o ts o ccu r as c o m p le x -c o n ju g a te p airs. S om e o f th e N ro o ts
m ay be id en tical, in w hich case w e have m u ltip Je -o rd e r ro o ts.
F o r th e m o m e n t, let us assum e th a t th e ro o ts a re d istin ct, th a t is, th e re are
n o m u ltip le -o rd e r ro o ts. T h e n th e m o st g e n eral so lu tio n to th e h o m o g e n e o u s
d iffe re n c e e q u a tio n in (2.4.14) is
yh(n) = C \ k \ + C2k 2 + • ■■+ C^ k ^ ,
(2.4.17)
w h ere C i. C 2.........C s are w eig h tin g coefficients.
T h e se co efficien ts are d e te rm in e d fro m th e in itial c o n d itio n s specified fo r th e
sy stem . Since th e in p u t jr(n) = 0. (2.4.17) can b e u sed to o b ta in th e z ero- i nput
response o f th e system . T h e follow ing ex am p les illu stra te th e p ro c e d u re .
Example 2.4.4
D eterm ine the hom ogeneous solution of the system described by the first-order dif­
ference equation
_v(n) + a\y(rt — 1) = x(n)
Solution
The assumed solution obtained by setting x(n) = 0 is
y*(n) = V
(2.4.18)
D iscrete-Time Signals and System s
102
Chap. 2
When we substitute this solution in (2.4.18), we obtain [with x(n) = 0]
X -f-12]An ^ = 0
A" *(A + ai) = 0
A = —O)
Therefore, the solution to the hom ogeneous difference equation is
= CA" = C ( - a ,)"
(2.4.19)
The zero-input response of the system can be determ ined from (2.4,18) and
(2.4.19). With x(n) = 0, (2.4.18) yields
v(0) =
-a, v ( - l )
On the other hand, from (2.4,19) we have
v*(0) = C
and hence the zero-input response of the system is
Vii(«) = (—£i)n+1 v{—1)
n >0
(2.4.20)
With a = —ai, this result is consistent with (2.4,11) for the first-order system, which
was obtained earlier by iteration of the difference equation.
Example 2AS
Determ ine the zero-input response of the system described by the homogeneous
second-order difference equation
y(«) - 3y(n - 1) - 4y(n - 2) = 0
(2.4.21)
Solution First we determ ine the solution to the homogeneous equation. We assume
the solution to be the exponential
yh(n) = X"
U pon substitution of this solution into (2.4.21), we obtain the characteristic equation
a" -
3xn- ' - 4 r ~ 2 = o
A"-2(X2 —3A —4) = 0
Therefore, the roots are X = - 1 , 4, and the general form of the solution to the
homogeneous equation is
}7i(n) = CjX" + C 2X2
(2.4.22)
= C1( - i r + C2(4)"
The zero-input response of the system can be obtained from the homogenous
solution by evaluating the constants in (2.4.22), given the initial conditions y (—1) and
y ( ~ 2). From the difference equation in (2.4.21) we have
y(0) = 3y(—1) + 4y(—2)
y(l) = 3v(0) + 4_y(—1)
= 3 [3 y (-l)+ 4y(-2)] + 4y(-l)
= 13y(—1) + l 2y( —2)
Sec. 2.4
Discrete-Time Systems Described by Difference Equations
103
On the other hand, from (2.4.22) we obtain
v(0> = C) + C;
V( 1 I = -C l +4C;
By equating these two sets of relations, we have
C i + C ; = 3_v( —1) + 4y(—2)
- C , + 4 C ; = 13 v{ —1) + 12 y( —2)
The solution of these two equations is
Ci = —7 v( —1 ) + ? v ( - 2 )
C; = X .V (-l) + T.v( -2 )
Therefore, the zero-input response of the system is
v,i(n) = [—5 v( —1 >- i v(-2)](-l)''
(2.4.23)
n >0
+ [ x . v ( - l » + t.v(-2)](4>"
For example, if v{—2) = 0 and y ( - l ) = 5. then C| = —1, C: = 16. and hence
y-,i(H) = ( -1 ) " * 1 + (4)"+:
n >0
T h e se e x am p les illu strate the m e th o d fo r o b ta in in g the h o m o g e n e o u s so lu tio n
and th e z e ro -in p u t re sp o n se o f th e system w hen the ch a ra c te ristic e q u a tio n co n tain s
d istin ct ro o ts. O n th e o th e r h an d , if th e c h a ra c te ristic e q u a tio n co n ta in s m u ltip le
ro o ts, th e form o f th e so lu tio n given in (2.4.17) m ust be m odified. F o r ex am p le, if
ai is a ro o t o f m u ltip licity m, th en (2.4.17) b eco m es
\h(ii) = Ci A.'.' + C2/iX’! + C v;:a',' + ■■• + Cm>7n'~lX'!
(2.4.24)
+ Cm+\)"m+\ + ■• • + C^Kl
The particular solution of the difference equation. T h e p a rtic u la r so ­
lu tio n \ p (n) is re q u ire d to satisfy th e d ifferen ce e q u a tio n (2.4.13) fo r th e specific
in p u t signal x (n ). n > 0. In o th e r w ords, y p(n) is any so lu tio n satisfying
N
M
' Y ^ a ky p (n - k) — ' Y ^ b kx ( n - k)
A-0
*=0
an — 1
(2.4.25)
T o solve (2.4.25). w e assu m e fo r yp (n), a fo rm th at d e p e n d s on th e fo rm o f th e
in p u t j:(h ). T h e fo llow ing ex am p le illu strates the p ro c e d u re .
Example 2.4.6
D eterm ine the particular solution of the first-order difference equation
y(n) + fliy(n - 1) = x(n).
| f l i | <l
when the input x(n) is a unit step sequence, that is.
x(n) = u (n )
(2,4.26)
104
Discrete-Time Signals and System s
Chap. 2
Solution Since the input sequence x(n) is a constant for n > 0. the form of the solu­
tion that we assume is also a constant. Hence the assumed solution of the difference
equation to the forcing function x(n), called the particular solution of the difference
equation, is
vr (n) = Ku(n)
where A- is a scale factor determined so that (2.4.26) is satisfied. U pon substitution
of this assumed solution into (2.4.26). we obtain
Ku i n ) + d]Ku{,n — 1) = u ( n )
To determine K, we must evaluate this equation for any n > 1. where none of the
terms vanish. Thus
K -t- O] K — 1
1
K = -------l+^i
Therefore, the particular solution to the difference equation is
v (n) = --------u(n)
'’
1 +fii
(2.4.27)
In th is ex am p le, th e in p u t * (« ). n > 0. is a c o n s ta n t an d th e fo rm assu m ed
for th e p a rtic u la r so lu tio n is also a c o n sta n t. If x{n) is an e x p o n e n tia l, w e w ould
assu m e th a t th e p a rtic u ia r so lu tio n is also an e x p o n e n tia l. If x {n) w e re a sin u so id ,
th e n >■/,(«) w o uld also be a sinusoid. T h u s o u r assu m ed fo rm fo r th e p a rtic u la r
so lu tio n tak es th e basic fo rm of th e signal x{n). T a b le 2.1 p ro v id e s th e g en eral
fo rm o f th e p a rtic u la r so lu tio n for sev eral ty p es of excitatio n .
Exam ple 2.4.7
Determ ine the particuiar solution of the difference equation
v(n) = | y(n - 1) - £ v(w - 2) + x(n)
when the forcing function x(n) = 2". n > 0 and zero elsewhere.
TABLE 2.1 GENERAL FORM OF THE PARTICULAR
SOLUTION FOR SEVERAL TYPES OF INPUT
SIGNALS
Input Signal,
x(n)
Particular Solution,
vr (n)
A (constant)
AM"
AnM
K
KM"
Kan” + K ^ " - ' 1 + . . . + KM
A cos Wf>n
A sin a>on
Kj cos toon -I- K2 sin won
AnnM
An(Ki)nM+ K\ttM 1
Sec. 2.4
Discrete-Time Systems Described by Difference Equations
Solution
105
The form of the particular solution is
yp (n ) = K2n
n >0
Upon substitution of yP(n) into the difference equation, we obtain
K2"u( n) =
-
1 ) - I K 2 n- 2u(n - 2 ) + 2 " « ( « )
To determ ine the value of K, we can evaluate this equation for any n > 2, where
none of the terms vanish. Thus we obtain
4A" = \ ( 2K) - i f f + 4
and hence K =
Therefore, the particular solution is
yn(n) = ^2"
n > 0
W e h av e n o w d e m o n s tra te d how to d e te rm in e th e tw o c o m p o n e n ts o f the
so lu tio n to a d ifferen ce e q u a tio n w ith c o n s ta n t coefficients. T h e se tw o c o m p o n e n ts
are th e h o m o g e n e o u s so lu tio n and th e p a rtic u la r so lu tio n . F ro m th e s e tw o co m ­
p o n e n ts, we c o n s tru c t the to tal so lu tio n fro m w hich w e can o b ta in th e ze ro -sta te
re sp o n se .
The total solution of the difference equation. T h e lin e a rity p ro p e rty o f
th e lin ear c o n stan t-co efficien t d ifferen ce e q u a tio n allow s u s to a d d th e h o m o g e ­
n e o u s so lu tio n an d the p a rtic u la r so lu tio n in o rd e r to o b ta in th e total solution. T h u s
v(«) = y h(n) + \ p( n)
T h e r e s u lta n t sum v(n) co n tain s th e c o n s ta n t p a r a m e te r s {C/} e m b o d ie d in th e
h o m o g e n e o u s so lu tio n c o m p o n e n t >v,(n). T h e se c o n s ta n ts can be d e te rm in e d to
satisfy th e in itial co n d itio n s. T h e follow ing ex a m p le illu stra te s th e p ro c e d u re ,
Exam ple 2.4.8
D eterm ine the total solution y(/i), n > 0, to the difference equation
y(n) +a i y ( n - 1) = x(n)
(2.4.28)
when x( n) is a unit step sequence [i.e., x(n) = «(«)] and y (—1) is the initial condition.
Solution
From (2.4,19) of Example 2.4.4, the hom ogeneous solution is
y*(n) = C(-fli)"
and from (2.4.26) of Example 2.4.6, the particular solution is
Consequently, the total solution is
y(n) = C ( - a i)" + — -—
1 + fli
n >0
where the constant C is determ ined to satisfy the initial condition y ( - l ) .
(2.4.29)
106
Discrete-Time Signals and Systems
Chap. 2
In particular, suppose that we wish to obtain the zero-siate response of the
system described by the first-order difference equation in (2.4.28). T hen we set
y( —1) = 0. To evaluate C. we evaluate (2.4.28) at n = 0 obtaining
y ( 0 ) + Oi y ( — 1 ) =
1
y(0) = 1
On the other hand, (2.4.29) evaluated at n = 0 yields
v(0) = C 4- —i —
1 + ax
Consequently.
1
C + -------- = 1
1 + «i
ai
1 + ai
c
Substitution for C into (2.4.29) yields the zero-state response of the system
l-f-fli)" * '
Vi*(/i) = ---- —-----—
1 + «i
n >0
If we evaluate the param eter C in (2,4.29) under the condition that y( —1) ^ 0. the
total solution will include the zero-input response as well as the zero-state response
of the system. In this case (2.4.28) yields
y(0) + a]V( —1) = 1
y (0) = —fliy( —1) 4- 1
On the other hand. (2.4.29) yields
]
1 + U\
By equating these two relations, we obtain
C + — ^— = —ai v(—1) 4- 1
1 4 - a,
C = -g ] v( 1) +
- —
1 4- a ]
Finally, if we substitute this value of C into (2.4.29). we obtain
y(n) = (—a , r +1y(—1) + -—
=
+
^ —
1 + a)
n >0
(2.4.30)
Vzs( n)
W e o b se rv e th a t th e system re sp o n se as given by (2.4.30) is c o n s iste n t w ith
th e re sp o n se y ( n) given in (2.4.8) for th e first-o rd e r system (w ith a = - o i ) . w hich
was o b ta in e d by solv in g th e d ifferen ce e q u a tio n iterativ e ly . F u rth e rm o r e , we n o te
th a t th e v alu e o f th e c o n s ta n t C d e p e n d s b o th o n th e initial co n d itio n y (—1) an d
o n th e ex citatio n fu n ctio n . C o n se q u e n tly , th e v alue o f C in flu en ces b o th th e zeroin p u t re sp o n se a n d th e z e ro -sta te re sp o n se . O n th e o th e r h a n d , if w e w ish to
Sec. 2.4
Discrete-Time Systems Described by Difference Equations
107
obtain the zero-state resp onse only, w e sim ply solve for C under th e con d ition
that y ( - l ) = 0, as d em onstrated in E xam ple 2.4.8.
W e further ob serve that the particular solution to the d ifferen ce equation can
b e o b tain ed from the zero-state resp onse o f the system . In d eed , if |o]| < 1, which
is the con d ition for stability o f the system , as w ill be show n in S ection 2.4.4, the
lim iting valu e o f >’zs(«) as n approaches infinity, is the particular solu tion , that is,
1
» ( n ) = lim >zs(«) = --------n-*Oo
1+ai
Since this com p onent o f the system response d o es not go to zero as n approaches
infinity, it is usually called the steady-state re spons e o f the system . T his response
persists as lon g as the input persists. T he com p onent that d ies out as n approaches
infinity is called the transient re sponse o f the system .
Example 2.4.9
D eterm ine the response y(n), n > 0, of the system described by the second-order
difference equation
v(n) - 3y(n - 1) - 4v(n - 2) = x( n ) + 2x(n - 1)
(2.4.31)
when the input sequence is
x(n) =4"u(n)
Solution We have already determ ined the solution to the homogeneous difference
equation for this system in Example 2.4.5. From (2.4.22) we have
y*(n) = C , ( - l ) n + C 2(4)n
(2.4.32)
The particular solution to (2.4.31) is assumed to be an exponential sequence of the
same form as x(n). Normally, we could assume a solution of the form
yp(n) = K(4,)au(n)
However, we observe that » ( n ) is already contained in the hom ogeneous solution,
so that this particular solution is redundant. Instead, we select the particular solution
to be linearly independent of the terms contained in the homogeneous solution. In
fact, we treat this situation in the same manner as we have already treated multiple
roots in the characteristic equation. Thus we assume that
yp(n) = Kn{4)"u(n)
(2.4.33)
Upon substitution of (2.4.33) into (2.4.31), we obtain
Kn(4)"u(n) - 3 K ( n - l)(4 ),' - 1u(n - 1) - 4 K{n - 2)(4)n- 2u(n - 2)
= (4)"u(n) + 2(4)"-1 m(/i —1)
To determ ine K , we evaluate this equation for any n > 2, where none of the
unit step term s vanish. To simplify the arithmetic, we select n = 2, from which we
obtain K = | . Therefore,
yP{n) « |rt(4)n«(/l)
(2.4.34)
108
Discrete-Time Signals and System s
Chap. 2
The total solution to the difference equation is obtained by adding (2.4.32) to
(2.4.34). Thus
y(«) — C ](—1)" + C;(4)n + |w(4)"
n > 0
(2.4.35)
where the constants C\ and C2 are determined such that the initial conditions are
satisfied. To accomplish this, we return to (2.4.31), from which we obtain
v(0) = 3 y (- l) + 4y (—2) + 1
y (1) = 3v(0) + 4_v(-l) + 6
= 1 3 y (- l) + 12y(-2) + 9
O n the other hand, (2.4.35) evaluated at n = 0 and n = 1 yields
y(0) = Ci + C2
y (l) = -C] + 4C2 + f
W e can now equate these two sets of relations to obtain C\ and C 2. In so doing, we
have the response due to initial conditions y (—1) and y (—2) (the zero-input response),
and the zero-state or forced response.
Since we have already solved for the zero-input response in Exam ple 2.4.5. we
can simplify the computations above by setting v (—1) = y (—2) = 0. Then we have
C, + C; = 1
- C l + 4 C2 + f
=9
Hence C : = —^ and C 2 =
Finally, we have the zero-statc response to the forcing
function
= (4)"«(n) in the form
vB(n) =
+5<4)" +H 4)"
nZ0
(2.4.36)
The total response of the system, which includes the response to arbitrary initial
conditions, is the sum of (2.4.23) and (2.4.36).
2.4.4 The Impulse Response of a Linear Time-Invariant
Recursive System
T h e im p u lse re sp o n se o f a lin e a r tim e -in v a ria n t sy stem w as p re v io u sly defin ed as
th e re sp o n se o f th e sy stem to a u n it sa m p le e x citatio n [i.e., x (n ) = <5(n)]. In th e
case o f a recu rsiv e sy stem , h ( n ) is sim ply e q u a l to th e z e ro -s ta te re sp o n s e o f the
system w h en th e in p u t j:(n ) = <5(n) an d th e system is initially re la x e d .
F o r ex am p le, in th e sim ple first-o rd e r re c u rsiv e system g iv en in (2.4.7), th e
z e ro -sta te re sp o n se g iv en in (2.4.8), is
n
(2.4.37)
*=o
W ith jr(n) = i ( « ) is substituted into (2.4.37), we obtain
n
yzs(n) = Y 2 a k8(n - k)
=
an
n > 0
Sec. 2.4
Discrete-Time Systems Described by Difference Equations
109
H e n c e th e im p u lse re sp o n se o f th e first-o rd e r rec u rsiv e system d e sc rib e d by
(2.4.7) is
h(n) = a nu(n)
(2.4.38)
as in d ic a te d in S ectio n 2.4.2.
In th e g e n e ra l case o f an a rb itra ry , lin e a r tim e -in v a ria n t rec u rsiv e system , the
z e ro -s ta te re sp o n s e ex p re sse d in te rm s o f th e co n v o lu tio n su m m a tio n is
>’zs(n) — ^ 2 h ( k ) x ( n — k)
n > 0
(2.4.39)
i=0
W h e n th e in p u t is an im p u lse [i.e., * (« ) = <5(«)], (2.4.39) re d u c e s to
Vzs(n) = h(n)
(2.4.40)
N o w , let us c o n s id e r th e p ro b le m o f d e te rm in in g th e im p u lse re sp o n se h i n ) given a
lin e a r c o n s tan t-co efficien t d ifferen ce e q u a tio n d e sc rip tio n o f th e system . In term s
o f o u r d isc u ssio n in th e p re c e d in g su b sectio n , we h av e e s ta b lis h e d th e fact th a t the
to ta l re sp o n s e o f th e sy stem to any e x citatio n fu n ctio n co n sists o f th e sum o f tw o
so lu tio n s o f th e d ifferen ce e q u atio n : th e so lu tio n to the h o m o g e n e o u s e q u atio n
p lu s th e p a rtic u la r so lu tio n to th e e x c itatio n fu n ctio n . In th e case w h ere the exci­
ta tio n is an im p u lse, th e p a rtic u la r so lu tio n is z e ro , since x ( n ) = 0 for n > 0. th at is.
yp (n) = 0
C o n s e q u e n tly , th e re sp o n se o f th e system to an im p u lse co n sists only o f th e so lu ­
tio n to th e h o m o g e n e o u s e q u a tio n , w ith the ( Q ) p a r a m e te r s e v a lu a te d to satisfy
th e in itial c o n d itio n s d ic ta te d by th e im pulse. T h e follow ing ex am p le illu strates
th e p ro c e d u re fo r o b ta in in g h(n) given th e d ifferen ce e q u a tio n fo r the system .
Example 2.4.10
D eterm ine the impulse response h(n) for the system described by the second-order
difference equation
y(n ) — 3v(n — 1) — 4 y (« — 2) = x ( n ) + 2 x i n — 1)
(2.4.41)
Solution W e have already determ ined in Example 2.4.5 that the solution to the
hom ogeneous difference equation for this system is
^ ( n ) = C, (-1 )" + C2(4)"
n > 0
(2.4.42)
Since the particular solution is zero when x(n) = 6(n), the impulse response of the sys­
tem is simply given by (2.4.42), where C] and C2 must be evaluated to satisfy (2.4.41).
For n = 0 and n = 1, (2.4.41) yields
v(0) = 1
y (1) = 3 y (0 )+ 2 = 5
where we have imposed the conditions y (—1) = y (—2) = 0. since the system must be
relaxed. On the other hand, (2.4.42) evaluated at n = 0 and n = 1 yields
y (0) = C, + C2
y (1) = —Ci + 4C2
Discrete-Time Signals and Systems
110
Chap. 2
By solving these two sets of equations for C] and C2, we obtain
=
C: = 5
Therefore, the impulse response of the system is
h{n) = [—i ( —1)" + f (4)-}u(n)
W e m ak e th e o b se rv a tio n th a t b o th th e sim ple firs t-o rd e r recu rsiv e system
an d th e se c o n d -o rd e r recu rsiv e system h av e im pulse re sp o n se s th a t a re infinite in
d u ra tio n . In o th e r w o rds, b o th o f th e s e recu rsiv e system s a re IIR system s. In
fact, d u e to th e recu rsiv e n a tu re of th e system , an y recu rsiv e sy stem d e sc rib e d by
a lin e a r co n stan t-co efficien t d iffe re n c e e q u a tio n is an I IR system . T h e co n v erse
is n o t tru e, h o w ev er. T h a t is, n o t ev ery lin e a r tim e -in v a ria n t I I R sy stem can be
d e sc rib e d by a lin e a r c o n s tan t-co efficien t d ifferen ce e q u a tio n . In o th e r w ords,
recu rsiv e sy stem s d e sc rib e d by lin ear co n sta n t-c o e ffic ie n t d iffe re n c e e q u a tio n s are
a su bclass o f lin e a r tim e -in v a ria n t I I R system s.
T h e ex ten sio n o f th e a p p ro a c h th a t w e h av e d e m o n s tra te d fo r d e te rm in ­
ing th e im p u lse re sp o n se o f th e first- a n d s e c o n d -o rd e r system s, g en e ra liz e s in a
stra ig h tfo rw a rd m a n n e r. W h e n th e system is d e sc rib e d by an A 'th -o rd e r lin e a r
d ifferen ce e q u a tio n o f th e ty p e given in (2.4.13), th e so lu tio n o f th e h o m o g e n e o u s
e q u a tio n is
*=i
w h en th e ro o ts {a*} o f th e c h a ra c te ristic p o ly n o m ial are d istin ct. H e n c e th e im pulse
re sp o n se o f th e sy stem is id en tical in fo rm , th a t is.
(2.4.43)
w h ere th e p a ra m e te rs {Ctl are d e te rm in e d by se ttin g th e initial c o n d itio n s v ( —1) =
. . . = y ( - N ) = 0.
T h is fo rm o f h{n) allow s us to easily re la te th e stab ility of a system , d escrib ed
by an N th -o rd e r d iffe re n c e e q u a tio n , to th e values o f th e ro o ts o f th e ch a ra c te ristic
p o ly n o m ial. In d e e d , since B IB O sta b ility re q u ire s th a t th e im p u lse re sp o n s e be
ab so lu te ly su m m ab le, th e n , fo r a causal system , w e h av e
x
oc
N
N
oo
oo
£ > (* )! = £
Y ^ C kXnk < £ | C * | £ | A * | "
«=0
n
=0
n=0 Ik=-l
k=\
n=0
N ow if |
j < 1 fo r all k, th e n
an d h en ce
OC
IM*)[ < oo
Sec. 2.5
im plementation of Discrete-Time Systems
111
O n th e o th e r h an d , if o n e o r m o re o f th e
> 1, h{n) is n o lo n g e r ab so lu te ly
su m m ab le, an d c o n se q u e n tly , th e sy stem is u n sta b le . T h e re fo re , a n ecessa ry an d
sufficient c o n d itio n fo r th e sta b ility o f a causal IIR system d e sc rib e d by a lin ear
c o n s tan t-co efficien t d iffe re n c e e q u a tio n , is th a t al! ro o ts o f th e c h a ra c te ristic p o ly ­
n o m ial b e less th a n u n ity in m a g n itu d e. T h e re a d e r m ay verify th a t th is c o n d itio n
ca rrie s o v er to th e case w h ere th e system h as ro o ts o f m u ltip lic ity m.
2.5 IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
O u r tre a tm e n t o f d isc re te -tim e sy stem s h as b e e n fo cu sed on th e tim e -d o m a in c h a r­
ac te riz a tio n an d analysis o f lin ear tim e -in v a ria n t system s d esc rib e d by c o n stan tco efficien t lin e a r d ifferen ce e q u a tio n s. A d d itio n a l an aly tical m e th o d s a re d e v e l­
o p e d in th e n ex t tw o c h a p te rs, w h ere we c h a ra c te riz e an d an aly ze L TI sy stem s in
th e fre q u e n c y d o m ain . T w o o th e r im p o rta n t to p ics th a t will b e tr e a te d la te r are
th e d esig n an d im p le m e n ta tio n o f th e se system s.
In p ractice, system design a n d im p le m e n ta tio n a re usually tr e a te d jo in tly
r a th e r th a n se p a ra te ly . O fte n , th e system design is d riv en by th e m e th o d o f
im p le m e n ta tio n an d by im p le m e n ta tio n c o n stra in ts, such as cost, h a rd w a re lim ­
itatio n s, size lim ita tio n s, a n d p o w e r re q u ire m e n ts . A t th is p o in t, we h av e n o t
as y e t d e v e lo p e d th e n ecessa ry analysis an d design tools to tr e a t such com plex
issues. H o w ev er, we h ave d e v e lo p e d sufficient b a c k g ro u n d to c o n sid e r so m e b a ­
sic im p le m e n ta tio n m e th o d s for re a liz a tio n s o f L TI sy stem s d escrib ed by lin ear
c o n s tan t-co efficien t d iffe re n c e e q u atio n s.
2.5.1 Structures for the Realization of Linear
Time-Invariant Systems
In th is su b sectio n we d esc rib e s tru c tu re s fo r th e re a liz a tio n o f system s d escrib ed
by lin e a r c o n s tan t-co efficien t d ifferen ce e q u a tio n s. A d d itio n a l stru c tu re s fo r th ese
sy stem s a re in tro d u c e d in C h a p te r 7.
A s a b eg in n in g , let us c o n sid e r th e first-o rd e r system
y(n) = —a iy (n — 1) + box ( n) + b \ x ( n - 1)
(2.5.1)
w hich is re a liz e d as in Fig. 2.32a. T h is re a liz a tio n uses s e p a ra te delays (m em o ry )
fo r b o th th e in p u t a n d o u tp u t signal sa m p le s a n d it is called a direct f o r m I structure.
N o te th a t th is sy stem can b e view ed as tw o lin e a r tim e -in v a ria n t system s in cascad e.
T h e first is a n o n re c u rsiv e , sy stem d e sc rib e d by th e e q u a tio n
u(n) = Z»oj;(n) + 6 j j ( n — 1)
(2.5.2)
w h e re a s th e se c o n d is a rec u rsiv e system d e sc rib e d by th e e q u a tio n
y ( n ) = - a i y ( n - 1) + t>{«)
(2.5.3)
H o w e v e r, as we h a v e se e n in S ectio n 2.3.4, if w e in te rc h a n g e th e o r d e r o f the
c a s cad ed lin e a r tim e -in v a ria n t system s, th e o v erall system re sp o n s e re m a in s th e
112
Discrete-Time Signals and Systems
Chap. 2
sam e. T h u s if w e in te rc h a n g e the o rd e r o f the recu rsiv e an d n o n re c u rsiv e system s,
we o b ta in an a lte rn a tiv e s tru c tu re fo r th e re a liz a tio n o f th e sv stem d e sc rib e d by
(2.5.1). T h e re su ltin g system is show n in Fig. 2.32b. F ro m th is figure we o b tain
th e tw o d ifferen ce e q u a tio n s
w( n) — —fl] w( n — 1) + jt(«)
(2-5.4)
v(n) = t>nw(n) + b\ w{ n - 1)
(2.5.5)
w hich p ro v id e an a lte rn a tiv e a lg o rith m for co m p u tin g th e o u tp u t o f the system
d e sc rib e d by th e single d ifferen ce e q u a tio n given in (2.5.1). In o th e r w o rd s, the
tw o d ifferen ce e q u a tio n s (2.5.4) an d (2.5.5) are e q u iv a le n t to th e single differen ce
e q u a tio n (2.5.1).
A close o b se rv a tio n o f Fig. 2.32 re v e a ls th at th e tw o d elay e le m e n ts co n tain
th e sam e in p u t w( n) an d h en c e the sam e o u tp u t w( n — 1). C o n s e q u e n tly , these
tw o e le m e n ts can be m e rg ed in to o n e delay, as show n in Fig. 2.32c. In c o n tra st
x(n)
b{,
r(n)
f
n
vim
-----------------~
(a)
(b.)
(c)
Figure 132 Steps in converting from the direct form I realization in (a) to the
direct form II realization in (c).
Sec. 2.5
Implementation of Discrete-Time Systems
113
to th e d ire c t fo rm I s tru c tu re , th is new re a liz a tio n re q u ire s o n ly o n e d elay fo r
th e au x iliary q u a n tity ui(n), an d h en c e it is m o re efficient in te rm s o f m em o ry
re q u ire m e n ts . It is called th e direct f o r m I I structure a n d it is u sed ex ten siv ely in
p ra c tic a l a p p licatio n s.
T h e se stru c tu re s can read ily be g e n e ra liz e d fo r th e g e n e ra l lin e a r tim ein v a ria n t recu rsiv e sy stem d escrib ed by th e d iffe re n c e e q u a tio n
N
M
y (n ) = - y ^ a ^ .y (« - k) +
*=l
- k)
(2.5.6)
k=0
F ig u re 2.33 illu stra te s th e d ire c t fo rm I s tru c tu re fo r th is sy stem . T h is stru c tu re
re q u ire s M + N d elay s a n d N + M + 1 m u ltip lic atio n s. I t c a n be v iew ed as the
c ascad e o f a n o n re c u rsiv e system
M
i;(«) = Y bkx{n - k)
(2.5.7)
i=U
a n d a recu rsiv e system
s
y(n) = -
v(n ~ k) + v(n)
(2.5.8)
By rev ersin g th e o r d e r o f th ese tw o system s as was p rev io u sly d o n e fo r th e
first-o rd e r sy stem , w e o b ta in th e d ire c t form II s tru c tu re sh o w n in Fig. 2.34 fo r
Figure 1 3 3
Direct form I structure of the system described by (2.5.6).
114
Discrete-Time Signals and Systems
w(n)
b0
Chap. 2
N
y(n )
-1
UJ{/1 - 1)
J
-1
Figure 2 3 4
w(n - 2)
b2
w(n - 3)
by
Direct form II structure for the system described by (2.5.6).
N > M. T h is stru c tu re is the cascad e o f a recu rsiv e system
N
w( n) = — ^
ai W(n - k ) -f x( n)
(2.5.9)
*=1
fo llo w ed by a n o n re c u rsiv e sy stem
M
v(n) =
vu(n - k )
(2.5.10)
W e o b serv e th a t if N > M . this s tru c tu re re q u ire s a n u m b e r o f d elay s e q u a l to
th e o rd e r N o f th e sy stem . H o w e v e r, if M > N , th e re q u ire d m e m o ry is specified
by M . F ig u re 2.34 can easily by m o d ified to h a n d le th is case. T h u s th e d ire c t form
II stru c tu re re q u ire s M + N + 1 m u ltip lic a tio n s a n d m ax{M , jV} d elay s. B e cau se it
re q u ire s th e m in im u m n u m b e r o f d elay s fo r th e re a liz a tio n o f th e system d e sc rib e d
by (2.5.6), it is so m e tim es ca lle d a canoni c f o r m .
Sec. 2.5
Implementation of Discrete-Time Systems
115
A special case o f (2.5.6) occurs if w e set th e system p a ra m e te rs ak — 0.
it = 1.........N. T h e n th e in p u t-o u tp u t re la tio n sh ip fo r th e sy stem re d u c e s to
M
v(n) =
- k)
(2.5.11)
k=0
w hich is a n o n re c u rsiv e lin e a r tim e -in v a ria n t system . T his sy stem view s only the
m o st re c e n t M + 1 in p u t signal sa m p les a n d , p rio r to a d d itio n , w eights each sam ple
by th e a p p ro p ria te co efficien t bk from th e set {i^}. In o th e r w ords, th e system
o u tp u t is b asically a weight ed m o v i n g average of th e in p u t signal. F o r this reaso n
it is so m e tim e s called a m o v i n g average ( M A ) syst em. S uch a system is an F IR
sy stem w ith an im p u lse re sp o n s e h (k ) e q u a l to th e coefficients bk, th at is.
* '* ) = { o ! '
o to rw l"
« '5 1 2 »
If w e r e tu rn to (2.5.6) a n d se t M = 0, th e g e n eral lin e a r tim e -in v a ria n t system
red u ces to a “p u re ly re c u rsiv e ” system d escrib ed by th e d ifferen ce e q u a tio n
N
y ( n ) = ~ Y a O ’(n ~ k) + bux(n)
i =1
(2.5.13)
In th is case th e sy stem o u tp u t is a w eig h ted lin ear c o m b in a tio n o f N past o u tp u ts
a n d th e p re s e n t in p u t.
L in e a r tim e -in v a ria n t system s d esc rib e d by a se c o n d -o rd e r d ifferen ce e q u a ­
tio n a re an im p o rta n t subclass o f th e m o re g e n eral system s d e sc rib e d by (2.5.6)
o r (2.5.10) o r (2.5.13). T h e re a so n fo r th e ir im p o rta n c e will be ex p lain ed later
w h en w e discuss q u a n tiz a tio n effects. Suffice to say at this p o in t th a t se c o n d -o rd e r
sy stem s a re u su ally u sed as b a sic b u ild in g b lo ck s fo r realizin g h ig h e r-o rd e r system s.
T h e m o st g e n e ra l se c o n d -o rd e r sy stem is d escrib ed by th e d ifferen ce e q u a tio n
y (n ) =
- a[v( n - 1) - a2v(n - 2) 4- b(,x(n)
(2.5.14)
+ b\ x ( n — 1) •+- b 2x ( n — 2)
w hich is o b ta in e d fro m (2.5.6) b y se ttin g N = 2 an d M = 2. T h e d irect form II
s tru c tu re fo r realizin g th is sy stem is sh o w n in Fig. 2.35a. If we se t a\ =
= 0.
th e n (2.5.14) re d u c e s to
y( n) = box(n) + b\x(rt — 1) + b 2x ( n - 2)
(2.5.15)
w hich is a sp ecial case o f th e F IR system d esc rib e d by (2.5.11). T h e stru c tu re
fo r realizin g th is sy stem is sh o w n in Fig. 2.35b. F inally, if w e set b\ = bz = 0
in (2.5.14), w e o b ta in th e p u re ly recu rsiv e se c o n d -o rd e r sy stem d esc rib e d by th e
d iffe re n c e e q u a tio n
y( n) = —a \ y ( n — 1) — a 2y(n - 2) 4- box(rt)
(2.5.16)
w hich is a sp e cial case o f (2.5.13). T h e stru c tu re fo r realizin g th is system is show n
in Fig. 2.35c.
Discrete-Time Signals and Systems
116
Chap. 2
(a)
-©---- -0
Y[n)
(b)
x i n)
—-^-1 H
—02
I _a'
-1
„-1
<c)
Figure 2.35 Structures for the realization of second-order systems: (a) general
second-order system; (b) FIR system; (c) “purely recursive system”
2.5.2 Recursive and Nonrecursive Realizations of FIR
Systems
W e h a v e a lread y m a d e th e d istin ctio n b etw e e n F IR an d IIR system s, b ased on
w h e th e r th e im p u lse re sp o n se h( n) o f th e system h a s a finite d u ra tio n , o r an infi­
n ite d u ra tio n . W e h av e also m a d e th e d istin c tio n b e tw e e n rec u rsiv e a n d n o n re c u r­
sive system s. B asically, a causal recu rsiv e system is d esc rib e d by an in p u t-o u tp u t
e q u a tio n o f th e fo rm
y(n) = F[v(^i — 1).........y{n — N ) , x ( n ) , ------x ( n - M)]
(2.5.17)
an d fo r a lin ear tim e -in v a ria n t system specifically, by th e d iffe re n c e e q u a tio n
N
U
y { n ) = - Y ak>’(n ~ k ) + Y , bkX(n ~ k )
*=i
*=o
(2.5.18)
Sec. 2.5
Implementation of Discrete-Time Systems
117
O n th e o th e r h a n d , cau sal n o n recu rsiv e system s d o n o t d e p e n d o n p a st values of
th e o u tp u t a n d h e n c e a re d escrib ed by an in p u t-o u tp u t e q u a tio n o f th e form
y (n) = F [x (n ), jc(n - 1).........x ( n — Mj ]
(2.5.19)
a n d fo r iin e a r tim e -in v a ria n t system s specifically, by th e d iffe re n c e e q u a tio n in
(2.5.18) w ith ak: = 0 fo r k = 1, 2 , . . . , N.
In th e case o f F IR system s, we h av e a lre a d y o b se rv e d th a t it is aiw ays possible
to re a liz e such sy stem s n o n recu rsiv ely . In fact, w ith at. = 0, k = 1, 2 , . . . , N , in
(2.5.18), w e h av e a sy stem w ith an in p u t-o u tp u t e q u a tio n
v(«) = ] p 6 * ;t( n ~
(2.5.20)
T h is is a n o n re c u rsiv e a n d F IR system . A s in d ic a te d in (2.5.12), th e im pulse
re sp o n s e o f th e sy stem is sim ply e q u a l to th e coefficients {&*). H e n c e every F IR
sy stem can b e re a liz e d n o n recu rsiv ely . O n th e o th e r h a n d , an y F IR system can
also b e realized recu rsiv ely. A lth o u g h the g e n eral p ro o f o f th is s ta te m e n t is given
la te r, w e shall give a sim ple ex am p le to illu stra te th e p o in t.
S u p p o se th a t w e h av e an F IR system o f th e form
1
m
y( n) = —— M + If-'
- k)
(2.5.21)
fo r co m p u tin g th e mo v i n g average of a signal jc(h). C learly , th is sy stem is F IR w ith
im p u lse re sp o n se
h( n) ~
1
0 < n < M
M + 1
F ig u re 2.36 illu stra te s th e stru c tu re of th e n o n re c u rsiv e re a liz a tio n o f th e system .
N ow , su p p o se th a t w e ex p ress (2.5.21) as
1
M
y (n ) = --------- x ( n — 1 — k)
M
m +
^ 1 ^Jt=0
+
. [x(n) ~ x ( n - 1 - M )]
M + 1
y(n - 1) +
Figure 2 3 6
1
M + V
[x(n) — x (n — 1 — Af)]
N onrecursive realization of an FIR moving average system.
(2.5.22)
118
Discrete-Time Signals and Systems
Chap. 2
N ow , (2.5.22) re p re se n ts a recu rsiv e re a liz a tio n of th e F IR system . T h e stru c tu re
o f th is recu rsiv e re a liz a tio n of th e m oving av erag e system is illu stra te d in Fig. 2.37.
In su m m ary , w e can th in k of th e te rm s F IR a n d IIR as g e n e ra l ch aracteristics
th a t distin g u ish a ty p e of lin e a r tim e -in v a ria n t system , and of th e term s recursive
an d nonrecursi ve as d e sc rip tio n s of th e stru c tu re s fo r realizin g o r im p lem en tin g
th e system .
vt/> - 1)
Figure 2 3 7
Recursive realization of an FIR moving averapc svstem.
2.6 CORRELATION OF DISCRETE-TIME SIGNALS
A m a th em atical o p e ra tio n th a t closely rese m b le s co n v o lu tio n is c o rre la tio n . Just
as in th e case o f c o n v o lu tio n , tw o signal se q u e n c e s are in v o lv ed in c o rre la tio n .
In c o n tra st to c o n v o lu tio n , h o w ev er, o u r o bjective in co m p u tin g th e c o rre la tio n
b etw een th e tw o signals is to m e a su re th e d e g re e to w hich th e tw o signals are
sim ilar an d th u s to e x tra c t so m e in fo rm a tio n th a t d e p e n d s to a large e x te n t on
th e ap p licatio n . C o rre la tio n o f signals is o ften e n c o u n te re d in ra d a r, so n a r, digital
co m m u n icatio n s, geolo g y, an d o th e r a re a s in science an d en g in eerin g .
T o b e specific, le t us su p p o se th a t we have tw o signal se q u e n c e s x ( n ) an d
y( n) th a t we w ish to co m p a re . In r a d a r an d active so n a r a p p lic a tio n s. x( n ) can
re p re s e n t th e sa m p le d v ersio n o f th e tra n s m itte d signal and y{n) can re p re s e n t th e
sam p led v ersio n o f th e receiv ed signal at th e o u tp u t o f the a n a lo g -to -d ig ita l (A /D )
c o n v erter. If a ta rg e t is p re se n t in th e space b e in g se a rc h e d by th e ra d a r o r so n a r,
th e receiv ed signal y( n) consists of a d e la y e d v ersio n o f th e tra n sm itte d signal,
reflec te d from th e ta rg e t, an d c o rru p te d b y a d d itiv e noise. F ig u re 2.38 d e p ic ts the
ra d a r signal re c e p tio n p ro b le m .
W e can re p re s e n t th e re c e iv e d signal se q u e n c e as
y ( n ) = a x ( n — D) + w( n )
(2.6.1)
w h ere a is som e a tte n u a tio n fa c to r re p re s e n tin g th e signal loss involved in th e
ro u n d -trip tran sm issio n o f th e signal x (n ), D is th e ro u n d -trip d elay , w hich is
Sec. 2.6
Correlation of Discrete-Time Signals
119
assum ed to be an integer m ultiple o f the sam pling interval, and w (n) represents
the additive n oise that is picked up by the antenna and any n oise generated by the
electron ic com p onents and amplifiers contained in the front end o f the receiver.
O n the other hand, if there is no target in the space searched by the radar and
sonar, the received signal y(n ) consists o f noise alone.
H avin g the tw o signal sequ en ces, x ( n ) , which is called the reference signal or
transm itted signal, and y ( n ) , the received signal, the problem in radar and sonar
d etection is to com p are y(n) and x ( n ) to determ ine if a target is present and, if
so, to determ in e the tim e d elay D and com p ute the distance to the target. In
practice, the signal x ( n — D) is h eavily corrupted by the additive n oise to the point
w here a visual inspection o f y ( n ) d oes n ot reveal the p resen ce or absence o f the
desired signal reflected from the target. Correlation provides us with a m eans for
extracting this im portant inform ation from y( n).
D igital com m unications is an oth er area w here correlation is o ften used. In
digital com m unications the inform ation to be transm itted from on e p oin t to an­
other is usually con verted to binary from , that is, a seq u en ce o f zeros and ones,
which are then transm itted to the in tend ed receiver. T o transm it a 0 w e can trans­
m it the signal seq u en ce xo(n) for 0 < n < L — 1, and to transm it a 1 w e can transmit
the signal seq u en ce jti(n) for 0 < n < L — 1, w here L is som e integer that d en otes
the num ber o f sam p les in each o f the tw o sequ en ces. V ery often , x\ (n) is selected
to be th e negative o f xo(n). T h e signal received by the in tend ed receiver m ay be
represented as
y (n ) = x,-(n) + w (n )
* = 0 ,1
0 < n < L —1
(2.6.2)
w here now the uncertainty is w hether x 0(n) or *](n ) is the signal com p on en t in
>(n), and w (n ) rep resents the additive n oise and other interferen ce inherent in
120
Discrete-Time Signals and Systems
Chap. 2
any co m m u n icatio n system . A g ain , such noise has its o rig in in th e e le ctro n ic
c o m p o n e n ts c o n ta in e d in th e fro n t e n d o f th e receiv er. In an y case, th e receiv er
k n o w s th e p o ssib le tra n sm itte d se q u e n c e s xo(n) a n d
(n) a n d is fa c e d w ith the task
o f co m p a rin g th e receiv ed signal y( n) w ith b o th xo(n) a n d Jti(n) to d e te rm in e w hich
o f th e tw o signals b e tte r m a tc h e s y(n). T h is c o m p a riso n p ro c e ss is p e rfo rm e d by
m ean s o f th e c o rre la tio n o p e ra tio n d e sc rib e d in th e follow ing su b se c tio n .
2.6.1 Crosscorrelation and Autocorrelation Sequences
S u p p o se th a t we h av e tw o real signal se q u e n c e s x ( n ) an d y ( n ) ea c h o f w hich has
finite en erg y . T h e crosscorrelation o f x ( n ) and v(n) is a se q u e n c e rxy(l), w hich is
d efin ed as
r.tv(l) —
~ 0
l = 0. ± 1 , ± 2 , . . .
(2.6.3)
riy(t) — Y , X ^n + 0 y ( « )
1 = 0, ± 1 , ± 2 , . . .
(2.6.4)
n —~ x
or, eq u iv alen tly , as
OC
n= -oc
T h e in d ex I is th e (tim e ) sh ift (o r lag) p a r a m e te r an d th e su b scrip ts x y on th e c ro ss­
c o rre la tio n se q u e n c e rxy(l) in d icate the se q u e n c e s b ein g c o rre la te d . T h e o rd e r of
th e su b scrip ts, w ith .x p re c e d in g y, in d ic a te s th e d ire c tio n in w h ich o n e se q u en ce
is sh ifted , re lativ e to th e o th e r. T o e la b o ra te , in (2.6.3), th e se q u e n c e x ( n ) is left
u n sh ifte d an d y (n ) is sh ifted by / u n its in tim e, to th e rig h t fo r / p o sitiv e a n d to
th e left for I n eg ativ e. E q u iv a le n tly , in (2.6.4), th e se q u en ce y (« ) is left u n sh ifted
a n d x{n) is sh ifted by I u n its in tim e, to th e left fo r / p o sitiv e a n d to th e rig h t fo r
/ n eg ativ e. B u t sh iftin g x (n ) to th e left by / u n its relativ e to y (n ) is e q u iv a le n t
to sh iftin g y(«) to th e rig h t by / u n its re lativ e to x ( n) . H e n c e th e c o m p u ta tio n s
(2.6.3) an d (2.6.4) yield id en tical c ro ssc o rre la tio n se q u en ces.
If w e rev erse th e ro les o f jr(«) an d y (n) in (2.6.3) an d (2.6.4) a n d th e re fo re
re v e rse th e o rd e r o f th e indices xy. we o b ta in th e c ro ss c o rre la tio n se q u e n c e
OC
ryx(I) =
y ( n ) x ( n — I)
(2.6.5)
y ( n + l ) x ( n)
(2.6.6)
n=—0C
o r, e q u iv alen tly ,
OC
ryx(l) = Y
n — —d c
By c o m p a rin g (2.6.3) w ith (2.6.6) o r (2.6.4) w ith (2.6.5), we c o n c lu d e th a t
rxy(l) = ryx( ~l )
(2.6.7)
T h e re fo re , r y i (l) is sim ply th e fo ld ed v ersio n o f rxy(l), w h e re th e fo ld in g is d o n e
w ith re sp e c t to / = 0. H e n c e , ryx(l) p ro v id e s exactly th e sam e in fo rm a tio n as rxv(l),
w ith re sp e c t to th e sim ilarity o f x ( n ) to y(n ).
Sec. 2.6
Correlation of Discrete-Time Signals
121
Exam ple 2.6.1
D eterm ine the crosscorrelation sequence rxv(l) of the sequences
x(n) = { ....0 .0 .2 . —1 .3 .7 .1 .2 . —3. 0. 0 ....}
t
v(n) = { . . . . 0 . 0 . 1 . - 1 . 2 . - 2 . 4 . 1 . - 2 . 5 .0 .0 ,...]
t
Solution
.(/). For I = 0 w e have
Let us use the definition in (2.6.3) to compute
r rv(0) = Y
x(ri)v(n)
The product sequence u()(n) = x ( n ) y ( r ) is
u„(n) = { ..., 0. 0. 2. 1. 6. -1 4 . 4. 2, 6, 0, 0. . . .)
t
and hence the sum over all values of n is
r,v(0) = 7
For I > 0, we simpiy shift v(«) to the right relaLive to a ' ( h ) hy / units, compute
the product sequence v/(n) = jr(n)_v(n — I), and finally, sum o v er all va lu es o f the
product sequence. Thus we obtain
r t ,( l) = 13,
rJV(2) = -1 8 .
r vv(3) = 16.
r,,(4 ) = - 7
rM5) = 5.
r ,v(6) = - 3 ,
rxy (/) = 0.
I >1
For / < 0, we shift y(n) to the left relative to jr(n) by / units, compute the product
sequence v/(n) = j(n ) v(n —I), and sum over all values of the product sequence. Thus
we obtain the values of the crosscorrelation sequence
rIV(—1) = 0,
riy( - 2 ) = 33.
rly{ - 3) = -1 4 .
rTV( - 4 ) = 36
rJV( -5 ) = 19,
fxv(-6) = - 9 ,
rIV( - 7) = 10,
rxv(l) = 0, I < - 8
Therefore, the crosscorrelation sequence of x{n) and y(n) is
r,AD = (10, - 9 ,1 9 , 36, -1 4 , 33,0, 7,13, -1 8 ,1 6 . - 7 , 5, - 3 )
t
T h e sim ilarities b e tw e e n th e c o m p u ta tio n o f th e c ro ss c o rre la tio n o f tw o se ­
q u e n c e s a n d th e c o n v o lu tio n o f tw o se q u e n c e s is a p p a re n t. In th e c o m p u ta tio n of
c o n v o lu tio n , o n e o f th e se q u e n c e s is fo ld ed , th e n sh ifted , th e n m u ltip lie d by th e
o th e r se q u e n c e to fo rm th e p ro d u c t se q u e n c e fo r th a t shift, a n d finally, th e values
o f th e p r o d u c t se q u e n c e a re su m m ed . E x c e p t fo r th e fo ld in g o p e ra tio n , th e co m ­
p u ta tio n o f th e c ro ss c o rre la tio n se q u e n c e involves th e sa m e o p e ra tio n s: shifting
o n e o f th e se q u e n c e s , m u ltip lic atio n o f th e tw o se q u e n c e s, an d su m m in g o v er all
v alu es o f th e p r o d u c t se q u e n c e . C o n se q u e n tly , if w e h av e a c o m p u te r p ro g ra m
th a t p e rfo rm s c o n v o lu tio n , w e can use it to p e rfo rm c ro ss c o rre la tio n by p ro v id in g
122
Discrete-Time Signals and Systems
Chap. 2
as in p u ts to th e p ro g ra m , th e se q u e n c e jc(«) an d th e fo ld ed se q u e n c e y ( —n). T h e n
th e c o n v o lu tio n o f x ( n ) w ith y ( —n) yields th e c ro ss c o rre la tio n r rv(/). th a t is,
rxv(D = x ( l ) * y ( - l )
(2.6.8)
In th e special case w h ere y( n) = x( n ) , we h av e th e aut ocorrel ati on o f *(« ),
w hich is d efined as th e se q u e n c e
OC
rXx(l)=
x(n)j:(n - 0
(2.6.9)
ft ~ —OC
o r, eq u iv alen tly , as
OO
rxx{l) =
^ 2 , x (n +
(2.6.10)
n= —oc
In d ealin g w ith fin ite -d u ra tio n se q u e n c e s, it is cu sto m a ry to ex p ress th e a u to ­
c o rre la tio n an d c ro ss c o rre la tio n in te rm s of th e finite lim its on th e su m m a tio n . In
p a rtic u la r, if x (« ) a n d v(n) a re causal se q u e n c e s o f le n g th N [i.e., .v(n) = y(n) = 0
for n < 0 an d n > N], th e c ro ssc o rre la tio n an d a u to c o rre la tio n se q u e n c e s m ay be
ex p ressed as
rxy(l) =
x(n)y(n-l)
^
(2.6.11)
an d
A'-1*1-1
Y
rxx( l ) =
(2.6.12)
n=f
w h ere i = I, k = 0 fo r / > 0, a n d / = 0, k = I fo r / < 0.
2.6.2 Properties of the Autocorrelation and
Crosscorrelation Sequences
T h e a u to c o rre la tio n an d c ro ssc o rre la tio n se q u e n c e s h av e a n u m b e r of im p o rta n t
p ro p e rtie s th a t w e n o w p re se n t. T o d e v e lo p th e s e p ro p e rtie s , le t us assu m e th a t
we h av e tw o se q u e n c e s x ( n ) a n d y (n) w ith finite en erg y from w hich w e fo rm the
lin e a r c o m b in a tio n ,
ax(n)
bv( n — I)
w h ere a an d b a re a rb itra ry c o n s ta n ts a n d I is so m e tim e shift. T h e en erg y in this
signal is
OC
Y
fj —-oc
OC
OC
[ax(n) + by( n - I)]2 = a 2 ^
x 2( n ) + b 2 ^
n = —oc
y 2{n - I)
/i“ —oc
-j l ^V""' x (in )\ y (tn — I)
n
+. 2ab
n = —oc
= a 2rxx(0) + b2r yy( 0) + l a b r xy{l)
(2 '6 -13)
Sec. 2.6
123
Correlation of Discrete-Time Signals
F irst, we n o te th a t rx x (0) = E x a n d /-vy(0) = £ v, w hich a re th e en e rg ie s o f
x{n) an d y (n ), resp ectiv ely . It is o b v io u s th a t
a 2rxx(0) -I- b 2r y>.(0) + 2abrxy(l) > 0
(2.6.14)
N ow , assu m in g th a t b ^ 0, w e can divide (2.6.14) by b 2 to o b ta in
r„ (0 )
(^)2
+ 2rxy(l) Q
+ r v_v(0) > 0
W e view th is e q u a tio n as a q u a d ra tic w ith coefficients r XJ(0), 2rxv(l), a n d ^ ,( 0 ) .
Since th e q u a d ra tic is n o n n e g a tiv e , it follow s th a t th e d isc rim in a n t of this q u a d ra tic
m u st b e n o n p o sitiv e , th a t is,
4 [ r ; v(/) - r , , ( 0 K v(0)] < 0
T h e re fo re , th e c ro ss c o rre la tio n se q u e n c e satisfies th e c o n d itio n th a t
|r,v(/>| < y r „ ( 0 ) r , v(0) =
(2.6.15)
In th e sp ecial case w h ere v(n) = x ( n ), (2.6.15) re d u c e s to
\rxx( D \ < r xx(0) = E x
(2.6.16)
T his m ean s th a t th e a u to c o rre la tio n se q u e n c e o f a signal a tta in s its m ax im u m value
at z e ro lag. T h is re su lt is c o n siste n t w ith th e n o tio n th a t a sig n a l m a tc h e s p erfe ctly
w ith itself at z e ro shift. In th e case o f th e c ro ssc o rre la tio n se q u en ce, th e u p p e r
b o u n d o n its v alu es is given in (2.6.15).
N o te th a t if an y o n e o r b o th o f th e signals involved in th e c ro ssc o rre la tio n
a re scaled , th e sh a p e o f th e c ro ss c o rre la tio n se q u e n c e d o e s n o t ch an g e, only the
a m p litu d e s o f th e c ro ss c o rre la tio n se q u e n c e a re sc aled accordingly. S ince scaling
is u n im p o rta n t, it is o ften d e s ira b le , in p ra c tic e , to n o rm a liz e th e a u to c o rre la tio n
a n d c ro ss c o rre la tio n se q u e n c e s to th e ran g e fro m - 1 to 1. In th e case o f the
a u to c o rre la tio n se q u e n c e , w e can sim ply d iv id e by ^ ( 0 ) . T h u s th e n o rm aliz ed
a u to c o rre la tio n se q u e n c e is d efin ed as
P. A D =
rixiO)
(2-6 -17)
Sim ilarly, we d efin e th e n o rm a liz e d c ro ssc o rre la tio n se q u e n c e
pXY(l) =
r ' v(l)
:
v/ r xx(0 )rvv(0)
(2.6.18)
N ow \pXI{l)\ < 1 a n d |/oXv(0! < 1, a n d h e n c e th ese se q u e n c e s a re in d e p e n d e n t of
signal scaling.
F in ally , as we h av e a lre a d y d e m o n s tra te d , th e c ro ss c o rre la tio n se q u e n c e sa t­
isfies th e p ro p e rty
r Xy ( l ) = f y x ( - 0
Discrete-Time Signals and Systems
124
Chap. 2
W ith y(n) = x ( n) , th is re la tio n resu lts in th e follow ing im p o rta n t p ro p e rty fo r the
a u to c o rre la tio n se q u en ce
(2.6.19)
r „ { l ) = rx s ( . - l )
H e n c e th e a u to c o rre la tio n fu n ctio n is an ev en fu n ctio n . C o n s e q u e n tly , it suffices
to co m p u te rx x (l) fo r / > 0.
Example 2.6.2
Compute the autocorrelation of the signal
x(n) = a"u(n), 0 < a < 1
Solution Since x (n) is an infinite-duration signal, its autocorrelation also has infinite
duration. We distinguish two cases.
If I > 0. from Fig. 2.39 we observe that
r t J (/) =
x ( n ) x ( n - /) =
' = a
n= l
n =i
1
n~l
Since a < 1, the infinite series coti crges and we obtain
r tl (/) ~
I> 0
^ -V
1 —a-
For / < 0 we have
=
n=(l
x(n)x(/i —/) = a -1'S~'(rr)" = ------- -ti~!
I - a-
/ < 0
n=(l
But when / is negative, c r 1 — a ' 1'. Thus the two relations for r i t ( ! ) can be combined
into the following expression:
rx,(!) =
■— ,,a ,t:
—oc < / < oc
1 —a~
The sequence rxx(l) is shown in Fig. 2.42(d). We observe that
(2.6.20)
r „ ( ~ / | = rxAD
and
rlt(0) =
1
1 —a2
Therefore, the normalized autocorrelation sequence is
r (/)
p s,(!) - — — — cr|,:
~ oc < I < oc
rxxW)
(2.6.21)
2.6.3 Correlation of Periodic Sequences
In S ectio n 2.6.1 w e d efin ed th e c ro ss c o rre la tio n an d a u to c o r re la tio n se q u en ces of
en erg y signals. In this se ctio n we co n sid e r th e c o rre la tio n se q u e n c e s o f p o w er
signals an d , in p a rtic u la r, p e rio d ic signals.
L et x( n ) a n d y(rc) b e tw o p o w e r signals. T h e ir c ro ss c o rre la tio n se q u e n c e is
d efin ed as
1
M
rx \ 0 ) —
lim
— ----- - Y ]
M -oc 2 M + 1
x(n)y(n~l)
( 2 .6 .2 2 )
Sec. 2.6
125
Correlation of Discrete-Time Signals
xi n)
) i'
in .
-2-10123
(a)
xin-I)
/ >o
I
o
(b)
x(n - I )
l< 0
(c)
r„(!) =
■■■ - 2 - 1 0
Figure 139
1 2
(d)
' , a1'1
1 - a2
/
Compulation of the autocorrelation o f the signal xin)
=
a",
0 < a < 1.
If x ( n ) = y(tt), we have the definition o f the autocorrelation seq u en ce of a
p ow er signal as
1
M
rxx(I) = iim . . . , 1 Y ] x ( n ) x ( n - l )
(2.6.23)
Af-oo 2M + 1
In particular, if x ( n ) and y ( n ) are two periodic seq u en ces, each with period
the averages indicated in (2.6.22) and (2.6.23) o ver the infinite interval, are identical
126
D iscrete-Time Signals and Systems
Chap. 2
to th e av erag es o v er a single p e rio d , so th a t (2.6.22) an d (2.6.23) re d u c e to
(2.6.24)
an d
(2.6.25)
It is c lear th a t r ry(l) an d rxx(l) are p e rio d ic c o rre la tio n se q u e n c e s w ith p e rio d N .
T h e fa c to r 1 / N can b e v iew ed as a n o rm a liz a tio n scale facto r.
In som e p ractical ap p licatio n s, c o rre la tio n is u se d to id en tify p erio d icitie s in
an o b se rv e d physical signal w hich m ay be c o rru p te d by ra n d o m in te rfe re n c e . F o r
ex am p le, c o n sid er a signal se q u e n c e y( n) o f th e form
y{n) = * (n ) + w(n)
(2.6.26)
w h ere jc(/i) is a p erio d ic se q u e n c e o f so m e u n k n o w n p e rio d N a n d w( n) re p re se n ts
an ad d itiv e ran d o m in te rfe re n c e . S u p p o se th a t w e o b se rv e M sa m p le s o f y(n ), say
0 < n < M — 1, w h ere M > > N. F o r all p ractical p u rp o se s, w e can assum e th a t
y( n) — 0 fo r n < 0 an d n > M. N ow th e a u to c o rre la tio n se q u e n c e of y(n), using
th e n o rm aliz atio n facto r o f \ / M . is
(2.6.27)
If we su b stitu te for y(n) fro m (2.6.26) in to (2.6.27) w e o b ta in
__1
ry.T(/) = ■
■j M- 1
— Y ] x ( n ) x i n - /)
M
j
M —J
H— - y ^ [ .r ( n ) itj (/2 — /) + w{ n) x{n — /)]
M “
(2.6.28)
T h e first fa c to r on th e rig h t-h a n d sid e of (2.6.28) is th e a u to c o rre la tio n se ­
q u e n c e o f x i n) . S ince x( n) is p e rio d ic , its a u to c o rre la tio n s e q u e n c e ex h ib its th e
sam e p erio d icity , th u s co n ta in in g relativ ely larg e p e a k s at / = 0, N , 2N , a n d so
on. H o w ev er, as th e shift I a p p ro a c h e s M , th e p e a k s a re re d u c e d in a m p litu d e
d u e to th e fact th a t w e h av e a finite d a ta re c o rd o f M sa m p le s so th a t m any o f th e
p ro d u c ts ;t(n)j:(rt — /) a re zero . C o n s e q u e n tly , w e sh o u ld av o id c o m p u tin g r VT(/)
fo r larg e lags, say, I > M f l .
Sec. 2.6
Correlation of Discrete-Time Signals
127
T h e c ro ss c o rre la tio n s rxu,(/) an d rwx(l) b etw e e n th e signal
a n d th e a d ­
ditiv e ra n d o m in te rfe re n c e a re e x p ected to be relativ ely sm all as a resu lt of the
e x p e c ta tio n th a t ;r(/i) a n d w ( n ) will be to ta lly u n re la te d . F in ally , the last term on
th e rig h t-h a n d sid e of (2.6.28) is th e a u to c o rre la tio n se q u e n c e o f th e ra n d o m se ­
q u e n c e w( n) . T h is c o rre la tio n se q u en ce will c e rta in ly c o n tain a p e a k a t / = 0, but
b ecau se o f its r a n d o m ch aracteristics, r ww(l) is ex p e c te d to d ecay rap id ly to w ard
zero . C o n s e q u e n tly , o nly rxx{l) is e x p e c te d to h av e large p e a k s fo r / > 0. T his
b e h a v io r allow s us to d e te c t th e p re se n c e of th e p e rio d ic signal a (/ j ) b u rie d in the
in te rfe re n c e u>(n) a n d to id e n tify its p e rio d .
A n e x am p le th a t illu stra te s th e use of a u to c o rre la tio n to identify a h id d e n
p e rio d ic ity in an o b se rv e d physical signal is show n in Fig. 2.40. T h is figure illus­
tra te s th e a u to c o rre la tio n (n o rm a liz e d ) se q u e n c e fo r th e W o lfe r su n sp o t n u m b e rs
fo r 0 < / < 20, w h ere an y v alu e o f / c o rre sp o n d s to o n e y ear. T h e se n u m b e rs are
given in T a b le 2.2 fo r th e 100-year p e rio d 1770-1869. T h e re is c lear ev id en ce in
th is fig u re th a t a p e rio d ic tre n d exists, w ith a p e rio d o f 10 to 11 years.
Example 2.6.3
Suppose that a signal sequence jr(n) = sin(7r/5)/i, for 0 < n < 99 is corrupted by
an additive noise sequence «Kn), where the values of the additive noise are selected
independently from sample to sample, from a uniform distribution over the range
TABLE 2.2
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
YEARLY WOLFER SUNSPOT NUMBERS
101
82
66
35
31
7
20
92
154
125
85
68
38
23
10
24
83
132
131
118
90
67
60
47
41
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
21
16
6
4
7
14
34
45
43
48
42
28
10
8
2
0
1
5
12
14
35
46
41
30
24
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
16
7
4
2
8
17
36
50
62
67
71
48
28
8
13
57
122
138
103
86
63
37
24
11
15
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
40
62
9X
124
96
66
64
54
39
21
7
4
23
55
94
96
77
59
44
47
30
16
7
37
74
128
Discrete-Time Signals and Systems
Chap. 2
Year
(a)
Figure 2.40 Identification of periodicity in the Wolfer sunspot numbers: (a) an­
nual Wolfer sunspot numbers; (b) autocorrelation sequence.
(—A /2, A /2), where A is a param eter of the distribution. The observed sequence is
y(n) = x(n) + w(n). D eterm ine the autocorrelation sequence rvt(n) and thus determine
the period of the signal x(rt).
Solution The assumption is that the signal sequence x(n) has some unknown period
that we are attem pting to determ ine from the noise-corrupted observations {y(n)).
Although x(n) is periodic with period 10, we have only a ftmte-duration sequence of
Sec. 2.6
129
Correlation of Discrete-Time Signals
length M = 100 [i.e.. 10 periods of jcm )]. The noise power level P„ in the sequence
u'(n) is determ ined by the param eter A. We simply state that Pu = A: /12. The signal
power level is P, = I. Therefore, the signal-to-noise ratio (SNR) is defined as
3
P,
6
~P~U ~ A : /1 2 “ A :
Usually, the SNR is expressed on a logarithmic scale in decibels (dB) as 101ogui
( PJPv) .
Figure 2.41 illustrates a sample of a noise sequence w(n), and the observed
sequence y(n) — x(n) + it’(n) when the SNR = 1 dB. The autocorrelation sequence
J '
I l l
♦ Lt
1 i ([Ijj-liii •
'
Tllrii!
1
ttIiIt.
j
i
i|
4
•
*
4
*
t
ill ..tI
1. I i i h i
p p T T J ljw ^
IT
* 4
1
1 •
la]
(c)
Figure 2.41
noise.
Use of autocorrelation to detect the presence of a periodic signal corrupted by
Discrete-Tim e Signals and Systems
130
Chap. 2
L T TTTT. Tt TJ IT.TTUTtTT ._L T .T. I .T. ttT .1.
pp V|Vvli
w(n)
'
|
(a)
(b)
r„M .
1
I
?
r![It
xij|il
rl It
i o1
r!lfit
F
1
tTT
1 [1* ?
(c)
Figure 2.42 Use of autocorrelation to detect the presence of a periodic signal
corrupted by noise.
r vv(/) is illustrated in Fig. 2.41c. W e observe that the periodic signal •*(/?), embedded
in y(n), results in a periodic autocorrelation function rZI(l) with period N — 10. The
effect of the additive noise is to add to the peak value at / = 0. but for I ^ 0, the
correlation sequence rwu,(l) = ^ 0 a s a result of the fact that values of w(ri) were gen­
erated independently. Such noise is usually called white noise. The presence of this
noise explains the reason for the large peak at I = 0. The smaller, nearly equal peaks
at I = ±10, ± 2 0 ,... are due the periodic characteristics of x(n).
Figure 2.42 illustrates the noise sequence w(n), the noise-corrupted signal y(n),
and the autocorrelation sequence r vv(/) for the same signal, within which is embedded
a signal at a smaller noise level. In this case, the SNR = 5 dB. E ven with this relatively
small noise level, the periodicity of the signal is not easily determ ined from observa­
tion of y( n). However, it is clearly evident from observation of the autocorrelation
sequence ryy(n).
2.6.4 Computation of Correlation Sequences
A s indicated on Section 2.6.1, the procedure for com puting the crosscorrelation
seq u en ce b etw een x ( n ) and y ( n ) in volves shifting on e o f the seq u en ces, say x(n),
Sec. 2.6
Correlation of Discrete-Time Signals
131
to o b ta in x( n - /), m u ltip ly in g th e sh ifted se q u e n c e by v(n) to o b ta in th e p r o d ­
u ct se q u e n c e y ( n ) x ( n - /), an d th en su m m in g all th e v alu es o f th e p ro d u c t se ­
q u en ce to o b ta in r yx(l). T his p ro c e d u re is r e p e a te d fo r d iffe re n t values o f th e
lag /. E x c e p t fo r th e fo ld in g o p e ra tio n th a t is involved in c o n v o lu tio n , th ese b a ­
sic o p e ra tio n s fo r co m p u tin g th e c o rre la tio n se q u e n c e a re id e n tic a l to th o se in
co n v o lu tio n .
T h e p ro c e d u re fo r co m p u tin g th e co n v o lu tio n is directly a p p licab le to c o m ­
p u tin g th e c o rre la tio n o f tw o se q u en ces. S pecifically, if w e fo ld y (n) to o b ta in
y ( —n), th e n th e c o n v o lu tio n o f x ( n ) w ith v ( —n) is id en tical to th e cro ssc o rre la tio n
o f x ( n ) w ith y(n). T h a t is.
rxy(l) = x(rt) * y ( —n ) |n=/
(2.6.29)
A s a c o n se q u e n c e , th e c o m p u ta tio n a l p ro c e d u re d escrib ed for c o n v o lu tio n can be
a p p lied d irectly to th e c o m p u ta tio n o f th e c o rre la tio n se q u en ce.
W e no w d e sc rib e an a lg o rith m th a t can b e easily p ro g ra m m e d to c o m p u te
th e c ro ss c o rre la tio n se q u e n c e of tw o fin ite -d u ra tio n signals * (« ), 0 < n < N ~ I,
an d y ( n) , 0 < n < M — 1.
T h e alg o rith m c o m p u te s rJV(/) fo r p o sitiv e lags. A cc o rd in g to th e re la tio n
rxy(—l) = r YX(I), th e v alu es o f rXY(l) fo r n eg ativ e lags can be o b ta in e d by using th e
sam e a lg o rith m fo r p o sitiv e lags, an d in te rc h a n g in g th e roles o f jt(« ) an d v(n). W e
o b se rv e th a t if M < N, rXY{i) can be c o m p u te d by th e re la tio n s
M - \ + l
Y ' , x ( n ) y ( n — /),
0 < 1 < N —M
(2.6.30)
O n th e o th e r h a n d , if M > N , th e fo rm u la fo r th e c ro ss c o rre la tio n becom es
rxy(l) = Y
x ( n ) v ( n - 1)
0 < 1< N -1
(2.6.31)
T h e fo rm u la s in (2.6.30) a n d (2.6.31) c a n b e c o m b in ed a n d c o m p u te d by m ean s
o f th e fo llo w in g sim p le alg o rith m illu stra te d in th e flow chart in Fig. 2.43. By
in terch an g in g th e ro les o f j ( « ) a n d y(n) an d re c o m p u tin g th e cro ssc o rre la tio n
se q u e n c e , w e o b ta in th e v alu es o f rXY(l) c o rre sp o n d in g to n eg ativ e shifts I.
If w e w ish to c o m p u te th e a u to c o rre la tio n se q u e n c e rxx(l), w e se t y (n ) = jr(n)
an d M = N in (2.6.31). T h e co m p u ta tio n o f rxx{l) can be d o n e by m e a n s of the
sa m e a lg o rith m fo r p o sitiv e shifts only.
2.6.5 Input-Output Correlation Sequences
In th is se ctio n w e d e riv e tw o i n p u t- o u tp u t re la tio n sh ip s fo r L T I system s in th e
“c o rre la tio n d o m a in .” L e t us assu m e th a t a signal x ( n ) w ith k n o w n a u to c o r re la ­
tio n rxx(I) is a p p lied to an L T I system w ith im p u lse re sp o n s e h( n), p ro d u c in g th e
132
Discrete-Tim e Signals and Systems
C j “lD
Chap. 2
Figure 2A3 Flowchart for software
implementation of crosscorrelation.
Sec. 2.6
133
Correlation of Discrete-Time Signals
o u tp u t signal
v(n) = h( n) * .x (/?) =
^
h(k)x(>i — k)
k=—oc
T h e c ro ss c o rre la tio n b e tw e e n th e o u tp u t a n d th e in p u t signal is
r VJ-(7) = y d ) * x ( - l ) — h ( l ) * [*(/) * * ( - / ) ]
or
ryx(l) = /;(/) * rxx(l)
(2.6.32)
w h ere w e h av e u se d (2.6.8) a n d th e p ro p e rtie s o f co n v o lu tio n . H e n ce th e c ro ssco r­
re la tio n b e tw e e n th e in p u t an d the o u tp u t of th e sy stem is th e co n v o lu tio n of the
im p u lse re sp o n s e w ith the a u to c o rre la tio n of th e in p u t se q u e n c e . A lte rn a tiv e ly .
ryx(l) m av b e view ed as th e o u tp u t of th e L T I sy stem w hen th e in p u t se q u en ce is
rxx (/). T h is is illu stra te d in Fig. 2.44. If we rep lace / by - / in (2.6.32), w e o b tain
r.tyU) = h ( - D * rxx(l)
T h e a u to c o rre la tio n of th e o u tp u t signal can be o b ta in e d by using (2.6.8) w ith
x(/i) = y(/i) an d th e p ro p e rtie s o f co n v o lu tio n . T h u s we have
r lv(/) = v(/) * v (—/)
= [/;(/) * * (/)] * [/j ( —/ J * * ( —/)]
(2.6.33)
= [/?(/) * h ( —/)] * [*(/) * * ( —/)]
= rhh(l) * rxx(l)
T h e a u to c o rre la tio n rhh(l) of th e im pulse resp o n se h(n) exists if th e system is stable.
F u rth e rm o re , th e stab ility in su res th a t th e system d o e s n ot ch an g e th e ty p e (en erg y
o r p o w e r) o f th e in p u t signal. By ev alu atin g (2.6.33) fo r / = 0 we o b ta in
CC
(2.6.34)
w hich p ro v id e s th e en e rg y (o r p o w er) of th e o u tp u t signal in te rm s o f a u to c o r re ­
latio n s. T h e se re la tio n sh ip s h o ld fo r b o th en erg y a n d p o w e r signals. T h e direct
d e riv a tio n o f th e s e re la tio n sh ip s for en erg y an d p o w e r signals, a n d th e ir e x ten sio n s
to co m p lex sig n als, are left as exercises fo r the stu d e n t.
Input
rxx<n )
LTI
SYSTEM
Output
ft(n)
rvr(n)
Figure 2.44 Input-output relation for
crosscorrelation ryx(n).
134
Discrete-Time Signals and Systems
Chap. 2
2.7 SUMMARY AND REFERENCES
T h e m a jo r th e m e o f th is c h a p te r is th e c h a ra c te riz a tio n o f d isc re te -tim e signals an d
sy stem s in th e tim e d o m ain . O f p a rtic u la r im p o rta n c e is th e class o f lin e a r tim ein v a ria n t (L T I) sy stem s w hich a re w idely u se d in th e design a n d im p le m e n ta tio n
o f d ig ital signal p ro cessin g system s. W e c h a ra c te riz e d L T I sy ste m s by th e ir u n it
sa m p le re sp o n se h{n) an d d e riv e d th e co n v o lu tio n su m m a tio n , w hich is a fo rm u la
fo r d e te rm in in g th e re sp o n se y( n) o f th e system c h a ra c te riz e d by h( n) to an y given
in p u t se q u e n c e x(n).
T h e class o f L T I system s c h a ra c te riz e d by lin e a r d ifferen ce e q u a tio n s w ith
c o n s ta n t coefficients is by fa r th e m o st im p o rta n t of th e L T I sy stem s in th e th e o ry
an d a p p licatio n o f d ig ital sig n a l p ro cessin g . T h e g e n e ra l s o lu tio n of a lin e a r dif­
fe re n c e e q u a tio n w ith c o n s ta n t coefficients w as d eriv e d in th is c h a p te r an d show n
to co n sist of tw o c o m p o n en ts: th e so lu tio n o f th e h o m o g e n e o u s e q u a tio n w hich
r e p re s e n ts th e n a tu ra l re sp o n se o f th e sy stem w h en th e in p u t is z e ro , an d th e p a r­
tic u la r so lu tio n , w hich re p re s e n ts th e re sp o n s e o f th e system to th e in p u t signal.
F ro m th e d ifferen ce e q u a tio n , w e also d e m o n s tra te d h o w to d e riv e th e u n it sam ple
re sp o n s e of th e L T I system .
L in e a r tim e -in v a ria n t sy stem s w ere g en erally su b d iv id ed in to F IR (finited u ra tio n im p u lse re sp o n se ) a n d I I R (in fin ite -d u ra tio n im pulse re sp o n s e ) d e p e n d ­
ing o n w h e th e r h(n) h as finite d u ra tio n o r infinite d u ra tio n , resp ec tiv ely . T he
re a liz a tio n s o f such system s w ere briefly d escrib ed . F u rth e rm o re , in the re a liz a ­
tio n o f F IR system s, we m a d e th e d istin ctio n b e tw e e n recu rsiv e a n d n o n recu rsiv e
re a lizatio n s. O n th e o th e r h a n d , w e o b se rv e d th a t I I R system s can be im p le m e n te d
recu rsiv ely , only.
T h e re are a n u m b e r o f te x ts o n d isc re te -tim e signals a n d system s. W e m e n ­
tio n as ex am p les th e b o o k s by M c G illem a n d C o o p e r (1984), O p p e n h e im a n d W illsky (1983), an d S ie b e rt (1986). L in e a r c o n sta n t-c o e ffic ie n t d iffe re n c e e q u a tio n s are
tr e a te d in d e p th in th e b o o k s by H ild e b ra n d (1952) a n d L evy a n d L essm an (1961).
T h e last to p ic in this c h a p te r, o n c o rre la tio n o f d isc re te -tim e signals, plays an
im p o rta n t ro le in d ig ital signal p ro cessin g , esp ecially in a p p lic a tio n s d ealin g w ith
digital c o m m u n ic a tio n s, ra d a r d e te c tio n a n d e stim a tio n , so n a r, a n d geophysics. In
o u r tr e a tm e n t o f c o rre la tio n se q u e n c e s, w e a v o id e d th e use o f sta tis tic a l concepts.
C o rre la tio n is sim ply d efin ed as a m a th e m a tic a l o p e r a tio n b e tw e e n tw o se q u en ces,
w hich p ro d u c e s a n o th e r se q u e n c e , called e ith e r th e crosscorrelat ion s e quence w hen
th e tw o se q u e n c e s a re d iffe re n t, o r th e aut ocorrel ati on sequence w h e n th e tw o se ­
q u e n c e s are id en tical.
In p ractical a p p lic a tio n s in w hich c o rre la tio n is u sed , o n e ( o r b o th ) o f th e
se q u e n c e s is (a re ) c o n ta m in a te d by n o ise a n d , p e rh a p s , by o th e r fo rm s o f in te rfe r­
en ce. In such a case, th e noisy se q u e n c e is c alled a r a n d o m se q u e n c e a n d is c h a r­
a c te riz e d in sta tistical term s. T h e c o rre sp o n d in g c o rre la tio n se q u e n c e b e c o m e s a
fu n ctio n o f th e sta tistical c h a ra c te ristic s of th e n o ise an d an y o th e r in te rfe re n c e .
T h e statistical c h a ra c te riz a tio n o f se q u e n c e s a n d th e ir c o rre la tio n is tr e a te d in
A p p e n d ix A . S u p p le m e n ta ry re a d in g o n p ro b a b ilis tic an d sta tistical c o n c e p ts deal-
Chap. 2
135
Problems
ing w ith c o rre la tio n can be fo u n d in th e b o o k s by D a v e n p o rt (1970). H e lstro m
(1990). P ap o u lis (1984). an d P eeb les (1987).
PROBLEMS
2.1 A discrete-time signal x(n) is defined as
I l + j,
—3 < n < —1
1.
0 < n < 3
0,
elsewhere
(a) Determ ine its values and sketch the signal .v(n).
(b) Sketch the signals that result if we:
(1) First fold x{n) and then delay the resulting signal by four samples.
(2) First delay xin) by four samples and then fold the resulting signal
(c) Sketch the signal x ( —n + 4 ).
(d) Com pare the results in parts (b) and (c) and derive a rule for obtaining the signal
,v(—n -t- k) from
(e) Can you express the signal ,r(n) in term s of signals S(n) and u{n)l
2.2 A discrete-time signal ,v(n) is shown in Fig. P2.2. Sketch and label carefully each of
the following signals.
.v(n)
J_
J_
L L _ ________ _
- 2 - 1 0 1 2 3 4
„
FigUre P2.2
(a ) x(n - 2)
(b) x(4 —n) ( c)x(n + 2) (d) x(n)u(2 — n)
(e) x(n - 1 )8{n - 3) (F) x( n2) (g) even part of x{n)
(h) odd part of x ( n )
2 3 Show that
(a ) &(n) = u(n) — u(n — 1)
(b) u(n) =
8(k) =
^
2.4 Show that any signal can be decomposed into an even and an odd component. Is the
decomposition unique? Illustrate your arguments using the stgnal
x(n) = {2. 3, 4. 5. 6)
t
2.5 Show that the energy (power) of a real-valued energy (power) signal is equal to the
sum of the energies (powers) of its even and odd components.
2.6 Consider the system
vf«) = T[x( n) ] = x ( n 2)
(a) Determ ine if the system is time invariant.
136
Discrete-Time Signals and Systems
Chap. 2
(b) To clarify the result in part (a) assume that the signal
_fl,
\ 0,
O 5 /1 < 3
elsewhere
is applied into the system.
(1) Sketch the signal *(«).
(2) D eterm ine and sketch the signal y ( n ) = T[x(n)},
(3) Sketch the signal y'2(n) = y(n — 2).
(4) D eterm ine and sketch the signal x2(n) = x(n - 2).
(5) Determ ine and sketch the signal
= T \ x 2(n)].
(6) Com pare the signals ^ ( n ) and y(n - 2). W hat is your conclusion?
(c) R epeat part (b) for the system
y(rt) = x(n) - x(n — 1)
Can you use this result to make any statem ent about the time invariance of this
system? Why?
(d) Repeat parts (b) and (c) for the system
y(rt) = T [ x ( n )] = nx(n)
2.7 A discrete-time system can be
(1) Static or dynamic
(2) Linear or nonlinear
(3) Time invariant or time varying
(4) Causal or noncausal
(5) Stable o r unstable
Examine the following systems with respect to the properties above.
(a) v(«) = cos[jc(n)]
(b) v(n) =
x(k>
(c) v(n) = x(n)cos(£^n)
(d) y(n) ~ x ( —n + 2)
(e) y(n) = Trun[;c(n)], where Trun[jc(n)] denotes the integer part of x(n), obtained
by truncation
(f) y(n) = Round[jc(n)], where Round[;c(n)] denotes the integer part of Jt(n) obtained
by rounding
Remark: The systems in parts (e) and (f) are quantizers that perform truncation and
rounding, respectively.
(g) y(B) = |*(n)|
(h) v(rt) = x(n)u(n)
(I) y(n) = x(n) + nx{n + 1)
(j) y ( n ) = x ( 2 n )
(I) y( n) = x ( - n )
(m) y(n) = sign[j:(n)]
(n) The ideal sampling system with input xaU) and output x(n) = x a(nT), —oc <
n < oo
2J8 Two discrete-time systems 7] and T2 are connected in cascade to form a new system
T as shown in Fig. P2.8. Prove or disprove the following statements.
Chap. 2
137
Problems
yin)
xin )
Ti
T;
T - T-. T 2
Figure P2.8
(a) If T\ and % are linear, then T is linear (i.e.. the cascade connection of two linear
systems is linear).
(b) If T\ and
are time invariant, then T is time invariant.
(c) If T[ and 7? are causal, then T is causal.
<d) If T] and T2 are linear and time invariant, the same holds for T .
(e) If 7] and T2 are linear and time invariant, then interchanging their order does not
change the system T.
(0 As in part (e) except that 7J, T2 are now time varying. (Hint: Use an example.)
(g) If 7] and T2 are nonlinear, then T is nonlinear.
(h) If T< and T2 are stable, then T is stable.
(i) Show by an example that the inverse of parts (c) and (h) do not hold in general.
2.9 Let T be an LTI, relaxed, and BIBO stable system with input x{n) and output y(n).
Show that:
(a) If x(n) is periodic with period N [i.e., jr(n) = x{n + N) for all n > 0], the output
y(n) tends to a periodic signal with the same period.
(b) If x(n) is bounded and tends 10 a constant, the output will also tend to a constant.
(c) If x{n) is an energy signal, the output y(n) will also be an energy signal.
2.10 The following in p u t-output pairs have been observed during the operation of a umeinvariam system:
x,(n) = {1.0,2} ^
y,(fi) = (0, 1.2}
t
t
x,(n) = {0.0,3} ^
v; (n) = (0, 1.0,2}
t
t
x-\(n) = {0. 0, 0. 1}
vj(n) = (1,2, 1}
t
t
Can you draw any conclusions regarding the linearity of the system. W hat is the
impulse response of the system?
2.11 The following input-output pairs have been observed during the operation of a linear
system:
Xi(n) = {-1. 2. 1}
t
x2{n) = {1, - 1 , -1}
y,(/i) = (1, 2. - 1 , 0. 1}
t
\'2in) = {-1. 1, 0, 2}
t
x 3(n) = {0, 1, 1)
t
t
y i ( n) = {1, 2. 1}
r
Can you draw any conclusions about the time invariance of this system?
2.12 The only available information about a system consists of N input-output pairs, of
signals y,(rc) = T[xj(n)], / = 1, 2........N.
Discrete-Time Signals and Systems
138
Chap. 2
(a) What is the class of input signals for which we can determ ine the output, using
the information above, if the system is known to be linear?
(b) The sam e as above, if the system is known to be tim e invariant.
2.13 Show that the necessary and sufficient condition for a relaxed LTI system to be BIBO
stable is
y .
i/i(«)i < Mh < oo
for some constant Mn.
2.14 Show that;
(a) A relaxed linear system is causal if and only if for any input x(n) such that
for n < no
jc(n) = 0 for n < no => y(n) = 0
(b) A relaxed LTI system is causal if and only if
h(n) = 0
for n < 0
2.15 (a) Show that for any real or complex constant a, and any finite integer num bers M
and N, we have
n
a M —a h
. if « * !
1 — <3
N - M + 1,
if a = 1
(b) Show that if |o| < 1 , then
y v
= - i -
l - a
2.16 (a) If y(n) = x (n) * h(n). show that £ v =
£ / .- where ^
_w
(b) Compute the convolution y(n) = x(n) * h(n) of the following signals and check
the correctness of the results by using the test in (a).
(1) jr(n) — (1,2, 4), /)(n) = (1 ,1 ,1 ,1 ,1 }
(2) x(n) = {1, 2, - 1 ) , h(n) = x (n)
(3) x(n) = (0,1, - 2 , 3. - 4 ) . h(n) = {£, i , 1, 1}
(4) jc(n )= :{1 .2 .3.4.5J.A (n) = {l)
(5) x(n) = (1, -2 ,3 } , h(n) = (0, 0 .1 .1 ,1 ,1 )
t
t
(6) x(n) = { 0 ,0 ,1 ,1 ,1 ,1 ), h(n) = { 1 ,-2 . 3}
t
t
(7) jr(u) = {0,1, 4, -31. h(n) = [1,0, - 1 , -1}
t
t
(8)
= [1,1,2], h(n) = u(n)
t
(9) jt(n) = [1,1. 0,1,11, h(n) = {1, - 2 , - 3 , 4}
t
t
(10) jc(n) = (1,2, 0,2 , l}/i(n) = Jt(n)
t
(11) *{n) = (i)"u(n), h(rt) = ( j ) nM(n)
2.17 Compute and plot the convolutions x(n) * h(n) and h(n) *x(n) for the pairs of signals
shown in Fig. P2.17.
Chap. 2
139
Problems
trln)
b TI f 0 12 3
0 l 2 3 4 5 b
n
■Iiit.
1 2 3
-3-2-10 I 2 3
n
hin)
x(n)
i
n
h\n)
j x(n)
..III!..
3 4 5 6
II
n
htnl
J (n )
]
111
!
II
—2-1
2 3 4 5
<di
2.18
Figure P2.17
Determ ine and sketch the convolution y(n) of the signals
-V(/l) =
h(n) -
0.
0 < Ji < 6
elsewhere
1,
0.
—2 < n < 2
elsewhere
(a) Graphicallv
(b) Analytically
2.19 Compute the convolution y(n) of the signals
x (n) =
h(n) =
o'".
—3 < n < 5
0.
elsewhere
1,
0,
0< n< 4
elsewhere
2.20 Consider the following three operations.
(a) Multiply the integer numbers: 131 and 122.
(b) Compute the convolution of signals: {1. 3.1) * (1,2. 2}.
(c) Multiply the polynomials: 1 4- 3; + z2 and 1 4- 2z 4- 2z2.
(d> Repeat part (a) for the numbers 1.31 and 12.2.
(e) Comment on your results.
2.21 Com pute the convolution y(n) = x ( n ) * h(n) of the following pairs of signals.
(a) x(n) = a"u(n), h(n) = b"u{n) when a ^ b and when a ~ b
(b) x ( n) =
1.
2,
n = —2, 0, 1
n = —1
. 0,
elsewhere
h ( n ) = S(n) — S (n — 1) + S(n — 4) + S(n —5)
140
Discrete-Time Signals and System s
Chap. 2
(c) x{n) = u(n + 1) —u(n —4) —<5(n —5)
h(n) = [u(n +2) — u(n - 3)] • (3 - |n|)
(d ) x ( n ) = u ( n ) - u( n - 5)
h(n) = u(n —2) —u(n —8) 4- u(n — 11) —u(n — 17)
2.22 Let x(n) be the input signal to a discrete-time filter with impulse response ht(n) and
let y,(n) be the corresponding output.
(a) Compute and sketch x(n) and \ j ( n ) in the following cases, using the same scale
in all figures.
x(n) = {1,4, 2. 3, 5, 3, 3. 4. 5. 7. 6. 9}
h](n) = (1,1)
h2(n) = {1,2.1}
]ij(n) = {i, j)
*4(») = {?• {■ j)
Sketch x(n), yi(n), y 2(n) on one graph and *(«). y3(n), y,j(n), y.s(n) on another
graph
(b) W hat is the difference between yi(«) and \’2(n). and between y^(n) and y^(n)?
(c) Comment on the smoothness of v2(/?) and v4(n). Which factors affect the sm ooth­
ness?
(d) Compare y4(n) with ysfn). What is the difference? Can you explain it?
(e) Let h(,(n) = {^, - j } . Com pute y’6<n). Sketch v(n), y 2(n), and yft(n) on the same
figure and comment on the results.
2.23 The discrete-time system
v(n) = ny(n — 1) + jr(n)
n > 0
is at rest [i.e., v(—1) = 0]. Check if the system is linear time invariant and BIBO stable.
2.24 Consider the signal y(n) = a"u(n), 0 < a < 1.
(a) Show that any sequence x{n) can be decomposed as
and express ck in terms of x(n).
(b) Use the properties of linearity and time invariance to express the output y(n) =
T[x(n)] in term s of the input x (n) and the signal g(n) = T[y(n)], where T [ ] is
an LTI system.
(c) Express the impulse response h(n) = T[B{n)} in terms of g(rt).
2.25 D eterm ine the zero-input response of the system described by the second-order dif­
ference equation
x(n) - 3y(n - 1) - 4y(n - 2) = 0
2.26 Determ ine the particular solution of the difference equation
y(n) = jv(ii - 1) - £y(n - 2) +x ( n)
when the forcing function is x(n) = 2"u(n).
Chap. 2
Problems
141
2 2 1 D eterm ine the response yin). n > 0. of the system described by the second-order
difference equation
yin) - 3v(n —1) —4y(/i —2) = xin) + 2x(n — 1)
to the input xin) = 4"w(n).
2.28 Determ ine the impulse response of the following causal system:
y(n) —3y(n — 1) —4v(n —2) = jr(n) + 2x(n — 1)
2.29 Let xin). A'i < n < N2 and h(n),
< n < M2 be two finite-duration signals.
(a) Determ ine the range L\ < n < L 2 of their convolution, in term s of N\, N2, M\
and M2.
(b) Determ ine the limits of the cases of partial overlap from the left, full overlap,
and partial overlap from the right. For convenience, assume that h(n) has shorter
duration than jc(«).
(c) Illustrate the validity of your results by computing the convolution of the signals
-2 < n < 4
elsewhere
-1 < n < 2
elsewhere
2.30 Determ ine the impulse response and the unit step response of the systems described
by the difference equation
(a) yin) = ().6y(;i - 1) - ().08v(n - 2) + xin)
(b) _v(« ) = 0.7y(;; - 1) - 0.1 yin - 2) -f 2xin) - xin - 2)
231 Consider a svstem with impulse response
A,«) = H r '
( 0.
° - n - 4
elsewhere
Determ ine the input xin) for 0 < n < S that will generate the output sequence
v(n) = 1 1 .2 .2 .5 .3 .3 .3 .2 .1 .0 ....}
t
232 Consider the interconnection of LTI systems as shown in Fig. P2.32.
(a) Express the overall impulse response in terms of h \ (n), h2(n), h^in). and h^in).
(b) D eterm ine h{n) when
M « ) = {j. 3 . 7 }
h2{n) — hy(n) = (n + 1 )u(n)
fi4(n) = S(n — 2)
Figure P 2J2
Discrete-Time Signals and Systems
142
Chap. 2
(c) Determ ine the response of the system in part (b) if
x(n) = &(n + 2) + 3S(n - 1) - 4 S(n - 3)
2 3 3 Consider the system in Fig. P2.33 with h(n) = a"u(n), —1 < a < 1. Determ ine the
response y(n) of the system to the excitation
*(n) = «(n + 5) —u(n — 10)
x(n)
Figure P233
2 3 4 Com pute and sketch the step response of the system
U_ 1
2 3 5 Determ ine the range of values of the param eter a for which the linear time-invariant
system with impulse response
(Hint: The solution can be obtained easily and quickly by applying the linearity and
tim e-invariance properties to the result in Exam ple 2.3.5.)
2 3 7 D eterm ine the response of the (relaxed) system characterized by the impulse response
h(n) = ( l ) ”u(n)
to the input signal
| 1,
x(n) = { ^
10,
0 < n < 10
~
otherwise
2 3 8 D eterm ine the response of the (relaxed) system characterized by the impulse response
h(n) = ( j ) Hu{n)
to the input signals
(a) x(n) = 2nu(n)
(b) x(n) = u ( - n )
Chap. 2
143
Problems
239 T hree systems with impulse responses h\(n) — 5(n) — &(n — 1). h2(rt) = h( n}. and
= u(n), are connected in cascade.
(a) W hat is the impulse response.
of the overall system?
(b ) Does the order of the interconnection affect the overall system?
2.40 (a) Prove and explain graphically the difference between the relations
x(n )5(n —no) =
—«o)
x(n) *&(n - n0) = x(n - nu)
and
(b ) Show that a discrete-time system, which is described by a convolution summation,
is LTI and relaxed,
(c) W hat is the impulse response of the system described by y(n) = x(n —n())?
2.41 Two signals 5(n) and u(n) are related through the following difference equations
j(n) + a\ j(n — 1) + ■
■
— N) = bi)v(n)
Design the block diagram realization of:
(a) The system that generates x(n) when excited by v(n).
(b ) The system that generates u(n) when excited by s(n).
(c) What is the impulse response of the cascade interconnection of systems in parts
(a) and (b)?
2.42 Com pute the zero-state response of the system described by the difference equation
y(n ) + ^ v(n — 1) = x(n ) + 2x{n —2)
to the input
xin) = (1.2. 3. 4, 2, 1)
T
by solving the difference equation recursively.
2 .43 Determ ine the direct form II realization for each of the following LTI systems.
(a) 2v(n) + y(n — 1 ) - 4 v(n — 3) = x(n) + 3x(n —5)
(b) y(n) = Jr(n) —x(n — 1) + 2x(n — 2) — 3x(n - 4)
2 .4 4 Consider the discrete-time system shown in Fig. P2.44.
Figure P2.44
(a) Com pute the 10 first samples of its impulse response.
(b ) Find the input-output relation.
(c) Apply the input x(n) = { 1 .1 .1 ....} and com pute the first 10 samples of the output,
t
144
Discrete-Time Signals and Systems
Chap. 2
(d) Com pute the first 10 samples of the output for the input given in part (c) by using
convolution.
(e) Is the system causal? Is it stable?
2 ^ 5 Consider the system described by the difference equation
y(n) = ay(n - 1) + bx(n)
(a) Determ ine b in terms of a so that
(b ) Com pute the zero-state step response s(n) of the system and choose b so that
j(oo) = 1.
(c) Com pare the values of b obtained in parts (a) and (b). W hat did you notice?
2*46 A discrete-time system is realized by the structure shown in Fig. P2.46.
(a) D eterm ine the impulse response.
(b) Determ ine a realization for its inverse system, that is, the system which produces
x( n) as an output when y(n) is used as an input.
x in )
-o
-0
■v(n)
0.8
Figure P2.46
2 .4 7 Consider the discrete-time system shown in Fig. P2.47.
v(n)
Figure P2^f7
(a) Com pute the first six values of the impulse response of the system.
(b ) Com pute the first six values of the zero-state step response of the system.
(c) Determ ine an analytical expression for the impulse response of the system.
1 4 8 D eterm ine and sketch the impulse response of the following systems for n — 0,
1........9.
(a) Fig. P2.48(a).
(b ) Fig. P 2 .4 8 (b ).
(c) Fig. P2.48(c).
Chap. 2
Problems
145
Cc)
Figure P2.48
(d) Classify the systems above as FIR or IIR.
(e) Find an explicit expression for the impulse response of the system in part (c).
2.49 Consider the systems shown in Fig. P2.49.
(a) Determ ine and sketch their impulse responses /i|(n), h2(n), and h3(n).
(b) Is it possible to choose the coefficients of these systems in such a way that
h\(n) = h2(n) = h3(n)
2.50 Consider the system shown in Fig. P2.50.
(a) Determ ine its impulse response h(n).
(b) Show that h(n) is equal to the convolution of the following signals.
h] (n) = 6(n) + 6(n - 1)
M " ) = (^)"u(n)
Discrete-Time Signals and Systems
146
Chap. 2
v (n )
yin)
2.51 Com pute the sketch the convolution y,(n) and correlation r,(n) sequences for the
following pair of signals and comment on the results obtained.
(a) *,{«) = (1.2.4)
A ,(n )= (1 ,1 .1 .1 . If
t
t
(b) x2(n) = (0,1. - 2 . 3, - 4 ]
h2(n) = U. 1. 2 . 1 , i}
t
(c) jt-j(b ) = (1.2. 3, 4}
t
(d) x4(n) = {1. 2, 3,4)
t
‘
A3 (u )
t
= (4. 3. 2, 1)
t
hA(n) = (1.2.3. 4)
t
2.52 The zero-state response of a causal LTI system to the input x ( n ) = {1,3, 3,1) is
y(n) = (1,4, 6 ,4 ,1 ). Determ ine its impulse response.
t
Chap. 2
Problems
147
2.53 Prove by direct substitution the equivalence of equations (2.5.9) and (2.5.10), which
describe the direct form II structure, to the relation (2.5.6), which describes the direct
form I structure.
2.54 D eterm ine the response y(n). n > 0 of the system described by the second-order
difference equation
y ( n) — 4 y(n - 1) + 4v(/i — 2) = x( n) — x( n — 1)
when the input is
x(n) = (-l)" u (n )
and the initial conditions are v(—1) = y ( -2 ) = 0.
2.55 D eterm ine the impulse response h(n) for the system described by the second-order
difference equation
v(ni — 4v(;i — 1 > + 4y(n — 2) = x(r?) — x(n — 1)
2.56 Show that any discrete-tim e signal x(n) can be expressed as
[.v(A') —x(k — 1)]u(n - k)
c(n) =
where «(/i - k ) is a unit step delayed by k units in time, that is,
u(/i -
1.
I 0,
k) =
.
n >k
otherwise
2.57 Show that the output of an LTI system can be expressed in term s of its unit step
response v(n) as follows.
i'(n) = y " ' [,v(A:) —x(k — 1)]jr(fl —k)
=
Y
[x(AQ - x ( k - l) ] s ( /i - k)
c
2.58 Com pute the correlation sequences rIX(l) and rtv(l) for the following signal sequences.
j _
P ■ nu - N < n < n {, + N
I 0,
f 1.
v(n)
=
1
„
10,
otherwise
-N <n < N
,
otherwise
2.59 D eterm ine the autocorrelation sequences of the following signals.
(a) x(n) = {1. 2.1.1)
t
(b) v(n) = il. 1.2.1}
t
W hat is your conclusion?
2.60 W hat is the normalized autocorrelation sequence of the signal x(n) given by
1,
x(n) = , „
0,
-jV < n < N
otherwise
Discrete-Time Signals and Systems
148
Chap. 2
2.61 An audio signal j(r) generated by a loudspeaker is reflected at two different walls
with reflection coefficients r ] and r2. The signal *(/) recorded by a microphone close
to the loudspeaker, after sampling, is
x (n) = s(n) + rxs(n — k\) + r2s(n — k2)
where kt and k2 are the delays of the two echoes.
(a) Determ ine the autocorrelation rzx(I) of the signal x(n).
Can we obtain ri, r2, k\, and k2 by observing r,s (1)1
(c) W hat happens if r2 = 0?
(b)
2.62* Time-delay estimation in radar Let xa(t) be the transm itted signal and yfl(r) be the
received signal in a radar system, where
y„(r) = axa(t - id) + vu(f)
and va(t) is additive random noise. The signals xa(t) and >■„(/) are sampled in the
receiver, according to the sampling theorem , and are processed digitally to deter­
mine the time delay and hence the distance of the object. The resulting discrete-time
signals are
jr(n) = xa(nT)
y(n) = y„(nT) = axu(nT - DT) + vu(nT)
= ax(n — D) + u(n)
(a) Explain how we can measure the delay D by computing the crosscorrelation r*,.(/).
(b) Let x(n) be the 13-point Barker sequence
X (n) =
(+ 1,+1,+1,+1,+1, -1, -l.+l.+l, -1,+1. -1,+1)
and u(n) be a Gaussian random sequence with zero mean and variance a 2 = 0.01.
Write a program that generates the sequence v(n), 0 < n < 199 for a = 0.9 and
D = 20. Plot the signals jt(«), y(n), 0 < n < 199.
(c) Compute and plot the crosscorrelation rTV(/), 0 < / < 59. Use the plot to estimate
the value of the delay D.
(d) Repeat parts (b) and (c) for a 2 = 0.1 and a 2 = 1.
<e) Repeat parts (b) and (c) for the signal sequence
jt(n) = j _ l , _ l , - l , + i , + i , + i . + i , - i ,
+ 1 . - l . + l . + l , - 1 ,- 1 ,+ 1 }
which is obtained from the four-stage feedback shift register shown in Fig. P2.62,
Figure P2.61
register.
Linear feedback shift
Chap. 2
Problems
149
Note that x(n) is just one period of the periodic sequence obtained from the
feedback shift register.
(f) Repeat parts (b) and (c) for a sequence of period N — 27 — 1, which is obtained
from a seven-stage feedback shift register. Table 2.3 gives the stages connected
to the modulo-2 adder for (maximal-length) shift-register sequences of length
N =2" —
TABLE 2.3 SHIFT-REGISTER
CONNECTIONS FOR GENERATING
MAXIMAL-LENGTH SEQUENCES
m
Stages Connected to Modu)o-2 Adder
1
1
2
1. 2
1. 3
1. 4
S. 4
L6
1. 7
1, 5, 6. 7
1.6
1. K
1. 10
i. 7. y, 12
1. H), 11. 13
1. 5. 9. 14
1. 15
1. 5. 14, 16
1. 15
3
4
5
6
7
H
y
id
li
12
13
14
15
16
17
2.63* Implementation o f L T I systems
by the difference equation
Consider the recursive discrete-time system described
_v(n) = —U\ v( n — 1) —a;v (n —2) 4- b^x(rt)
where a\ - —0.8, u? = 0.64. and b() = 0.866.
(a) Write a program to compute and plot the impulse response h{n) of the system
for 0 < n < 49.
(b) Write a program to com pute and plot the zero-state step response s(n) of the
system for 0 < n < 100.
(c) Define an F IR system with impulse response ^ fir (h ) given by
,
, .
1 h(n),
10.
0 < n < 19
elsewhere
where h(n) is the impulse response computed in part (a). W rite a program to
compute and plot its step response.
(d) Com pare the results obtained in parts (b) and (c) and explain their similarities
and differences.
150
Discrete-Time Signals and Systems
Chap. 2
2jS4* W rite a com puter program that computes the overall impulse response h(n) of the sys­
tem shown in Fig. P2.64 for 0 < n < 99. The systems TU T2, T 3, and % are specified by
Ti : hi(n) = {1 . 5 .
t
g.
55)
T2 : h 2(n) = {1,1,1,1,11
t
Ts :
+ 5*(" - 1) +
- 2)
T4 : y(n) = 0.9y(n — 1) —0.81y{n —2) + v(n) + u(n —1)
Plot h(n) for 0 < n < 99.
Figure P2.64
3
The Z -Transform and Its
Application to the Analysis of
LTI Systems
T ra n s fo rm te c h n iq u e s a re an im p o rta n t tool in the analysis o f signals a n d lin­
e a r tim e -in v a ria n t (L T I) system s. In this c h a p te r w e in tro d u c e th e ^ -tran sfo rm ,
d ev elo p its p ro p e rtie s , and d e m o n s tra te its im p o rta n c e in th e analysis an d c h a ra c ­
te riz a tio n o f lin ear tim e -in v a ria n t system s.
T h e : -tra n sfo rm plays th e sam e role in th e analysis o f d isc re te -tim e signals
an d L T I sy stem s as th e L ap lace tran sfo rm d o e s in th e analysis o f c o n tin u o u s-tim e
signals a n d L T I svstem s. F o r ex am p le, w e sh a d see th a t in th e ^ -d o m ain (com plex
z -p lan e) th e c o n v o lu tio n of tw o tim e-d o m ain signals is e q u iv a le n t to m u ltip lic atio n
o f th e ir c o rre sp o n d in g ^ -tran sfo rm s. T h is p ro p e rty g reatly sim plifies the analysis
o f th e re sp o n s e o f an L TI system to v ario u s signals. In ad d itio n , th e c-tran sfo rm
p ro v id es us w ith a m ean s of ch a ra c te riz in g an L T I system , a n d its resp o n se to
v ario u s signals, by its p o le - z e r o locations.
W e b eg in th is c h a p te r by defining th e c-tran sfo rm . Its im p o rta n t p ro p e rtie s
a re p re s e n te d in S ectio n 3.2. In S ection 3.3 the tra n sfo rm is u se d to ch aracterize
signals in te rm s o f th e ir p o le - z e r o p a tte rn s. S ection 3.4 d escrib es m e th o d s fo r
in v ertin g th e z-tra n sfo rm o f a signal so as to o b ta in th e tim e -d o m a in re p re s e n ta ­
tio n o f th e signal. T h e o n e-sid ed :-tra n s fo rm is tr e a te d in S ectio n 3.5 a n d used
to solve lin e a r d ifferen ce e q u a tio n s w ith n o n z e ro in itial co n d itio n s. T h e c h a p te r
c o n clu d es w ith a d iscussion o n th e use of th e z -tra n sfo rm in th e analysis o f L TI
system s.
3.1 THE Z-TRANSFORM
In th is sectio n w e in tro d u c e th e z -tra n sfo rm of a d isc re te -tim e signal, in v estig ate
its c o n v e rg e n c e p ro p e rtie s , an d briefly discuss th e in v erse z-tran sfo rm .
151
152
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
3.1.1 T h e D ire c t z - T r a n s f o r m
T h e ^ -tran sfo rm o f a d isc re te -tim e signal jt(fi) is d efin ed as th e p o w e r series
OC
X(z) =
n= —oo
x { n ) z ~'‘
(3-u )
w h ere z is a co m p lex v ariab le. T h e re la tio n (3.1.1) is so m e tim e s called th e direct
z - t ransform b ecau se it tra n sfo rm s th e tim e -d o m a in signal x ( n ) in to its co m p lex p la n e re p re s e n ta tio n X (z). T h e in v erse p ro c e d u re [i.e., o b ta in in g x ( n ) fro m X (z)]
is called th e i nverse z- t ransf orm a n d is e x a m in e d briefly in S e ctio n 3.1.2 a n d in
m o re d etail in S ectio n 3.4.
F o r c o n v en ien ce , th e z -tra n sfo rm o f a signal x ( n ) is d e n o te d by
X (z) - Z U (« )}
(3.1.2)
w h ereas th e re la tio n sh ip b e tw e e n x ( n ) a n d X ( z ) is in d ic a te d by
jc(«)
< -^ * (;)
(3.1.3)
S ince th e z-tra n sfo rm is an infinite p o w e r se ries, it exists only fo r th o se v alu es of
z fo r w hich th is se ries co n v erg es. T h e region o f conver gence ( R O C ) o f X (z) is th e
set o f all v alu es o f z fo r w hich X ( z ) a tta in s a finite value. T h u s an y tim e w e cite
a z-tra n sfo rm w e sh o u ld also in d icate its R O C .
W e illu strate th e se co n c e p ts by so m e sim ple ex am p les.
Example 3.1.1
D eterm ine the ^-transforms of the following finite-duration signals.
(a) jf,(n) = (1 ,2 .5 .7 ,0 ,1 }
(b) jf■,(«) = ( I . 2 . 5 . 7 . 0 . 1)
t
(c) jc3(n) = (0 ,0 ,1 ,2 , 5, 7,0.1}
(d) *4(h) = (2 .4 ,5 .7 .0 ,1 )
t
(e) x j ( n ) = S( n)
(f) x$(n) = <S(n —k), k > 0
(g) x j ( n ) = &(n + k) , k > 0
Solution
From definition (3.1.1), we have
(a) X](z) = 1 + 2z~' + 5z~2 + 7 z '3 + z~5, ROC: entire z-plane except z = 0
(b) X 2(z) = z2 + 2z + 5 + 7c-1 + z-3, ROC: entire z-plane except z = 0 and z = oo
(c) Xj(z) = z~2 + 2z-3 + 5z-4 + 7z-5 -I- z-7, ROC: entire z-plane except z = 0
(d) X4(z) = 2z2 -I- 4z -I- 5 4- 7z_1 -I- z-3, ROC: entire z-plane except z = 0 and z = oo
(e) X;(z) = l[i.e„ S(n)
*■ 1], ROC: entire z-plane
(f) Xb(z) = z_ t[i.e„ &(n — k) «— ►z_t], k > 0, ROC: entire z-plane except z = 0
(g) Xy(z) = zk[i.e„ &(n + k)
z*], k > 0, ROC: entire z-plane except z = oo
Sec. 3.1
153
The z-Transform
F ro m th is ex a m p le it is easily se en th a t th e R O C o f a f i ni t e-durat i on signal
is th e e n tire ;- p la n e , e x cep t p ossibly th e p o in ts z = 0 a n d /o r z — oo. T h e se p o in ts
a re e x c lu d ed , b e c a u s e z t (k > 0} b eco m es u n b o u n d e d fo r z = oc an d z ~ k01 > 0)
b ec o m e s u n b o u n d e d fo r z = 0.
F ro m a m a th e m a tic a l p o in t of view th e z-tra n sfo rm is sim p ly an a lte rn a tiv e
r e p re s e n ta tio n o f a signal. T h is is nicely illu stra te d in E x a m p le 3.1.1, w h e re w e
see th a t th e co effic ie n t o f z~", in a given tra n sfo rm , is th e v a lu e o f th e signal at
tim e n. In o th e r w o rd s, th e e x p o n e n t o f z co n tain s th e tim e in fo rm a tio n w e n eed
to id en tify th e sa m p le s o f th e signal.
In m an y cases w e can ex p ress th e sum of th e finite o r infinite se rie s fo r th e
z-tra n s fo rm in a c lo sed -fo rm e x p ressio n . In such cases th e z -tra n s fo rm o ffers a
co m p act a lte rn a tiv e r e p re s e n ta tio n of th e signal.
Example 3.1.2
D eterm ine the z-transform of the signal
Jf(n) = (5
Solution
The signal jc(n) consists of an infinite num ber of nonzero values
x(n)= ( l . a u ^ . U )'1....
The z-transform of x(n) is the infinite power series
X( z ) = 1 + U ” ' + ( ^) 2z - 2 + (| ) "z"" + ---
ft=<)
nail
This is an infinite geometric series. We recall that
1 + A + ,42 + A3 - t - - - - = —
1 —A
Consequently, for | 1
if | A 1 < 1
< 1, or equivalently* for \z\ >
X(z) =
J
X(z) converges to
ROC: (zl > |
jZ
We see that in this case, the z-transform provides a compact alternative representation
of the signal x(n).
L e t us e x p ress th e co m p lex v a ria b le z in p o la r fo rm as
z = r e }6
w h ere r = |z| a n d 6 = i^z- T h e n X ( z ) can be e x p ressed as
OC
X ( z ) \ t- r ' » = Y
x ( n ) r ~ ne - j en
n*-00
(3.1.4)
154
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
In th e R O C o f X U ). |X ( ;) | < oc. B ut
(3.1.5)
5
\ M n ) r - ne - ' * n \ =
£
\ x( n) r ~n\
H e n c e |X (z)| is finite if th e se q u en ce x ( n ) r ~ ' ’ is a b s o lu te ly su m m a b le .
T h e p ro b le m o f finding th e R O C fo r X ( z ) is e q u iv a le n t to d e te rm in in g the
ra n g e o f v alu es o f r fo r w hich th e se q u e n c e x ( n ) r ~ " is a b so lu te ly su m m ab le. T o
e la b o ra te , let us e x p ress (3.1.5) as
(3.1.6)
If X ( z ) co n v erg es in so m e region of the co m p lex p la n e , b o th su m m a tio n s in (3.1.6)
m u st be finite in th a t region. If the first sum in (3.1.6) co n v erg es, th e re m ust exist
v alu es o f r sm all e n o u g h such th at the p ro d u c t se q u e n c e x ( —n) r" . 1 < /; < oc, is
a b s o lu te ly su m m ab le. T h e re fo re , the R O C for th e first sum co n sists o f all p o in ts
in a circle of so m e rad iu s r^, w here /■] < oc, as illu stra te d in Fig. 3.1a. O n the
o th e r h a n d , if th e seco nd sum in (3.1.6) co n v erg es, th e re m u st exist v alu es o f r
larg e en o u g h such th a t th e p ro d u c t se q u e n c e x ( n ) / r " . 0 < n < oc, is a b so lu te ly
su m m ab le. H en ce th e R O C fo r the se co n d sum in (3.1.6) co n sists o f all p o in ts
o u tsid e a circle o f rad iu s r > r2. as illu stra te d in Fig. 3.1b.
Since th e co n v erg en ce of X (c) re q u ire s th a t b o th sum s in (3.1.6) b e finite, it
follow s th a t th e R O C o f X ( z ) is g en erally specified as th e a n n u la r region in th e
;- p la n e , r: < r < r\. w hich is the co m m o n reg io n w h e re b o th su m s are finite. T his
reg io n is illu stra te d in Fig. 3.1c. O n th e o th e r h a n d , if
> r\, th e r e is no co m m o n
reg io n o f co n v erg en ce fo r th e tw o sum s an d h en ce X ( ;) d o es n o t exist.
T h e follow ing ex am p les illu strate th e se im p o rta n t co n cep ts.
Example 3.1.3
Determ ine the e-transform of the signal
Solution
From the definition (3.1.1) we have
If \az 11 < 1 or equivalently, |z| > |a |, this power series converges to 1/(1 - a ; -1).
Sec. 3.1
The z-Transform
155
Im(z)
Im(;)
Im(z)
Figure 3.1 R egion of convergence for
X (z) and its corresponding causal and
anticausa! components.
156
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
lm(;)
Figure 3.2 The exponential signal xi n) = txnu{n) (a), and the ROC of its
transform (b).
Thus we have the z-transform pair
x(n) =a" u( n )
X (;) =
1
ROC: |z| > |cr|
1 - a ;-1
(3.1.7)
The R O C is the exterior of a circle having radius |a |. Figure 3.2 shows a graph of the
signal .x(n) and its corresponding ROC. Note that, in general, or need not be real.
If we set or = 1 in (3.1.7), we obtain the z-transform of the unit step signal
x(n) - u(n)
X(z) =
1
ROC: |zl > 1
1
(3.1.6
Example 3.1.4
Determ ine the z-transform of the signal
x(n) = —a"u( —n — 1) =
Solution
n >0
n < -1
0,
From the definition (3.1.1) we have
-I
oc
n—---oc
/= i
where / = —n. Using the formula
A + A: + A3 + • ■■= A(1 + A + A2 + ■■■) =
A
1 - A
when | >1f < 1 gives
* ( ;) = - -
1
1 —a ~ lz
l - a ; '1
provided that |cr_1z| < 1 or, equivalently, |z| < jar j. Thus
1
(3.1.9)
ROC: [z| < |a |
1-azThe R O C is now the interior of a circle having radius |a|. This is shown in Fig. 3.3.
x(n) = —a"u( —n — 1)
X(Z) = -
Sec. 3.1
The z-Transform
157
Im(c)
Figure 3.3 Anticausal signal jt(h) = -crnu( —n - 1) (a), and the ROC of its
transform (b).
E x a m p le s 3.1.3 an d 3.1.4 illu stra te tw o very im p o rta n t issues. T h e first c o n ­
cern s th e u n iq u e n e s s o f th e "-tran sfo rm . F ro m (3.1.7) an d (3.1.9) we see that
th e cau sal signal a nu ( n ) a n d th e an ticau sal signal —a " u ( —n — 1) have id en tical
clo sed -fo rm e x p re ssio n s fo r th e ^ -tran sfo rm , th a t is,
Z ( o " w ( « ) ) = Z { —a nu ( —n -
1) ) -
------ --------
1 —a c ' 1
T h is im p lies th a t a clo sed -fo rm e x p ressio n fo r th e z-tra n sfo rm d o e s n o t u n iq u ely
specify th e signal in th e tim e do m ain . T h e am b ig u ity can b e reso lv ed only if
in a d d itio n to th e c lo sed -fo rm ex p ressio n , th e R O C is specified. In su m m ary , a
discrete-time signal x ( n ) is uni quel y d et er mi ned b y its z- t rans f orm A' (;) a n d the
region o f conver gence o f X( z ) . In this te x t th e te rm “z -tra n s fo rm " is u se d to re fe r
to b o th th e clo sed -fo rm e x p ressio n an d th e c o rre sp o n d in g R O C . E x a m p le 3.1.3
also illu stra te s th e p o in t th a t the R O C o f a causal signal is the exterior o f a circle
o f s o m e radius r 2 whi l e the R O C o f an ant icausal signal is the interior o f a circle o f
s o m e radi us rj. T h e fo llow ing e x am p le co n sid ers a se q u en ce th a t is n o n z e ro for
—00 < n < 00.
Example 3.1.5
D eterm ine the z-transform of the signal
x (n) = a"u(n) + bnu( —n — 1)
Solution
From definition (3.1.1) we have
b"z " =
X(z) =
n=0
n = —oc
+ Y ib -'z)1
n=0
i= l
The first pow er series converges if locz-11 < 1 or |z| > |a |. The second power series
converges if \b~xz\ < 1 or |z| < |6j.
In determ ining the convergence of XCz), we consider two different cases.
158
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
Case 1 |b| < |a |:
In this case the two R O C above do not overlap, as shown
in Fig. 3.4(a). Consequently, we cannot find values of z for which both power series
converge simultaneously. Clearly, in this case, X(c) does not exist.
Case 2 |f>| > |a |:
In this case there is a ring in the z-plane where both power
series converge simultaneously, as shown in Fig. 3.4(b). Then we obtain
X(z) =
1 —ccz 1
1 —bz 1
b —a
(3.1.10)
a + b — z — abz~l
The R O C of X(z) ts |cr| < \z\ < \b\.
T h is e x am p le show s th a t i f there is a R O C f o r an infinite durat i on two-si ded
signal, it is a ring ( annul ar region) in the z-plane. F ro m E x a m p le s 3.1.1, 3.1.3, 3.1.4,
an d 3.1.5. w e see th a t th e R O C of a signal d e p e n d s o n b o th its d u ra tio n (finite
or in fin ite) an d o n w h e th e r it is cau sal, a n tic a u sa l, o r tw o -sid ed . T h e se facts are
su m m a riz e d in T a b le 3.1.
O n e special case of a tw o -sid ed signal is a signal th a t h as infinite d u ra tio n
on th e rig h t sid e b u t n o t on th e left [i.e., x( n ) = 0 fo r n < « (l < 0], A sec­
on d case is a signal th a t has infinite d u ra tio n o n th e left side b u t n o t on the
•plane
Ifcl < tot
X(z) does not exisl
Im(;)
krl < IAI
ReU)
ROC for X(z)
Figure 3.4 R O C fo r z-transform in
E xam ple 3.1.5.
Sec. 3.1
159
The ^-Transform
TABLE 3.1
CH A R A C TE R IS TIC FAM ILIES O F SIGN ALS W IT H T H E IR
C O R R E S P O N D IN G ROC
Signal
ROC
Finite-Duration Signals
Two-sided
. . TTT l i t , —
n
0
Jnfinite-Duration
Causal
l l T t*
-
Anltcausal
T] T1
Two-sided
I . I t.....
rig h t [i.e., x{ n ) = 0 fo r n > n\ > 0]. A th ird sp ecial case is a signal th a t has
finite d u ra tio n o n b o th th e left a n d rig h t sides [i.e., x ( n ) = 0 fo r n < no < 0
a n d n > n\ > 0]. T h e se ty p e s o f signals a re so m e tim e s c alled right-sided, left­
sided, a n d finite-d uration two-sided, signals, resp ec tiv ely . T h e d e te rm in a tio n o f th e
R O C fo r th e s e th r e e ty p es o f signals is left as an e x ercise fo r th e r e a d e r (P r o b ­
lem 3.5).
F in ally , w e n o te th a t th e z -tra n sfo rm d efin ed b y (3.1.1) is so m e tim e s re fe rre d
to as th e tw o-sided o r bilateral z-transform , to d istin g u ish it fro m th e o ne-sided o r
160
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
unilateral z- t ransf orm given by
X~(z) = ^ * ( « ) z ~ "
(3.1.11)
T h e o n e-sid ed z -tra n sfo rm is ex am in ed in S ection 3.5. In this tex t w e use the
ex p ressio n z-tra n sfo rm exclusively to m ean th e tw o -sid ed z -tra n sfo rm d efined by
(3.1.1). T h e te rm ‘'tw o -sid e d ’' will b e u se d oniy in cases w h ere w e w an t to resolve
any am b ig u ities. C learly , if x ( n ) is causal [i.e., * ( 72) = 0 for n < 0], th e o n e-sid ed
and tw o -sid ed z-tran sfo rm s a re eq u iv a le n t. In any o th e r case, th e y a re d ifferen t.
3.1.2 The Inverse z-Transform
O ften , we h av e th e z-tra n sfo rm X ( z ) of a signal a n d w e m ust d e te rm in e th e signal
se q u en ce. T h e p ro c e d u re fo r tra n sfo rm in g from th e z-d o m ain to th e tim e dom ain
is called th e inverse z- t ransform. A n in v ersio n fo rm u la for o b ta in in g x( n) from
X (z) can b e d eriv ed by using th e Cau c hy integral t h e o r e m , w hich is an im p o rta n t
th e o re m in th e th e o ry o f co m p lex variables.
T o b egin, we h av e th e z-tra n sfo rm defin ed by (3.1.1) as
S u p p o se th a t we m u ltip ly b o th sides o f (3.1.12) by z"~' an d in te g ra te both sides
o v er a closed c o n to u r w ithin the R O C o f X ( z ) w hich en clo ses th e origin. Such a
c o n to u r is illu stra te d in Fig. 3.5. T h u s we have
(3.1.13)
w h e re C d e n o te s th e clo sed c o n to u r in th e R O C o f A'(z). ta k e n in a c o u n te rc lo c k ­
w ise d irectio n . Since th e se rie s c o n v erg es on th is c o n to u r, w e can in te rc h a n g e
th e o rd e r o f in te g ra tio n an d su m m atio n o n th e rig h t-h a n d side o f (3,1.13). T h u s
im (o
Figure 3.5
(3.1.13).
Contour C for integral in
Sec. 3.2
Properties of the z-Transform
161
(3.1.13) b eco m es
£ x ( z ) z n- ' d z =
t= —00
x ( k ) ( h z n- l~kd z
*
(3.1.14)
N o w w e can in v o k e th e C a u ch y in te g ra l th e o re m , w hich s ta te s th a t
ftM5)
w h ere C is an y c o n to u r th a t en clo ses th e origin. B y ap p ly in g (3.1.15), th e righth a n d side o f (3.1.14) re d u c e s to 2 n j x ( n ) a n d h en c e th e d e sire d in v e rsio n fo rm u la
x ( n ) = ^ - ^ X ( z ) z n_I d z
(3.1.16)
A lth o u g h th e c o n to u r in te g ra l in (3.1.16) p ro v id es th e d e s ire d in v ersio n fo r­
m u la fo r d e te rm in in g th e se q u e n c e * (« ) fro m th e z-tra n sfo rm , w e sh all n o t use
(3.1.16) d irectly in o u r e v a lu a tio n o f in v erse z -tran sfo rm s. In o u r tre a tm e n t w e d eal
w ith signals a n d sy stem s in th e z-d o m ain w hich h av e ra tio n a l z -tra n s fo rm s (i.e., ztra n sfo rm s th a t a re a ra tio o f tw o p o ly n o m ials). F o r such z -tra n sfo rm s w e d e v e lo p a
s im p ler m e th o d fo r in v ersio n th a t ste m s from (3.1.16) and e m p lo y s a ta b le lo o k u p .
3.2 PROPERTIES OF THE Z-TRANSFORM
T h e z -tra n sfo rm is a v ery p o w erfu l to o l fo r th e stu d y o f d isc re te -tim e signals an d
system s. T h e p o w e r o f th is tra n sfo rm is a c o n s eq u en ce o f so m e v ery im p o rta n t
p ro p e rtie s th a t th e tra n sfo rm possesses. In th is sectio n w e e x am in e som e o f th e se
p ro p e rtie s .
In th e tre a tm e n t th a t follow s, it sh o u ld b e re m e m b e re d th a t w h e n w e c o m b in e
sev eral z -tra n sfo rm s, th e R O C o f th e o v erall tra n sfo rm is, at least, th e in te rse c tio n
o f th e R O C o f th e in d iv id u al tra n sfo rm s. T h is will b eco m e m o re a p p a re n t la te r,
w h en w e discu ss specific ex am p les.
Linearity.
If
Jt,(n)
Xi ( z )
an d
x 2(n) < -U X 2(z)
th e n
x( n) = a \ x \ ( n ) + a 2x 2(n)
X( z ) = a j Xj ( z ) -h o2X 2(z)
(3.2.1)
fo r an y c o n s ta n ts a i a n d a 2. T h e p ro o f o f th is p ro p e rty follow s im m e d ia te ly from
th e d efin itio n o f lin e a rity a n d is left as an ex ercise fo r th e re a d e r.
T h e lin e a rity p ro p e rty can easily be g e n e ra liz e d fo r an a r b itra r y n u m b e r o f
signals. B asically , it im p lies th a t th e z -tra n sfo rm o f a lin e a r c o m b in a tio n o f signals
is th e sa m e lin e a r c o m b in a tio n o f th e ir z-tran sfo rm s. T h u s th e lin e a rity p ro p e rty
h e lp s u s to find th e z -tra n s fo rm o f a signal by ex p ressin g th e signal as a su m o f
e le m e n ta ry signals, fo r ea c h o f w hich, th e z -tra n sfo rm is a lre a d y k n o w n .
162
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
Example 3.2.1
Determ ine the z-transform and the R O C of the signal
*(«) = [3(2") -4 (3 ")]« (« )
Solution
If we define the signals
jti(«) = 2"u(n)
and
x2(n) = 3"u(n)
then jc(«) can be written as
x(n) = 3xi(n) — 4 x2(n)
According to (3.2.1), its z-transform is
X(;) = 3 X ,( c ) - 4 X 2(z)
From (3.1.7) we recall that
a'‘u(n)
1 - az
-
ROC: |z| > |a|
(3.2.2)
By setting a = 2 and or = 3 in (3.2.2). we obtain
Ai(n) = 2”u(n)
X,(z) = ^
x2(n) = 3nu(ri)
X 2(z) ~ ^ __■
ROC: |zj > 2
,
ROC: |zl > 3
The intersection of the ROC of X,(z) and X2(:) is |z| > 3. Thus the overall transform
X (z) is
ROC: |;| > 3
Example 3.2.2
Determ ine the z-transform of the signals
(a) x(n) = (COSo»on)u(n)
(b) x(n) = (sin w^n)u(n)
Solution
(a) By using E uler's identity, the signal .t(n) can be expressed as
x (n) = (cosa>on)u(n) = \ e Jaln"u(.n) + ^e~JUJn”u(n)
Thus (3.2.1) implies that
X( z ) = \ Z { e ^ u ( n ) ) + ^ { g - ^ u i n ) }
Sec. 3.2
Properties of the z-Transform
163
If we set a = e^^O ori = \e±i<u°\ = 1) in (3.2.2), we obtain
eJUV"u(fi)
---------------
R O C : |z[ > 1
e - ,wnnu(n) ^
--------— T T T
R O C : l; l >
and
1-
1
Thus
X(z) = -------- ----------h I ------- - ------r
2 1 —^ “f z -1
2 1 —e~J‘^lz~I
RO C : |;1 > 1
A fter some simple algebraic manipulations we obtain the desired result, namely,
:
1 — C_ ! COSW()
(coscO(}f?)u(n) <— ►-— -— ;---------------- 1 - 2z~1cos ton +
,
ROC: |-1 > 1
,, ,, ,
L r2.j)
(b) From E uler’s identity,
x(n) = (sino<o«)«(n) = ^-\eiun" u(n) — e~,w""u(»)]
2j
Thus
X(-) = ^ - f - ------1------- - ------- ------- - |
2j \ 1 —
r _1
1 —
ROC: |:| > 1
and finally.
z sm OH)
(sin coi,n)u(n) x - ^ ----- -— -----------------1 - 2z_1 cos wo + z~-
Time shifting.
ROC: |;| > 1
(3.2.4)
If
x( n)
X (c)
th e n
x{n - k ) z ~ kX{ z )
(3.2.5)
T h e R O C o f z ~kX ( z ) is th e sam e as th a t o f X ( z ) e x c e p t fo r j = 0 if k > 0 and
z = oo if k < 0. T h e p r o o f o f th is p ro p e rty follow s im m e d ia te ly fro m the definition
o f th e ^ -tra n sfo rm given in (3.1.1)
T h e p r o p e rtie s o f lin e a rity an d tim e shifting a re th e k ey fe a tu re s th a t m ake
th e z -tra n s fo rm e x tre m e ly u se fu l fo r th e analysis o f d isc re te -tim e L T I system s.
Exam ple 3*23
By applying the time-shifting property, determ ine the z-transform of the signals .T2(n,i
and xi(n) in Exam ple 3.1.1 from the Z'transform of Jti(n).
Solution
It can easily be seen that
x 2( n ) = * ](« + 2)
164
The ^-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
and
*i(n) = j:i(n - 2)
Thus from (3.2.5) we obtain
X2(z) = z2X, (z) = r + 2; + 5 + 7z ' 1 + z“3
and
X 3(z) = r 2X,(z) = z"2 + 2z"3 + 5z-4 + 7z"s + z-7
Note that because of the multiplication by z2, the R O C of X2(z) does not include the
point z = oc, even if it is contained in the R O C of A^i(z).
E x am p le 3.2.3 p ro v id es a d d itio n a l insight in u n d e rs ta n d in g th e m e an in g of
th e shiftin g p ro p e rty . In d e e d , if we recall th a t th e coefficient o f z ~ n is th e sam ple
v alu e at tim e n, it is im m e d ia te ly se en th a t d elay in g a signal by k( k > 0) sam ples
[i.e., x ( n )
x ( n — A')] c o rre sp o n d s to m u ltip ly in g all te rm s o f th e z -tra n sfo rm by
z~ k. T h e co efficien t o f z~" b ec o m e s th e coefficient o f ~~tn+k).
Example 3.2.4
Determ ine the transform of the signal
f 1,
' ‘" H o .
0 < n < jV - 1
„ „ ,
e lse w h e r e
<316>
Solution We can determ ine the z-transform of this signal by using the definition
(3.1.1). Indeed,
* -i
f N,
if z = l
X(z) = ^ l • z ^ = l + ; - ' + - - - + z - (Af- l l =
l - r *
-f . , ,
(3.2.7)
»*=<'
I 1 - z-1' 1
Since .v(n) has finite duration, its R O C is the entire z-plane, except z — 0.
Let us also derive this transform by using the linearity and time shifting prop­
erties. Note that x in) can be expressed in terms of two unit step signals
x(n) = u(n) —u(n — N)
By using (3.2.1) and (3.2.5) we have
X(z) = Z{u(n)} - Z{u(n - N)) = (1 - z"*')Z{u («)}
(3.2.8)
However, from (3.1.8) we have
Z[u(n)) = ^
ROC: jzl > 1
which, when combined with (3.2.8), leads to (3.2.7).
E x a m p le 3.2,4 h e lp s to clarify a v ery im p o rta n t issue re g a rd in g th e R O C
o f th e c o m b in a tio n o f se v era l z-tran sfo rm s. If th e lin ear c o m b in a tio n of several
sig n als h as finite d u ra tio n , th e R O C o f its z -tra n sfo rm is exclusively d ic ta te d by the
fin ite -d u ra tio n n a tu re o f th is signal, n o t by th e R O C o f th e in d iv id u al tran sfo rm s.
Scaling in the z-domain.
x{n)
If
X{z )
ROC: ri < [z| < r2
Sec. 3.2
165
Properties of the z-Transform
th en
a nx ( n )
►X (c ~ ’z)
R O C : \a\r\ < |z| < \a\r2
(3.2.9)
for a n y c o n s ta n t a, re a l o r com plex.
Proof. F ro m th e d efin itio n (3.1.1)
OC
Z{a" x( n) } =
Y
OC
a nx ( n ) z " =
^
x( n) ( a ‘z) n
\o\r i < I;I < \a\r2
T o b e tte r u n d e rsta n d th e m ean in g a n d im plicatio n s o f th e scaling p ro p e rty ,
w e e x p re ss a a n d z in p o la r form as a = rae->a", z = r e i<0, a n d w e in tro d u c e a new
co m p lex v a ria b le w = a~^z- T h u s Z { x ( n ) ) = X ( z ) an d Z{a" x( n) } = X (if). It can
easily b e seen th a t
T h is ch a n g e o f v a ria b le s re su lts in e ith e r sh rin k in g (if r 0 > 1) o r e x p a n d in g (if
r 0 < 1) th e z -p lan e in co m b in a tio n w ith a ro ta tio n (if too # 2i-jr ) o f th e z-p lan e
(see Fig. 3.6). T h is e x p lain s w hy w e h av e a ch an g e in th e R O C o f th e n ew tra n sfo rm
w h e re |a | < 1. T h e case \a\ = 1, th a t is, a = e^w" is o f special in te re st b e c a u se it
c o rre sp o n d s o n ly to ro ta tio n o f th e z-p lan e.
Exam ple 3.2.5
Determ ine the z-transforms of the signals
(a) x(n) = a"(cosu\in)u(n)
(b) x(n) = tf"(sinio»n)u(n)
r-planc
w-plane
Im(-)
(W-C^o
0
Figure 3.6
o _ 1 Z,
Re(z)
0
Re(w)
Mapping of the r-plane to the u -plane via the transformation ui =
a —roe^.
166
The z-Transform and Its Application to the Analysis of LTI S ystem s
Chap. 3
Solution
(a) From (3.2.3) and (3.2.9) we easily obtain
1 - a ; -1 cos wn
a
(C O S a i |,« ) u ( r t )
(3.2.10)
*-------*■ --------- ---------- ;--------------------------
1 — l a z r 1 co s ton -f a
(b) Similarly, (3.2.4) and (3.2.9) yield
a r -1 sin to<i
1 —2az~ cos to,, + a
a (sin wii/i)u(n) *— ► --- ---- ;---------
Time reversal.
|z! > la I
(3.2.11)
If
a (//) *-—> X ( z )
R O C : r\ <
< r;
th en
j r ( - n ) <-i-> X { z ~ ])
R O C : — < |z < r2
(3.2.12)
r\
P r o o f F ro m th e d efin itio n (3.1.1), w e have
Z{a (—/;)) =
Y
h= - x.
-x { - !i ) z~" — Y
■'‘ ( h ( z ~ t )~l — Ar (z~ ')
/= ->;
w h ere th e ch an g e o f v ariab le / = —n is m ad e. T h e R O C o f A '( ;_1) is
< |c —1f < r-i
o r e q u iv alen tly
— < ];| < —
r2
n
N o te th a t th e R O C fo r x( n) is the inverse o f th a t fo r x ( —n). T h is m ean s th a t if co
b elo n g s to th e R O C o f x( n) , th en l/" o is in th e R O C fo r x ( —n).
A n in tu itiv e p ro o f o f (3.2.12) is th e follow ing. W h en w e fold a signal, the
co efficien t o f z ~n b e co m es th e coefficient o f z n. T h u s, fo ld in g a signal is eq u iv alen t
to rep lacin g ; by
in th e z-tra n sfo rm fo rm u la. In o th e r w o rd s, reflectio n in the
tim e d o m ain c o rre sp o n d s to inversion in th e z-d o m ain .
Exam ple 3.2.6
Determ ine the z-transform of the signal
x i n ) = u( —n)
Solution
It is known from (3.1.8) that
u(n) <-U i
ROC: | z l > 1
By using (3.2.12), we easily obtain
u(—n)
*■ ------
Differentiation in the z-domain.
ROC: |z < 1
If
(3.2.13)
167
Properties of the z-Transform
Sec. 3.2
then
n x { n ) <— >- —z ^ ^ y —
dz
(3.2.14)
Proof . B y differentiating both sid es o f (3.1.1), w e have
dX{z)
n~—oc
n=—oc
— - z ~ l Z{nx(n)}
N o te that b oth transform s have the sam e RO C .
Example 3.2.7
D eterm ine the z-transform of the signal
jr(n) = na"u(n)
Solution The signal j:(n) can be expressed as njcitn), where Xj(n) = a"u(n). From
(3.2.2) we have that
jri(rt) = aKu(n) < - X,(z) = -— ----- 1 —az~'
ROC: |z| > \a\
Thus, by using (3.2.14), we obtain
n<j"ii(n)
dXji z)
a : '1
X(z) = - z — ^ - = ----------- r-r
az
(1 - az~' V
ROC: |z| > |a|
(3.2.15)
If we set a = 1 in (3.2.15), we find the z-transform of the unit ram p signal
2
ntt(n)
ROC: jz| > 1
(3.2.16)
Example 3.2.8
D eterm ine the signal x(n) whose z-transform is given by
X(z) = log(l -t- o z '1)
|z! > |o(
Solution By taking the first derivative of X(z), we obtain
dX{z)
- a z ~2
dz “ 1 + az~l
Thus
dX(z)
dz
= az
1 - (- a ) z ->
> \a]
The inverse z-transform of the term in brackets is (-a) ". The multiplication by
z _1 implies a time delay by one sample (time shifting property), which results in
( - a ) " _ 1u(n — 1). Finally, from the differentiation property we have
nx(n ) = a ( —a)n~lu(n — 1)
168
The z -Transfomn and Its Application to the Analysis of LTI Systems
Chap. 3
x ( n) = ( - 1 } " +1 — u(n - 1)
n
Convolution of two sequences.
If
Xi (rt)
X^z)
x 2(n)
X 2 {z)
th e n
x( n) = jfi(n) * x 2 (n)
X ( z ) = X \ ( z ) X 2 (z)
(3.2.17)
T h e R O C o f X (;) is, at least, th e in te rse c tio n of th a t fo r ATi(c) an d AS(z).
Proof. T h e co n v o lu tio n o f x i(n ) a n d x 2 (n) is d efin e d as
OC
x(n) =
Y
x i ( k ) x 2(n - k)
i' = —oc
T h e z-tra n sfo rm o f x( n ) is
AT{;) =
oc
oc
Y 2 x( n) z ~ " =
Y ,
ac
n—- x
n= - o c |_Jt = -cc
x \ ( k ) x 2(n - k)
U p o n in te rch an g in g th e o rd e r o f the su m m a tio n s a n d ap p ly in g th e tim e-sh iftin g
p ro p e rty in (3.2.5). we o b tain
X{z) =
Y
x 2(n — k) z "
x ' {k)
= X 2 (z) Y
^ ( k ) z ~ k = X 2 ( z ) X ](z)
Example 3.2.9
Compute the convolution x(rt) of the signals
jt,(n) = { 1 .-2 ,1 )
_ f 1.
0 < n < 5
2
10 ,
elsewhere
Solution From (3.1.1), we have
* ,( ; ) = 1 - 2 ; “ ' + z ~ 2
X 2(z) = I + z ~ ' + z~2 + j ' 3 + z"4 +
According to (3.2.17), we carry out the multiplication of X\(z) and X 2(z). Thus
X(z) = X 1 (z)X 1 (z) = 1 - z "1 - z ~6 + z "7
Hence
x ( n) = { 1 . - 1 . 0 , 0, 0, 0 , - 1 , 1 }
t
Sec. 3.2
Properties of the z-Transform
169
The same result can also be obtained by noting that
X,U) = (1 - ; ~ 'r
x 2w
1 - ; -6
= 3— p
Then
- z "6 + :" 7
X(z) = (1 - ; - ‘)(l - z~*) = 1 -
The reader is encouraged to obtain the same result explicitly by using the convolution
summation formula (tim e-domain approach).
T h e co n v o lu tio n p ro p e rty is o n e o f th e m o st p o w erfu l p ro p e rtie s o f th e ztra n sfo rm b ec a u se it c o n v e rts th e co n v o lu tio n o f tw o signals (tim e d o m a in ) to
m u ltip lic a tio n o f th e ir tra n sfo rm s. C o m p u ta tio n of th e c o n v o lu tio n o f tw o signals,
using th e z -tra n sfo rm , re q u ire s th e follow ing steps:
1. C o m p u te th e z -tra n s fo rm s o f th e signals to be co n v o lv ed .
X i(z) = Z{ x \ ( n ) \
(tim e d o m ain — *■ -.-dom ain)
X 2 (z) = Z { x 2 (n) ]
2 . M u ltip ly th e tw o z -tran sfo rm s.
X (z) = X ,(z )X 2(;)
(z-d o m ain )
3. F in d th e in v e rse z -tra n s fo rm o f X (z).
x ( n ) = Z _ , {X(z))
(z-d o m ain — ►tim e d o m a in )
T h is p ro c e d u re is, in m a n y cases, c o m p u ta tio n a lly e a s ie r th a n th e d irect e v a l­
u a tio n o f th e co n v o lu tio n su m m atio n .
Correlation of two sequences.
If
xj(n)
X i(z)
x 2 (n)
X 2 (z)
th e n
OC
Rx,J2 (z) = X 1 (Z)X 2(Z"1)
OC
Proof . We recall that
rXix2U) = x\ ( l ) * x 2( - l )
(3.2.18)
170
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
U sing th e co n v o lu tio n an d tim e-rev ersa l p ro p e rtie s , we easily o b ta in
= Z{x] ( / ) }Z{xz ( —/)} = X l ( z ) X 2 ( z ~1)
T h e R O C o f R X}X2(z) is at least the in te rse c tio n o f th a t fo r X i(c ) an d X 2("- 1 )A s in th e case o f co n v o lu tio n , th e c ro ssc o rre la tio n o f tw o signals is m o re
easily d o n e via p o ly n o m ial m u ltip lic atio n acco rd in g to (3.2.18) a n d th en inverse
tra n sfo rm in g th e resu lt.
Exam ple 3.2.10
Determ ine the autocorrelation sequence of the signal
x (n ) =
a " u {n ). — 1 < a
< 1
Since the autocorrelation sequence of a signal is its correlation with itself,
(3.2.18) gives
S o lu tio n
Rxs(z) = Z{rxJ I ) ) = X (z)X (;~l )
From (3.2.2) we have
ROC: |z| > \a\
Ale) = ---------- J - az~'
(causal signal)
and by using (3.2.15). we obtain
1
| = -.... ■
1 - az
1
ROC: |;| < —
(a|
(anticausal signal)
Thus
= -— = ------------ ------------ 1 —az 1 1 - az
1 —a (; + ; ) + fl
ROC: \a\ < |;| < —
|c|
Since the R O C of Rxx(z) is a ring, rIX{i) is a two-sided signal, even if x(n) is causal.
To obtain rsJ l ) , we observe that the ^-transform of the sequence in Exam­
ple 3.1.5 with b = 1 /a is simply (1 —a1)Rxx(z). Hence it follows that
rXI(l) = — -—- a ^
— oc < I < oo
\ — a-
The reader is encouraged to compare this approach with the time-dom ain solution of
the same problem given in Section 2.6.
Multiplication of two sequences.
If
x \ ( n)
ATi(z)
x 2 (n)
X 2 {z)
th e n
•r(n) = x i ( n ) x 2 (n)
X(z) =
( “ ) v~ ^ v
(3.2.19)
w h ere C is a clo sed c o n to u r th a t e n clo ses th e o rig in a n d lies w ith in th e reg io n of
co n v erg en ce c o m m o n to b o th X](i>) a n d ^ ( l / u ) .
Sec. 3.2
171
Properties of the z-Transform
Proof . T h e z -tra n sfo rm o f J 3 (n) is
OC
X(z) = Y
OC
x( n)z ~n = Y
x ] ( n) x 2 ( n) z ~n
n = —oc
ft—
L e t us s u b s titu te th e in v erse tra n sfo rm
* i(n ) =
fo r
in th e z -tra n sfo rm X (z) a n d in te rc h a n g e th e o r d e r o f su m m a tio n an d
in te g ra tio n . T h u s w e o b ta in
1
/
" oc
X (z)= 2
v~ ^ v
T h e su m in th e b ra c k e ts is sim ply th e tra n sfo rm X 2 (:) e v a lu a te d a t z / v . T h e re fo re ,
X(z) = ^ - 6 x ^ x 2
2 n j Jc
w hich is th e d e s ire d result.
T o o b ta in th e R O C o f X (z) w e n o te th a t if X i(u ) co n v e rg e s fo r ru < |u| < r\
an d X 2 iz) c o n v erg es fo r r2! < |z| < r2ll, th e n th e R O C o f X 2 {z/v) is
H e n c e th e R O C fo r A"(z) is at least
r\ir2i < |z[ < r u r2u
(3.2.20)
A lth o u g h this p ro p e rty will n o t be used im m e d ia te ly , it will p ro v e u se fu l later,
esp ecially in o u r tr e a tm e n t o f filter design b ased o n th e w in d o w te c h n iq u e , w h ere
w e m u ltip ly th e im p u lse re sp o n s e o f an IIR system by a fin ite -d u ra tio n “w in d o w ”
w h ich se rv es to tru n c a te th e im p u lse re sp o n se o f th e I I R sy stem .
F o r c o m p le x -v a lu e d se q u e n c e s .ti(rt) an d x 2 (n) w e c a n define th e p ro d u c t
s e q u e n c e as x( n ) = Jti (nj x^i n) . T h e n th e c o rre sp o n d in g c o m p lex co n v o lu tio n
in te g ra l b eco m es
x ( n ) = x i ( n ) x 2 {n)
X(z) =
v~ldv
(3.2.21)
T h e p r o o f o f (3.2.21) is left as an ex ercise fo r th e re a d e r.
Parseval’s relation.
y
If jti(n ) an d x 2 (n) a re c o m p le x -v a lu e d se q u en ces, th e n
* i(n )x 2 (n) = l j ( j ^ X i ( v ) X 2
v ~ 'd v
(3.2.22)
p ro v id e d th a t r ^ r y < 1 < n ur2u, w h e re ry < |z| < r \ u a n d r ^ < |z| < r2u a re th e
R O C o f X ](z) a n d X 2(z). T h e p ro o f o f (3.2.22) follow s im m e d ia te ly by ev alu atin g
X ( z ) in (3.2.21) at z = 1.
172
The z-Transform and Its Application to the Analysis of LTI Systems
The Initial Value Theorem.
Chap. 3
If j ( ;i ) is causal [i.e.. x ( n ) = 0 fo r n < 0], th e n
.r(0) = lim X ( z )
:—*•sc
(3.2.23)
Proof. Since x{n) is causal. (3.1.1) gives
X (z) — y x {n )z ” — x (0) + x (1 )z 1 + x (2)z
«=o
+ ■■•
O b v iou sly , as z —►oc. z ~" —►0 since n > 0 an d (3.2.23) follow s.
A ll th e p ro p e rtie s o f th e z -tra n sfo rm p r e s e n te d in th is s e c tio n are su m m arized
in T a b le 3.2 fo r easy re fe re n c e . T h ey are listed in th e sam e o rd e r as th ey have
b een in tro d u c e d in th e tex t. T h e c o n ju g atio n p ro p e rtie s a n d P a rse v a l's relatio n
are left as ex ercise s fo r th e re a d e r.
W e have now' d e riv e d m o st o f th e z -tra n sfo rm s th a t are e n c o u n te re d in m any
p ractical ap p licatio n s. T h e se z -tra n sfo rm pairs a re su m m a riz e d in T ab le 3.3 for
easy re fe re n c e . A sim ple in sp e ctio n o f th is ta b le show s th a t th e s e z-tran sfo rm s
are all rational f u n c t i o n s (i.e., ratio s o f p o ly n o m ials in z _1). A s will soon b ecom e
a p p a re n t, ratio n al z -tra n sfo rm s are e n c o u n te re d n o t only as th e z-tran sfo rm s of
v ario u s im p o rta n t signals b u t also in th e c h a ra c te riz a tio n of d isc re te -tim e lin ear
tim e -in v a ria n t sy stem s d esc rib e d by c o n s tan t-co efficien t d iffe re n c e e q u a tio n s.
3.3 RATIONAL Z-TRANSFORMS
A s in d ic a te d in S ectio n 3.2, an im p o rta n t fam ily o f z-tra n sfo rm s a re th o se fo r w hich
X ( z ) is a ra tio n a l fu n ctio n , th a t is. a ra tio of tw o p o ly n o m ials in z _l (o r z). In
this sectio n w e discuss som e very im p o rta n t issues re g a rd in g th e class o f ra tio n a l
z-tran sfo rm s.
3.3.1 Poles and Zeros
T h e zeros of a z -tra n sfo rm X (z) a re th e v alu es of z fo r w hich X (z) = 0. T h e pol es
o f a z-tra n sfo rm are th e v alu es o f z fo r w hich X (z) = oc. If X ( z ) is a ratio n al
fu n ctio n , th e n
J K ,) .
D( z)
ao + a i z 1 + ----- \ - aNz ~ h
= ™ --------*
(3.3.1)
k=o
If ao / 0 an d bo ^ 0, w e can avoid th e n eg ativ e p o w e rs o f z by fa cto rin g o u t the
te rm s boz~M a n d a$z~N as follow s:
X(z) =
N( z)
b 0z ~ M z M + {bx/ bo) z M~ x + • • ■+ b M/ b 0
D( z)
a0z N Z N + ( a \ / a a) z fJ~ l H--------\ - aN/ a Q
TABLE 3.2
PROPERTIES OF THE Z-TRANSFORM
Property
Time Domain
z-Domain
R OC
Notation
Linearity
*(n)
xt (n)
■^(n)
aixy(n) + a2x 2(n)
X(z)
Xi(z)
X 2(z)
<t\* l(z ) + « 2X 2(z)
Time shifting
x{n - k)
z~kX(z)
Scaling in the z-domain
a"x(n)
X(a~' z)
ROC: r2 < |z) < r\
ROC,
ROC2
At least the intersection of R O Q
and R O C 2
T hat of X (z), except z = 0 if k > 0
and z = oo if k < 0
\a\r2 < |z| < |a|r[
Time reversal
x(~rj)
X(z~')
-
Conjugation
Real part
Imaginary part
x'(rt)
Relx(n)}
lm{x(n))
X' ( z ' )
ROC
Includes ROC
Includes ROC
Differentiation in the
z-domain
Convolution
nx(n)
x i ( n ) * x 2(n)
Jf|(z)X 2(z)
Correlation
rx,x2(l) = * i(0 * x 2 (~l)
/ W z ) = Xi<z)* 2<z_l)
Initial value theorem
If x(n) causal
.r(O) = lim X(z)
Multiplication
Xi(n)x 2(n)
2~J^X,(v)X2^ J v - ' d v
Parseval’s relation
i[X (z) + X*U*)]
i[X (z )-* ♦ (;• )]
dX( z )
Z dz
= 2? i i
5X ,( t)) ^ (l/tJ ’) v - ' dv
1
r1
1
< |z| < —
r2
r2 < |z| < r.
A t least, the intersection of ROCi
and R O C 2
A t least, the intersection of ROC of
X,(z) and -K^z” 1)
A t least rj/ry < |z| < r]„r2 a
174
The z-Transform and Its Application to the Analysis of LTI Systems
TABLE 3.3
SOME COMMON Z-TRANSFORM PAIRS
Signal. x(n)
1
u(n)
3
a"u(n)
A ll ;
1 —az~l
a:" 1
na"u(n )
( l - a z ^ )2
—a”u(—n — 1)
6
—na'’u(—n — 1)
(cos aion)u(n)
(sin a>nn)u(n)
( a" cosiH ) n ) u ( n )
10
1
1 - r-1
1
5
8
ROC
1
2
7
: -Transform, X (c)
5{n)
4
Chap. 3
(a'1sin£i*in)u{n)
1:1 > 1
\z\ > \a\
Izt > !a|
1
1 —a : -1
a z~l
(1 - c : - 1)2
1 - Z '1 COSf^o
1 - 2; _1 costal + z ~2
sin
2 ;" 1 cos tun + z ~2
1 —az~] cos a*)
1-
k l < la|
|;| < N
kl > 1
\z\ > 1
1 — 2az~‘ cos wo + a 2z ~2
a ; -1 sinaiii
1 - 2a ; " 1 cos a*, + a 2z ~2
Since N ( z ) and D( z ) are polynom ials in z, they can be expressed in factored form as
X (-) —
— b() - - M + N
D( z )
~
~ Z2) ■■■ (Z ~ Z m )
(z - p i ) ( z - P i ) ■• ■(z - p n )
u
n « " z*}
(3.3.2)
X ( z ) = G z N~ M^A ------------
n<* ~ pjt)
*=i
w here G = 6o/ao- Thus X (r) has M finite zeros at z = z\, zi , ■. •, z m (th e roots o f
the num erator p olyn om ial), N finite p oles at z = p \ , p i .........P n (th e roots o f the
d enom inator p olyn om ial), and \N — M\ zeros (if N > M ) or p o le s (if N < M ) at
the origin z = 0. P o les or zeros m ay also occur at z = 0 0 . A zero exists at z = oc if
X ( 0 0 ) = 0 and a p o le exists at z = oc if X ( 0 0 ) = oc. If w e count the p oles and zeros
at zero and infinity, w e find that X (z) has exactly the sam e num ber o f p o les as zeros.
W e can represent X ( z ) graphically by a p o l e - z e r o p l o t (or pat t ern) in the
com plex plane, which show s the location o f p oles by crosses ( x ) and the location
o f zeros by circles (o). T he m ultiplicity of m ultipie*order p o les or zeros is indicated
by a num ber clo se to the corresponding cross or circle. O b viou sly, by definition,
the R O C o f a z-transform should not contain any poles.
Sec. 3.3
175
Rational z-Transform s
Exam ple 33.1
D eterm ine the pole-zero plot for the signal
a >0
x(n) = a"u(n)
Solution
From Table 3.3 we find that
1_____
X(z) =
ROC; \z\ > a
- a
1 —a ; -1
Thus
has one zero at n = 0 and one pole at pi = a. The pole-zero plot is
shown in Fig. 3.7. Note that the pole p\ = a is not included in the R O C since the
z-transform does not converge at a pole.
Re(;)
Figure 3.7
Pole-zero plot for the
causal exponential signal ,v(«) = a"imu
Exam ple 3 3 .2
D eterm ine the pole-zero plot for the signal
1 0,
Of n < W - 1
elsewhere
where a > 0.
Solution
From the definition (3.1.1) we obtain
^
X(z) = X W
"
ln
l - t a ; ' 1) "
z“ - a u
1 - az 1 = Sz T T t--------T
(z - a)
1}" = - r r ^ ------ r
Since a > 0, the equation z M = a M has M roots at
Z t =
a e ' 1* * ' "
it
=
0 , 1 . . . . . . M
-
1
The zero zo = a cancels the pole at z = a. Thus
w
,
(Z - Z\ ) ( Z - Zl ) ■’ ■(Z - Z j t f - i )
X(z) = ---------------- ^ - ----------------
which has M —1 zeros and M - 1 poles, located as shown in Fig. 3.8 for M = 8. Note
th at the R O C is the entire z-plane except z = 0 because of the M - 1 poles located
at the origin.
176
The z-Transfonm and Its Application to the Analysis of LTI Systems
Chap. 3
Im(c)
Red)
Figure 3.8 Pole-zero patlem for
the finite-duration signal x(n) = a",
0 < n < M — l(a > 0). for M = 8.
C learly , if we are given a p o le - z e r o p lo t, w e can d e te rm in e X ( ’ ), by using
(3.3.2), to w ithin a scaling fa c to r G. T his is illu stra te d in th e follow ing ex am p le.
Example 3.3.3
Determine the c-transform and the signal that corresponds to the pole-zero plot of
Fiji. 3.9.
Solution There are two zeros (M = 2) at Z] = 0, Z2 = r cosojo and two poles (N = 2)
al p] — reJ , Pz = re~iw". By substitution of these relations into (3.3.2), we obtain
X(z) = (
Z(Z — r C O S ttH i)
(; - Pi)(c - pi)
= G(c - re}wu)(z - re~>w")
ROC:
A fter some simple algebraic manipulations, we obtain
1-
C O S a><)
Xiz) = G-.
1 - 2 r - ~ l costou + r1
ROC: 1-1 > r
From Table 3.3 we find that
x(n) = G(r" cosa>on)u(n)
F ro m E x a m p le 3.3.3, w e see th a t th e p r o d u c t (; — p \ ) ( z — P 2 ) resu lts in a
p o ly n o m ial w ith re a l coefficients, w h en p\ a n d p 2 a re c o m p lex co n ju g ates. In
Im(2)
Figure 3.9 P ole-zero p attern for
Exam ple 3.3.3.
Sec. 3.3
177
Rational z - T ransforms
g e n e ra l, if a p o ly n o m ial h a s re a l coefficients, its ro o ts a re e ith e r real o r o ccu r in
c o m p lex -co n ju g ate pairs.
A s w e h av e seen , th e ; -tran sfo rm X ( z ) is a com plex fu n c tio n of th e com plex
v ariab le z = R e ( z ) + j Im (r). O b v io u sly , |X (c )|, th e m a g n itu d e o f X ( z ), is a real
and p o sitiv e fu n ctio n o f c. S ince : re p re se n ts a p o in t in th e co m p lex p la n e , |X ( ’ )|
is a tw o -d im e n sio n a l fu n c tio n an d d esc rib e s a “su rfa c e .” T h is is illu stra te d in
Fig. 3.10(a) fo r th e z -tra n sfo rm
7 - 1
_
- - 2
(a)
Figure 3.10 Graph of |X (;)| for the
;;-Transform in (3.3.3). [Reproduced with
permission fr om Introduction to Systems
Analysis, by T. H. Glisson, © 1985 by
McGraw-Hill B ook Company.]
178
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
w hich h as o n e z e ro a t z\ = 1 a n d tw o p o le s a t p \ , p 2 = 0. 9e±J" /4. N o te the
high p e a k s n e a r th e sin g u larities (p o les) a n d th e d e e p v alley close to th e zero.
F ig u re 3.10(b) illu stra te s th e g rap h o f lX (z)j fo r z = eJO>.
3.3.2 Pole Location and Time-Domain Behavior for
Causal Signals
In th is su b sectio n w e c o n sid e r th e re la tio n b e tw e e n th e z -p la n e lo catio n o f a pole
p a ir a n d th e fo rm (sh a p e ) o f th e c o rre sp o n d in g signal in th e tim e d o m ain . T h e dis­
cu ssio n is b a s e d g en erally o n th e collectio n o f z -tra n s fo rm p a irs given in T a b le 3.3
an d th e resu lts in th e p re c e d in g su b sectio n . W e d e a l exclusively w ith real, causal
signals. In p a rtic u la r, w e se e th a t th e c h a ra c te ristic b e h a v io r o f cau sal signals d e­
p e n d s o n w h e th e r th e p o les o f th e tra n sfo rm a re c o n ta in e d in th e reg io n |z| < 1 ,
o r in th e reg io n |z| > 1, o r on th e circle |z| = 1. S ince th e circle jz| = 1 h as a
ra d iu s o f 1 , it is called th e unit circle.
If a real signal h as a z-tran sfo rm w ith o n e p o le , th is p o le h a s to b e real. T he
o n ly su ch signal is th e re a l e x p o n e n tia l
x ( n ) = a nu{n) ^
X ( z ) = — ------ r
1 — az
R O C : |z] > |o|
h av in g o n e z e ro a t zi = 0 an d o n e p o le a t pi — a on th e re a l axis. F ig u re 3.11
x (n)
11
o
x(n )
1 1 ! TTt
l
i
t
,
x(n
x(n)
0
i l l
rt
I I I
x(n)
'1
0
n
n i
1
-oi
'
Figure 3.11 Time-domain behavior of a single-real pole causal signal as a function
of the location of the pole with respect to the unit circle.
Sec. 3.3
179
Rational z-Transform s
illu stra te s th e b e h a v io r o f th e signal w ith re sp e c t to th e lo c a tio n o f th e p o le r e l­
ative to th e u n it circle. T h e signal is d ecay in g if th e p o le is inside the unit
circle, fixed if th e p o le is o n th e u n it circle, an d gro w in g if th e p o le is o u t­
side th e u n it circle. In a d d itio n , a n eg ativ e p o le resu lts in a signal th a t a lte r­
n a te s in sign. O b v io u sly , causal signals w ith p o les o u tsid e th e u n it circle b e ­
co m e u n b o u n d e d , cau se overflow in d igital system s, a n d in g e n e ra l, sh o u ld be
av o id ed .
A cau sal re a l signal w ith a d o u b le re a l p o le h as th e fo rm
x ( n ) = n a nu ( n )
(see T a b le 3.3) an d its b e h a v io r is illu stra te d in Fig. 3.12. N o te th a t in c o n tra s t to
th e sin g le-p o le signal, a d o u b le real p o le o n th e u n it circle re su lts in an u n b o u n d e d
signal.
F ig u re 3.13 illu stra te s th e case of a p a ir o f c o m p le x -c o n ju g a te poles. A c c o rd ­
ing to T a b le 3.3, th is co n fig u ratio n of p o le s resu lts in an e x p o n e n tia lly w eig h ted
sin u so id al signal. T h e d istan c e r of th e p o les fro m th e o rigin d e te rm in e s th e e n v e ­
lope o f th e sin u so id al signal an d th e ir angle w ith th e real p o sitiv e axis, its relative
freq u e n cy . N o te th a t th e a m p litu d e o f th e signal is gro w in g if r > 1, c o n stan t if
r — 1 (sin u so id a l sig n als), a n d d ecaying if r < 1 .
x(n)
<■
T T ....
T
j
.
xin)
T
0
!
1
l
Figure 3.12 Time-domain behavior of causal signals corresponding to a double (m = 2) real
pole, as a function of the pole location.
n
180
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
Figw e 3.13 A pair of complex-conjugate poles corresponds to causal signals with
oscillatory behavior.
Finally, Fig. 3.14 show s the b eh avior o f a causal signal w ith a d ou b le pair of
p o les on the unit circle. T his reinforces the corresponding results in Fig. 3.12 and
illustrates that m ultiple p o les on the unit circle should b e treated with great care.
T o sum m arize, causal real signals w ith sim ple real p oles or sim ple com plexconjugate pairs o f p oles, w hich are in sid e or on the unit circle are always bounded
in am plitude. Furtherm ore, a signal with a p o le (or a com p lex-con ju gate pair
o f p o les) near the origin decays m ore rapidly than on e a ssociated with a pole
n ear (but inside) the unit circle. Thus the tim e behavior o f a signal depends
strongly on the location o f its p o les relative to th e unit circle. Z eros also af­
fect the behavior o f a signal but n ot as strongly as p oles. F or exam ple, in the
Sec. 3.3
181
Rational z-Transform s
Figure 3.14 Causal signal corresponding to a double pair of complex-conjugate
poles on the unit circle.
case o f sin u so id al signals, th e p resen c e an d lo catio n o f ze ro s affects only th e ir
p h ase.
A t this p o in t, it sh o u ld be stressed th a t e v e ry th in g w e h av e said a b o u t causal
signals ap p lies as w ell to causal L TI system s, since th e ir im p u lse re sp o n se is a causal
signal. H en ce if a p o le of a system is o u tsid e the unit circle, th e im pulse resp o n se
of th e system b eco m es u n b o u n d e d and. co n se q u e n tly , th e sy stem is u n sta b le .
3.3.3 The System Function of a Linear Time-Invariant
System
In C h a p te r 2 w e d e m o n s tra te d th a t th e o u tp u t o f a (re la x e d ) lin e a r tim e -in v a ria n t
sy stem to an in p u t se q u e n c e x( n) can be o b ta in e d by co m p u tin g th e co n v o lu tio n
of jc(h) w ith th e u n it sa m p le resp o n se o f th e system . T h e co n v o lu tio n p ro p e rty ,
d eriv e d in S ectio n 3.2, allow s us to ex p ress th is re la tio n sh ip in th e z-d o m ain as
Y( z ) = m z ) X ( z )
(3.3.4)
w h ere Y( z ) is th e z -tra n sfo rm o f th e o u tp u t se q u en ce v(n), X (z) is th e z-tra n sfo rm
o f th e in p u t se q u e n c e x ( n ) an d H( z ) is th e z -tra n sfo rm o f th e u n it sa m p le resp o n se
h{n).
If w e k n o w h( n) an d x( n ) , w e can d e te rm in e th e ir c o rre sp o n d in g z-tra n sfo rm s
H ( z ) a nd X ( z ) , m u ltip ly th e m to o b ta in Y(z), a n d th e r e fo r e d e te rm in e y( n) by
e v a lu a tin g th e in v erse z -tra n sfo rm of K(z). A lte rn a tiv e ly , if w e k now x ( n ) an d we
o b se rv e th e o u tp u t y( n) of th e system , w e can d e te rm in e th e u n it sa m p le resp o n se
by first solv in g fo r H( z ) fro m th e re la tio n
a n d th e n e v a lu a tin g th e in v erse z -tra n sfo rm o f H( z ) .
Since
OC
(3.3.6)
182
The z-Transform and Its Application to the Analysis of LTI System s
Chap. 3
it is c le a r th a t H ( z ) re p re s e n ts th e z -d o m ain c h a ra c te riz a tio n o f a system , w h ereas
h{n) is th e c o rre sp o n d in g tim e -d o m a in c h a ra c te riz a tio n of th e system . In o th e r
w o rd s, H{z ) an d h( n) are e q u iv a le n t d e s c rip tio n s o f a system in th e tw o d o m ain s.
T h e tra n sfo rm H{ z ) is called th e s yst em f unct i on.
T h e re la tio n in (3.3.5) is p a rtic u la rly u sefu l in o b ta in in g H{ z ) w h en th e system
is d e s c rib e d b y a lin e a r c o n s tan t-co efficien t d iffe re n c e e q u a tio n o f th e fo rm
M
N
(3.3.7)
In th is case th e sy stem fu n ctio n can be d e te rm in e d d irectly fro m (3.3.7) by com ­
p u tin g th e z -tra n s fo rm o f b o th sid es o f (3.3.7). T h u s, by a p p ly in g th e tim e-sh iftin g
p ro p e rty , w e o b ta in
M
y u ) =
- J 2 a*Y ^ z ~k + E
b iX (z )r‘
M
Y(z)
X(z)
o r, eq u iv alen tly ,
£ > z-*
(3.3.8)
T h e re fo re , a lin e a r tim e -in v a ria n t system d e s c rib e d by a c o n s ta n t-c o e ffic ie n t dif­
fere n ce e q u a tio n h a s a ra tio n a l system fu n ctio n .
T h is is th e g e n e ra l fo rm fo r th e system fu n c tio n o f a sy ste m d e sc rib e d by a
lin e a r c o n s tan t-co efficien t d ifferen ce e q u a tio n . F ro m th is g e n e ra l fo rm w e o b tain
tw o im p o rta n t sp e cial fo rm s. F irst, if ajt = 0 fo r 1 < k < N , (3.3.8) re d u c e s to
M
H ( z ) = J 2 bkz ~k
k=0
1
»
(3.3.9)
In th is case, H ( z ) c o n ta in s M z ero s, w h o se valu es a re d e te r m in e d by the
system p a ra m e te rs {£*}, a n d an M th -o rd e r p o le a t th e o rigin z = 0. S ince the
sy stem c o n ta in s o n ly trivial p o le s (a t z = 0) an d M n o n triv ia l z e ro s, it is called
Sec. 3.3
183
Rational z-Transform s
an all -zero syst em. C learly, such a system h as a fin ite -d u ra tio n im p u lse resp o n se
(F IR ), a n d it is called an F IR system o r a m oving av erag e (M A ) system .
O n th e o th e r h an d , if bk — 0 fo r 1 < k < M, th e system fu n c tio n re d u c e s to
A'
1
*= 1
k
(3.3.10)
boz"
<io = l
at*.
In this case H ( z ) co n sists of N poles, w hose v alu es are d e te r m in e d by th e system
p a ra m e te rs {a*} a n d an /V th-order zero at th e o rig in z = 0- W e usually do not
m a k e re fe re n c e to th e se trivial zeros. C o n se q u e n tly , th e sy stem fu n c tio n in (3.3.10)
c o n ta in s o n ly n o n triv ia l p o les an d th e c o rre sp o n d in g sy stem is called an all-pole
syst em. D u e to th e p re se n c e o f poles, th e im p u lse re sp o n s e o f such a system is
infinite in d u ra tio n , a n d h e n c e it is an I IR system .
T h e g e n e ra l fo rm o f th e sy stem fun ctio n given by (3.3.8) c o n ta in s b o th poles
a n d zero s, and h e n c e th e c o rre sp o n d in g sy stem is called a p o l e - z e r o s y s t e m , with
N p o le s a n d M z ero s. P o les a n d /o r zero s at c = 0 an d z = oc a re im p lied b u t are
n o t c o u n te d ex p licitly. D u e to th e p re se n c e o f p o les, a p o le - z e r o system is an IIR
system .
T h e fo llo w in g ex am p le illu strates th e p ro c e d u re fo r d e te rm in in g th e system
fu n ctio n a n d th e u n it sa m p le resp o n se from the d iffe re n c e e q u a tio n .
E xam ple 3.3.4
D eterm ine the system function and the unit sample response of the system described
by the difference equation
v(n) =
Solution
1y(n -
1) + 2 x ( n )
By computing the -transform of the difference equation, we obtain
Y(z) = ± z - >Y(z) + 2X(z)
H ence the system function is
xt:l
I - Jz -1
This system has a pole at z = \ and a zero at the origin. Using Table 3.3 we obtain
the inverse transform
h(n) = 2(i)"u(«)
This is the unit sample response of the system.
184
The z-T ransform and Its Application to the Analysis of LTI Systems
Chap. 3
W e h av e n o w d e m o n s tra te d th a t ra tio n a l z -tra n s fo rm s a re e n c o u n te re d in
c o m m o n ly u se d sy stem s a n d in th e c h a ra c te riz a tio n o f lin e a r tim e -in v a ria n t sys­
tem s. In S ectio n 3.4 w e d esc rib e se v e ra l m e th o d s fo r d e te rm in in g th e inverse
z -tra n sfo rm o f ra tio n a l fu n ctio n s.
3.4 INVERSION OF THE Z-TRANSFORM
A s w e saw in S ectio n 3.1.2, th e in v erse z -tra n s fo rm is fo rm ally given by
x ( n ) = - — < £ x ( z ) z n~ 1d z
2
njjt
(3.4.1)
w h ere th e in te g ra l is a c o n to u r in te g ra l o v e r a clo sed p a th C th a t en clo ses the
o rig in an d lies w ith in th e reg io n o f c o n v e rg e n c e o f ^ ( z ) . F o r sim plicity, C can be
ta k e n as a circle in th e R O C o f X (z) in th e z-p lan e.
T h e re a re th re e m e th o d s th a t a re o fte n u se d fo r th e e v a lu a tio n o f th e inverse
z-tran sfo rm in practice:
1. D ire c t e v a lu a tio n o f (3.4.1), by c o n to u r in te g ra tio n .
2. E x p a n sio n in to a se rie s o f te rm s, in th e v a ria b le s z, an d z _1.
3. P a rtia l-fra c tio n ex p an sio n a n d ta b le lo o k u p .
3.4.1 The Inverse z-Transform by Contour Integration
In th is se ctio n w e d e m o n s tra te th e use o f th e C au ch y re sid u e th e o r e m to d e te rm in e
th e in v erse z -tra n sfo rm d irectly fro m th e c o n to u r in teg ral.
Cauchy residue theorem. L e t / ( z ) b e a fu n c tio n o f th e c o m p lex v ariab le
z an d C b e a clo sed p a th in th e z -p lan e. If th e d e riv a tiv e d f ( z ) / d z exists o n and
inside th e c o n to u r C a n d if / ( z ) has no p o le s a t z = zo, th e n
- L (f) J ! ± d z = |^ <Zl,)'
2njjcz-zo
10,
(3.4.2)
if zo is ou tsid e C
M o re g en erally , if th e (k + l ) - o r d e r d e riv a tiv e o f / ( z) exists a n d / ( z ) h a s n o p o les
a t z = zo, th e n
1
»-!>'
0,
d k- ' f ( z )
C
04.3)
if zo is o u ts id e C
T h e v alu es o n th e rig h t-h a n d sid e o f (3.4.2) a n d (3.4.3) a re c a lle d th e re sid u e s of
th e p o le a t z = zo- T h e re su lts in (3.4.2) a n d (3.4.3) a re tw o fo rm s o f th e Cauc hy
residue t heorem.
W e can ap p ly (3.4.2) a n d (3.4.3) to o b ta in th e v a lu e s o f m o re g en eral c o n to u r
in teg rals. T o b e specific, su p p o s e th a t th e in te g ra n d o f th e c o n to u r in te g ra l is
Sec. 3.4
Inversion of th e 2 - T ra n s fo rm
185
P(z) = f ( z ) / g ( z ) ~ w h e re f ( z ) h a s no p o les inside th e c o n to u r C an d g (z) is a
p o ly n o m ial w ith d istin ct (sim p le) ro o ts c i, ^ 2 . ___-n inside C. T h e n
(3.4.4)
n
1=1
w h ere
f(z)
(3.4.5)
A l (z) = ( z - z i ) P{ z ) = ( z - z l) -J - 1x
g( z)
T h e v alu es (A, (-;,)} a re re sid u e s o f th e c o rre sp o n d in g p o le s at z =
/ = 1, 2 , . . . . n.
H e n c e th e v alu e o f th e c o n to u r in te g ra l is e q u a l to th e sum o f th e resid u es o f all
th e p o le s in sid e th e c o n to u r C.
W e o b se rv e th a t (3.4.4) w as o b ta in e d by p e rfo rm in g a p a rtia l-fra c tio n e x p a n ­
sion o f th e in te g ra n d an d ap p ly in g (3.4.2). W h en g(z) has m u ltip le -o rd e r ro o ts
as w ell as sim p le ro o ts inside th e c o n to u r, th e p a rtia l-fra c tio n e x p a n sio n , w ith a p ­
p ro p ria te m o d ifica tio n s, an d (3.4.3) can b e used to e v a lu a te th e resid u es at th e
c o rre sp o n d in g p o les.
In th e case o f th e in v erse z -tra n sfo rm , w e h ave
[resid u e of X (z )z n 1 a t z
a ll p o l e s
U t)
in s id e
(3.4.6)
C
p ro v id e d th a t th e p o les {z,} a re sim ple. If X (z )r "_1 has no p o le s inside th e c o n to u r
C fo r o n e o r m o re v alu es o f n, th e n x (n) = 0 fo r th e s e values.
T h e fo llo w in g ex am p le illu stra te s th e e v a lu a tio n o f th e in v erse z-tra n sfo rm
by u se o f th e C a u ch y re sid u e th e o re m .
Exam ple 3.4.1
Evaluate the inverse z-transform of
X(z) = ---------- r
1 —az~
using the complex inversion integral.
Solution
kl > kil
We have
where C is a circle at radius greater than |a|. We shall evaluate this integral using
(3.4.2) with f ( z ) = z". We distinguish two cases.
186
The ^-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
L If n > 0, f ( z ) has only zeros and hence no poles inside C. T he only pole inside
C is z = a. Hence
*(*) =• /(zo) = a n
n > 0
2. If n < 0, / ( z ) = z" has an nth-order pole at z = 0, which is also inside C. Thus
there are contributions from both poles. For n = - 1 we have
x ( - l ) -=
1 S)
1
dz =
1
2 nj j c z ( z — a)
z —a
1
+ -
= 0
If n = —2, we have
2)
2 n j § z H z - a ) dZ
dz ( z - f l )
= 0
By continuing in the same way we can show that *(n) = 0 for n < 0. Thus
x (n) = a"u(n)
3.4.2 The Inverse z-Transform by Power Series
Expansion
The basic idea in this m eth od is the follow ing: G iven a z-transform X ( z ) with its
corresponding R O C , w e can expand X (z) into a p ow er series o f the form
OO
X(z) =
c»z~n
£
(3.4.7)
co
w hich con verges in the given R O C . T h en , by the u n iqu en ess o f the z-transform,
x ( n ) = c„ for all n. W hen X ( z ) is rational, the exp an sion can b e perform ed by
long division.
T o illustrate this tech n iqu e, w e w ill invert som e z-transform s involving the
sam e expression for X ( z ) , but different R O C . T his w ill also serve to em phasize
again the im portance o f the R O C in d ealing with z-transform s.
Exam ple 3A 2
D eterm ine the inverse z-transform of
1 —1.5z_1 + 0.5z “2
when
(a) ROC: |z| > 1
(b) ROC: |z| < 0.5
Solution
(a) Since the R O C is the exterior of a circle, we expect x(n) to be a causal signal.
Thus we seek a power series expansion in negative powers of z. By dividing
Sec. 3.4
187
Inversion of the z-Transform
the num erator of X{z) by its denom inator, we obtain the power series
A’UI = ! _ 3 _ -; +
+
= 1+
^
+ TE; "4 + " '
By com paring this relation with (3.1.1), we conclude that
t
Note that in each step of the long-division process, we eliminate the lowestpower term of c~*.
(b) In this case the ROC is the interior of a circle. Consequently, the signal x(n)
is anticausal. To obtain a power series expansion in positive powers of c. we
perform the long division in the following way:
2: 2 + 6c3 + 14c4 + 30cs + 62c* + ■• ■
+ ill
1 - 3: + 2c:
3c - 2z 2
3c - 9c: + 6c3
l z 2 - 6c3
7 r - 21 c3 + 14c4
15c3 - 14c4
15c3 - 45c4 + 30cs
31c4 - 30c5
Thus
X (c) =
1_
,
1------:------= 2c: + 6c3 + 14c4 + 30c5 + 62cfi + ■• •
In this case x(n) = 0 for n > 0. By comparing this result to (3.1.1), we conclude
that
Jt(n) = {
62. 30. 14.6,2, 0. 0}
t
We observe that in each step of the long-division process, the lowest-power
term of c is eliminated. We emphasize that in the case of anticausal sig­
nals we simply carry out the long division by writing down the two poly­
nomials in “reverse” order (i.e., starting with the most negative term on the
left).
F ro m th is e x a m p le w e n o te th a t, in g e n eral, th e m e th o d o f long d ivision will
n o t p ro v id e a n sw ers fo r x( n) w h en n is larg e b e c a u se th e lo n g division b eco m es
ted io u s. A lth o u g h , th e m e th o d p ro v id es a d irect e v a lu a tio n o f x( n ) , a clo sed -fo rm
so lu tio n is n o t p o ssib le , ex cep t if th e resu ltin g p a tte r n is sim p le e n o u g h to infer
th e g e n e ra l te rm x ( n ) . H e n c e th is m e th o d is used only if o n e w ish e d to d e te rm in e
th e v a lu e s o f th e first few sa m p le s o f th e signal.
188
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
Example 3.4.3
Determ ine the inverse z-transform of
X(z) = log(l + az_1)
Solution
|zj > |a|
Using the power series expansion for log(l + x ) , with |jc| < 1, we have
^
( - r v r "
Thus
n < 0
0,
Expansion of irrational functions into power series can be obtained from tables.
3.4.3 The Inverse z-Transform by Partial-Fraction
Expansion
In th e tab le lo o k u p m e th o d , w e a tte m p t to e x p ress th e fu n c tio n X (z) as a linear
c o m b in a tio n
X ( z ) = a j X] (;) + 02 X 2 ( 1 ) + • - - + c/Kx fc(z)
(3.4.8)
w h ere X] ( ; ) , . . . . X k (z ) a re ex p ressio n s w ith in v erse tra n sfo rm s x \ ( n ) , . . . , x k W
av ailab le in a ta b le o f z-tra n sfo rm p airs. If such a d e c o m p o s itio n is possible,
th e n x( n) , th e in v erse z -tra n sfo rm o f X ( z ) , can easily b e fo u n d using th e lin earity
p ro p e rty as
x ( n) = at]Xi(n) + a 2x 2 (n) H--------\ - a Kx K (n)
(3.4.9)
T h is a p p ro a c h is p a rtic u la rly useful if X ( z ) is a ra tio n a l fu n ctio n , a s in (3.3.1). W ith ­
o u t loss o f g e n erality , w e assu m e th a t ao = 1, so th a t (3.3.1) can b e e x p ressed as
_ t o
D(z)
_ b +b,r' + -
+ t„ r«
l + f l i Z - 1 H----- + a u z ~ N
N o te th a t if a 0 ^ 1. w e can o b ta in (3.4.10) fro m (3.3.1) by div id in g b o th n u m e ra to r
an d d e n o m in a to r by aoA ra tio n a l fu n ctio n o f th e form (3.4.10) is called p r o p e r if a N ^ 0 an d M < N.
F ro m (3.3.2) it fo llo w s th a t th is is e q u iv a le n t to saying th a t th e n u m b e r o f finite
zero s is less th a n th e n u m b e r o f fin ite p o les.
A n im p ro p e r ra tio n a l fu n ctio n ( M > N ) can alw ays b e w ritte n as th e sum of
a p o ly n o m ial an d a p r o p e r ra tio n a l fu n ctio n . T h is p ro c e d u re is illu stra te d by the
fo llow ing ex am p le.
Example 3.4.4
Express the im proper rational transform
l + 3 z - ' + n z - 2 + l 2-3
1
5
1 + 6Z
+
in terms of a polynomial and a proper function.
Sec. 3.4
189
Inversion of the z-Transform
Solution First, we note that we should reduce the num erator so that the term s ; -2
and c- *' are eliminated. Thus we should carry out the long division with these two
polynomials written in reverse order. We stop the division when the order of the
rem ainder becomes
Then we obtain
= 1 + 2: - i +
^
In g e n e ra l, an y im p ro p e r ra tio n a l fu n ctio n (M > N ) can b e e x p ressed as
X ( ;) = W
i = Co + C i:" 1 + ' ' ' + Cu~n Z ~im~N) +
(3 A ll)
T h e in v erse z -tra n s fo rm o f the p o ly n o m ial can easily b e fo u n d by in sp ectio n .
W e fo cu s o u r a tte n tio n on th e inversion o f p r o p e r ra tio n a l tra n sfo rm s, since any
im p ro p e r fu n c tio n can b e tra n sfo rm e d in to a p ro p e r fu n c tio n by using (3.4.11).
W e carry o u t th e d e v e lo p m e n t in tw o steps. F irst, we p e rfo rm a p a rtia l fra c ­
tio n e x p a n sio n o f th e p r o p e r ra tio n a l fu n ctio n a n d th e n w e in v ert each o f th e
term s.
L e t A"(c) b e a p ro p e r ra tio n a l fu n ctio n , th a t is,
* (.-) = —
= - " - - ' " T ,1 +—
+buZ^
D( z)
1 + ^ iC
+ • *• -t~
(3.4.12)
w h ere
aN ^ 0
M < N
an d
T o sim p lify o u r discu ssion w e e lim in ate n eg ativ e p o w ers of c by m ultip ly in g b o th
th e n u m e r a to r a n d d e n o m in a to r of (3,4,12) by z N. T h is resu lts in
L
„N
I
L
_ y v -l
I
I
L
„ N -M
X{z) =
CA 4- a \ z N
w hich c o n ta in s only p o sitiv e p o w e rs o f
(3.4.13,
+ ------(- <a/v
Since N > M , th e fu n ctio n
,N -2 _i_____ l
(3,4.14)
;
: w + tiic w- 1 + - - - + flW
is also alw ays p ro p e r.
O u r ta sk in p e rfo rm in g a p a rtia l-fra c tio n ex p an sio n is to ex p ress (3.4.14)
o r, e q u iv a le n tly , (3.4.12) as a sum o f sim ple fractio n s. F o r th is p u rp o se w e first
fa c to r th e d e n o m in a to r p o ly n o m ial in (3.4.14) in to facto rs th a t c o n tain th e poles
P i, p 2, . . . , p n o f X (z). W e d istinguish tw o cases.
Distinct poles. S u p p o se th a t th e p o le s p \ , p 2 ........ p/v a re all d iffe re n t (dis­
tin ct). T h e n w e se e k an ex p an sio n of th e fo rm
z
z — p\
z — P2
+ -. - +
( 3 .
z — Pn
4
.
1
5
)
T h e p ro b le m is to d e te rm in e th e coefficients A i , A 2 , . - . , A s - T h e re a re tw o w ays
to so lv e th is p ro b le m , as illu stra te d in th e follow ing exam p le.
190
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
Example 3.4 .5
D eterm ine the partial-fraction expansion of the proper function
(3'‘U 6 ,
Solution First we elim inate the negative powers, by multiplying both num erator and
denom inator by z2. Thus
.2
X i z ) = zV2 - 11.5z
< +i 0n.5
The poles of X(z) are p\ = 1 and P2 = 0.5. Consequently, the expansion of the form
(3.4,15) is
X(z)
z
Cz —l)(z —0.5)
A\
z
1
A2
z —0.5
(3.4.17)
A very simple method to determ ine A[ and A 2 is to multiply the equation by the
denom inator term (z - l)(z - 0.5). Thus we obtain
z = ( z - 0 .5 M i + ( z - 1 ) A 2
(3.4.18)
Now if we set z = p\ = 1 in (3.4.18), we eliminate the term involving A2. Hence
1 = (1 -0 .5 )A ,
Thus we obtain the result A i = 2. Next we return to (3.4.18) and set z = p 2 = 0.5,
thus eliminating the term involving Ai, so we have
0.5 = ( 0 .5 - 1)A2
and hence Ai = —1. Therefore, the result of the partial-fraction expansion is
X(z)
2
z- 1
1
z - 0.5
(3.4.19)
T h e exam ple given ab ove suggests that w e can determ ine the coefficients A \,
A i , . . . , Afj , by m ultiplying b oth sides o f (3.4.15) by each o f the term s (z - Pk).
k = 1 , 2 , , . . , N , and evaluating the resulting exp ression s at the corresp on d ing pole
p osition s, p \ , p i .........P n ■ T h u s w e have, in general,
(Z- Pl)X(;) = (z-wM.i+ ... + /lt+,..+ fa-P»)^
z
z - PI
(3420)
z - Pn
C onsequently, w ith z = Pk, (3.4.20) yield s the Jtth coefficient as
Ak =
( z ~ ^ )X (; )i
z
k = 1, 2. . N
(3.4.21)
\z-pl
Exam ple 3.4.6
Determ ine the partial-fraction expansion of
1 17- ^ 0 .5 ,-
,3A22)
Sec. 3.4
191
inversion of the z-Transform
Solution To eliminate negative powers o f ; in (3.4.22), we multiply both num erator
and denom inator by
Thus
X(z)
z+ 1
The poles of X(z) are complex conjugates
Px =
\ + J i
Pi =
\ ~ j \
z2 - ; + 0.5
and
Since p\ ^ p 2- we seek an expansion of the form (3.4.15). Thus
* (:)
; + l
A]
Ai
z
(z-p i)(:-p 2)
z-P\
z-pi
To obtain A , and A2, we use the formula (3,4.21), Thus we obtain
(z-pi)X(z)
At = -----------
; + 1 I
- P
( z - p 2)X(z)
A- = ------------
2 \ ^ P1
Pi
!
1
; + l
-
?+
3+ J5-5+./5
1
:=P2
T h e ex p a n sio n (3,4.15) a n d the fo rm u la (3.4.21) h o ld fo r b o th real a n d c o m ­
p lex p o les. T h e o n ly c o n s tra in t is th a t all p o les be d istin ct. W e also n o te th at
A; = A*. It can b e easily seen th a t th is is a c o n s e q u e n c e o f th e fact th a t p 2 = p ' .
In o th e r w o rd s, comp l ex- conj ugat e pol es result in c o mpl ex- con j ugat e coefficients in
the part ial-fraction expansi on. T h is sim ple resu lt will p ro v e v ery u se fu l la te r in o u r
d iscu ssio n .
M u ltip le - o rd e r p o l e s .
If X U) h a s a p o le o f m u ltip lic ity /, th a t is, it co n ta in s
in its d e n o m in a to r th e fa c to r (z - pk)1, th e n th e e x p a n s io n (3.4.15) is n o lo n g er
tru e. In th is case a d iffe re n t ex p an sio n is n e e d e d . F irst, w e in v e stig a te th e case of
a d o u b le p o le (i.e., 1 = 2).
Exam ple 3.4.7
D eterm ine the partial-fraction expansion of
Solution
First, we express (3.4,23) in terms of positive powers of
in the form
* ( ;) =
z2
:
(z + 1)(; - l ) 2
X (z) has a simple pole at p\ = - 1 and a double pole pi = p$ = 1. In such a case the
appropriate partial-fraction expansion is
™
= ______ t ______ =
+
z
(z + l)(z - 1)2
z+ 1 z - l
(z -1 )2
The problem is to determ ine the coefficients A 1, A2, and A3.
(3424)
192
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
We proceed as in the case of distinct poles. T o determ ine Aj, we multiply both
sides of (3.4.24) by (z + 1) and evaluate the result at z = - 1 . Thus (3.4.24) becomes
(z + l)X (z)
z+ 1
z+ 1
z
z-1
( z - 1)2
------------------ = Ai + ------ - A 2 + ------- r - z A i
which, when evaluated at z = - 1 , yields
1
4
(z + 1)X(z)
A i = ------------------
z
Next, if we multiply both sides of (3.4.24) by (z - l ) 2, we obtain
=
Z
+
+
<3.4.25,
z+ 1
Now, if we evaluate (3.4.25) at z = 1, we obtain A 3. Thus
(z — l) 2X(z) I
= 1
*
Li
2
The remaining coefficient Az can be obtained by differentiating both sides of
(3.4.25) with respect to z and evaluating the result at z = 1. Note that it is not
necessary formally to carry out the differentiation of the right-hand side of (3.4.25),
since all term s except A2 vanish when we set z = 1. Thus
d
A2 = —
dz
(z — l ) 2X(z)
T h e gen eralization o f the p rocedure in the exam p le ab ove to the case o f an
/th-order p ole ( z — p k)! is straightforward. T h e partial-fraction expansion must
contain the terms
Mk
■Alt
Z - Pk
( z - Pk)2
(z - Pk)1
T h e coefficients {A,*} can b e evalu ated through d ifferen tiation as illustrated in
E xam p le 3.4.7 for / = 2.
N ow that w e have perform ed the partial-fraction exp an sion , w e are ready to
take the final step in the inversion o f X (z). First, let us con sid er th e case in which
X ( z ) contains distinct poles. F rom the partial-fraction exp an sion (3.4.15), it easily
follow s that
X ( z ) — A \ -------------r + A i -------------- + ■■•-)- A n --------------1 -
P\Z~1
1 - PlZ
(3.4.27)
1 - P n Z~ 1
T he inverse z-transform , x ( n ) = Z ~ i { X( z ) ) , can b e ob tain ed by inverting each
term in (3.4.27) and taking the corresponding linear com b ination . From T ab le 3.3
it fo llo w s that these term s can be inverted using th e form ula
( p k)nu{n),
- ( P k ) nu ( - n - 1),
if RO C : |*| > Ip*J
(causal signals)
if RO C: |z| < \P k \
(anticausal sign als)
„
Sec. 3.4
Inversion of the ^-Transform
193
If th e signal jc(n) is cau sal, the R O C is |r | > p max- w h e re p max = m ax{|/>i|,
\p2\.........IpnII- In th is case all te rm s in (3.4.27) re su lt in causal signal c o m p o n e n ts
an d th e signal x ( n ) is given by
jr(/i) = (A i + A -Pz --------f" A Np nN )u(n)
(3.4.29)
If all p o le s a re real. (3.4.29) is th e d e sire d e x p ressio n fo r th e signal j («). T h u s a
causal sig n al, h av in g a ; -tra n s fo rm th a t co n tain s real a n d d istin ct p o les, is a linear
c o m b in a tio n o f re a l e x p o n e n tia l signals.
S u p p o se n o w th a t all p o les a re d istin ct b u t som e o f th e m a re co m p lex . In
this case so m e o f th e te rm s in (3.4.27) re su lt in com plex e x p o n e n tia l c o m p o n en ts.
H o w e v e r, if th e signal x ( n) is real, w e sh o u ld b e ab le to re d u c e th e se te rm s in to
real c o m p o n e n ts. If x ( n ) is real, th e p o ly n o m ials a p p e a rin g in X (z) have real co ­
efficients. In th is case, as w e h av e seen in S ectio n 3.3, if pj is a p o le , its com plex
c o n ju g a te p j is also a p o le. A s w as d e m o n s tra te d in E x am p le 3.4.6, th e c o rre s p o n d ­
ing coefficien ts in th e p a rtia l-fra c tio n e x p an sio n a re also co m p lex co n ju g ates. T h u s
th e c o n trib u tio n o f tw o c o m p lex -co n ju g ate p o les is of th e form
x k (n) = \ A k {pt )n + A H p l )"]«(«)
(3.4.30)
T h e se tw o te rm s can be c o m b in ed to form a real signal c o m p o n e n t. F irst,
we ex p re ss Aj an d Pj in p o la r form (i.e., a m p litu d e an d p h a se ) as
A* = \ Ak \eja'
(3.4.31)
Pi: =
(3.4.32)
w h ere a k an d fik a re th e p h a s e c o m p o n e n ts o f A k a n d p k. S u b stitu tio n o f th ese
re la tio n s in to (3.4.30) gives
x k{n) = IAi
’ + e - -'(A"+“‘ l]u(n)
or, eq u iv alen tly ,
x k(n) = 2|A *|r" c o s($ tn + a k)u(n)
(3.4.33)
T h u s w e co n clu d e th a t
Z - 1 ( - — — — r + -— ~ — r ) = 2 \ A k \rR
k c o s ( f tn + a k) u(n)
\i - p kz ~ l
i - p;z~ v
(3.4.34)
if th e R O C is |zj > \ pk \ = rk .
F ro m (3.4.34) we o b se rv e th a t ea c h p a ir o f c o m p le x -c o n ju g a te p o le s in th e
z -d o m ain resu lts in a causal sin u so id al signal c o m p o n e n t w ith an e x p o n e n tia l e n ­
v elo p e. T h e d ista n c e rk o f th e p o le fro m th e o rig in d e te rm in e s th e e x p o n e n tia l
w eig h tin g (g ro w in g if r k > 1, d ecay in g if r k < 1, c o n s ta n t if rk = 1). T h e angle of
th e p o le s w ith re sp e c t to th e p o sitiv e re a l axis p ro v id e s th e fre q u e n c y o f th e sin u ­
so id a l signal. T h e zero s, o r e q u iv alen tly th e n u m e ra to r o f th e ra tio n a l tran sfo rm ,
affect o n ly in d ire c tly th e a m p litu d e an d th e p h a se o f x k (n) th ro u g h A k.
In th e case o f mul t i pl e p o les, e ith e r re a l o r co m p lex , th e in v erse tra n sfo rm
o f te rm s o f th e fo rm A j ( z — p k)n is re q u ire d . In th e case o f a d o u b le p o le the
194
The 2 -Transform and Its Application to the Analysis of LTI Systems
Chap. 3
fo llow ing tra n sfo rm p a ir (see T a b le 3.3) is q u ite useful:
pz~x
(1 - p z ] )2
= n p nu( n)
(3.4.35)
p ro v id e d th a t th e R O C is |z| > \p\. T h e g e n e ra liz a tio n to th e case o f p o le s w ith
h ig h e r m u ltip licity is left as an exercise fo r th e re a d e r.
Example 3.4.8
Determ ine the inverse z-transform of
X U) =
1
1 - 1 .5 ;-' +0.5z~ 2
(a) RO C III > 1
(b) RO C Izl < 0.5
(c) RO C 0.5 < |z| < 1
Solution This is the same problem that we treated in Exam ple 3.4.2. The partialfraction expansion for X(z) was determined in Example 3.4.5. The partial-fraction
expansion of X(z) yields
=
<3-4-36>
To invert X(z) we should apply (3.4,28) for pi — 1 and p 2 = 0.5. However, this
requires the specification of the corresponding ROC.
(a) In case when the R O C is |z| > 1, the signal x(n) is causal and both term s in
(3.4.36) are causal terms. According to (3.4.28), we obtain
x(n) = 2 (l)n«(n) —(0.5)"u(n) = (2 — 0.5 ”)u(n)
(3.4.37)
which agrees with the result in Example 3.4.2(a).
(b) When the R O C is |z| < 0.5, the signal x(n) is anticausal. Thus both term s in
(3.4.36) result in anticausal components. From (3.4.28) we obtain
x(n) = [—2 + (0.5)'I]u(—n — 1)
(3.4.38)
(c) In this case the ROC 0.5 < |z| < 1 is a ring, which implies that the signal x(n) is
two-sided. Thus one of the term s corresponds to a causal signal and the other
to an anticausal signal. Obviously, the given ROC is the overlapping of the
regions (z| > 0.5 and |z| < 1. Hence the pole p 2 = 0.5 provides the causal part
and the pole p\ = 1 the anticausal. Thus
x(n) = -2 (1 ) "u( - n - 1) - (0.5)"«(n)
Example 3.4.9
D eterm ine the causal signal jc(n) whose z-transform is given by
(3.4.39)
Sec, 3.4
Inversion of the z-Transtorm
Solution
195
In Exam ple 3.4.6 we have obtained the partial-fraction expansion as
where
A, = Al = j - j
and
Pi = p ’ = i
Since we have a pair of complex-conjugate poles, we should use (3.4.34). The
polar forms of Aj and p, are
Hence
Example 3.4.10
D eterm ine the causal signal x(n) having the ;-transiorm
X(z) =
Solution
(1 + ; - ’)(]
From Example 3.4.7 we have
X(Z)
3
+
+ T
41
41
.-1
+ 2 ( 1 - ; - 1)2
By applying the inverse transform relations in (3.4.28) and (3.4.35), we obtain
1
3
1
f 1
x( n ) = - ( —l)"«(n) + t« (« ) + - n u ( n ) =
t(-1)
4
4
2
4
3
w*l
+ - + u(n)
4
2
3.4.4 Decomposition of Rational z-Transforms
A t th is p o in t it is a p p ro p ria te to discuss som e a d d itio n a l issues c o n c e rn in g th e
d e c o m p o s itio n o f ra tio n a l z-tran sfo rm s, w hich will p ro v e v ery u se fu l in th e im p le­
m e n ta tio n o f d isc re te -tim e system s.
S u p p o se th a t w e h ave a ra tio n a l z-tran sfo rm X ( z ) e x p re ss e d as
X( z) =
(3.4.40)
196
The z-T ransform and Its Application to the Analysis of LTI Systems
Chap. 3
w h ere, fo r sim plicity, w e h av e a ssu m ed th a t ao = 1. If M > N [i.e., X ( z ) is
im p ro p e r], w e c o n v e rt X (z) to a sum o f a p o ly n o m ial a n d a p r o p e r fu n c tio n
M-H
X ( z ) = J 2 c t z ~ k + X pA z )
k=o
(3.4.41)
If th e p o le s o f X pr( z ) are d istin ct, it can b e e x p a n d e d in p a rtia l fra c tio n s as
pT(z) = A \ - ------------- + A 2 ------------- r + --- + ^ jv -------------- r
l-p \z~ l
l ~ P 2Z~l
1 - PnZ~1
(3.4.42)
A s w e h av e a lre a d y o b se rv e d , th e re m ay b e so m e c o m p le x -c o n ju g a te p airs of
p o le s in (3.4.42). S in ce we u su a lly d eal w ith real signals, w e sh o u ld av o id com plex
co efficien ts in o u r d eco m p o sitio n . T h is can b e ach iev ed by g ro u p in g a n d co m b in in g
te rm s co n ta in in g co m p lex -co n ju g ate p o les, in th e follow ing w ay:
A — A p * z ~ l + A* - A * p z ~ l
1 - pz~]
1 - p z 1 - p*z ~' + PP*z ~2
1 — /5*z-1
(3.4.43)
bo + b -iZ ~ l
_
1 + a i z ~ l + 02 z ~2
w h ere
£>o = 2 R e ( A ) ,
a \ = —2 R e ( p )
b\ = - 2 Re (Ap*),
a 2 = \ p \2
(3.4.44)
a re th e d e sire d co efficients. O b v io u sly , a n y ra tio n a l tra n sfo rm o f th e fo rm (3.4.43)
w ith co efficien ts given by (3.4.44), w hich is th e case w h en a 2 — 4 a 2 < 0, can be
in v e rte d using (3.4.34). B y c o m b in in g (3.4.41), (3.4.42), a n d (3.4.43) w e o b ta in a
p a rtia l-fra c tio n e x p a n sio n fo r th e z -tra n s fo rm w ith distinct p o les th a t co n ta in s real
coefficients. T h e g e n e ra l re su lt is
M—N
Xiz) =
K\
1
££ 3 C Z - * + E
1.
1 l.
—1
+ Et l l +, a/ u z 1f'l+ a u Z - 22
<3 A 4 S >
w h ere K\ + 2 AS = N . O b v io u sly , if Af = TV, th e first te rm is ju s t a c o n stan t,
a n d w h e n M < N , th is te rm v an ish es. W h e n th e r e a re also m u ltip le p oles, som e
a d d itio n a l h ig h e r-o rd e r te rm s sh o u ld b e in clu d ed in (3.4.45).
A n a lte rn a tiv e fo rm is o b ta in e d by ex p ressin g X ( z ) as a p r o d u c t o f sim ple
te rm s as in (3.4.40). H o w e v e r, th e co m p le x -c o n ju g a te p o le s a n d z e ro s sh o u ld be
co m b in e d to av o id co m p lex co efficien ts in th e d e c o m p o sitio n . S u ch c o m b in atio n s
re su lt in se c o n d -o rd e r ra tio n a l te rm s o f th e follow ing form :
(1 - Z*Z-1 )(1 - z*kz ~ ' )
1 + b\kZ~l + b u z ~2
(1 - />*z- 1 ) ( l - p*kz ~ l )
1 -I- a u z -1 + a u z ~2
(3.4.46)
w h ere
b\ k = - 2 R t ( z k ) ,
flu = —2 R e (p * )
b u = jz i l ,
02* = \Pk\
(3.4.47)
Sec. 3.5
The One-sided 2 -Transform
197
A ssu m in g fo r sim p licity th a t M = N, w e se e th a t X (z) can be d e c o m p o s e d in the
follow ing way:
1 + a kz 1
1 + a u z 1+ axz 2
<3 -4 -48>
w h ere N = K\ + 2 / ^ * W e will r e tu rn to th e s e im p o rta n t fo rm s in C h a p te rs 7 a n d 8.
3.5 THE ONE-SIDED Z-TRANSFORM
T h e tw o -sid ed z -tra n sfo rm re q u ire s th a t th e c o rre sp o n d in g signals be specified
fo r th e e n tire tim e ran g e —oo < n < oo. T his re q u ire m e n t p re v e n ts its u se for
a v ery u se fu l fam ily o f p ra c tic a l p ro b lem s, n am ely th e e v a lu a tio n o f th e o u tp u t
o f n o n re la x e d system s. A s we recall, th e s e system s a re d e sc rib e d by d ifferen ce
e q u a tio n s w ith n o n z e ro initial c o n d itio n s. Since th e in p u t is a p p lie d a t a finite
tim e, say n ()l b o th in p u t a n d o u tp u t signals are specified fo r n > no, b u t by no
m e a n s a re z e ro fo r n < no- T h u s th e tw o -sid ed z-tran sfo rm c a n n o t b e used. In this
se c tio n w e d e v e lo p th e o n e -sid e d z-tra n sfo rm w hich can be u se d to solve differen ce
e q u a tio n s w ith in itial c o n d itio n s.
3.5.1 Definition and Properties
T h e one- si ded o r unilateral z -tra n sfo rm o f a signal x ( n ) is d efin ed by
CC
* + (;) n=0
W e also u se th e n o ta tio n s Z +{x(n)} a n d
*{n )
X + (z)
T h e o n e -sid e d z -tra n sfo rm d iffers fro m th e tw o -sid ed tra n sfo rm in th e low er
lim it o f th e su m m a tio n , w hich is aiw ays z e ro , w h e th e r or n o t th e signal x ( n ) is zero
fo r n < 0 (i.e., cau sal). D u e to th is choice o f lo w er lim it, th e o n e -sid e d z-tran sfo rm
h as th e fo llow ing ch aracteristics:
1. It d o e s n o t c o n ta in in fo rm a tio n a b o u t th e signal jc(n) fo r n e g a tiv e v alu es o f
tim e (i.e., fo r n < 0).
2. It is un i q u e o n ly fo r ca u sa l signals, becau se only th ese signals a re z e ro for
n < 0.
3. T h e o n e -sid e d z -tra n s fo rm A,+(z) o f x( n ) is id en tical to th e tw o -sid ed ztra n s fo rm o f th e signal x( n) u( n) . S ince x ( n ) u ( n ) is causal, th e R O C o f its
tra n sfo rm , a n d h en c e th e R O C of X + {z), is alw ays th e e x te rio r o f a circle.
T h u s w h en w e d e a l w ith o n e-sid ed z-tran sfo rm s, it is n o t n ecessa ry to re fe r
to th e ir R O C .
198
The /-T ran sform and Its Application to the Analysis of LTI Systems
Chap. 3
Example 3.5.1
Determ ine the one-sided z-transform of the signals in Exam ple 3.1.1.
Solution
From the definition (3.5.1), we obtain
x,(n) = {1, 2 ,5 ,7 ,0 ,1 }
t
X f ( z ) = 1 + 2 ; ' 1 + 5z~2 + 7z~3 + ;~ 5
x 2 (n) = {1.2, 5, 7, 0,1}
t
Jt:+ (z) = 5 + 7z ~l + z ~3
x 3(n) = (0 ,0 ,1 ,2 , 5,7, 0,1}
t
xA(n) = {2,4, 5, 7 ,0 ,1 )
t
X^(z ) = z ~2 + 2z"3 + 5z~4 + l z ' s + z~7
X4+ (z) = 5 + 7 ; ' 1 + z’ 3
x 5(n) ~ S(n) -*-1- *• X+(z) = 1
x6(n) = 6(n - A),
k > 0 -e—*•
*700 = <S(k -I- k),
k > 0
(z) = z~k
Xy (z) = 0
Note that for a noncausal signal, the one-sided z-transform is not unique. Indeed,
X 2 (z) = X^(z) but X2 (n) ^ x 4(n). Also for anticausal signals,
(z) is always zero.
A lm ost all properties we have studied for the tw o-sid ed z-transform carry o ver to
the on e-sid ed z-transform with the excep tion of the shi ft i ng property.
Shifting Property
C a se 1: T im e D e la y
If
x(n) «
X +(z)
then
k
X (n - k ) X * z~*[Ar+(z) + ] [ ] j c ( - n ) z n]
k> 0
(3.5.2)
n=l
In case jc(n) is causal, then
x{n ~ k ) «
z~k X +(z)
(3.5.3)
Proof. F rom the definition (3.5.1) w e have
Z +{x(n - k)} = z~
f S ( 0 z - ' + Y t x( l ) z -k
By changing the in d ex from / to n = — t he result in (3.5.2) is easily obtained.
Sec. 3.5
199
The One-sided z-Transform
Example 3.5.2
Determ ine the one-sided ^-transform of the signals
(a) x( n ) = a nu(n)
(b) Ai(n) = xin — 2) where x( n) = a "
Solution
(a) From (3.5.1) we easily obtain
(b) We will apply the shifting property for k = 2. Indeed, we have
Z +{ x ( n ~ 2)1 = ; " 2[X + ( ; ) + j : ( - 1 ) ; + j : ( - 2 ) - 2]
= r 2^ + (:) + J T ( - l) ;- 1 + * ( -2 )
Since x( —1) = a ~ '. x ( —2) = a~2. we obtain
^
1 - az ~l
+ <T2
T h e m e a n in g o f th e shifting p ro p e rty can be in tuitively ex p la in e d if w e w rite (3.5.2)
as follow s:
Z +[x (?i — &)} = [jc (—k)
x ( —£ + 1 ); ' +
+
1 ); *+* ]
(3.5.4)
+ :.~kX +(.z)
k> 0
T o o b ta in x { n - k ) ( k > 0) fro m ;t(r?), w e sh o u ld shift x ( n ) by k sa m p le s to th e right.
T h e n k “ n e w ” sa m p le s, x ( - k ) , x { —k + 1), — * ( - 1 ) , e n te r th e p o sitiv e tim e axis
w ith x ( —k) lo c a te d at tim e zero . T h e first te rm in (3.5.4) sta n d s fo r th e z-tran sfo rm
o f th e s e sam p les. T h e “o ld ” sa m p les o f x( n — k) a re th e sa m e as th o se o f .r(n)
sim ply sh ifted by k sa m p le s to th e right. T h e ir z-tra n sfo rm is o b v io u sly z _i’X + (z),
w hich is th e se co n d te rm in (3.5.4).
Case 2: Time advance
If
xin)
X +(z)
th e n
X+( z ) - Y x ( n ) z - n k > 0
x ( n + k) «— ►z
(3.5.5)
Proof . F ro m (3.5.1) we h ave
oc
Z +[x(n + *)} =
oc
+ k)z~n = zk Y m z ~ l
n=0
l= k
w h e re w e h a v e c h a n g e d th e in d ex o f su m m atio n fro m n t o 1 = n + k. N o w , from
200
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
(3.5.1) w e o b ta in
* + (z) = j r * ( / ) : “ ' = Y x { l ) z ~l +
/=o
i=0
i=t
By co m b in in g th e last tw o re la tio n s, w e easily o b ta in (3.5.5).
Exam ple 3.5.3
With x(n), as given in Example 3.5.2, determ ine the one-sided ^-transform of the
signal
xi (n) = x(n + 2 )
Solution
obtain
We will apply the shifting theorem for k = 2. From (3.5.5), with k = 2, we
Z +{x(n + 2)| = -2X+(z) - x (0 );2 - *(1);
But jc(0) = 1, jc(1) = a. and X + (z) = 1/(1 - a ; -1). Thus
,2
Z +{x(n + 2)) = -— ----- - - z 2 - az
1 —az~
T h e case o f a tim e ad v an ce can be in tu itiv e ly e x p la in e d as follow s. T o o b tain
x( n- i - k) , k > 0, we sh o u ld shift jc (n) by k sa m p le s to th e left. A s a re su lt, th e sam ples
* (0 ). * ( 1 ) , ___x ( k — 1) “ le a v e ” th e p o sitiv e tim e axis. T h u s w e first re m o v e th e ir
co n trib u tio n to th e X +(z), an d th e n m u ltip ly w h at re m a in s by z k to co m p e n sa te
fo r th e shifting o f th e signal by k sam p les.
T h e im p o rta n c e o f th e shifting p ro p e rty lies in its a p p lic a tio n to th e so lu tio n
of d ifferen ce e q u a tio n s w ith c o n s ta n t co efficien ts a n d n o n z e ro in itial co n d itio n s.
T h is m a k e s th e o n e-sid ed z -tra n sfo rm a v ery useful to o l fo r th e an aly sis o f recu rsiv e
lin e a r tim e -in v a ria n t d isc re te -tim e system s.
A n im p o rta n t th e o re m useful in th e analysis o f signals a n d system s is th e
final v alu e th e o re m .
F in al V a lu e T h e o re m .
If
x{n)
X +{z)
th e n
lim x( n ) = lim (z - l ) X + (z)
n-» 00
7-»l
(3.5.6)
T h e lim it in (3.5.6) exists if th e R O C o f (z - l)A '+ (z) in clu d es th e u n it circle.
T h e p ro o f o f th is th e o re m is left as an ex ercise fo r th e re a d e r.
T h is th e o re m is u sefu l w h en w e a re in te re s te d in th e a s y m p to tic b e h a v io r of
a signal x( n) a n d w e k n o w its z -tra n sfo rm , b u t n o t th e signal itself. In such cases,
esp ecially if it is co m p licated to in v e rt X + (z), w e can u se th e final v alue th e o re m
to d e te rm in e th e lim it o f x{n) as n goes to infinity.
Sec. 3.5
201
The One-sided z-Transform
Example 3.5.4
The impulse response of a relaxed linear time-invariant system is k(n) = a"u(n),
|« | < 1. D eterm ine the value of the step response of the system as n —►oo.
Solution
The step response of the system is
y{n) = h(n) * x(n)
where
Jt(n) = u(n)
Obviously, if we excite a causal system with a causal input the output will be causal.
Since h(n), x(n), v(n) are causal signals, the one-sided and two-sided z-transforms are
identical. From the convolution property (3.2.17) we know that the z-transforms of
h(n) and *(n) must be multiplied to yield the z-transform of the output. Thus
= .
■— , .
- , = 7----- tt? ------- r
1 - az 1 1 - z 1
(z - l ) ( z - cr)
ROC: |z| > |ar|
Now
(z - l)y (z) -
Z —a
ROC: |z| > |a|
Since |a | < 1 the R O C of (z - l)K(z) includes the unit circle. Consequently, we can
apply (3.5.6) and obtain
lim v(n) = lim —-— = ———
n—oc'
1 —a
3.5.2 Solution of Difference Equations
T he on e-sid ed z-transform is a very efficient tool for the solu tion of d ifference
eq u a tio n s w ith n on zero initial conditions. It ach ieves that by reducing the dif­
feren ce eq u ation relating the tw o tim e-d om ain signals to an eq u ivalen t algebraic
eq u ation relating their on e-sid ed z-transform s. T h is eq u ation can b e easily solved
to obtain the transform o f the desired signal. T he signal in the tim e dom ain is
o b ta in ed by inverting the resulting z-transform . W e will illustrate this approach
with tw o exam ples.
Exam ple 3-5.5
The well-known Fibonacci sequence of integer num bers is obtained by computing
each term as the sum of the two previous ones. The first few terms of the sequence are
1 ,1 ,2 , 3,5. 8 ,...
D eterm ine a closed-form expression for the n th term of the Fibonacci sequence.
Solution Let y(n) be the nth term of the Fibonacci sequence. Clearly, y(n) satisfies
the difference equation
y (n ) = y(n - 1) + y(n - 2)
(3.5.7)
202
The z -Transform and Its Application to the Analysis of LTI System s
Chap. 3
with initial conditions
v(0) = v(—11 + v(—2) = 1
(3.5.8a)
y (l) = y(0) + V(-1) = 1
(3.5.8b)
From (3.5.8b) we have y (—1) = 0. Then (3.5.8a) gives v(—2) = 1. Thus we have to
determ ine y(n), n > 0, which satisfies (3.5.7), with initial conditions y (—1) = 0 and
y(—2) = 1.
By taking the one-sided ^-transform of (3.5.7) and using the shifting property
(3.5.2). we obtain
y+(z) = [ ; - 'y +(-) + y (—1) ] + [z ~2Y ^ ( z ) + y (-2 ) + . v ( - l ) ; - 1]
or
1
K + (c) =
"
(3.5.9)
where we have used the fact that y{ —1) = 0 and v(—2) = 1.
We can invert K+(;) by the partial-fraction expansion m ethod. The poles of
y'+(;) are
1 -I- V5
P2 =
Pi -
l-V s
and the corresponding coefficients are A i = p i/V 5 and A 2 = -/> ;/V 5 . Therefore,
v(n) =
'l + V5 / 1 + v'S \ "
1 - V5 / I - n / 5 \ ’"
2V5
u(rt)
2 ^5
or, equivalently.
u{n)
(3.5.10)
Example 3.5.6
Determ ine the step response of the system
v(n) = ay(n — 1) + x(rt)
—
1< a < 1
(3.5.11)
when the initial condition is y (—1) = 1.
Solution
By taking the one-sided ^-transform of both sides of (3.5.11), we obtain
K+(;) = a [ ;- 'y * (c ) + v (- D ] + X +(z)
Upon substitution for v ( - l ) and X+(;) and solving for y + (;). we obtain the result
Y+(z) =
1 —a ; -1
1
(1 —a ; - l )(l - ; _l)
(3.5.12)
By performing a partial-fraction expansion and inverse transform ing the result, we
have
y(n) = a"
,u(n) + —;-------1 —or"+I
u(n)
1 —a
( l - a " +2) u ( n )
1 —a
( 3 .5 . 13)
Sec. 3.6
Analysis of Linear Tim e-Invariant Systems in the z-D om ain
203
3.6 ANALYSIS OF LINEAR TIME-INVARIANT SYSTEMS IN THE
Z-DOMAIN
In S e ctio n 3.4.3 w e in tro d u c e d th e sy stem fu n ctio n o f a lin e a r tim e -in v a ria n t sys­
tem a n d re la te d it to th e u n it sa m p le re sp o n se a n d to th e d iffe re n c e e q u a tio n
d e sc rip tio n o f sy stem s. In th is sectio n w e d escrib e th e use o f th e sy stem fu n c­
tio n in th e d e te rm in a tio n o f th e re sp o n s e of th e system to so m e ex c ita tio n signal.
F u rth e rm o re , w e e x te n d th is m e th o d o f a nalysis to n o n re la x e d sy stem s. O u r a tte n ­
tio n is fo c u se d o n th e im p o rta n t class o f p o le - z e r o sy stem s r e p re s e n te d by lin e a r
c o n s tan t-co efficien t d iffe re n c e e q u a tio n s w ith a rb itra ry in itial co n d itio n s.
W e also c o n s id e r th e to p ic o f sta b ility o f lin e a r tim e -in v a ria n t sy stem s an d
d esc rib e a test fo r d e te rm in in g th e sta b ility o f a sy stem b a s e d o n th e co efficien ts
o f th e d e n o m in a to r p o ly n o m ial in th e system fu n c tio n . F in ally , w e p ro v id e a
d e ta ile d an aly sis o f se c o n d -o rd e r system s, w hich fo rm th e b asic b u ild in g b lo ck s in
th e re a liz a tio n o f h ig h e r-o rd e r system s.
3.6.1 Response of Systems with Rational System
Functions
L e t us c o n s id e r a p o le - z e r o system d e s c rib e d by th e g e n e ra l lin e a r c o n s ta n tc o efficien t d iffe re n c e e q u a tio n in (3.3.7) a n d th e c o rre sp o n d in g system fu n ctio n
in (3.3.8). W e r e p re s e n t H ( z ) as a ra tio o f tw o p o ly n o m ials B ( z ) / A ( z ) , w h ere
B (z) is th e n u m e r a to r p o ly n o m ia l th a t c o n ta in s th e z e ro s o f H( z ) , a n d A ( z ) is the
d e n o m in a to r p o ly n o m ia l th a t d e te rm in e s th e p o le s o f H ( z ) . F u rth e rm o r e , let us
a ssu m e th a t th e in p u t signal x ( n ) h as a ra tio n a l z -tra n sfo rm X (z) o f th e fo rm
X(z) = —
Q( z)
(3.6.1)
T h is a s su m p tio n is n o t o v erly restrictiv e, since, as in d ic a te d p rev io u sly , m o st signals
o f p ra c tic a l in te re s t h av e ra tio n a l z -tran sfo rm s.
If th e sy stem is in itially re lax ed , th a t is, th e in itia l c o n d itio n s fo r th e d iffe re n c e
e q u a tio n are z e ro , y ( —1) = y ( —2) = • ■■ = y ( —N ) = 0, th e z -tra n s fo rm o f th e
o u tp u t o f th e sy stem h a s th e fo rm
Y(z) = H ( z ) X ( z ) =
Mz)Q(z)
(3.6.2)
N o w su p p o s e th a t th e sy stem c o n ta in s sim ple p o le s p \ , p j .........p s a n d th e ztra n sfo rm o f th e in p u t signal co n ta in s p o le s <71, qt, ■■■, q u w h e re p t ^ qm fo r all
it = 1, 2
a n d m = 1, 2 , . . . , L. In a d d itio n , w e a ssu m e th a t th e z e ro s o f
th e n u m e r a to r p o ly n o m ia ls B( z ) a n d N ( z ) d o n o t co in cid e w ith th e p o les {p t } an d
{<7i}, so th a t th e re is n o p o le - z e r o c a n c e lla tio n . T h e n a p a rtia l-fra c tio n ex p an sio n
o f K(z) yield s
t o 1 - PkZ 1
*-* 1 - qkZ 1
204
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
T h e in v erse tra n sfo rm o f ^ (z ) y ield s th e o u tp u t signal fro m th e sy stem in th e form
N
L
y (n ) = Y A t ( P k ) n u ( n ) + 5 2 £2*0?*)'1n (n)
*=i
*-i
(3.6.4)
W e o b se rv e th a t th e o u tp u t se q u e n c e y( n) can b e su b d iv id e d in to tw o p a rts. T he
first p a r t is a fu n ctio n o f th e p o les {pt} o f th e system an d is called th e natural
response o f th e sy stem . T h e in flu e n ce o f th e in p u t signal o n th is p a r t o f th e
re sp o n s e is th ro u g h th e scale facto rs {A*}. T h e se c o n d p a r t o f th e re sp o n se is a
fu n ctio n o f th e p o le s {qk} o f th e in p u t signal an d is called th e f o r c e d response of
th e sy stem . T h e in flu e n ce o f th e sy stem o n th is re sp o n s e is e x e rte d th ro u g h th e
scale facto rs { Qk } W e sh o u ld e m p h asize th a t th e scale fa c to rs {A*} an d { Q k } a re fu n ctio n s o f
b o th se ts o f p o le s {pk } a n d { ^ ) . F o r e x am p le, if AXz) = 0 so th a t th e in p u t is
zero , th e n K(z) = 0, a n d c o n s e q u e n tly , th e o u tp u t is z e ro . C lea rly , th e n , th e
n a tu r a l re sp o n se o f th e sy stem is zero . T h is im p lies th a t th e n a tu ra l re sp o n se of
th e sy stem is d iffe re n t fro m th e z e ro -in p u t re sp o n se .
W h en X (z) a n d H ( z ) h av e o n e o r m o re p o les in c o m m o n o r w h e n X (z)
a n d /o r H{ z ) c o n ta in m u ltip le -o rd e r p o les, th e n K(z) will h av e m u ltip le -o rd e r poles.
C o n se q u e n tly , th e p a rtia l-fra c tio n ex p a n sio n o f Y( z ) will c o n ta in fa c to rs o f th e form
1/(1 - p / z ~ l )t , k = 1, 2 , . . . , m, w h ere m is th e p o le o rd e r. T h e in v ersio n o f th ese
facto rs will p ro d u c e te rm s o f th e fo rm n k~ xp* in th e o u tp u t y( n ) o f th e system , as
in d ic a te d in S ectio n 3.4.2.
3.6.2 Response of Pole-Zero Systems with Nonzero
Initial Conditions
S u p p o se th a t th e sig n al x ( n ) is a p p lie d to th e p o le - z e r o sy stem at n = 0. T hus
th e sign al x ( n ) is a ssu m ed to be cau sal. T h e effe cts o f all p re v io u s in p u t signals to
th e system a re reflec te d in th e in itial c o n d itio n s y ( —1), y ( ~ 2 ) .........y ( —N ) . Since
th e in p u t x ( n ) is ca u sa l a n d since w e a re in te re s te d in d e te rm in in g th e o u tp u t y ( n )
fo r n > 0, w e can u se th e o n e -sid e d z-tra n s fo rm , w hich allow s us to d e a l w ith th e
in itial co n d itio n s. T h u s th e o n e -sid e d z -tra n s fo rm o f (3.4.7) b e c o m e s
■ f t) —
E
akz
Y +{z) + Y j y (' - n ) z ''
+ £ f c * z - * X + (z)
(3.6.5)
S ince x ( n) is cau sal, we can se t X + (z) = X (z). In an y case (3.6.5) m ay b e e x p ressed
as
52 ^
Y +(z) =
*=0
s
-X(z)-
1+52<:
= H(z)X(z) +
akz
No(z)
A (z)
(3.6.6)
Sec. 3.6
Analysis of Linear Tim e-Invariant Systems in the z-D om ain
205
w h e re
N
k
No(z) = - 5 Z a *z_,t
k= 1
n=l
(3.6.7)
F ro m (3.6.6) it is a p p a r e n t th a t th e o u tp u t o f th e sy stem w ith n o n z e ro initial
c o n d itio n s can b e su b d iv id e d in to tw o p a rts. T h e first is th e z e ro -sta te re sp o n se of
th e sy stem , d e fin ed in th e z -d o m ain as
Ya (z) = H ( z ) X ( z )
(3.6.8)
T h e se c o n d c o m p o n e n t c o rre sp o n d s to th e o u tp u t re su ltin g fro m th e n o n z e ro initial
co n d itio n s. T h is o u tp u t is th e z e ro -in p u t re sp o n se o f th e system , w hich is defined
in th e z -d o m ain as
(z) = TTT
A(z)
a 6 ’9)
H e n c e th e to ta l re sp o n se is th e su m o f th ese tw o o u tp u t c o m p o n e n ts , w hich can
b e e x p ressed in th e tim e d o m a in by d e te rm in in g th e in v erse z -tra n s fo rm s o f Kzs(;)
a n d Y A z ) s e p a ra te ly , a n d th e n ad d in g th e resu lts. T h u s
y ( n) = >zs(«) + >'zi(n)
(3.6.10)
S ince th e d e n o m in a to r of Fz|( z ) , is A(z), its p o le s are p i , P 2 ........ Ps- C o n se ­
q u e n tly , th e z e ro -in p u t re sp o n se h a s th e form
N
>'zi(n) =
52
(3.6.11)
T h is can b e a d d e d to (3.6.4) a n d th e te rm s involving th e p o les {/?*} can be co m b in ed
to yield th e to ta l re sp o n s e in th e form
y( n) =
N
L
k= 1
k=\
52 A k' ( Pk) nu( n) + 51 Q k i q k T u ( n )
(3.6.12)
w h e re , by d efin itio n ,
A'k =
+ Dk
(3.6.13)
T h is d e v e lo p m e n t in d icates clearly th a t th e effect o f th e initial co n d itio n s
is to a lte r th e n a tu ra l re sp o n s e o f th e system th ro u g h m o d ifica tio n o f th e scale
f a c to rs {Ak}. T h e re a re n o n ew p o le s in tro d u c e d by th e n o n z e ro in itial co n d itio n s.
F u rth e rm o r e , th e r e is no effect on th e fo rce d re sp o n s e o f th e system . T h ese
im p o rta n t p o in ts a re re in fo rc e d in th e follow ing e x am p le.
Exam ple 3.6.1
D eterm ine the unit step response of the system described by the difference equation
y{n) = 0.9y(n —1) - 0.81y(n - 2) + x( n)
under the following initial conditions:
(a) y ( - l ) = > (-2 ) = 0
(b) y ( - 1) = y ( - 2 ) = 1
206
The z-Transform and Its Application to the Analysis of LTI Systems
Solution
Chap. 3
The system function is
H {z) =
1
1 - 0 . 9 : - 1 + 0 .8 1 ;- 2
This system has two complex-conjugate poles at
P 2 = G. 9e- jn/i
p l = Q.9e^°
The z-transform of the unit step sequence is
1
X(z) =
1-z -1
Therefore,
1
(1 - 0 . 9 e j ”^ z - l K l - 0 . 9 e - j ”'1z -'l ) ( l - z ~ ' )
0.542 - /0 .0 4 9
1 - 0 .9 e ^ f h ~ l
0.542 + y0.049
+ :---- — r +
1 - 0 .9e->nPz - 1
1.099
1 - z~l
and hence the zero-state response is
y^(n) =
£l.099 + 1.088(0.9)" cos ( ~ n - 5.2C) J u ( n )
(a) Since the initial conditions are zero in this case, we conclude that v(n) = >'is(fl).
(b) For the initial conditions v ( - l ) = v (-2 ) = 1, the additional com ponent in the
z-transform is
Mi(z)
A(z)
I'ziW
0.09 - 0.S ir " 1
1 - 0 . 9 ; - ' + 0.81r“2
0.026 + j0.4936 i 0.026 - jO.4936
“
1 - 0 H e i ' f i z - 1 + 1 - 0 . 9 e - '* P z ~ '
Consequently, the zero-input response is
yzi(n) = 0.988(0.9)'' cos ^ ~ n + 87°^J u(n)
In this case the total response has the z-transform
y (z ) =
J ^ w + y^ u )
1.099
0.568 + y'0.445
0.568 - /0.445
+ :---- „ r +
1 - z "1
1 - Q.9eJ*Vz- 1
1 - 0.9e“' ff/3; -1
The inverse transform yields the total response in the form
y(n) = 1.099u(n) + 1.44(0.9)" cos ( ^ n + 3 8 ^ u (n)
3.6.3 Transient and Steady-State Responses
A s w e h av e seen fro m o u r p re v io u s d iscussion, th e re sp o n se o f a sy ste m to a given
in p u t can b e se p a ra te d in to tw o c o m p o n e n ts , th e n a tu r a l re sp o n s e a n d th e fo rce d
Sec. 3.6
Analysis of Linear Tim e-Invariant Systems in the z-Dom ain
207
re sp o n se . T h e n a tu ra l re sp o n s e o f a cau sal system h a s th e fo rm
N
>’nr(n) =
52 A k i p k ) nu{ n)
(3.6.14)
*=1
w h e re {pk), k = 1, 2,
N a re th e p o le s o f th e sy stem a n d {A*} a re sc ale fac­
to rs th a t d e p e n d o n th e in itial c o n d itio n s an d on th e c h a ra c te ristic s o f th e in p u t
se q u en ce.
If |/>*| < 1 fo r all k, th e n , y nT(n) d ecay s to z e ro as n a p p ro a c h e s infinity. In
such a case w e re fe r to th e n a tu ra l re sp o n se of th e system as th e t ransient response.
T h e ra te a t w hich >'nr(n) d ecay s to w a rd z e ro d e p e n d s on th e m a g n itu d e o f th e p o le
p o sitio n s. I f all th e p o le s h av e sm all m a g n itu d e s, th e d ec a y is v ery ra p id . O n the
o th e r h a n d , if o n e o r m o re p o le s a re lo c a te d n e a r th e u n it circle, th e c o rre sp o n d in g
te rm s in >’nr(n) w ill d e c a y slow ly to w a rd z e ro a n d th e tra n s ie n t will p ersist fo r a
relativ ely lo n g tim e.
T h e fo rc e d re sp o n s e o f th e system h as th e fo rm
i
Vfr(«) = 5 2 <2*(<?*)"«(«)
k=\
(3.6.15)
w h e re {^t), k = 1, 2 , . . . , L a re th e p o le s in th e forcing fu n c tio n a n d { Qk } a re
scale fa c to rs th a t d e p e n d o n th e in p u t se q u e n c e an d o n th e c h a ra c te ristic s o f th e
system . If all th e p o le s o f th e in p u t signal fall in sid e th e u n it circle, ^ ( n ) will d ecay
to w a rd z e ro as n a p p ro a c h e s infinity, ju st as in th e case o f th e n a tu ra l resp o n se.
T h is sh o u ld n o t b e su rp risin g since th e in p u t signal is also a tra n s ie n t signal. O n
th e o th e r h a n d , w h e n th e cau sal in p u t signal is a sin u so id , th e p o le s fall o n th e unit
circle a n d c o n s e q u e n tly , th e fo rce d re sp o n se is also a sin u so id th a t p ersists fo r all
n > 0. In th is case, th e fo rce d re sp o n se is called th e steady-state respons e o f th e
system . T h u s, fo r th e sy stem to sustain a ste a d y -s ta te o u tp u t fo r n > 0, th e in p u t
sig n al m u st p e rsist fo r all n > 0.
T h e fo llo w in g ex a m p le illu stra te s th e p re se n c e o f th e s te a d y -s ta te resp o n se.
Exam ple 3*6.2
D eterm ine the transient and steady-state responses of the system characterized by
the difference equation
>{n) = 0.5;y(n - 1) + jc(n)
when the input signal is x ( n ) = 10cos(jrn/4)u(n). The system is initially at rest (i.e.,
it is relaxed).
Solution
The system function for this system is
and therefore the system has a pole at z = 0.5. The z-transform of the input signal is
(from Table 3.3)
10(1 — ( l/y / 2 ) z ~ 1)
X (z ) ---------------f ---------------1 - y / 2 £_1 + Z~1
208
The z-Transform and Its Application to the Analysis of LTI S ystem s
Chap. 3
Consequently.
K (o = H (:)X (z )
10(1 - ( l / v '2 ) ; - 1)
(1 —0.5~-‘ )(1 - e - w 4; - 1)!!
6.78e~J2&1
6.3
1 - 0.5;- 1
‘ ,—"T
1-
i)
6.7&ej2K1
1 - e-J*iAz~x
The natural or transient response is
ynr(n) = 6.3(0.5)"u(«)
and the forced or steady-state response is
Vfr (n) =
+6. 1&eJ2*J e - inn!A]u(},)
= 13.56cos (^~^n —2 8 . 7 w(«)
Thus we see that the steady-state response persists for all n > 0. just as the input
signal persists for all n > 0.
3.6.4 Causality and Stability
A s d efin ed p rev io u sly , a causal lin e a r tim e -in v a ria n t system is o n e w hose unit
sa m p le resp o n se h (n ) satisfies the co n d itio n
h (n ) = 0
n < 0
W e h av e also show n th at the R O C o f th e ;;-tra n sfo rm o f a cau sal se q u e n c e is the
e x te rio r o f a circle. C o n se q u e n tly , a lin e a r t im e -in v a r ia n t sy ste m is c a u s a l i f an d
o n ly i f the R O C o f the system f u n c t io n is the e x te rio r o f a c ir c le o f ra d iu s r < 00,
in c lu d in g the p o in t z — d o .
T h e stab ility o f a lin ear tim e -in v a ria n t system can also be e x p re ss e d in term s
o f th e ch a ra c te ristic s o f th e system fu n ctio n . A s w e recall fro m o u r prev io u s
d iscu ssio n , a n ecessa ry an d sufficient co n d itio n fo r a lin e a r tim e -in v a ria n t system
to b e B IB O sta b le is
52
n=- x
In tu rn , this c o n d itio n im plies th a tt H ( z ) m
u st c o n ta in th e u n it circle w ith in its R O C .
musi
In d e e d , since
OC
H (z) =
h (n)z
it follow s th a t
OC
OC
n ——oc
n= —oc
W h en e v a lu a te d o n th e u n it circle (i.e., |z| = 1),
OC
\ h (z )\ <
52
Sec. 3.6
Analysis of Linear Tim e-Invariant Systems in the 7 -Domain
209
H en ce, if the system is B IB O stable, the unit circle is con tained in the R O C o f
H(z)- T h e con verse is also true. T h erefore, a linear tim e-in va ria n t sy stem is B IB O
stable i f a n d o n ly i f th e R O C o f the sy stem fu n c tio n includes the u n it circle.
W e should stress, how ever, that the con d ition s for causality and stability are
d ifferent and that o n e d oes not im ply the other. F or exam p le, a causal system
m ay b e stable or unstable, just as a noncausal system m ay b e stable or unstable.
Sim ilarly, an unstable system m ay be eith er causal or n oncausal, just as a stable
system m ay be causal or noncausal.
For a causal system , h ow ever, the con d ition on stability can be narrowed
to so m e exten t. In d eed , a causal system is characterized by a system function
H ( z ) having as a R O C the exterior o f som e circle o f radius r. For a stable
system , the R O C m ust include the unit circle. C on sequ en tly, a causal and sta­
ble system m ust have a system function that con verges for |z| > r < 1. Since
the R O C cannot contain any p oles o f H ( z ) , it follow s that a causal linear tim ein va ria n t sy stem is B I B O stable i f a n d o n ly i f all the p o le s o f H ( z ) are inside the
u n it circle.
Example 3.63
A linear tim e-invariant system is characterized by the system function
3 - 4 z-'
H(z) ~ 1 - 3.5z-> + 1.5: ' 2
1
"
2
l - ' i z - 1 + 1 - 3 z- 1
Specify the R O C of H(z) and determ ine h(n) for the following conditions:
(a) The system is stable.
( b ) The system is causal.
(c) The system is anticausal.
Solution
T he system has poles at z = 5 and z = 3.
(a) Since the system is stable, its R O C must include the unit circle and hence it is
\ < \z\ < 3. Consequently, h(n) is noncausal and is given as
h(n) = (i)"n(«) - 2(3)-it ( - n - 1)
(b) Since the system is causal, its R O C is jz| > 3. In this case
/i(n) = ( i r « ( n)+ 2 (3 )"u (n )
This system is unstable.
(c) If the system is anticausal, its R O C is |z| < 0.5. Hence
h{n) = - { ( \ y + 2 ( 3 T ) u ( - n - l )
In this case the system is unstable.
210
The z - T ransform and Its Application to the Analysis of LTI Systems
Chap. 3
3.6.5 Pole-Zero Cancellations
W h en a z -tra n sfo rm h as a p o le th a t is a t th e sam e lo catio n as a z e ro , th e pole
is ca n c e le d by th e z e ro an d , c o n s e q u e n tly , th e te rm c o n ta in in g th a t p o le in the
in v erse z-tra n sfo rm v an ish es. S uch p o le - z e r o c a n ce llatio n s a re v ery im p o rta n t in
th e an alysis o f p o le - z e r o system s.
P o le -z e ro c a n ce llatio n s can o c c u r e ith e r in th e sy stem fu n c tio n itself o r in
th e p ro d u c t o f th e sy stem fu n ctio n w ith th e z -tra n sfo rm o f th e in p u t signal. In the
first case w e say th a t th e o r d e r o f th e sy stem is re d u c e d by o n e . In th e la tte r case
w e say th a t th e p o le o f th e system is su p p re s se d by th e z e ro in th e in p u t signal,
o r vice v ersa. T h u s, by p ro p e rly se lectin g th e p o sitio n o f th e ze ro s o f th e in p u t
signal, it is p o ssib le to su p p re ss o n e o r m o re sy stem m o d es (p o le fa c to rs) in th e
re sp o n se o f th e sy stem . S im ilarly, by p r o p e r se le c tio n o f th e z e ro s o f th e system
fu n ctio n , it is p o ssib le to su p p re s s o n e o r m o re m o d e s o f th e in p u t signal fro m the
re sp o n se o f th e system .
W h e n th e z e ro is lo c a te d v ery n e a r th e p o le b u t n o t ex actly a t th e sa m e loca­
tio n , th e te rm in th e re sp o n se h as a v ery sm all a m p litu d e . F o r e x a m p le , n o n ex act
p o ie - z e r o c an ce llatio n s can o ccu r in p ra c tic e as a re su lt o f in su ffician t n u m erical
p recisio n u sed in re p re se n tin g th e co efficien ts o f th e system . C o n s e q u e n tly , one
sh o u ld n o t a tte m p t to stab ilize an in h e re n tly u n sta b le system by p lacing a z e ro in
th e in p u t signal at th e lo catio n o f th e pole.
Example 3.6.4
Determ ine the unit sample response of the system characterized by the difference
equation
v(n) = 2.5 v(n —1) —>’(n —2) + jr(n) —5jr(n — 1) + 6x(n — 2)
Solution
The system function is
1 - 5*-1 + 6z ' 2
(1 - i z - ') ( l - 2 z - ‘
This system has poles at p\ = 2 and p x = ~, Consequently, at first glance it appears
that the unit sample response is
Y(z) = H( z) X( z) =
1 - 5 z ~ ' + 6z“2
(1 - j z _ ,)(l - 2z_l
By evaluating the constants at z = j and z = 2, we find that
A = *
B =0
The fact that 5 = 0 indicates that there exists a zero at z = 2 which cancels
the pole at z = 2. In fact, the zeros occur at z = 2 and z = 3. Consequently, H{z)
Sec. 3.6
Analysis of Linear Tim e-Invariant Systems in the z-Dom ain
211
reduces to
H(z) =
1 - 3 : -1
1- k -1
2 .5 ;-1
= 1and therefore
h(rt) = S{n) -
2 .5 ( 5)"
~ ^
The reduced*order system obtained by canceling the common pole and zero is char­
acterized by the difference equation
y(n) = iy(n - 1) + x(n) - 3x{n - 1)
Although the original system is also BIBO stable due to the pole-zero cancellation,
in a practical im plementation of this second-order system, we may encounter an
instability due to imperfect cancellation of the pole and the zero.
Example 3.6.5
D eterm ine the response of the system
v(n) = jjy(>i - 1) - £y(n - 2) + x(n)
to the input signal x{n) = S(n) — ^&(n — 1).
Solution
The system function is
1
WU) =
(1 -
(1 - 1 c-1)
This system has two poles, one at z = ^ and the other at : =
the input signal is
The ^transform of
X {z ) = 1 - j i ' 1
In this case the input signal contains a zero at ; = i which cancels the pole at ; =
Consequently,
Y(z) = HU) X( z )
m
-
i r p
and hence the response of the system is
y(n) = ( 5)n«(n)
Clearly, the m ode ( |)" is suppressed from the output as a result of the pole-zero
cancellation.
3.6.6 Multiple-Order Poles and Stability
A s w e h a v e o b se rv e d , a n ecessa ry an d sufficient c o n d itio n fo r a causal lin e a r tim ein v a ria n t sy stem to b e B IB O sta b le is th a t all its p o le s lie in sid e th e u n it circle.
T h e in p u t sig n al is b o u n d e d if its z -tra n s fo rm c o n ta in s p o le s {qk}, k = 1 , 2 — , L,
212
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
w hich satisfy th e c o n d itio n \qk \ < 1 fo r all k. W e n o te th a t th e fo rc e d re sp o n se of
th e system , given in (3.6.15). is also b o u n d e d , ev en w h e n th e in p u t signal contains
o n e o r m o re d istin ct p o les on th e u n it circle.
In view o f th e fact th a t a b o u n d e d in p u t signal m ay h av e p o le s on th e unit
circle, it m ight a p p e a r th a t a sta b le system m ay also h av e p o le s on th e u n it circle.
T his is n o t th e case, h o w ev er, since such a sy stem p ro d u c e s an u n b o u n d e d resp o n se
w h en ex cited by an in p u t signal th a t also has a p o le a t th e sa m e p o sitio n o n the
u n it circle. T h e follow ing ex a m p le illu stra te s th is p o in t.
Example 3.6.6
D eterm ine the step response of the causal system described by the difference equation
v(n) = y{n - 1) + x{n)
Solution
The system function for the system is
We note that the system contains a pole on the unit circle at c = 1. The ^-transform
of the input signal x(n) = u{n) is
which also contains a pole at ; = 1. Hence the output signal has the transform
K(=) = H {z)X {z)
1
“
(1 - ; - 1)2
which contains a double pole at z = 1.
The inverse z-transform of Y(z) is
>-(«)= (n + l)n(B)
which is a ramp sequence. Thus y(n) is unbounded, even when the input is bounded.
Consequently, the system is unstable.
E x am p le 3.6.6 d e m o n s tra te s clearly th a t B IB O stab ility r e q u ire s th a t th e sys­
tem p o le s b e strictly in side th e u n it circle. If th e sy stem p o les a re all inside th e unit
circle an d th e e x c ita tio n se q u e n c e x ( n ) c o n ta in s o n e o r m o re p o le s th a t coincide
w ith th e p o les o f th e sy stem , th e o u tp u t Y(z ) w ill c o n ta in m u ltip le -o rd e r poles. As
in d ic a te d p rev io u sly , such m u ltip le -o rd e r p o les re su lt in an o u tp u t se q u e n c e th at
co n tain s term s o f th e form
A kn b( p k)nu(n)
w h ere 0 < b < m — 1 an d m is th e o r d e r o f th e p o le. If |p*| < 1, th e s e te rm s decay
to z e ro as n a p p ro a c h e s infinity b e c a u s e th e e x p o n e n tia l f a c to r (pk) n d o m in ates
th e te rm n b. C o n se q u e n tly , n o b o u n d e d in p u t signal can p r o d u c e an u n b o u n d e d
o u tp u t signal if th e sy stem p o les a re all in sid e th e u n it circle.
Sec. 3.6
Analysis of Linear Tim e-Invariant Systems in the z - D o m a in
213
F in ally , w e sh o u ld s ta te th a t th e o n ly u sefu l system s w h ich c o n ta in p o les
on th e u n it circle a re th e d ig ital o sc illato rs discussed in C h a p te r 4. W e call such
sy stem s mar gi nal l y stable.
3.6.7 The Schur-Cohn Stability Test
W e h av e sta te d p rev io u sly th a t th e sta b ility o f a sy stem is d e te rm in e d by th e
p o sitio n o f th e p o les. T h e p o les o f th e system are th e ro o ts o f th e d e n o m in a to r
p o ly n o m ia l o f H ( z ), n am ely ,
A(z) = 1 + a \ z 1 + Q-2Z 2 + ■• • + QpfZ ^
(3.6.16)
W h en th e sy stem is cau sal all th e ro o ts o f A (z) m u st lie inside th e u n it circle fo r
th e sy stem to b e sta b le .
T h e re a re se v e ra l c o m p u ta tio n a l p ro c e d u re s th a t aid u s in d e te rm in in g if any
o f th e ro o ts o f A (z) lie o u tsid e th e u n it circle. T h e se p ro c e d u re s a re called stability
criteria. B e io w w e d esc rib e th e S c h u r-C o h n test p ro c e d u re fo r th e sta b ility o f a
sy stem c h a ra c te riz e d by th e system fu n c tio n H( z ) = B ( z ) / A { z).
B e fo re w e d e sc rib e th e S c h u r-C o h n te st w e n e e d to e sta b lish so m e useful
n o ta tio n . W e d e n o te a p o ly n o m ia l o f d e g re e m by
Am(z) = £ a m(* )z - A
i=0
a m(0) = l
(3.6.17)
T h e reci procal o r reverse p o l y n o m i a l Bm(z) o f d e g ree m is d e fin e d as
Bm(z) =
z-')
(3.6.18)
UmV"‘ —
k
k*=0
W e o b se rv e th a t th e coefficients o f B m( z) are th e sa m e a s th o se o f Am(z), b u t
in re v e rse o rd e r.
In th e S c h u r-C o h n sta b ility te s t, to d e te rm in e if th e p o ly n o m ia l A (z) has all
its r o o ts in sid e th e u n it circle, w e c o m p u te a set o f coefficients, called reflection
coefficients, K \ , K i .........K n fro m th e p o ly n o m ials A m(z). F irst, w e set
A N (z) = A( z )
and
(3.6.19)
ATjV = a # ( N )
T h e n w e c o m p u te th e lo w e r-d e g re e p o ly n o m ials Am(z), m = N , N — 1, N — 2 , . . . , 1,
a c c o rd in g to th e rec u rsiv e e q u a tio n
A
^
^m (z) — K mB m(z)
A m- i ( z ) = --------1
--------
^ c
(3.6.20)
w h e re th e c o effic ie n ts K m a re d e fin ed as
Km = am{m)
(3.6.21)
214
The z-Transform and Its Application to the Analysis of LTI Systems
Chap, 3
T h e S c h u r-C o h n sta b ility test sta te s th a t the p o l y n o m i a l A (;) gi en by (3.6.16)
has all its roots inside the unii circle i f a n d onl y i f the coefficients K m satisfy the
condit i on \Km \ < 1 f o r all m — 1, 2 ........ N.
W e shall n o t p ro v id e a p r o o f o f th e S c h u r-C o h n te s t at th is p o in t. The
th e o re tic a l ju stificatio n fo r th is te s t is given in C h a p te r 11. W e illu stra te th e com ­
p u ta tio n a l p ro c e d u re w ith th e follow ing exam ple.
Example 3.6.7
Determ ine if the system having the system function
is stable.
Solution
We begin with A ;(;), which is defined as
A 2 (z ) =
1 -
lz~] -
k " 2
Hence
Now
#>(:) =
and
/\2(r) 1 - K;
Therefore.
Ki = - I
Since |ATi | > 1 il follows that the system is unstable. This fact is easily estab­
lished in this example, since the denom inator is easily factored to yield the two poles
at pi = —2 and p 2 —
However, for higher-degree polynomials, the Schur-Cohn
test provides a simpler test for stability than direct factoring of //(- ) .
T h e S c h u r-C o h n sta b ility te s t can b e easily p ro g ra m m e d in a d igital c o m p u ter
an d it is very efficien t in te rm s o f a rith m e tic o p e ra tio n s. S pecifically, it req u ires
o n ly N 2 m u ltip lic a tio n s to d e te rm in e th e co efficien ts {Km}, m = 1 , 2 .........N. The
recu rsiv e e q u a tio n in (3.6.20) can b e e x p ressed in te rm s o f th e p o ly n o m ial coef­
ficients by e x p a n d in g th e p o ly n o m ia ls in b o th sides o f (3.6.20) a n d e q u a tin g the
co efficien ts c o rre sp o n d in g to e q u a l p o w ers. In d e e d , it is easily e s ta b lis h e d that
(3.6.20) is e q u iv a le n t to th e follow ing alg o rith m : Set
a N (k) = ak
it
= 1 ,2 .........N
(3.6.23)
Kfj = a f j ( N)
T h e n , fo r m = N , N — 1 , . . . , 1, c o m p u te
K m = a m(rn)
(3.6.22)
<jm_ i(0 ) = l
Sec. 3.6
Analysis of Linear Tim e-Invariant Systems in the z-D om ain
215
it = 1, 2 , . . . , m — 1
(3.6.24)
an d
w h ere
bm(k) = am(m — k)
k = 0,1,...,m
(3.6.25)
T h is recu rsiv e a lg o rith m fo r th e c o m p u ta tio n o f th e c o effic ie n ts {ATm} finds
a p p lic a tio n in v a rio u s signal p ro cessin g p ro b le m s, especially in sp e e c h signal p r o ­
cessing.
3.6.8 Stability of Second-Order Systems
In th is se c tio n w e p ro v id e a d e ta ile d an aly sis o f a sy stem h av in g tw o p o les. A s
w e sh all se e in C h a p te r 7, tw o -p o le system s fo rm th e b asic b u ild in g b lo ck s fo r th e
r e a liz a tio n o f h ig h e r-o rd e r system s.
L e t us c o n s id e r a cau sal tw o -p o le sy stem d e s c rib e d by th e s e c o n d -o rd e r dif­
fe re n c e e q u a tio n
y (n ) = - a ^ y ( n - I) - a 2y ( n - 2) + b 0x ( n )
(3.6.26)
T h e sy stem fu n ctio n is
X (z)
1 4- a i r 1 + a 2z ~ :
(3.6.27)
boz2
z 2 + a \z + «2
T h is sy stem h a s tw o z ero s at th e origin a n d p o les a t
(3.6.28)
T h e sy stem is B IB O sta b le if th e p o le s lie in sid e th e u n it circle, th a t is, if
|P i| < 1 a n d \ p2\ < 1. T h e se c o n d itio n s can b e re la te d to th e v alu es o f th e
co effic ie n ts a\ a n d a 2. In p a rtic u la r, th e ro o ts o f a q u a d ra tic e q u a tio n satisfy th e
re la tio n s
fil = —(P5 + pi )
(3.6.29)
(3.6.30)
F ro m (3.6.29) a n d (3.6.30) w e easily o b ta in th e c o n d itio n s th a t a\ a n d a 2 m u st
satisfy fo r sta b ility . F irst, aj m u st satisfy th e c o n d itio n
\ai\ = \ p\ pi \ - \p\Wpi\ < 1
(3.6.31)
T h e c o n d itio n fo r a\ can b e e x p re sse d as
1 1 < 1 + «2
(3.6.32)
216
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
T h e c o n d itio n s in (3.6.31) a n d (3.6.32) can also be d e riv e d fro m th e S c h u rC o h n stab ility te st. F ro m th e recu rsiv e e q u a tio n s in (3.6.22) th r o u g h (3.6.25), we
find th a t
(3.6.33)
an d
K 2 = 02
(3.6.34)
T h e sy stem is sta b le if a n d o n iy if [AT]| < 1 an d lA^I < 1. C o n s e q u e n tly ,
—1 < «2 < 1
o r eq u iv a le n tly \a%\ < 1, w hich a g rees w ith (3.6.31). A lso,
or, eq u iv alen tly ,
ai < 1 ~b 02
d\ > —1 — a 2
w hich a re in a g re e m e n t w ith (3.6.32). T h e re fo re , a tw o -p o le sy stem is sta b le if and
only if th e co efficien ts a\ a n d a ; satisfy th e c o n d itio n s in (3.6.31) a n d (3.6.32).
T h e stab ility c o n d itio n s given in (3.6.31) an d (3.6.32), d efin e a reg io n in the
co efficien t p lan e (o i. ai), w hich is in th e fo rm o f a tria n g le , as sh o w n in Fig. 3.15.
T h e sy stem is sta b le if an d only if th e p o in t (o j, q t) lies inside th e tria n g le , w hich
w e call th e stability triangle.
T h e ch a ra c te ristic s o f th e tw o -p o le system d e p e n d o n th e lo catio n o f the
p o les o r. eq u iv alen tly , on th e lo catio n o f th e p o in t (e i, 02) in th e sta b ility triangle.
T h e p o le s o f th e sy stem m ay be re a l o r co m p lex c o n ju g a te , d e p e n d in g on the
v alu e o f th e d isc rim in a n t A = a* — Aa2- T h e p a ra b o la a 2 = a \ f 4 splits th e stability
Figmre 3 .15 Region of stability
(stability triangle) in the ( a i, a?)
coefficient plane for a second-order
system.
Sec. 3.6
Analysis of Linear Tim e-Invariant Systems in the ^-D om ain
217
tria n g le in to tw o reg io n s, as illu stra te d in Fig. 3.15. T h e re g io n b elo w th e p a ra b o la
(,a\ > 4 d 2 ) c o rre sp o n d s to re a l an d d istin ct poles. T h e p o in ts o n th e p a ra b o la
( af = 4*22) re su lt in re a l a n d e q u a l (d o u b le ) poles. F inally, th e p o in ts ab o v e th e
p a ra b o la c o rre sp o n d to co m p lex -co n ju g ate poles.
A d d itio n a l in sig h t in to th e b e h a v io r o f th e system can be o b ta in e d fro m th e
u n it sa m p le re sp o n s e s fo r th e s e th re e cases.
Real and distinct poles (af = 4a2)- Since p 1 ,
system fu n c tio n can b e ex p re sse d in th e fo rm
A\
,
1 - Piz 1
pi are re a l a n d p\ ^ p2. th e
A2
(3.6.35)
1 - P 2Z' 1
w h ere
koPi
Ai ~
-boP2
(3.6.36)
P 1 - P2
Pi - P2
C o n se q u e n tly , th e u n it sa m p le re sp o n se is
b°
(3.6.37)
' ( P T ' - P ? l )u(n)
Pi — P 2
T h e re fo re , th e u n it sa m p le re sp o n se is th e d ifferen ce of tw o d e c ay in g e x p o n e n tia l
se q u en ces. F ig u re 3.16 illu stra te s a ty p ical g rap h for h( n) w h en th e p o les are
distinct.
Real and equal poles (af = 4a2).
In th is case p\
P 2 — p = —a \ / 2. T he
system fu n ctio n is
( 1 - p z - 1)2
(3.6.38)
an d h e n c e th e u n it sa m p le re sp o n se o f th e system is
h{n) = b0(n + 1 ) p nu(n)
h(n)
Figure 3.16 Plot of h(n) given by (3.6.37) with p\ = 0.5, pz = 0.75; h(n) =
~ P2)](P”+1 - P 2 +I)u(n).
(3.6.39)
218
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
h(n)
Figure 3.17
Plot of h(rr) given by (3.6.39) with p =
ti(n) = (n + 1
W e o b serv e th a t h( n) is th e p ro d u c t o f a ra m p se q u e n c e a n d a real decaying
e x p o n e n tia l se q u en ce. T h e g ra p h o f h( n) is show n in Fig. 3.17.
Complex-conjugate poles (af < 4a2). Since the p o le s a re co m p lex c o n ­
ju g a te , th e system fu n ctio n can be fa c to re d and ex p re sse d as
A
A’
1 — p : -1
+
A
1 - re^'z' 1
1 — p*z~l
(3.6.40)
A'
+
1 - r e ~ )tUi' z ~x
w h ere p = r e Jai an d 0 < coq < tt. N o te th a t w hen th e p o le s are co m p lex co n ju g ates,
th e p a ra m e te rs a\ a n d 02 a re re la te d to r an d a>o acco rd in g to
a\ = —2 r cos coo
(3.6.41)
az = r
T h e c o n stan t A in th e p a rtia l-fra c tio n ex p an sio n o f H( z ) is easily sh o w n to be
A =
bo p
boreJW"
p — p*
r ( e i ™0 — e~JW0)
b 0eJ<I*
(3.6.42)
j 2 sin ti>o
C o n seq u en tly , th e u n it sa m p le re sp o n se o f a system w ith c o m p ie x -c o n ju g a te poles
is
h{n) =
sin coo
born
sin coo
2j
-«(n)
(3.6.43)
sin(rt -t- l)cuou(n)
In this case h ( n ) h a s an o sc illato ry b e h a v io r w ith an e x p o n e n tia lly decaying
e n v elo p e w hen r < 1. T h e an g le wo o f th e p o les d e te rm in e s th e fre q u e n c y of
o scillatio n an d th e d istan c e r o f th e p o le s from th e origin d e te rm in e s th e ra te of
Sec. 3.7
Summ ary and References
219
ft(n)
Figure 3.18
Plot of h(n) given by (3.6.43) with bo = 1, a*> = n/4, r = 0.9;
sin[(n + l)twn)u(n).
h(n) = [fcor"/(sin
decay . W h e n r is clo se to u n ity , th e d ecay is slow . W h en r is close to th e origin,
th e d ecay is fast. A typical g ra p h o f h(n) is illu stra te d in Fig. 3.18.
3.7 SUMMARY AND REFERENCES
T h e z -tra n s fo rm p lay s th e sa m e ro le in d isc re te -tim e signals a n d sy stem s as th e
L a p la c e tra n sfo rm d o e s in c o n tin u o u s-tim e signals a n d system s. In th is c h a p te r we
d e riv e d th e im p o rta n t p ro p e rtie s o f th e z-tra n sfo rm , w hich a re e x tre m e ly useful in
th e an aly sis o f d isc re te -tim e system s. O f p a rtic u la r im p o rta n c e is th e co n v o lu tio n
p ro p e rty , w hich tra n sfo rm s th e c o n v o lu tio n o f tw o se q u e n c e s in to a p ro d u c t o f
th e ir z-tran sfo rm s.
In th e c o n te x t o f L T I system s, th e co n v o lu tio n p ro p e rty re su lts in th e p ro d u c t
o f th e z -tra n s fo rm X ( z ) o f th e in p u t signal w ith th e sy stem fu n c tio n H ( z ) , w h e re
th e la tte r is th e z -tra n s fo rm o f th e u n it sa m p le re sp o n s e o f th e sy stem . T his
re la tio n sh ip allo w s u s to d e te rm in e th e o u tp u t o f an L T I sy stem in re s p o n s e to an
in p u t w ith tra n s fo rm X ( z ) b y c o m p u tin g th e p ro d u c t Y ( z ) = H ( z ) X ( z ) a n d th e n
d e te rm in in g th e in v e rse z -tra n sfo rm of Y ( z ) to o b ta in th e o u tp u t se q u e n c e y(n).
W e o b se rv e d th a t m an y signals o f p ra c tic a l in te re s t h a v e r a tio n a l z-tran sfo rm s.
M o re o v e r, L T I sy stem s c h a ra c te riz e d b y c o n s tan t-co efficien t lin e a r d ifferen ce
220
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
e q u a tio n s , also p o ssess r a tio n a l system fu n ctio n s. C o n s e q u e n tly , in determ in in g
th e in v erse z -tran sfo rm , w e n a tu ra lly e m p h a siz e d th e in v ersio n o f ra tio n a l trans­
fo rm s. F o r such tra n sfo rm s, th e p a rtia l-fra c tio n ex p an sio n m e th o d is relatively
easy to apply, in c o n ju n ctio n w ith th e R O C , to d e te rm in e th e c o rre sp o n d in g se­
q u e n c e in th e tim e d o m ain . T h e o n e-sid ed z -tra n sfo rm w as in tro d u c e d to solve for
th e resp o n se o f cau sal system s ex cited by causal in p u t signals w ith n o n z e ro initial
co n d itio n s.
F inally, w e c o n s id e re d th e c h a ra c te riz a tio n of L T I sy stem s in th e z-transform
d o m a in . In p a rtic u la r, w e re la te d th e p o le - z e r o lo catio n s of a sy stem to its timed o m a in c h a racteristics an d re sta te d th e re q u ire m e n ts fo r sta b ility a n d cau sality of
L T I sy stem s in te rm s o f th e p o le lo catio n s. W e d e m o n s tra te d th a t a causal system
h as a system fu n ctio n H( z ) w ith a R O C |z| >
w h e re 0 < r\ < oc. In a stable
an d cau sal system , th e p o les o f H( z ) lie inside th e u n it circle. O n th e o th e r hand,
if th e system is n o n cau sal, th e c o n d itio n for stab ility re q u ire s th a t th e u n it circle be
c o n ta in e d in th e R O C o f H( z ) . H en ce a n o n cau sal sta b le L T I sy stem has a system
fu n c tio n w ith p o les b o th in sid e an d o u tsid e th e u n it circle w ith an a n n u la r R O C
th a t in clu d es th e u n it circle. T h e S c h u r-C o h n test fo r th e sta b ility o f a causal LTI
sy stem w as d escrib ed and th e stab ility o f se c o n d -o rd e r system w as co n sid e re d in
so m e d etail.
A n ex cellen t c o m p reh en siv e tre a tm e n t o f the z -tra n s fo rm a n d its application
to th e analysis o f L TI system s is given in the tex t by Ju ry (1964). T h e S churC o h n test for sta b ility is tr e a te d in se v era l texts. O u r p r e s e n ta tio n w as given in
th e c o n tex t o f reflec tio n co efficien ts w hich are used in lin e a r p re d ic tiv e cod in g of
sp e ech signals. T h e te x t by M a rk e l an d G ra y (1976) is a go o d re fe re n c e for the
S c h u r-C o h n te st a n d its a p p licatio n to sp e ech signal processing.
PROBLEMS
3.1 Determ ine the z-transform of the following signals,
(a) x(n) = {3. 0. 0. 0, 0, 6, 1. —4}
(} )\ » > 5
0.
n <4
3.2 D eterm ine the z-transforms of the following signals and sketch the corresponding
pole-zero patterns.
(a) x ( n ) = (1 + n ) u ( n )
(b) x ( n ) = (a" + a ' ”) u( n) , a real
(b) x(n) =
(c)
W)
(e )
(I)
x ( n ) = ( —1 )n2 -n u (r)
x ( n ) = ( n a " sina»on)w(rc)
x ( n ) = (na" CQSwon) u( n)
x (n) = Ar " c o s (w^n + <j>)u(n). 0 < r < 1
(g) *(n) = j( n : i ~
1)
(h) jr(rt) = ( i ) n[«(n) - u(n - 10)]
Chap. 3
221
Problems
33 Determine the z-transforms and sketch the ROC of the following signals.
f(i)\
n>
_0
“ 1 ( f) " " .
"<
(a) x,(n)
( i ) B- 2 \
0,
(c) *j(«) = x 1(n + 4 )
<d) x4(n) = j t , ( - n )
(b) x 2(n) =
n >0
n < 0
3.4 D eterm ine the z-transform of the following signals.
(a) x(n) = n(—l)"w(n)
(b) x(n) = n2u(n)
(c) x(n) = —nanu( —n — 1)
(d) x(n) = (-1 )" (cos ~n) u(n)
<e) x(n) = (—l)"u(n)
(f) jr(n) = ( 1 ,0 .- 1 ,0 . 1 ,- 1 , . . . }
t
3.5 D eterm ine the regions of convergence of right-sided, left-sided, and finite-duration
two-sided sequences.
3.6 Express the z-transform of
y(n) = Y x(k)
k=*—oc
in term s of X (-). [Him: Find the difference y(n) - y(n - 1).]
3.7 Com pute the convolution of the following signals by m eans of the z-transform.
*i(n)
= f (£)".
n >0
1 (£)"".
n<0
*2 (n) = ( j ) nH(n)
3.8 Use the convolution property to:
(a) Express the z-transform of
y(n) = 5 2 x
tt-oc
in terms of X(z).
(b) Determine the z-transform of x(n) = (n + l)u(n). [Hint. Show first that x ( n ) =
u(n) * n(n).]
3.9 The z-transform X(z) of a real signal x(n) includes a pair of complex-conjugate zeros
and a pair of complex-conjugate poles. What happens to these pairs if we multiply
x(n) by eJW°nl (Hint. Use the scaling theorem in the z-domain.)
3.10 Apply the final value theorem to determine jt( oo ) for the signal
1,
10,
if n is even
otherwise
3.11 Using long division, determine the inverse z-transform of
1 + 2Z-1
1 - 2z 1 + z 2
if (a) jr(n) is causal and (b) x(n) is anticausal.
222
The z-Transform and Its Application to the Analysis of LTI Systems
3.12 Determ ine the causal signal
Chap. 3
having the z-transform
1
*(z) = ( l - Z z - ' H l - z - 1)2
3.13 Let ;t(n) be a sequence with z-transform X(z)- Determ ine, in terms of X(z), the
z-transforms of the following signals.
i,n
(a) j:i(n) =
even
II
if n odd
0,
(b) x2(n) = x(2n)
3.14 D eterm ine the causal signal x (n) if its z-transform X(z) is given by:
l+ 3 z -’
(a)
1 + 3 z-‘ + 2 z-2
1
(b)
1 - z - ' + ^ z -2
II
*
><
II
(d)
><
II
z -6 + z^7
(c)
(e) X(z) =
(0 X(z) =
1
1 + 6z“ ‘ + z " 2
4 (1 - 2 ; - 1 + 2 z '2)(l - O.Sz"1)
2
— 1.5z-1
1 - 1.5--1 + C.5z^:
1 + 2z“ ‘ + z-2
II
*
(g)
11 + 2z-2
1+ :-2
II
(j)
*
II
1 + 4z_1 + 4z-2
(h) X(z) is specified by a pole-zero p atten
1- k -1
(i)
1 - a z -1
Figure P3.14
3.15 Determ ine all possible signals x(n) associated with the z-transform
X U ) = {1 - 2 z - ') ( 3 - z - ])
3.16 Determ ine the convolution of the following pairs of signals by means of the ztransform.
Chap. 3
223
Problems
(a) x\(ft) =
(b) X i ( n ) = u ( n ) ,
- 1), x 2(n) = [1 +
jc2(n) = 5(n) + (i)"u (n )
(c) x\ (n) = ( i ) nM(n),
(d) ari(n) = nu(n),
*2(71) = c-Osnnu(n)
x 2(n) = 2 "u(n — 1)
3.17 Prove the final value theorem for the one-sided z-transform.
3.18 If X(z) is the z-transform of x(n), show that:
(a) Z { x m(n)} = X ' ( z ' )
(b) Z{Re[jr(n}]} = j[X (z) + X*(z*)]
(c) Z{Im[jr(«)]l = |[X (z) - **(=*)]
(d) If
* * („ )= { * (? )•
1 0,
if " A integer
otherwise
then
X t U) = X (z k)
(e)
= X(ze~Jw°)
3.19 By first differentiating X(z) and then using appropriate properties of the z-transform.
determ ine x(n) for the following transforms.
(a) X(z) = l o g ( l —2 z),
\z\<{
(b) X(z) = log(l - z-1), |z! > 5
3.20 (a) Draw the pole-zero pattern for the signal
jcj(n) = (r" sin a>on)u(fi)
0 < r < 1
(b) Com pute the z-transform A^tz), which corresponds to the pole-zero pattern in
part (a).
(c) Com pare X]{z) with X 2(z). Are they indentical? If not. indicate a m ethod lo
derive Xi(z) from the pole-zero pattern.
3.21 Show that the roots of a polynomial with real coefficients are real or form complexconjugate pairs. The inverse is not true, in general.
3.22 Prove the convolution and correlation properties of the z-transform using only its
definition.
3.23 D eterm ine the signal x(n) with z-transform
X (z) = e: + e l/z
|z|^0
3.24 D eterm ine, in closed form, the causal signals x(n) whose z-transforms are given by:
(a) X(z) = T T T 5 ^ T
o3 P
(b) X(Z) = 1 - 0.5z~! + 0.6z “2
Partially check your results by computing *(0), x (l), *(2), and jt( oo) by an alternative
m ethod.
3.25 D eterm ine all possible signals that can have the following z-transforms.
1
224
The z - T ransform and Its Application to the Analysis of LTI Systems
Chap. 3
3.26 Determ ine the signal x(n) with z-transform
XU) =
+ z- 2
1-
if X(z) converges on the unit circle.
3 .2 7 Prove the complex convolution relation given by (3.2.22).
3.28 Prove the conjugation properties and Parse val’s relation for the z-transform given in
Table 3.2.
3.29 In Example 3.4.1 we solved for .r(n), n < 0, by perform ing contour integrations for
each value of n. In general, this procedure proves to be tedious. It can be avoided by
making a transform ation in the contour integral from z-plane to the uj = 1/z plane.
Thus a circle of radius R in the z-plane is mapped into a circle of radius 1/ R in the wplane. As a consequence, a pole inside the unit circle in the z-plane is mapped into a
pole outside the unit circle in the m-plane. By making the change of variable w = 1/z
in the contour integral, determ ine the sequence x ( n ) for n < 0 in Example 3.4.1,
3.30 Let *(n), 0 < n < N — 1 be a finite-duration sequence, which is also real-valued and
even. Show that the zeros of the polynomial X(z) occur in mirror-image pairs about
the unit circle. That is. if z = rej/> is a zero of X(z), then z = (1 / r ) e J" is also a zero.
3.31 Compute the convolution of the following pair of signals in the time domain and by
using the one-sided z-transform.
(a) .v,(n) = {1. 1. 1. 1. 1).
x 2(n) = (1. 1. 1)
t
t
(b) Xi (n) = ( j)"u(h).
x 2(n) = (i)"u(n)
(c) .ii(n) = (1.2, 3.4}.
x 2(n) = (4, 3, 2. 1}
t
t
(d) *,(«) = {1.1. 1.1.1}.
Jr2 ( « )
= {1.1,1}
t
t
Did you obtain the same results by both methods? Explain.
3.32 D eterm ine the one-sided z-transform of the constant signal x ( n ) = 1. —oo < n < 00 .
3 .3 3 Prove that the Fibonacci sequence can be thought of as the impulse response of the
system described by the difference equation ,y(/i) = v(n - 1) + y ( n - 2) 4- x(n). Then
determ ine h(n) using z-transform techniques.
3 .3 4 Use the one-sided z-transform to determ ine y(n), n > 0 in the following cases.
(a) y(n) + \ y( n - 1) - \ y( n - 2) = 0;
y ( - l ) = y (-2 ) = 1
(b) y(n) - 1.5y(n - 1) + 0.5y(n - 2) = 0;
y ( - l ) = 1. v(—2) = 0
(c) v(n) = \ y ( n - 1) + x ( n )
x ( n ) = (±)"u(rt).
y(-l) = 1
(d) _v(r) = j v(n - 2) + x(n)
x ( n ) = u{n)
>’(—1) = 0;
y(—2) = 1
3.3 5 Show that the following systems are equivalent.
(a) y ( n ) = 0 .2 y (n - 1) + x { n ) - 0.3.r(n - 1) + 0.02jt(n - 2)
(b) _y{«) = x ( n ) - 0 .1 x (n - 1)
Chap. 3
225
Problems
3.36 Consider the sequence xin) = ci"uin). —1 < a < 1. D eterm ine at least two sequences
that are not equal to xin) but have the same autocorrelation.
3.37 Com pute the unit step response of the system with impulse response
f3\
n < 0
n >0
3.38 Com pute the zero-state response for the following pairs of systems and input signals.
(a) hin) =
,v(h) = <j)"^cos w r j ;<('?)
(b) h(n) = ( | Y‘u Oi ). xi n) = { \ ) ’‘u{n) + ( \ r " u ( - n - 1)
(c) yin) = —0.1 yin — 1) + 0.2y(>i - 2) + xin) + xin - 1)
,v ( / i) =
( ^ V m (i i )
(d) yin) = |.v(») -
— 1)
x{n) = lO^Cos —n'ju(n)
(e) yin) = —yin —2 ) + lOjr(n)
,v(/7) = lO^cos —n^jnin)
(f) h{n) = i ^ Y ’ii(n). x i n ) — i/ln) — it in — 7)
(g) hin) = (|V'if(H). xin) = ( —1)", —3C < n < x
(h) hin) = U) ' ‘i
f
= in + 1)(j)"h<h)
3.39 Consider the system
1 - 2 r ' + 2:~: ----------;---- ----------- :-----------------(
1
- 0 . 5 ; - ‘) ( l - 0. 2: - ' )
H(Z) =
ROC: 0.5 < |d < l
(a) Sketch the pole-zero pattern. Is the system stable?
(b) D eterm ine the impulse response of the system.
3.40 Com pute the response of the system
ytn) = 0.7v(n - 1) — 0.12y(n - 2) + x(n — 1) + xin —2)
to the input v(/?) = nuin), Is the system stable?
3.41 D eterm ine the impulse response and the step response of the following causal systems.
Plot the pole-zero patterns and determ ine which of the systems are stable.
(a ) y i n ) = j v(« - 1) - jr v( n - 2) + x ( n )
(b) y(n) = vin — 1) —0.5y(n - 2) + x(n) + xin — I )
; ~ ' f l
+
(c> ™ =
(d) v(n) = 0.6y(n —1) —0.08v(n —2) -f- x(n)
(e) v(«) = 0.7y(n — 1) —0.1y(/3 —2) + 2xin) — x(n —2)
3.42 Let xin) be a causal sequence with ;-transform X(z) whose pole-zero plot is shown
in Fig. P3.42. Sketch the pole-zero plots and the R O C of the following sequences;
(a) Xt(n) = x ( ~ n -)- 2)
(b) x 2(n) = eim/iu,x(n)
226
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
Im(z)
Re(;)
Figure P3.42
3.43 W e want to design a causal discrete-time LTI system with the property that if the
input is
x( n ) = ( l ) nu ( n) -
- 1)
then the output is
V(lt) = (|)"tt(n)
(a) D eterm ine the impulse response h(n) and the system function H(z) of a system
that satisfies the foregoing conditions.
(b) Find the difference equation that characterizes this system.
(c) D eterm ine a realization of the system that requires the minimum possible amount
of memory.
(d) D eterm ine if the system is stable.
3 .4 4 D eterm ine the stability region for the causal system
H{z) =
1
1 + o ir -1 + a2z~2
by computing its poles and restricting them to be inside the unit circle.
3.45 Consider the svstem
H{z) =
_L J?i-r-2
I1 _ l --1 +
Determine:
(a) The impulse response
(b) The zero-state step response
(c) The step response if y(—1) = 1 and y ( -2 ) = 2
3 .4 6 D eterm ine the system function, impulse response, and zero-state step response of the
system shown in Fig P3.46.
3 .4 7 Consider the causal system
y in ) = - a i y i n - 1) + b0x(n) + b^xin - 1)
Determine:
(a) The impulse response
Chap. 3
227
Problems
Xi.fl)
y(n)
a
Figure P3.46
(b) The zero-state step response
(c) The step response if y ( —1) = A / 0
(d) The response to the input
x (n ) ~
CO S a> u f!
0 <
n
<
oo
3.48 Determ ine the zero-state response of the system
y(n) = i v(h - 1) + 4*(n) + 3x(n — 1)
to the input
jt(n) = em 'nu{n)
W hat is the steady-state response of the system?
3.49 Consider the causal system defined by the pole-zero pattern shown in Fig. P3.49.
(a) Determ ine the system function and the impulse response of the system given that
W U )U i = 1.
(b) Is the system stable?
(c) Sketch a possible im plementation of the system and determ ine the corresponding
difference equations.
Im(;)
3.50 An FIR LTI system has an impulse response h(n), which is real valued, even, and
has finite duration of 2/V + 1. Show that if ;i = rejaJa is a zero of the system, then
d = ( l / r ) e j a *> is also a zero.
3.51 Consider an LTI discrete-time system whose pole-zero pattern is shown in Fig. P3.51.
(a) D eterm ine the R O C of the system function H(z) if the system is known to be
stable.
228
The z-Transform and Its Application to the Analysis of LTI Systems
Chap. 3
Im(;)
Re<z)
-0 .5
Figure P3.51
(b ) It is possible for the given pole-zero plot to correspond to a causal and stable
system? If so, what is the appropriate ROC?
(c) How many possible systems can be associated with this pole-zero pattern?
3.52 Let x(n) be a causal sequence.
(a) What conclusion can you draw about the value of its z-transform A’(c) at ; = oo?
(b ) Use the result in part (a) to check which of the following transform s cannot be
associated with a causal sequence.
3.53 A causal pole-zero system is BIBO stable if its poles are inside the unit circle. Con­
sider now a pole-zero system that is BIBO stable and has its poles inside the unit
circle. Is the system always causal? [Hint: Consider the systems h\(n) = anu(n) and
ti2 (n) = anu{n + 3), |a| < 1.]
3 3 4 Let *(/i) be an anticausal signal [i.e., x (n) = 0 for n > 0]. Form ulate and prove an
initial value theorem for anticausal signals.
3.55 The step response of an LTI system is
J(w) =
+ 2)
(a) Find the system function H(z) and sketch the pole-zero plot.
(b ) D eterm ine the impulse response ft(n).
(c) Check if the system is causal and stable.
3 3 6 Use contour integration to determ ine the sequence jc(n) whose z-transform is given
Chap. 3
229
Problems
; —a
1 —a:
1-
(c) X'U) = --------<d) A'C)
3.57 Let ,v(h) be a sequence with c-transform
X d = —--- --------- —
(1 - t;r)(l - a ; " 1)
R O C : a < |;| < 1ja
with 0 < a < 1. Determ ine ,v(«) by using contour integration
3.58 The ^-transform of a sequence .v(n) is given by
X(;> =
(: - i ) ( ; - 2)5<- + ?): (r + 3)
Furthermore it is known that X (:) converges for |;| = 1.
(a ) Determ ine the R O C of X(c).
(b) Determ ine xin ) at n = -18. {Hint: Use contour integration.)
34
Frequency Analysis of Signals
and Systems
T h e F o u rie r tra n sfo rm is o n e o f se v e ra l m a th e m a tic a l to o ls th a t is useful in the
an alysis an d d esign o f L T I system s. A n o th e r is th e F o u rie r se ries. T h ese signal
re p re se n ta tio n s basically involve th e d e c o m p o sitio n o f th e sig n als in te rm s o f sinu­
so id al (o r co m p lex e x p o n e n tia l) c o m p o n e n ts. W ith such a d e c o m p o s itio n , a signal
is said to be re p re se n te d in th e f r e q u e n c y domai n .
A s w e shall d e m o n s tra te , m ost signals o f p ractical in te re st can be d eco m p o sed
in to a sum o f sin u so id al signal c o m p o n e n ts. F o r th e class of p e rio d ic signals, such
a d eco m p o sitio n is called a Fouri er series. F o r th e class o f finite e n e rg y signals, the
d eco m p o sitio n is called th e Fouri er t ransf orm. T h e se d e c o m p o s itio n s are ex trem ely
im p o rta n t in th e an aly sis o f L T I system s becau se th e re sp o n se o f a n L T I system to
a sin u so id al in p u t signal is a sin u so id o f th e sam e freq u e n c y b u t o f d iffe re n t am pli­
tu d e a n d p h ase. F u rth e rm o re , th e lin e a rity p ro p e rty o f th e L T I sy stem im plies th a t
a lin ear sum o f sin u so id al c o m p o n e n ts at th e in p u t p ro d u c e s a sim ilar lin e a r sum
o f sin u so id al c o m p o n e n ts a t th e o u tp u t, w hich d iffer only in th e a m p litu d e s and
p h ases fro m th e in p u t sin u so id s. T h is c h a ra c te ristic b e h a v io r o f L T I sy stem s ren ­
d e rs th e sin u so id al d e c o m p o sitio n o f signals v ery im p o rta n t. A lth o u g h m an y o th er
d e c o m p o sitio n s o f signals a re p o ssib le, only th e class o f sin u so id al (o r co m p lex ex­
p o n e n tia l) signals p o ssess th is d e sira b le p ro p e rty in passin g th ro u g h an L T I system .
W e begin o u r stu d y o f fre q u e n c y analysis o f signals w ith th e re p re se n ta tio n
o f co n tin u o u s-tim e p e rio d ic a n d a p e rio d ic signals by m e a n s of th e F o u rie r series
an d th e F o u rie r tra n sfo rm , resp ec tiv ely . T h is is follow ed by a p a ra lle l tre a tm e n t
o f d isc rete-tim e p erio d ic a n d a p e rio d ic signals. T h e p r o p e rtie s o f th e F o u rie r
tra n sfo rm a re d e sc rib e d in d e ta il an d a n u m b e r of tim e -fre q u e n c y d u alities are
p re se n te d .
4.1 FREQUENCY ANALYSIS OF CONTINUOUS-TIME SIGNALS
It is w ell k n o w n th a t a prism can be u sed to b re a k u p w h ite light (su n lig h t) in to the
co lo rs o f th e ra in b o w (see Fig. 4.1a). In a p a p e r su b m itte d in 1672 to th e R oyal
Society, Isaac N e w to n u se d th e te rm spe ct r um to d escrib e th e c on t i n u o u s b an d s
230
Sec. 4.1
231
Frequency Analysis of Continuous-Time Signals
Glass prism
Figure 4.1
(a) Analysis and
( b ) sy n th e sis o f th e w h ite light (s u n lig h t)
u sin g glass p rism s.
o f co lo rs p ro d u c e d by this a p p a ra tu s. T o u n d e rsta n d this p h e n o m e n o n , N ew to n
p laced a n o th e r p rism u pside-dow rn w ith resp ec t to th e first, an d sh o w ed th a t th e
co lo rs b le n d e d back into w hite light, as in Fig. 4.1b, By in se rtin g a slit b etw een
the tw o p rism s a n d blo cking o n e o r m o re colors from h ittin g the se co n d prism ,
he sh o w ed th a t th e rem ix ed light is no lo n g er w hite. H e n c e th e light passing
th ro u g h th e first p rism is sim ply an aly zed into its c o m p o n e n t co lo rs w ith o u t any
o th e r ch an g e. H o w e v e r, o n ly if we mix again all o f th ese c o lo rs d o we o b ta in the
o rig in a l w hite light.
L a te r, Jo s e p h
F r a u n h o f e r ( 1 7 8 7 - 1 8 2 6 ) . in m a k in g m e a s u r e m e n ts o f lig h t
e m itt e d b y th e su n a n d s ta r s , d is c o v e r e d th a t th e s p e c tr u m o f th e o b s e r v e d lig h t
c o n s is ts o f d is tin c t c o l o r lin e s . A fe w y e a r s la te r ( m i d - ] 8 0 0 s ) G u s ta v K i r c h h o f f a n d
R o b e r t B u n s e n fo u n d th a t e a c h c h e m ic a l e le m e n t, w h e n h e a te d to in c a n d e s c e n c e ,
r a d ia te d its o w n d is tin c t c o lo r o f lig h t. A s a c o n s e q u e n c e , e a c h c h e m ic a l e le m e n t
c a n b e id e n tifie d b y its o w n lin e sp e ctru m .
F r o m p h y s ic s w e k n o w th a t e a c h c o lo r c o r r e s p o n d s to a s p e c ific f r e q u e n c y o f
th e v is ib le s p e c tr u m . H e n c e th e a n a ly sis o f lig h t in to c o lo r s is a c tu a lly a f o r m o f
fr e q u e n c y a n a ly s is .
F r e q u e n c y a n a ly s is o f a s ig n a l in v o lv e s th e r e s o lu tio n o f th e s ig n a l in to its
fr e q u e n c y ( s in u s o id a l) c o m p o n e n ts .
In s te a d o f lig h t, o u r s ig n a l w a v e fo r m s a re
b a s ic a lly fu n c tio n s o f tim e . T h e r o le o f th e p ris m is p la y e d b y th e F o u r i e r a n a ly sis
to o ls th a t w e w ill d e v e lo p :
th e F o u r ie r s e r ie s a n d th e F o u r i e r t r a n s f o r m .
The
r e c o m b in a tio n o f th e s in u s o id a l c o m p o n e n ts to r e c o n s tr u c t th e o r ig in a l s ig n a l is
b a s ic a lly a F o u r i e r s y n th e s is p r o b le m . T h e p r o b le m o f s ig n a l a n a ly s is is b a s ic a lly
th e s a m e fo r th e c a s e o f a s ig n a l w a v e fo rm a n d f o r th e c a s e o f th e lig h t f r o m h e a te d
c h e m ic a l c o m p o s itio n s .
J u s t a s in th e c a s e o f c h e m ic a l c o m p o s itio n s , d if f e r e n t
s ig n a l w a v e fo r m s h a v e d iffe r e n t s p e c tr a . T h u s th e s p e c tr u m p r o v id e s a n “ id e n tity ”
232
Frequency Analysis of Signals and Systems
Chap. 4
o r a s ig n a tu r e f o r th e s ig n a l in th e s e n s e t h a t n o o t h e r s ig n a l h a s th e s a m e sp e c tru m .
A s w e w ill s e e , th is a ttr ib u te is r e la te d to th e m a th e m a tic a l t r e a t m e n t o f fre q u e n c y d o m a in te c h n iq u e s .
I f w e d e c o m p o s e a w a v e fo r m in to s in u s o id a l c o m p o n e n ts , in m u c h th e sa m e
w ay t h a t a p ris m s e p a r a te s w h ite lig h t in to d iffe r e n t c o lo r s , th e su m o f th e s e
s in u s o id a l c o m p o n e n ts re s u lts in th e o r ig in a l w a v e fo r m . O n th e o t h e r h a n d , if any
o f th e s e c o m p o n e n ts is m is s in g , th e r e s u lt is a d if f e r e n t s ig n a l.
I n o u r tr e a t m e n t o f fr e q u e n c y a n a ly s is , w e w ill d e v e lo p th e p r o p e r m a th e ­
m a tic a l to o ls ( “ p r is m s ” ) f o r th e d e c o m p o s itio n o f s ig n a ls ( “ l i g h t ” ) in to s in u so id a l
f r e q u e n c y c o m p o n e n ts ( c o l o r s ) . F u r t h e r m o r e , th e t o o ls ( “ in v e r s e p r is m s " ) f o r sy n ­
th e s is o f a g iv e n s ig n a l f r o m its f r e q u e n c y c o m p o n e n ts w ill a ls o b e d e v e lo p e d .
T h e b a s ic m o t iv a tio n f o r d e v e lo p in g th e fr e q u e n c y a n a ly s is to o ls is to p ro v id e
a m a th e m a tic a l a n d p ic to r ia l r e p r e s e n t a t io n f o r th e f r e q u e n c y c o m p o n e n ts th a t a re
c o n ta in e d in a n y g iv e n s ig n a l. A s in p h y s ic s , th e te r m s pe ct r um is u s e d w h e n r e f e r ­
rin g t o th e fr e q u e n c y c o n t e n t o f a s ig n a l. T h e p r o c e s s o f o b ta in in g th e s p e c tru m
o f a g iv e n sig n a l u sin g th e b a s ic m a th e m a tic a l t o o ls d e s c r ib e d in th is c h a p te r is
k n o w n a s f re q u e n c y o r spectral analysis.
In c o n tr a s t, th e p r o c e s s o f d e te r m in in g
th e s p e c tr u m o f a s ig n a l in p r a c tic e , b a s e d o n a c tu a l m e a s u r e m e n ts o f th e sig n a l,
is c a lle d spect rum estimation.
T h is d is tin c tio n is v e r y im p o r ta n t.
In a p r a c tic a l
p r o b le m th e s ig n a l to b e a n a ly z e d d o e s n o t le n d it s e lf to a n e x a c t m a th e m a tic a l
d e s c r ip tio n . T h e s ig n a l is u s u a lly s o m e i n f o r m a tio n - b e a r in g s ig n a l fr o m w h ich we
a r e a tte m p tin g t o e x t r a c t th e r e le v a n t in f o r m a tio n . I f th e in f o r m a tio n th a t w e wish
to e x t r a c t c a n b e o b ta in e d e i t h e r d ir e c tly o r in d ir e c tly fr o m th e s p e c tr a l c o n te n t o f
th e s ig n a l, w e c a n p e r fo r m spe ct rum est imation o n th e in f o r m a t io n - b e a r in g sig n a l,
a n d th u s o b ta in a n e s tim a te o f th e s ig n a l s p e c tr u m . In f a c t, w e c a n v iew s p e c tra l
e s tim a tio n as a ty p e o f s p e c tr a l a n a ly s is p e r fo r m e d o n s ig n a ls o b t a i n e d f r o m p h y si­
ca l s o u r c e s (e .g ., s p e e c h , E E G , E C G , e t c .) . T h e in s tr u m e n ts o r s o f tw a r e p r o g ra m s
u se d t o o b ta in s p e c tr a l e s tim a te s o f s u c h s ig n a ls a r e k n o w n a s sp e c t r u m analyzers.
H e r e , w e w ill d e a l w ith s p e c tr a l a n a ly s is . H o w e v e r , in C h a p t e r 12 w e sh all
t r e a t th e s u b je c t o f p o w e r s p e c tr u m e s tim a tio n .
4.1.1 The Fourier Series for Continuous-Time Periodic
Signals
In th is s e c tio n w e p r e s e n t th e f r e q u e n c y a n a ly s is to o ls fo r c o n tin u o u s - tim e p e ­
r io d ic s ig n a ls .
E x a m p le s o f p e r io d ic s ig n a ls e n c o u n te r e d in p r a c t i c e a r e s q u a re
w a v e s , r e c ta n g u la r w a v e s , tr ia n g u la r w a v e s , a n d o f c o u r s e , s in u s o id s a n d c o m p le x
e x p o n e n tia ls .
T h e b a s ic m a th e m a tic a l r e p r e s e n ta tio n o f p e r io d ic s ig n a ls is th e F o u r i e r s e ­
r ie s , w h ic h is a lin e a r w e ig h te d su m o f h a r m o n ic a lly r e la te d s in u s o id s o r c o m p le x
e x p o n e n tia ls . J e a n B a p t is t e J o s e p h F o u r i e r ( 1 7 6 8 - 1 8 3 0 ) , a F r e n c h m a th e m a tic ia n ,
u se d s u c h t r ig o n o m e t r ic s e r ie s e x p a n s io n s in d e s c r ib in g th e p h e n o m e n o n o f h ea t
c o n d u c tio n a n d te m p e r a tu r e d is tr ib u tio n th r o u g h b o d ie s . A lth o u g h h is w o r k was
m o t iv a te d b y th e p r o b le m o f h e a t c o n d u c tio n , th e m a th e m a tic a l te c h n iq u e s th a t
Sec. 4.1
Frequency Analysis of C ontinuous-Tim e Signals
233
h e d e v e lo p e d d u rin g th e e a r ly p a r . o f th e n in e t e e n t h c e n tu r y n o w fin d a p p lic a ­
tio n in a v a r ie ty o f p r o b le m s e n c o r r .r \ is s in g m a n y d if f e r e n t f ie ld s , in c lu d in g o p tic s ,
v ib r a tio n s in m e c h a n ic a l s y s te m s , s y s t e m th e o r y , a n d e le c t r o m a g n e t ic s .
F r o m C h a p t e r 1 w e r e c a ll t h .i : a lin e a r c o m b in a t io n o f h a r m o n ic a lly r e la te d
c o m p le x e x p o n e n tia ls o f th e fo r m
x {r) =
Y
cke j 2 * kF»‘
( 4 .1 .1 )
i = -3C
is a p e r io d ic s ig n a l w ith f u n d a m e n t a l p e r io d Tp =
1/Fo.
H e n c e w e c a n th in k o f
th e e x p o n e n t ia l s ig n a ls
{ e i i x k p k = Q ± i i± 2 l
a s th e b a s ic “ b u ild in g b lo c k s ” f r o m
w h ic h w e c a n c o n s t r u c t p e r io d ic s ig n a ls o f
v a r io u s ty p e s b y p r o p e r c h o ic e o f t h e fu n d a m e n ta l f r e q u e n c y a n d th e c o e f f ic ie n ts
( q ). F o d e te r m in e s th e f u n d a m e n ta l p e r i o d o f x ( t ) a n d th e c o e f f ic ie n t s { }
s p e c ify
th e s h a p e o f th e w a v e fo rm .
S u p p o s e th a t w e a r e g iv e n a p e r io d i c s ig n a l x { i) w ith p e r io d Tp .
W e ca n
r e p r e s e n t th e p e r io d ic sig n a l by t h e s e r ie s ( 4 .1 .1 ) , c a lle d a F o u r ie r series, w 'here
th e f u n d a m e n ta l fr e q u e n c y Fo is s e l e c t e d to b e th e r e c ip r o c a l o f th e g iv e n p e r io d
Tp . T o d e te r m in e th e e x p r e s s io n t o r th e c o e f f i c i e n t s ( q ) , w e firs t m u ltip ly b o th
sid e s o f ( 4 .1 .1 ) b y th e c o m p le x e x p o n e n t i a l
Fltl!
w h e r e I is a n in t e g e r an d th e n in t e g r a t e b o th s id e s o f th e r e s u ltin g e q u a tio n o v e r
a s in g le p e r io d , s a y fr o m 0 to T r , o r m o r e g e n e r a lly , fr o m fo to r0 + T p, w h e r e i o is
a n a r b itr a r y b u t m a th e m a tic a lly c o n v e n i e n t s ta r t in g v a lu e . T h u s w e o b ta in
fh>+Tr
J'
r'o+Tr
x ( t ) e - j2*IF,''dt = J'
I
g- j t oi Kt /
oc
\
cke+J2nkF"' J di
(4.1.2)
T o e v a lu a te th e in te g r a l o n th e r ig h t- h a n d s id e o f ( 4 .1 .2 ) , w e in te r c h a n g e th e o r d e r
o f th e s u m m a tio n a n d in te g r a tio n a n d c o m b in e th e tw o e x p o n e n tia ls . H e n c e
rl
sc
£
,, + r .
c*
I
OC
ei2* F»{k-'"dt = £
i = —oc
k= —oc
p j27rF < M -hl
Ck
- |'n + r ,
J l n F o i k - /)_
( 4 .1 .3 )
F o r k ^ I, th e r ig h t-h a n d s id e o f ( 4 .1 .3 ) e v a lu a te d a t th e lo w e r a n d u p p e r lim its , Zo
a n d t0 + Tp , r e s p e c tiv e ly , y ie ld s z e r o . O n th e o t h e r h a n d , if k = /, w e h a v e
fJtn
di — i
Consequently, (4.1.2) reduces to
'■‘o+Tp
fJ /ft
x ( t ) e - j2 * ,Fo'd t = c ,T p
Frequency Analysis of Signals and System s
234
Chap. 4
a n d t h e r e f o r e th e e x p r e s s io n f o r th e F o u r i e r c o e f fic ie n ts in t e r m s o f th e g iv e n
p e r io d ic s ig n a l b e c o m e s
fto+TP
I
c, = —
x ( l ) e - jlw,F"'dt
Tp
S in c e fo is a r b itr a r y , th is in te g r a l c a n b e e v a lu a te d o v e r a n y i n te r v a l o f le n g th Tp,
th a t is, o v e r a n y in te r v a l e q u a l t o th e p e r io d o f th e s ig n a l jr ( r ) . C o n s e q u e n tly , th e
in te g r a l fo r th e F o u r i e r s e r ie s c o e f f ic ie n t s w ill b e w r itte n as
c, = — f
x ( t ) e ~ j2nlF,>'d t
( 4 .1 .4 )
Tp J t„
A n im p o r ta n t is s u e th a t a r is e s in th e r e p r e s e n ta tio n o f th e p e r io d ic s ig n a l
x ( t ) b y th e F o u r ie r s e r ie s is w h e t h e r o r n o t th e s e r ie s c o n v e r g e s t o * ( f ) f o r e v e ry
v a lu e o f r, th a t is, if th e s ig n a l x (t ) a n d its F o u r ie r s e r ie s r e p r e s e n t a t io n
OC
c ke j2 * kFtt'
£
( 4 .1 .5 )
A=-oc
a r e e q u a l a t e v e r y v a lu e o f t.
T h e s o -c a lle d D ir ic h le t c o n d it io n s g u a r a n te e th a t
th e s e r ie s ( 4 .1 .5 ) w ill b e e q u a l to x ( t ), e x c e p t a t th e v a lu e s o f i f o r w h ich jc (r ) is
d is c o n tin u o u s .
A t th e s e v a lu e s o f r, ( 4 .1 .5 ) c o n v e r g e s t o th e m id p o in t (a v e r a g e
v a lu e ) o f th e d is c o n tin u ity . T h e D i r ic h l e t c o n d itio n s a r e :
1 . T h e s ig n a l .r ( f ) h a s a fin ite n u m b e r o f d is c o n tin u itie s in a n y p e r io d .
2 . T h e sig n a l x ( t ) c o n ta in s a fin ite n u m b e r o f m a x im a an d m in im a d u rin g an y
p e rio d .
3 . T h e s ig n a l x (t ) is a b s o lu te ly in te g r a b le in a n y p e r io d , t h a t is.
f
\x (t )\d t < oo
( 4 .1 .6 )
J Tr
A lt p e r io d ic s ig n a ls o f p r a c tic a l i n te r e s t s a tis fy th e s e c o n d itio n s .
T h e w e a k e r c o n d itio n , t h a t th e s ig n a l h a s fin ite e n e r g y in o n e p e r io d ,
j
\x ( t ) \2d i < o c
( 4 .1 .7 )
J tp
g u a r a n te e s th a t th e e n e r g y in th e d if f e r e n c e s ig n a l
OC
e (t) = x { t ) -
Ckej2”kF"'
k=—oc
is z e r o , a lth o u g h * ( ; ) a n d its F o u r i e r s e r ie s m a y n o t b e e q u a l f o r a ll v a lu e s o f t.
N o te th a t ( 4 .1 .6 ) im p lie s ( 4 .1 .7 ) , b u t n o t v ic e v e r s a .
A l s o , b o t h ( 4 .1 .7 ) a n d th e
D ir i c h le t c o n d itio n s a r e s u f fic ie n t b u t n o t n e c e s s a r y c o n d itio n s ( i .e ., th e r e a r e s ig ­
n a ls t h a t h a v e a F o u r ie r s e r ie s r e p r e s e n t a t i o n b u t d o n o t s a tis fy t h e s e c o n d itio n s ) .
In s u m m a r y , i f x ( t ) is p e r io d ic a n d s a tis fie s th e D i r ic h l e t c o n d itio n s , it ca n
b e r e p r e s e n te d in a F o u r i e r s e r ie s a s in ( 4 .1 .1 ) , w h e r e th e c o e f f i c i e n t s a r e s p e c ifie d
b y ( 4 .1 .4 ) . T h e s e r e la t io n s a r e s u m m a r iz e d b e lo w .
Sec. 4.1
Frequency Analysis of Continuous-Time Signals
235
FR E Q U E N C Y A N A LY S IS O F C O N T IN U O U S -TIM E PE R IO D IC S IG N A LS
Synthesis equation
-vU) = ^
Analysis equation
ct = y j
1r Jrp
(4.1.8)
cke,2~kl'"'
(4.1.9)
In g e n e r a l, th e F o u r i e r c o e ff ic ie n ts c k a r e c o m p le x v a lu e d .
e a s ily sh o w n t h a t if th e p e r io d ic s ig n a l is r e a l, c k a n d
M o r e o v e r , it is
a r e c o m p le x c o n ju g a te s .
A s a r e s u lt, if
ct = \ck\e}b‘
th e n
C-U ~~
C o n s e q u e n tly , th e F o u r i e r s e r ie s m ay a ls o b e r e p r e s e n te d in t h e fo r m
rv
a (M =
2
(-•(, +
k i I c o s 0 - 7 1 k F()t +
6k )
( 4 .1 .1 0 )
n-i
w h e r e t,, is r e a l v a lu e d w h e n x ( i) is r e a l.
F in a lly , w e s h o u ld in d ic a te th a t y e t a n o th e r fo r m f o r th e F o u r ie r s e r ie s ca n
b e o b ta in e d by e x p a n d in g th e c o s in e fu n c tio n in ( 4 .1 .1 0 ) as
c o s ( 2 jt A / v + 0k) — c o s 2 n k F o i c o s 0t — s i n l n k F ^ r s in fy
C o n s e q u e n tly , w e c a n re w r ite ( 4 .1 .1 0 ) in th e fo rm
3t
ji ( r ) — ao + Y 2 ( a ic c o $ 2 n k F o t — bk s i n 2 -n k F o i)
( 4 .1 .1 1 )
1=1
w h ere
<3() = Co
fl* = 2|C|- [ c o s 0k
bk = 2|q| sin # *
T h e e x p r e s s io n s in ( 4 .1 .8 ) , ( 4 ,1 .1 0 ) , a n d ( 4 .1 ,1 1 ) c o n s t itu te t h r e e e q u iv a le n t fo r m s
f o r th e F o u r i e r s e r ie s r e p r e s e n ta tio n o f a re a l p e r io d ic s ig n a l.
4.1.2 Power Density Spectrum of Periodic Signals
A p e r io d ic s ig n a l h a s in fin ite e n e r g y a n d a fin ite a v e r a g e p o w e r , w h ic h is g iv e n a s
Px = ~
f
P •'T .
\x ( t ) \2d t
( 4 .1 .1 2 )
Frequency Analysis of Signals and Systems
236
Chap. 4
I f w e ta k e th e c o m p le x c o n ju g a te o f ( 4 .1 .8 ) an d s u b s t itu te f o r * * ( / ) in ( 4 .1 .1 2 ) . we
o b ta in
OC
( 4 .1 .1 3 )
DC
=
E
k~ —3C
T h e r e f o r e , w e h a v e e s ta b lis h e d th e r e la tio n
( 4 .1 .1 4 )
w h ich is c a lle d P a r s e v a l's re la tio n fo r p o w e r s ig n a ls .
T o illu s tr a te th e p h y s ic a l m e a n in g o f ( 4 .1 .1 4 ) , s u p p o s e th a t .v(r) c o n s is ts o f a
s in g le c o m p le x e x p o n e n tia l
x (t) = c , e j27!tFn'
In th is c a s e , all th e F o u r ie r s e r ie s c o e f f ic ie n ts e x c e p t c* a r e z e r o .
C o n s e q u e n tly ,
th e a v e r a g e p o w e r in th e sig n a l is
It is o b v io u s th a t |q |: r e p r e s e n ts th e p o w e r in th e Ath h a r m o n ic c o m p o n e n t o f th e
sig n a l. H e n c e th e to ta l a v e r a g e p o w e r in th e p e r io d ic sig n a l is s im p ly th e su m o f
th e a v e r a g e p o w e r s in a ll th e h a r m o n ic s .
I f w e p lo t th e |q|2 as a fu n c t io n o f th e f r e q u e n c ie s kF o, k = 0 , ± 1 , ± 2 ..........th e
d ia g r a m th a t w e o b ta in sh o w s h o w th e p o w e r o f th e p e r io d ic s ig n a l is d is tr ib u te d
a m o n g th e v a r io u s f r e q u e n c y c o m p o n e n ts .
T h is d ia g r a m , w h ic h is illu s tr a te d in
F ig . 4 .2 , is c a lle d th e p o w e r d e n sity sp e ctru m * o f th e p e r io d ic s ig n a l x ( t ). S in c e th e
Pow er density spectrum
- 4 F 0 - 3 F 0 —2Fn —F0
Figure 4.2
lct P
0
Fn
2F0 3F0 4F0
Frequency. F
Pow er density spectrum of a continuous-tim e periodic signal.
‘ This function is also called the power spectral density or. simply, the power spectrum.
237
Frequency Analysis of Continuous-Time Signals
Sec. 4.1
p o w e r in a p e r io d ic s ig n a l e x is ts o n ly at d is c r e te v a lu e s o f f r e q u e n c ie s ( i .e .. F — 0.
± F o . ± 2 F q . . . . ) . th e s ig n a l is s a id to h a v e a lin e sp e ctru m . T h e s p a c in g b e tw e e n
tw o c o n s e c u tiv e s p e c tr a l lin e s is e q u a l to th e r e c ip r o c a l o f th e fu n d a m e n ta l p e rio d
Tp . w h e r e a s th e s h a p e o f th e s p e c tr u m ( i.e .. th e p o w e r d is tr ib u tio n o f th e s ig n a l),
d e p e n d s o n th e tim e -d o m a in c h a r a c t e r is t ic s o f th e sig n a l.
A s in d ic a te d in th e p r e c e d in g s e c t i o n , th e F o u r ie r s e r ie s c o e f f ic ie n ts ( q ) a re
c o m p le x v a lu e d , th a t is. th e y c a n b e r e p r e s e n te d as
Ck = \ck\eJhL
w h ere
6k = 4-Q
In s te a d o f p lo ttin g th e p o w e r d e n sity s p e c tr u m , w e c a n p lo t th e m a g n itu d e v o lta g e
s p e c tr u m {|ot|} a n d th e p h a s e s p e c tr u m {(?*} as a fu n c tio n o f f r e q u e n c y . C le a r ly , th e
p o w e r s p e c tr a l d e n s ity in th e p e r io d ic s ig n a l is s im p ly th e s q u a r e o f th e m a g n itu d e
s p e c tr u m . T h e p h a s e in fo r m a tio n is to ta lly d e s tr o y e d ( o r d o e s n o t a p p e a r ) in th e
p o w e r s p e c tr a l d e n s ity .
I f th e p e r io d ic sig n a l is re a l v a lu e d , th e F o u r ie r s e r ie s c o e f f ic ie n ts { c * } sa tis fy
th e c o n d itio n
c -k =
C o n s e q u e n tly . Ki|: = |q|: . H e n c e th e p o w e r s p e c tru m is a s y m m e tr ic fu n c tio n o f
f r e q u e n c y . T h i s c o n d itio n a ls o m e a n s th a t th e m a g n itu d e s p e c tr u m is s y m m e tr ic
( e v e n f u n c t io n ) a b o u t th e o rig in an d th e p h a s e s p e c tru m is a n o d d fu n c tio n .
As
a c o n s e q u e n c e o f th e s y m m e tr y , it is s u ffic ie n t to s p e c ify th e s p e c tr u m o f a re a l
p e r io d ic sig n a l f o r p o s itiv e f r e q u e n c ie s o n ly . F u r th e r m o r e , th e to ta l a v e r a g e p o w e r
c a n b e e x p r e s s e d as
Px —
+ 2
1q |
( 4 .1 .1 5 )
( 4 .1 .1 6 )
w h ich fo llo w s d ir e c t ly fro m th e r e la tio n s h ip s g iv e n in S e c t io n 4 .1 .1 a m o n g { aa },
{bn }. a n d ( q ) c o e f fic ie n ts in th e F o u r ie r s e r ie s e x p r e s s io n s .
Example 4.1.1
Determ ine the Fourier series and the power density spectrum of the rectangular pulse
train sienal illustrated in Fie. 4.3.
x(t)
-T„
Figure 4 J Continuous-time periodic
tram of rectangular pulses.
238
Chap. 4
Frequency Analysis of Signals and System s
Solution The signal is periodic with fundam ental period Tp and. clearly, satisfies the
Dirichlet conditions. Consequently, we can represent the signal in the Fourier series
given by (4.1.8) with the Fourier coefficients specified by (4.1.9).
Since x(t) is an even signal [i.e.. x(t) = * ( - ;) ] , it is convenient to select the
integration interval from ~ T p/2 to Tp/2. Thus (4.1.9) evaluated for k = 0 yields
(4.1.17)
The term c(I represents the average value (dc com ponent) of the signal .r (r). For k ^ 0
we have
Tp l ~j2TTkFn\ _ jri
ej*yf-i,r _
A
tt F»kTp
A t sin TrkF^r
Tp
(4.1.18)
)2
k = ± 1 .± 2 . ...
n k F\\T
It is interesting to note that the right-hand side of (4.1.18) has the form (sin 4>)/4>,
where <p = n k F ^ . In this case <t>takes on discrete values since F(, and r are fixed and
the index k varies. However, if we plot (sin $)/</> with 0 as a continuous param eter
over the range —oc < <t> < oc, we obtain the graph shown in Fig. 4.4. We observe
that this function decays to zero as 0 —<■ ± x . has a maximum value of unity at <p = 0,
and is zero at multiples of tt (i.e., at <
p= mn. m = ±1, ± 2 ,...) . It is clear that the
Fourier coefficients given by (4.1.18) are the sample values of the (sin <p)/<p function
for <$>— xkFut and scaled in amplitude by A t / T p.
Since the periodic function x{t) is even, the Fourier coefficients c* are real.
Consequently, the phase spectrum is either zero, when c* is positive, or t t when c k is
negative. Instead of plotting the m agnitude and phase spectra separately, we may sim­
ply plot |c*} on a single graph, showing both the positive and negative values ck on the
graph. This is commonly done in practice when the Fourier coefficients {c*} are real.
Figure 4.5 illustrates the Fourier coefficients of the rectangular pulse train when
Tp is fixed and the pulse width t is allowed to vary. In this case Tp = 0.25 second, so
that F() = \ j T p = 4 Hz and t = 0.057),, t = 0.17},, and r = 0.27),. We observe that
the effect of decreasing r while keeping Tp fixed is to spread out the signal power
over the frequency range. The spacing between adjacent spectral lines is F{) = 4 Hz,
independent of the value of the pulse width r.
sin <p
- I n —6n -57T —4n —3jt —2n —n
n
2n
3n
0
Figure 4.4
The function (sin <p)/<p.
4k
5n
bn
In
<j>
Sec. 4.1
ck
,mTrnit»
239
Frequency Analysis of Continuous-Time Signals
“J lU jii "
vfffilllll I l k ,
t = 0.1 Tp
u -itrnrm-i.
JJiLLUTT
j ct
F
T = 0.05
Tr
..................rrnTfniniiin|fnijnnfiiifiii........... ...
F
0
and the pulse width r varies.
On the other hand, it is also instructive to fix t and vary the period Tp when
Tp > r. Figure 4 .6 illustrates this condition when T,, ~ 5r. Tr = lOr. and Tp = 2 0 t .
In this case, the spacing between adjacent spectral lines decreases as Tp increases. In
the limit as Tr
oc, the Fourier coefficients q approach zero due to the factor of
Tp in the denom inator of (4 .1 .1 8 ). This behavior is consistent with the faci that as
Tp —<■ oc and r remains fixed, the resulting signal is no longer a power signal. Instead,
Cl
...... ...................IMn 1’'
..,.((111111 lllllllli,..
0
Tp = 20r
'M illin '"
F
Figure 4.6 Fourier coefficient of a rectangular pulse train with fixed pulse width
t and varying period Tp .
240
Frequency Analysis of Signals and Systems
Chap. 4
it becomes an energy signal and its average power is zero. The spectra of finite energy
signals are described in the next section.
We also note that if k / 0 and sm(7ikFnx) = 0. then ct = 0. The harmonics
with zero power occur at frequencies kF0 such that n{kFG)r = m n , m = ±1, ± 2 ,.
o r at JtF0 = m jx. For example, if F() = 4 Hz and t = Q.2TP, it follows that the spectral
components at ±20 Hz, ± 40 H z , . .. have zero power. These frequencies correspond
to the Fourier coefficients
k = ±5, ±10, ± 15........O n the other hand, if r = 0.1Tp,
the spectral components with zero power are k = ±10, ±20, ± 3 0 ........
The power density spectrum for the rectangular pulse train is
4.1.3 The Fourier Transform for Continuous-Time
Aperiodic Signals
In Section 4.1.1 w e d evelop ed the Fourier series to represent a periodic signal
as a linear com bination o f harm onically related com p lex exp on en tials. A s a con­
seq u en ce o f the periodicity, w e saw that these signals possess line spectra with
equidistant lines. T h e line spacing is equal to the fundam ental frequency, which
in turn is the inverse o f the fundam ental period of the signal. W e can view the
fundam ental period as providing the num ber o f lin es per unit o f frequency (line
d en sity), as illustrated in Fig. 4.6.
W ith this interpretation in m ind, it is apparent that if w e allow the period to
increase w ithout limit, the line spacing tends toward zero. In the limit, when the
period b eco m es infinite, the signal b eco m es aperiodic and its spectrum becom es
continuous. This argum ent suggests that the spectrum of an ap eriod ic signal will
b e the en v elo p e o f the line spectrum in the corresponding p eriod ic signal obtained
by repeating the aperiodic signal with som e period Tp.
L et us consider an aperiodic signal x ( t ) with finite duration as show n in
Fig. 4.7a. From this aperiodic signal, w e can create a periodic signal * , , ( 0 with p e­
riod Tp, as shown in Fig. 4.7b. Clearly, x r (t) = x ( t ) in the limit as Tp
oo, that is,
x{t) =
lim x p(t)
This interpretation im plies that w e should be able to obtain the spectrum of * (/)
from the spectrum o f x p(i) sim ply by taking the limit as Tp -*■ oo.
W e begin with the Fourier series representation o f x p(t).
xp{t) =
£
ckej l ^ ' 1
F0 = y
(4.1.20)
where
(4.1.21)
Sec. 4.1
241
Frequency Analysis o1 Continuous-Time Signals
,r(n
- T„
-TJ2
TJ2
0
TJ2
i
Figure 4.7
(a) Aperiodic signal ,v(/)
and (b) periodic signal xr U) constructed
bv repeating x(t) with a period Tr .
(h)
S in c e x p (t) — x (/ ) f o r ~ T r f 2 < t < Tp/ 2 , ( 4 .1 .2 1 ) c a n b e e x p r e s s e d as
1 'Tr /2
= — /
x ( t ) e ~ )27' kF'', d t
Tp J-Trfl
( 4 .1 .2 2 )
It is a ls o tr u e th a t .* (r ) = 0 fo r |r| > Tpj2 . C o n s e q u e n tly , th e lim its o n th e in te g r a l
in ( 4 .1 .2 2 ) c a n b e r e p la c e d b y —o c an d o c . H e n c e
ct = —
f
x { t ) e ~ j2 * kF"’ d t
( 4 .1 .2 3 )
Tp J-oc
L e t us n o w d e fin e a fu n c tio n X ( F ) , c a lle d th e F o u r ie r tra n sfo rm o f * { / ) , as
X (F) =
f
x (t)e ~ lln F , dt
( 4 .1 .2 4 )
J-oc
A' ( F )
F q.
is a fu n c t io n o f th e c o n tin u o u s v a r ia b le F .
I t d o e s n o t d e p e n d o n Tp o r
H o w e v e r , i f w e c o m p a r e ( 4 .1 .2 3 ) a n d ( 4 .1 .2 4 ) , it is c l e a r th a t th e F o u r ie r
c o e f f ic ie n t s ct c a n b e e x p r e s s e d in te r m s o f X ( F ) as
c* = ^ X ( k F o )
1p
o r e q u iv a le n tly .
Tpc k = X ( k F 0) = X
( 4 A .2 5 )
T h u s th e F o u r i e r c o e ff ic ie n ts a r e s a m p le s o f X ( F ) ta k e n a t m u ltip le s o f f o a n d
s c a le d b y F 0 (m u ltip lie d b y \ / T p).
S u b s titu tio n f o r c t f r o m ( 4 .1 .2 5 ) in to ( 4 .1 .2 0 )
y ie ld s
V O
= ^r 'jh x ( ^ r ) er M
(4 .1 .2 6 )
242
Frequency Analysis of Signals and Systems
W e w ish to ta k e th e lim it o f ( 4 .1 .2 6 ) a s T r a p p r o a c h e s in fin ity .
Chap, 4
F i r s t , w e d e fin e
A F = 1/7},. W ith th is s u b s t itu tio n , ( 4 .1 .2 6 ) b e c o m e s
xpU ) =
Y2
X (k & F )e
JlxkAFi
( 4 .1 .2 7 )
k=-x
I t is c l e a r th a t in th e lim it as Tp a p p r o a c h e s in fin ity , x p {t) r e d u c e s to jc(/ ). A ls o , A F
b e c o m e s th e d iff e r e n tia l d F a n d k A F b e c o m e s th e c o n tin u o u s f r e q u e n c y v a ria b le
F.
In tu r n , th e s u m m a tio n in ( 4 .1 .2 7 ) b e c o m e s a n in te g r a l o v e r th e fr e q u e n c y
v a r ia b le F . T h u s
( 4 .1 .2 8 )
T h is in te g r a l r e la tio n s h ip y ie ld s x ( t ) w h e n X ( F )
is k n o w n , a n d it is c a lle d th e
in erse F o u r ie r tra n sfo rm .
T h is c o n c lu d e s o u r h e u r is tic d e r iv a tio n o f th e F o u r i e r tr a n s f o r m p a ir g iv e n
b y ( 4 .1 .2 4 ) an d ( 4 .1 .2 8 ) f o r an a p e r io d ic s ig n a l x ( t ).
A lth o u g h th e d e r iv a tio n is
n o t m a th e m a tic a lly rig o r o u s , it le d to th e d e s ir e d F o u r i e r t r a n s f o r m re la tio n s h ip s
w ith r e la tiv e ly s im p le in tu itiv e a r g u m e n ts . In s u m m a r y , th e f r e q u e n c y a n a ly s is o f
c o n tin u o u s -tim e a p e r io d ic s ig n a ls in v o lv e s th e fo llo w in g F o u r i e r tr a n s f o r m p a ir.
FR EQ U EN C Y ANALYSIS O F C O N T IN U O U S -T IM E A P ER IO DIC SIGN ALS
Synthesis equation
inverse transform
(4 .1 .2 9 )
Analysis equation
direct transform
( 4 .1 .3 0 )
I t is a p p a r e n t th a t th e e s s e n tia l d if f e r e n c e b e tw e e n th e F o u r i e r s e r ie s a n d th e
F o u r i e r tr a n s fo r m is th a t th e s p e c tr u m in th e l a t t e r c a s e is c o n tin u o u s a n d h e n c e
th e s y n th e s is o f an a p e r io d ic s ig n a l fr o m its s p e c tr u m is a c c o m p lis h e d b y m e a n s o f
in te g r a tio n in s te a d o f s u m m a tio n .
F in a lly , w e w ish to in d ic a te th a t th e F o u r ie r t r a n s f o r m p a ir in ( 4 .1 .2 9 ) and
( 4 .1 .3 0 ) c a n b e e x p r e s s e d in te r m s o f th e r a d ia n fr e q u e n c y v a r ia b le Q =
2nF.
S in c e d F = d S l / l n . ( 4 .1 .2 9 ) a n d ( 4 .1 .3 0 ) b e c o m e
( 4 .1 .3 1 )
( 4 .1 .3 2 )
T h e s e t o f c o n d itio n s th a t g u a r a n te e th e e x is t e n c e o f th e F o u r i e r tr a n s f o r m is th e
Sec. 4.1
Frequency Analysis of Continuous-Time Signals
243
D ir ic h le t c o n d it io n s , w h ich m a y b e e x p r e s s e d as:
1 . T h e s ig n a l v (r) h as a fin ite n u m b e r o f fin ite d is c o n tin u itie s .
2 . T h e s ig n a l x ( t ) h a s a fin ite n u m b e r o f m a x im a a n d m in im a .
3 . T h e s ig n a l x ( t ) is a b s o lu te ly in te g r a b le , th a t is.
\x (t )\d t < o c
i:
( 4 .1 .3 3 )
T h e th ir d c o n d itio n fo llo w s e a s ily fr o m th e d e fin itio n o f th e F o u r i e r tr a n s f o r m ,
g iv e n in ( 4 .1 .3 0 ) . In d e e d .
\ X(F)\ =
jj
x ( t ) e - J2” F ,dt
< j
\x ( t )\d t
H e n c e ] X ( F ) ! < o c if ( 4 .1 .3 3 ) is s a tis fie d .
A w e a k e r c o n d itio n f o r th e e x is te n c e o f th e F o u r i e r t r a n s f o r m is th a t x {t)
h a s fin ite e n e r e v ; th a t is.
|.v(/)|^r < o c
( 4 .1 .3 4 )
N o te th a t if a s ig n a l x ( i) is a b s o lu te ly in te g r a b le . it w ill a ls o h a v e fin ite e n e rg y .
T h a t is. if
£
|.v(/)!^r < o c
J —CM
th e n
rx
|.v(/)i“^/ < o c
H o w e v e r , th e c o n v e r s e is n o t tr u e .
( 4 .1 .3 5 )
T h a t is. a s ig n a l m a y h a v e fin ite e n e r g y b u t
m a y n o t b e a b s o lu te ly in te g r a b le . F o r e x a m p le , th e s ig n a l
sin 2 n t
x {t) = — -—
7
( 4 .1 .3 6 )
Tt
is s q u a r e in te g r a b le b u t is n o t a b s o lu te ly in te g r a b le .
T h is s ig n a l h a s th e F o u r i e r
tr a n s fo r m
f1
* ( F ) = { o ;
\FI < 1
(4 -1 .3 7 )
S in c e th is s ig n a l v io la te s ( 4 .1 .3 3 ) , it is a p p a r e n t th a t th e D i r i c h l e t c o n d itio n s a re
s u ffic ie n t b u t n o t n e c e s s a r y f o r th e e x is te n c e o f th e F o u r i e r tr a n s f o r m . In a n y c a s e ,
n e a r ly all fin ite e n e r g y s ig n a ls h a v e a F o u r ie r tr a n s f o r m , s o t h a t w e n e e d n o t w o rry
a b o u t th e p a th o lo g ic a l s ig n a ls , w h ich a r e s e ld o m e n c o u n t e r e d in p r a c tic e .
4.1.4 Energy Density Spectrum of Aperiodic Signals
L e t x (t ) b e a n y fin ite e n e r g y s ig n a l w ith F o u r i e r t r a n s fo r m X ( F ) . Its e n e r g y is
244
Frequency Analysis of Signals and Systems
Chap. 4
w hich, in turn, may be expressed in term s o f X ( F ) as follows:
Ex -
f
x ( t ) x '( i) d t
J —OC
1oo
X '( F ) d F \
r /*oc
/
x { t ) e ~ j2 n F 'd t
'OC
\ X ( F ) \ 2d F
T h erefore, w e con clu d e that
(4.1.38)
This is P a r s e v a l’s re la tio n for aperiodic, finite energy signals and expresses the
principle o f conservation of energy in the tim e and frequency dom ains.
T he spectrum X ( F ) o f a signal is in general, com p lex valued. C onsequently,
it is usually expressed in polar forms as
X ( F ) = | X ( F ) | ^ W(f)
where |X ( F ) | is the m agnitude spectrum and © (F ) is the phase spectrum .
© (F ) - i U ( F )
O n the other hand, the quantity
SXX( F ) = \X ( F ) |2
(4.1.39)
which is the integrand in (4.1.38), represents the distribution of en ergy in the signal
as a function o f frequency. H en ce SXX( F ) is called the e n e rg y d e n sit y s p e ctru m of
x ( t ). T he integral o f S XX( F ) over all freq u en cies gives the total en ergy in the signal.
V iew ed in another way, the energy in the signal x ( t ) over a band o f frequencies
F \ < F < F [ + A F is
From (4.1.39) w e observe that S XX( F ) d o es not con tain any p h ase inform ation
[i.e., SXX( F ) is purely real and n on n egative]. Since the phase spectrum of ;t(r) is
not contained in SXX( F ) , it is im possible to reconstruct the signal given S XX( F ) .
Finally, as in the case of Fourier series, it is easily show n that if the signal
x (t ) is real, then
\X ( -F ) \ =
± X { -F )
\X ( F ) \
(4.1.40)
= -* X (F )
(4.1.41)
Sec. 4.1
Frequency Analysis of Continuous-Time Signals
245
By com bining (4.1.40) and (4.1.39), w e obtain
(4.1.42)
SXX( ~ F ) = S xA F )
In other words, the energy density spectrum o f a real signal has even sym m etry.
Exam ple 4.1.2
D eterm ine the Fourier transform and the energy density spectrum of a rectangular
pulse signal defined as
and illustrated in Fig. 4.8(a).
Solution Clearly, this signal is aperiodic and satisfies the Dirichlet conditions. Hence
its Fourier transform exists. By applying (4.1.30), we find that
(4.1.44)
We observe that X( F) is real and hence it can be depicted graphically using only
one diagram, as shown in Fig. 4.8(b). Obviously, X( F) has the shape of the (sin0)/<?
function shown in Fig. 4.4. Hence the spectrum of the rectangular pulse is the en­
velope of the line spectrum (Fourier coefficients) of the periodic signal obtained by
A
0
T
r
~>
2
(a)
X<F)
Ar
F
(b)
Figure 4.8
(a) R ectangular pulse and (b) its F ourier transform .
246
Frequency Analysis of Signals and System s
Chap. 4
periodically repeating the pulse with period Tp as in Fig. 4.3. In other words, the
Fourier coefficients ck in the corresponding periodic signal xp{t) are simply samples
of X( F) at frequencies kF0 = k/ Tp. Specifically,
From (4.1.44) we note that the zero crossings of X( F ) occur at multiples of 1 /r.
Furtherm ore, the width of the main lobe, which contains most of the signal en­
ergy, is equal to 2/z. As the pulse duration t decreases (increases), the main
lobe becomes broader (narrow er) and m ore energy is moved to the higher (lower)
frequencies, as illustrated in Fig. 4.9. Thus as the signal pulse is expanded (com­
pressed) in time, its transform is compressed (expanded) in frequency. This be­
havior, between the time function and its spectrum, is a type of uncertainty
principle that appears in different forms in various branches of science and engi­
neering.
Finally, the energy density spectrum of the rectangular pulse is
, /sm 7 rj
S „ ( F ) = ( A t )1 ( " ~ ' ^ T )
} ( ttF-
(4.1.46)
*(r)
X 0 I
2
2
x (r)
X(F)
A
- L
0
1
2
2
V
X(l)
A
Figure 4.9
F o u rier transform of a re d a n g u la r pulse for various w idth values.
Sec. 4.2
Frequency Analysis of Discrete-Time Signals
247
4.2 FREQUENCY ANALYSIS OF DISCRETE-TIME SIGNALS
In Section 4.1 we d ev elop ed the Fourier series representation for continuous-tim e
periodic (pow er) signals and the Fourier transform for finite energy aperiodic
signals. In this section we repeat the d evelop m en t for the class o f discrete-tim e
signals.
A s we have ob served from the discussion o f Section 4.1, the Fourier series
representation o f a con tinu ou s-tim e periodic signal can consist o f an infinite num ­
ber o f frequency com p onents, where the frequency spacing b etw een tw o successive
harm onically related freq u en cies is 1 / T p, and w here Tp is the fundam ental period.
Since the frequency range for continuous-tim e signals exten d s from —oo to oc, it
is p ossib le to have signals that contain an infinite num ber o f frequency com p o­
nents. In contrast, the frequency range for discrete-tim e signals is unique over the
interval ( - r t . T i ) or (0 .2 ^ ). A discrete-tim e signal o f fundam ental period N can
consist o f frequency co m p on en ts separated by 2n / N radians or / = 1/jV cycles.
C onsequently, the Fourier series representation o f the discrete-tim e periodic signal
will contain at m ost N frequency com ponents. This is the basic d ifference b etw een
the Fourier series representations for continuous-tim e and discrete-tim e periodic
signals.
4.2.1 The Fourier Series for Discrete-Time Periodic
Signals
Suppose that w e are given a periodic sequ en ce a (/ i ) with period N , that is, x i n ) =
x(n + N ) for all n. T he Fourier series representation for x(h) consists o f N har­
m onically related exp on en tial functions
k = 0 A ........ N — 1
and is expressed as
A'-l
X („) = Y ^ c kejlKkn/N
*=o
(4.2.1)
where the {<:*) are the coefficients in the series representation.
T o derive the expression for the Fourier coefficients, w e use the follow ing
formula:
= (
I 0,
1
y '1
n=0
* = 0. ± N , ± 2 N, . . .
otherw ise
2 2
N o te the sim ilarity o f (4.2.2) with the con tinu ou s-tim e counterpart in (4.1.3). T he
p roof o f (4.2.2) follow s im m ediately from the application of the geom etric sum ­
m ation form ula
jv-i
f
a = l
248
Frequency Analysis of Signals and System s
Chap. 4
T h e e x p r e s s io n f o r th e F o u r i e r c o e f f ic ie n ts c* c a n b e o b t a i n e d b y m u ltip ly in g
b o th s id e s o f ( 4 .2 .1 ) b y th e e x p o n e n tia l e ~ j2nin//v a n d s u m m in g th e p r o d u c t fro m
« = 0 t o n
= yV — 1. T h u s
N —1 A’- ]
A'-l
^ 2 x ( n ) e ~ il!rln /N
c ke J2n<k~ l)n/N
n=(J
(4 .2 .4 )
n=0 t=0
I f w e p e r fo r m th e s u m m a tio n o v e r n first, in th e r ig h t-h a n d s id e o f (4 .2 .4 ) ,
w e o b ta in
y ^ (fj2.TU-/)n/A' _
k - I = 0, ± N , ± 2 N , . . .
o th e r w is e
| N,
I 0,
w h e r e w e h a v e m a d e u se o f ( 4 .2 .2 ) .
(4 2 5)
T h e r e f o r e , th e rig h t-h a n d s id e o f ( 4 .2 .4 )
r e d u c e s to N c / a n d h e n c e
j A'-l
q = -
J ' x ( n ) e ~ J2*‘" /tl
1 = 0. 1 .......... N - 1
(4 .2 .6 )
/T = (J
T h u s w e h a v e th e d e s ir e d e x p r e s s io n fo r th e F o u r i e r c o e f f ic ie n t s in te r m s o f th e
s ig n a l x ( « ) .
T h e r e la tio n s h ip s ( 4 .2 .1 ) a n d ( 4 .2 .6 ) fo r th e f r e q u e n c y a n a ly s is o f d is c r e te *
tim e s ig n a ls a r e s u m m a r iz e d b e lo w .
FR E Q U E N C Y ANALYSIS O F D IS C R E TE -TIM E P E R IO D IC SIGN ALS
Synthesis equation
iW l
-1 %
II
Analysis equation
A'-l
X(/! ) = ^ Ck(j2jTt"!r'
*=0
(4.2.7)
(4.2.8)
E q u a t i o n ( 4 .2 .7 ) is o ft e n c a lle d th e d isc re te -tim e F o u r ie r s e rie s ( D T P S ) . T h e
F o u r i e r c o e ff ic ie n ts {c* }. k =
0 . 1 .......... N — 1 p r o v id e th e d e s c r ip tio n o f jc ( « ) in
th e f r e q u e n c y d o m a in , in th e s e n s e th a t c k r e p r e s e n t s th e a m p litu d e a n d p h a s e
a s s o c ia t e d w ith th e fr e q u e n c y c o m p o n e n t
sk(n) = e ^ kn^ ' = e jWtB
w h e r e to* = 2 n k / N .
W e re c a ll f r o m S e c tio n 1 .3 .3 th a t th e f u n c t io n s s k (n ) a r e p e r i o d i c w ith p e r io d
N . H e n c e sk(n) = sk {n + N ) . In v iew o f th is p e r io d ic ity , it fo llo w s t h a t th e F o u r ie r
c o e f f ic ie n ts c k, w h e n v ie w e d b e y o n d th e r a n g e k ~
0 , 1 , ____A ' - l , a ls o s a tis fy a
p e r io d ic ity c o n d itio n . In d e e d , fro m ( 4 .2 .8 ) , w h ich h o ld s f o r e v e r y v a lu e o f k , w e
have
Ck+ N = ^ J ^ x ( n ) e - ^ (k+N)n/N = ~ J 2 x { n ) e - ^ kn/N = c t
n=0
™ n=0
( 4 .2 .9 )
Sec. 4.2
249
Frequency Analysis of Discrete-Time Signals
T h e r e f o r e , th e F o u r i e r s e r ie s c o e f f ic ie n ts { q } fo r m a p e r io d ic s e q u e n c e w h e n e x ­
te n d e d o u ts id e o f t h e r a n g e k = 0 , 1 ..........jV — 1. H e n c e
Ck+N = <^k
th a t is , { c t } is a p e r io d ic s e q u e n c e w ith f u n d a m e n ta l p e r io d N.
T hu s the s p e ctru m
o f a signa l x(n ), w h ich is per iod ic with p e r i o d N , is a p er io d ic s e qu en ce with p e r io d
N . C o n s e q u e n tly , a n y N c o n s e c u tiv e s a m p le s o f th e s ig n a l o r its s p e c tr u m p r o v id e
a c o m p le t e d e s c r ip tio n o f th e s ig n a l in th e tim e o r f r e q u e n c y d o m a in s .
A lth o u g h th e F o u r i e r c o e f fic ie n ts fo r m a p e r io d ic s e q u e n c e , w e w ill fo c u s o u r
a tte n tio n o n th e s in g le p e r io d w ith ra n g e k = 0 , 1 .......... N — 1. T h i s is c o n v e n ie n t,
s in c e in th e f r e q u e n c y d o m a in , th is a m o u n ts to c o v e r in g th e f u n d a m e n ta l ra n g e
0 < a>t =
2 n k /N
< 2 n , fo r 0 < k < N — 1.
In c o n t r a s t , th e f r e q u e n c y ra n g e
—it < a)* = 2 7 i k / N
< j t , c o r r e s p o n d s to —N / 2 < k < N / 2 , w h ic h c r e a t e s an
in c o n v e n ie n c e w h e n N is o d d . C le a r ly , i f w e u se a s a m p lin g f r e q u e n c y F s , th e
r a n g e 0 < k < N — 1 c o r r e s p o n d s to th e f r e q u e n c y ra n g e 0 < F < F , .
Example 4.2.1
D eterm ine the spectra of the signals
(a) jr(») = cos -Ji nn
(b) x(n) = cos nn/ 3
(c) x( n ) is periodic with period N = 4 and
x(n) = {1, 1.0.0}
T
Solution
(a) For m = J i n , we have /« = 1j - Jl . Since f , is not a rational number, the signal
is not periodic. Consequently, this signal cannot be expanded in a Fourier series.
N evertheless, the signal does possess a spectrum. Its spectral content consists
of the single frequency component at id = wo = -Jin.
(b) In this case / (l = | and hence x{n) is periodic with fundam ental period N = 6.
From (4.2.8) we have
5
It = 0 . 1 ........5
However, x(n) can be expressed as
x(n) = cos —
" r— =
+ \ e~i2*nib
6
*
which is already in the form of the exponential Fourier series in (4.2.7). In
comparing the two exponential terms in x{n) with (4.2.7), it is apparent that
ci = j. The second exponential in x(n) corresponds to the term Jt = —1 in
(4.2.7), However, this term can also be written as
- j ’2 j r n / 6 _
^ j2jr (5 n> /6
which means that c_i = c$. But this is consistent with (4.2.9), and our previous
observation that the Fourier series coefficients form a periodic sequence of
250
Chap. 4
Frequency Analysis of Signals and Systems
period N . Consequently, we conclude that
= C\ = f 4 = 0
C(, =
c, = ^
i
(c) From (4.2.8). we have
1
A-= 0 . 1 ,2 ,3
Ck
or
Ct = l ( l + e - i ^ )
A-= 0 . 1.2.3
For k = 0, 1, 2, 3 we obtain
f'l — j ( l — j )
C'd = S
f'2 = 0
Cl = j { l + j }
The m agnitude and phase spectra are
K'd I
ki
4-Co = 0
4 f| = --
2tr- = undefined
4_o = —
4
4
Figure 4.10 illustrates the spectral content of the signals in (b) and (c).
4.2.2 Power Density Spectrum of Periodic Signals
T h e a v e r a g e p o w e r o f a d is c r e te - tim e p e r io d ic sig n a l w ith p e r io d N w as d efin ed
in ( 2 .1 .2 3 ) as
P, = -
£
M » )|
( 4 .2 .1 0 )
W e s h a ll n ow d e riv e a n e x p r e s s io n fo r P x in te r m s o f th e F o u r i e r c o e f f ic ie n t {c<J.
I f w e u se th e r e la tio n ( 4 .2 .7 ) in ( 4 .2 .1 0 ) , w e h a v e
j v -i
Pi =
— T
x(rt)x*(n)
n = ()
A'-l
N
N o w . w e c a n in te r c h a n g e t h e o r d e r o f th e tw o s u m m a tio n s a n d m a k e u s e o f ( 4 .2 .8 ) ,
o b ta in in g
p * = Y , ct
*=<j
IV—1
A'-l
, A-i
- y x ( n ) e - j2”kn/N
-i A —I
( 4 . 2 . 11 )
Sec. 4.2
Frequency Analysis of Discrete-Tim e Signals
251
(a)
Zc t
Jr
"4
-3
5
1
- 2
- 1
0
... t
2 3 4
71
4
Figure 4.10 Spectra of the periodic
signals discussed in Example 4.2.1 (b)
and (c).
(c)
which is the desired exp ression for the average p ow er in the p eriod ic signal. In
other w ords, the average pow er in the signal is th e sum o f th e pow ers o f the
individual frequency com p onents. W e view (4.2.11) as a P arseval’s relation for
d iscrete-tim e period ic signals. T h e seq u en ce | a |: for k = 0, 1 , — N - 1 is the
distribution o f p ow er as a function o f freq u en cy and is called the p o w e r density
spectrum o f the periodic signal.
If w e are interested in the energy o f th e se q u en ce Jt(n) o v er a single period,
(4.2.11) im plies that
/V—I
N -l
n=0
k—()
(4.2.12)
which is consistent with our p reviou s results for con tin u ou s-tim e p eriodic signals.
If the signal x ( n ) is real [i.e., x ‘ (n) = jr(n)], then, p roceed in g as in S ec­
tion 4.2.1, w e can easily sh ow that
ct = c . k
(4.2.13)
252
Frequency Analysis of Signals and Systems
Chap. 4
or equivalently,
|c_*| =
|c * f
- 4 c_t =
(even sym m etry)
(4.2.14)
(odd sym m etry)
(4.2.15)
T h ese sym m etry properties for the m agnitude and phase spectra o f a periodic sig­
nal, in conjunction with the periodicity property, have very im portant im plications
on the frequency range o f discrete-tim e signals.
In d eed , by com bining (4.2.9) with (4.2.14) and (4.2.15), w e obtain
Iq
I =
4-c*
=
(4.2.16)
and
—
(4.2.17)
M ore specifically, w e have
=
\c N p \
~
k jV /2 |,
A -c N[2
k (/V -l)/2
=
k ( A f + l)/2 l<
4-C (N -1)/2
*
O
II
|c il
4 .c 0 =
4 - c' l
=
- % - c' n
-
0
— % -C N -l
(4.2.18)
=
if N is even
o
=
~ 4 - C ( /V + l)/2
if N is odd
Thus, for a real signal, the spectrum c*, k = 0, 1 ,. . . , N [ 2 for N even, or
k = 0. 1........ ( N — l ) / 2 for N odd, com p letely specifies the signal in the frequency
dom ain. Clearly, this is consistent with the fact that the highest relative frequency
that can be represented by a discrete-tim e signal is eq u al to n . In d eed , if 0 < a>k =
2 n k / N < jt, then 0 < k < N f 2.
By m aking use o f these sym m etry p roperties o f the Fourier series coefficients
o f a real signal, the F ourier series in (4.2.7) can also be exp ressed in the alternative
forms
x ( n ) = co + 2 £
= ao + ^
lc*l cos
(4.2.19)
( ^ k cos — kn - b k sin — kn 'j
(4.2.20)
w here a0 = c0. <*k = 2 [ct|co s0 * . bk = 2|c*|sin 0*, and L = N [2 if N is even and
L — ( N ~ l ) / 2 if N is odd.
F inally, we n o te that as in the case o f continuous-tim e signals, the power
density spectrum |ct |2 d oes n ot contain any phase inform ation. Furtherm ore, the
spectrum is discrete and periodic with a fundam ental period eq u al to that o f the
signal itself.
Example 4.2.2 Periodic “Square-Wave” Signal
Determ ine the Fourier series coefficients and the power density spectrum of the
periodic signal shown in Fig. 4.11.
Sec. 4.2
253
Frequ ency A nalysis of Discrete-Time Signals
0
L
Figure 4.11 Discrete-time periodic
square-wave signal.
n
A'
Solution By applying the analysis equation (4.2.8) to the signal shown in Fig. 4.11.
we obtain
=
1 L~l
= -L Y Ae- P^n/
AJ i—/
1
A j—
k = 0. 1.
N - 1
which is a geom etric sum mation. Now we can use (4.2.3) to simplify the summation
above. T hus we obtain
AL
TT’
A 1 - e- j2*kL/s
I N 1'
k= 0
k = 1.2.
N - 1
The last expression can be simplified further if we note that
1
_
e -j2xkL;S
1 - <-
-jzk L tN
I.ikl/K
e -ink;'\
( rrl. \ _ ,-jxkl.\
_
-jxHt.-u.'K sin(xkL. /N )
s in (7 r£ /A /
T herefore.
AL
k = 0. +A'. ± 2 N ,
A
, s i ni n kL/ N)
sin(7zk/N)
I A'
(4.2.21)
otherwise
The pow er density spectrum of this periodic signal is
/ AL_
k*l" =
k = 0, +N. ± 2 A'.
V N
A V / sinnkL/N
Ar / V s i n n k / N
(4.2.22)
otherwise
Figure 4.12 illustrates the plots of jct |2 for L = 5 and 7, A' = 40 and 60. and A = 1.
4.2.3 The Fourier Transform of Discrete-Time Aperiodic
Signals
Just as in the case o f con tinu ou s-tim e aperiodic energy signals, the frequency anal­
ysis o f d iscrete-tim e aperiodic finite-energy signals in volves a Fourier transform of
the tim e-d om ain signal. C on sequ en tly, th e d evelop m en t in this section parallels
to a large ex ten t, that given in Section 4.1.3.
254
Frequency Analysis of Signals and Systems
Chap. 4
L = 5, N = 40
I 1II I 1. . I I I I ■
_xlJ
-2 0
-1 0
0
10
20
L = 7, N = 60
LJ i l
-3 0
- 20
-1 0
0
10
20
30
= 5. N =
-3 0
- 20
—10
0
10
20
30
Figure 4.12 Plot of the pow er density
spectrum given by (4.2.22).
T he Fourier transform o f a finite-energy discrete-tim e signal x ( n ) is defined as
OC
X (co) =
x ( n ) e ~ Jwr
(4.2.23)
FI — “ OC
Physically, X(co) represents the frequency con ten t o f the signal x ( n ) . In other
words. X{(o) is a d ecom p osition o f x ( n ) into its frequency com p onents.
W e observe tw o basic d ifferen ces b etw een the Fourier transform of a discrete­
tim e finite-energy signal and the F ourier transform o f a finite-energy analog signal.
First, for continuous-tim e signals, the Fourier transform , and h en ce the spectrum
o f the signal, have a frequency range o f {—0 0 , 0 0 ). In contrast, the frequency
range for a discrete-tim e signal is unique over the frequency interval o f (—n , n)
or, equivalently, (0. 2 tt). T h is property is reflected in the Fourier transform o f the
Sec. 4.2
Frequency Analysis of Discrete-Time Signals
255
signal. In d eed . X(co) is period ic with period 2 n . that is.
X(a> + 2 7 T k )=
22
j{u>^-27ik)u
x { n ) e 'i{w^ 7' k)n
—cc
oc
=
J 2 x ( n ) e - ' ame - JZ7,kn
(4.2.24)
/l = —OC
OC
=
2 2
x ( n ) e - , Mn
= X(w)
H en ce X ( t o ) is p eriod ic with period 2 t t . B u t this property is just a con seq u en ce of
the fact that the frequency range for any discrete-tim e signal is lim ited to (—n , n )
or (0 , 27r). and any frequency outside this interval is eq u ivalen t to a frequency
within the interval.
T h e secon d basic d ifference is also a con seq u en ce o f the discrete-tim e nature
o f the signal. Since the signal is discrete in tim e, the Fourier transform o f the
signal in v o lv es a sum m ation o f term s instead o f an integral, as in the case of
continuous-tim e signals.
Since X (w ) is a periodic function o f the frequency variable a), it has a Fourier
series exp an sion , provided that the con d ition s for the existen ce o f the Fourier
series, described previously, are satisfied. In fact, from th e definition o f the
Fourier transform X (co) o f the sequ en ce x(n), given by (4.2.23), w e ob serve that
X (a >) has the form o f a Fourier series. T he Fourier coefficients in this series
expansion are the valu es o f the seq u en ce x ( n ) .
T o dem onstrate this point, let us evaluate the seq u en ce x ( n ) from X(co). First,
we m ultiply both sides (4.2.23) by ej<um and integrate over the interval ( —tt, t t ) .
Thus w e have
(4.2.25)
The integral on the right-hand side o f (4.2.25) can be evaluated if w e can inter­
change the order o f sum m ation and integration. This interchange can be m ade if
the series
N
X N (a>) = 2 2 x ( n ) e - llon
n=-N
con verges uniform ly to X(o)) as A1 ->• oc. U niform con vergen ce m ean s that, for
every w, Xh((d) —*■ X(ca), as /V -*• oo. T he con vergen ce o f the Fourier transform
is discussed in m ore detail in the follow in g section. For the m om en t, let us as­
sum e that the series con verges uniform ly, so that w e can interchange the order of
sum m ation and integration in (4.2.25). Then
256
Frequency Analysis of Signals and Systems
Chap.
2:Tx(m).
0.
(4.2.26)
C onsequently,
£
x{n)
rj = — OC
n'da) =
fJ —71
By com bining (4.2.25) and (4.2.26). w e obtain the desired result that
1
x (n ) — —
fn
I
±.7T J —jt
X ( o j) e J
do)
(4.2.27)
If w e com pare the integral in (4.2.27) with (4,1.9), we n o te that this is just
the expression for the Fourier series coefficient for a function that is periodic with
period 2 n . The only difference b etw een (4.1.9) and (4.2.27) is the sign on the
exp on en t in the integrand, which is a con sequ en ce o f our definition o f the Fourier
transform as given by (4.2.23). T h erefore, the Fourier transform of the sequence
x ( n ) , defined by (4.2.23), has the form o f a Fourier series expansion.
In summary, the Fourier tra nsform p a ir f o r discrete-time signals is as follows.
FR EQ U EN C Y ANALYSIS OF DtS C R E TE -TIM E AP ER IO D IC S IG N A LS
Synthesis equation
inverse transform
x(
Analysis equation
direct transform
X {io )=
(4 .2 .2 8 )
X < w ) i ’ lu ' ' ' d w
OC
2 2
(4.2.29)
x ( n ) e ~ , w '1
4.2.4 Convergence of the Fourier Transform
In the derivation o f the inverse transform given by (4.2.28), w e assum ed that the
series
Xjv(cw) =
22 xWe~
(4.2.30)
converges uniform ly to X(a>), given in the integral o f (4.2.28), as N -* oc. By
uniform convergence w e m ean that for each
lim { s u p X (oj) ~ X s (>>)'} = 0
(4.2.31)
N—> * <*>
U niform convergence is guaranteed if ;c(/j) is ab solu tely sum m able. Indeed, if
DC
2 2 |j c )i < oc
(4.2.32)
n=-oc
then
|X M l =
22
x (n )e '
<
2 2
<
00
H en ce (4.2.32) is a sufficient condition for the existen ce o f the d iscrete-tim e Fourier
transform. W e n o te that this is the discrete-tim e counterpart of th e third Dirich-
Sec. 4.2
257
Frequency Analysis of Discrete-Time Signals
let condition for the Fourier transform o f con tinu ou s-tim e signals. T he first two
conditions d o n ot apply due to the discrete-tim e nature o f [*(«)}.
S om e se q u en ces are not absolutely sum m able, but they are square sum m able.
That is, they have finite energy
OC
Ex =
^
|jc(n ) |2 < oo
(4.2.33)
n = —DC
which is a w eak er condition than (4.2.32). W e w ould like to d efine the Fourier
transform o f finite-energy sequ en ces, but w e m ust relax the con d ition o f uniform
con vergen ce. For such seq u en ces w e can im pose a m ean-square con vergen ce co n ­
dition:
lim f |X M - X N ((ti)\2dto = 0
Af—<* J-71
(4.2.34)
Thus the energy in the error X(a>) - X/v(oj) tends toward zero, but the error
|X (w ) - Xw(a))j d o es not necessarily tend to zero. In this way w e can include
finite-energy signals in the class o f signals for which the Fourier transform exists.
Let us con sid er an exam p le from the class o f finite-energy signals. Suppose
that
X M = ( 1’
[ U.
H ^
‘|< 7 r
(4-2.35)
< ja>| < 7T
T he reader should rem em ber that X(a>) is periodic with p eriod 2 n . H en ce (4.2.35)
represents only on e period o f X(co). The inverse transform o f X { cj) results in the
sequence
i
jr(«) = —
=
1
-~
In
r
I
X { w ) e ja>nda)
I
eJ
sin<u,-n
dm = ------------
n^O
Jin
For n = 0, w e have
*( 0 )
H en ce
x (n ) =
n
(j)c sin a)cn
n
cocn
n = 0
(4.2.36)
n
0
This transform pair is illustrated in Fig. 4.13.
S om etim es, the seq u en ce {*(«)) in (4.2.36) is exp ressed as
sin cocn
x ( n ) = ----------
— oo < n < oo
(4.2.37)
258
Frequency Analysis of Signals and Systems
Chap. 4
X(w)
1
1
-JT
— UJt
0
(i),
7T
(b)
Figure 4.13
Fourier transform pair in (4.2.35) and (4.2.36j.
with the understanding that at n = 0, x ( n ) = w j n . W e should em p hasize, however,
that (sina)(7i)/jrfi is not a con tinu ou s function, and hence L ’H osp ital's rule cannot
be used to determ ine jr(0 ).
N ow iet us consider the determ ination o f the Fourier transform o f the se­
q uence given by (4.2.37), T he seq u en ce {jr(n)j is not absolutely sum m able. H ence
the infinite series
(4.2.38)
does not converge uniform ly for all w. H ow ever, the seq u en ce {jc(«)} has a finite
energy E x = o)c/ tc as will be show n in Section 4.3. H en ce the sum in (4.2.38) is
guaranteed to converge to the X (a>) given by (4.2.35) in the m ean-square sense.
T o elaborate on this point, let us con sid er the finite sum
X N (a>)=
(4.2.39)
Figure 4.14 show s the function X N{w) for several values of N . W e n ote that there
is a significant oscillatory oversh oot at co = coc, in d ep en dent o f the value o f N . As
Sec. 4.2
259
Frequency Analysis of Discrele-Ttme Signals
x , s(w)
^50<“)
A"7o(ul)
Figure 4.14 Illustration of convergence of the F ourier transform and the G ibbs
phenom enon at the point of discontinuity
N increases, the oscillation s b ecom e m ore rapid, but the size o f the ripple rem ains
the sam e. O ne can sh ow that as N -*■ oo, the oscillations con verge to the point
o f the discontinuity at to — a>t . but their am plitude d oes not go to zero. H ow ever,
(4.2.34) is satisfied, and therefore
converges to X ( w ) in the m ean-square
sense.
T he oscillatory behavior o f the approxim ation X^ic o) to the function X(a>) at
a point o f discontinuity o f
is called the G ib b s p h e n o m e n o n . A sim ilar effect
is ob served in the truncation o f the F ourier series o f a con tinu ou s-tim e periodic
signal, given by the syn th esis eq u ation (4.1.8). For exam ple, the truncation o f the
Fourier series for the period ic square-w ave signal in E xam ple 4.1.1, gives rise to
the sam e oscillatory behavior in the finite-sum approxim ation o f x ( t ) . T he G ibbs
ph en om en on will be en cou n tered again in the design o f practical, discrete-tim e
FIR system s considered in C hapter 8 .
260
Frequency Analysis of Signals and Systems
Chap. 4
4.2.5 Energy Density Spectrum of Aperiodic Signals
R ecall that the energy o f a discrete-tim e signal x ( n ) is defined as
OC
Ex = 2 2 l*(n )l2
n=—oc
(4.2.40)
Let us n o w e x p r e s s th e e n e r g y E x in te r m s o f th e s p e c tr a l c h a r a c t e r i s t i c X (w). F irst
we have
Ex =
00
22 x(n)x*(n)=
n ------- -v
I" J rJT
2 2 x(n) —
X*
n—
—-v _
J —JT
If we interchange the order o f integration and sum m ation in the equation above,
we obtain
dw
|X (co)\~du>
T h erefore, the energy relation b etw een x ( n ) and X(a>) is
E,=
oc
\
2 2 \x^)\2 = —
n=-oc
fn
\X (c o )l2dco
(4.2.41)
This is Parseval's relation for discrete-tim e aperiodic signals with finite energy.
T he spectrum X(a>) is, in general, a com p lex-valu ed function of frequency.
It may be expressed as
X(u>) = \X ( c u ) \e J* M
(4.2.42)
where
Q(co) - ^ X ( c o )
is the phase spectrum and \X (a > )\ is the m agnitude spectrum .
A s in the case o f continuous-tim e signals, the quantity
S„(o>) = | X M |2
(4.2.43)
represents the distribution of energy as a function o f frequency, and it is called
the ener gy density sp ectrum o f x ( n ) . Cleariy, Sxx(a>) d oes not contain any phase
inform ation.
Suppose now that the signal x ( n ) is real. T hen it easily follow s that
X*(o>) = X ( - oj)
(4.2.44)
or equivalently,
|X (—ti>)| = |X(tt>)|
(even symmetry)
(4.2.45)
Sec. 4.2
261
Frequency Analysis of Discrete-Time Signals
(4.2.46)
(4.2.47)
From these sym m etry properties we con clu d e that the frequency range of
real d iscrete-tim e sign als can be lim ited further to the range 0 < co < n (i.e.,
on e-half o f the p erio d ). Indeed, if we know X ( a >) in the range 0 < w < n , we
can d eterm in e it for the range - n < co < 0 using the sym m etry p roperties given
above. A s w e have already ob served , sim ilar results hold for discrete-tim e periodic
signals. T h erefo re, the frequency-dom ain description o f a real discrete-tim e signal
is co m p letely sp ecified by its spectrum in the frequency range 0 < co < n .
U su a lly , w e w ork with the fundam ental interval 0 < a > < 7r o r 0 < F < F J 2,
exp ressed in H ertz. W e sketch m ore than half a period only w hen required by the
specific application.
Example 4.2.3
D eterm ine and sketch the energy density spectrum 5,,r (w) of the signal
= a"u(/i)
—1 < a < 1
Solution Since \a\ < 1. the sequence x(n) is absolutely summable, as can be verified
by applying the geometric summation formula.
H ence the Fourier transform of x(n) exists and is obtained by applying (4.2.29). Thus
Since \ae~J,Jl = |a| < 1. use of the geom etric summation formula again yields
The energy density spectrum is given by
Sl x(w) = |X(w)|2 = X(a>)X'(w) =
(1 —ae~Ju,)( 1 —aeJW)
or, equivalently, as
1
1 —2a cos a>+ a■
Note that S „(-o> ) = Ss i (w) in accordance with (4.2.47).
Figure 4.15 shows the signal *(«) and its corresponding spectrum for a = 0.5
and a = -0 .5 . Note that for a = -0 .5 the signal has more rapid variations and as a
result its spectrum has stronger high frequencies.
Frequency Analysis of Signals and Systems
262
Chap. 4
x(n) - (-0.5)"u(«!
Figure 4.15
spectra.
(a) Sequence .\in) =
and mii I = i —
Figure 4.16
pulse.
(b) ih eir energy density
D iscrete-tim e rectangular
Example 4.2.4
Determ ine the Fourier transform and the energy density spectrum of the sequence
xin) =
0 < n <L otherwise
(4.2.48)
which is illustrated in Fig. 4.16.
Solution
Before computing the Fourier transform, we observe that
Hence xin) is absolutely summable and its Fourier transform exists. Furthermore,
we note that Jt(n) is a finite-energy signal with
= \A\: L,
The Fourier transform of this signal is
Sec. 4.2
Frequency Analysis of Discrete-Time Signals
263
1 - e~’uL
= Ae- ' ,u/2'a - 11
s m( w / 2 )
(4.2.49)
For w = 0 the transform in (4.2.49) yields X(0) = AL, which is easily established
by setting w = 0 in the defining equation for X(a>), or by using L ’H ospital’s rule in
(4.2.49) to resolve the indeterm inate form when w = 0.
The m agnitude and phase spectra of ;t(n) are
f \A\L,
|X<«)I =
w= 0
sin(«L /2)
sin(aj/2)
otherwise
w
2
sin(ctiZ./2)
'■
sin(w/2)
(4-2.50)
and
$X (a>) = $ A - - ( L - l ) + $
(4.2.51)
where we should rem em ber that the phase of a real quantity is zero if the quantity is
positive and n if it is negative.
The spectra |X(o>)| and ^.X(w) are shown in Fig. 4.17 for the case A — 1 and
L = 5. The energy density spectrum is simply the square of the expression given in
(4.2.50).
T here is an interesting relationship that exists b etw een the F ourier transform
o f the constant am plitude pulse in E xam p le 4.2.4 and the period ic rectangular
lX(w)l
Figure 4.17 Magnitude and phase of
Fourier transform of the discrete-time
rectangular pulse in Fig. 4.16.
264
Frequency Analysis of Signals and Systems
Chap. 4
w ave considered in E xam ple 4.2.2. If w e evaluate the F ourier transform as given
in (4.2.49) at a set o f equally spaced (harm onically related) freq u en cies
w e obtain
(4.2.52)
If w e com pare this result with the expression for the F ourier series coefficients
given in (4.2.21) for the period ic rectangular w ave, w e find that
k = 0. 1 , . . . , N — 1
(4.2.53)
T o elab orate, w e have established that the Fourier transform o f the rectangular
p ulse, which is identical with a single p eriod o f the period ic rectangular pulse
train, evaluated at the freq u en cies co = I n k / N , k = 0, 1 , ___ N — 1, which are
identical to the harm onically related frequency com p onents u sed in the Fourier
series representation o f the periodic signal, is sim ply a m ultiple o f the Fourier
coefficients f a ) at the corresponding frequencies.
T he relationship given in (4.2.53) for the F ourier transform o f the rectangular
pulse evaluated at co — 2 n k / N , k = 0. 1........ A ' - l , and the F ourier coefficients
o f the corresponding periodic signal, is not only true for these tw o signals but, in
fact, holds in general. This relationship is d evelop ed further in Chapter 5.
4.2.6 Relationship of the Fourier Transform to the
z-Transform
T h e z-transform o f a seq u en ce * («) is defined as
OC
RO C: r2 < \z\ < r\
w here ri < |;) < rj is the region o f con vergen ce o f X (z).
com plex variable z in polar form as
(4.2.54)
L et us express the
(4.2.55)
where r = |z| and co = 4 ;. T h en , w ithin the region o f con vergen ce of X (z), we
can substitute z = r e jai into (4.2.54), T his yields
(4.2.56)
From the relationship in (4.2.56) w e n ote that X ( z ) can be interpreted as
the Fourier transform o f the signal seq u en ce x ( n ) r ~ ”. T he w eigh tin g factor r ~ n is
growing with n if r < 1 and decaying if r > 1 . A ltern atively, if X ( z ) con verges for
Sec. 4.2
Frequency Analysis of Discrete-Time Signals
265
1-1 = 1 , then
X (:)U , = X H =
2 2 x { , ! ) e - 1,0,1
(4.2.57)
T h erefore, the Fourier transform can be view ed as the z-transform o f the sequ en ce
evaluated on the unit circle. If X ( z ) d oes not converge in the region |z| — 1 [i.e.. if
the unit circle is n o t contained in the region o f con vergen ce o f X (z)], the Fourier
transform X ( i o ) d o es not exist.
W e should n o te that the existen ce of the z-transform requires that the s e ­
quence {jr (« )r—"} be ab solu tely sum m able for som e value o f r. that is.
OC
22
| < oc
(4.2.58)
n~-oc
H en ce if (4.2.58) con verges only for values o f r > ro > 1. the z-transform exists,
but the Fourier transform d oes not exist. This is the case, for exam p le, for causal
seq u en ces o f the form jr(rc) = a " u(n ), where jo > 1 .
T here are seq u en ces, how ever, that do not satisfy the requirem ent in (4.2.58).
for exam p le, the seq u en ce
sin avrc
x ( n ) — ----------
~ oo < n < oc
(4.2,5^)
Tin
This seq u en ce d o es not have a .--transform. Since it has a finite energy, its Fourier
transform con verges in the m ean-square sense to the d iscon tin uou s function X {(jo ).
defined as
M <*v
[ (J,
(4 2 6 0 )
<w( < \
a>\ < 71
In con clu sion , the existen ce of the z-transform requires that (4,2.58) be sat­
isfied for so m e region in the z-plane. If this region contains the unit circle, the
Fourier transform X(cu) exists. H ow ever, the existen ce o f the Fourier transform,
which is defined for finite energy signals, d oes not n ecessarily ensure the existence
o f the z-transform .
4.2.7 The Cepstrum
Let us con sid er a seq u en ce {jc(/j)} having a z-transform X (z). W e assum e that
(jr(n)) is a stable seq u en ce so that X ( z ) con verges on the unit circle. The c o m p l e x
ce pstru m o f the seq u en ce {jc(«)} is defined as the seq u en ce (cr(n)}, which is the
inverse z-transform o f Cjj(z), w here
Cjciz) = In X (z)
(4.2.61)
T h e com p lex cepstrum exists if C^tz) con verges in the annular region n <
\z\ < n , w here 0 < r\ < 1 and r2 > 1. W ithin this region o f con vergen ce, C t (z)
can be represen ted by the Laurent series
Cx(z) = In X (z) =
c^
z ~n
(4.2.62)
266
Frequency Analysis of Signals and Systems
Chap. 4
where
cAn) =
f In X ( z ) z n' 1d z
Jc
(4.2.63)
C is a closed contour about th e origin and lies w ithin the region o f convergence.
Clearly, if C, (z) can be rep resen ted as in (4.2.62), the com plex cepstrum sequence
{cr(rt)} is stable. F urtherm ore, if the com p lex cepstrum exists, Cx (z) converges on
the unit circle and hence w e have
CC
CAat) = lnX(cu) =
2 2 c A n ) e ~ JU,n
(4.2.64)
n—-OC
where (cv(/i)} is the sequ en ce ob tain ed from the inverse F ourier transform of
In X(to). that is,
1 r
c r(n) = — /
ln X (a >)eJwnd w
2tt J _ n
(4.2.65)
If w e express X(co) in term s of its m agnitude and p h ase, say
X(a>) = \X (co)\eJf>iw)
(4.2.66)
in X(co) = In |X(ct))| + jd(a>)
(4.2.67)
th en
By substituting (4.2.67) into (4.2.65), w e obtain the com plex cepstrum in the form
1
fn
cx (n) — — I [In |X ( a >)| + j6(cL>)]eJU,ndco
2jt J _ n
(4,2.68)
W e can separate the inverse F ourier transform in (4.2.68) into the inverse Fourier
transforms o f In |X (w )| and 9(a>)\
cm(n) = 2 - j
ln\X(a>)\eJl^dcL>
(4.2.69)
ce(n) = - ^ J
6(co)eJa>ndco
(4.2.70)
In som e applications, such as sp eech signal processing, only the com p onent c„(n)
is com puted. In such a case the phase o f X (a>) is ignored. T h erefore, the sequence
{*(«)} cannot b e recovered from {cm(n)j. That is, the transform ation from (jr(n)}
to {cm(n)} is not invertible.
In speech signal p rocessing, the (real) cepstrum has b een used to separate
and thus to estim ate the spectral con ten t o f the sp eech from the pitch frequency
of the speech. T he com plex cepstrum is used in practice to separate signals that
are con volved . T h e process o f separating tw o con volved signals is called d e c o n ­
volution and the use o f the com p lex cepstrum to perform the separation is called
h o m o m o r p h i c d econ vo lutio n. T his topic is discussed in Section 4.6.
Sec. 4.2
Frequency Analysts of Discrete-Time Signals
267
4.2.8 The Fourier Transform of Signals with Poles on the
Unit Circle
A s was show n in Section 4.2.6, the Fourier transform o f a seq u en ce jc(h) can be
determ ined by evaluating its z-transform X (z) on the unit circle, provid ed that the
unit circle lies within the region o f con vergen ce o f X (z). O therw ise, the Fourier
transform d o es not exist.
T h ere are som e aperiodic sequ en ces that are neither ab solu tely sum m able
nor square sum m able. H en ce their Fourier transform s do not exist. O n e such
seq u en ce is the unit step seq u en ce, which has the z-transform
A n oth er such seq u en ce is the causal sinusoidal signal seq u en ce x ( n ) = (coscoon)
u(n). T his seq u en ce has the z-transform
y , , _
1 - Z ~ ‘ COS (at)
1 - 2 z _ i cosojo + z -2
N o te that both o f these seq u en ces have p oles on the unit circle.
For seq u en ces such as these tw o exam ples, it is som etim es useful to extend
the Fourier transform representation. T his can be accom plished, in a m ath em ati­
cally rigorous w ay, by allow ing the Fourier transform to contain im pulses at certain
frequencies corresponding to the location o f the p oles o f X ( z ) that lie on the unit
circle. T he im pu lses are functions o f the con tinu ou s frequency variable co and
have infinite am plitude, zero width, and unit area. A n im pulse can be view ed as
the lim iting form o f a rectangular pulse of height 1 /a and width a, in the limit
as a -+ 0. Thus, by allow ing im pulses in the spectrum of a signal, it is p ossible
to exten d the Fourier transform representation to som e signal seq u en ces that are
neither absolutely sum m able nor square sum m able.
T h e follow in g exam ple illustrates the exten sion of the F ourier transform rep­
resentation for three sequ en ces.
Exam ple 4.2.5
D eterm ine the Fourier transform of the following signals.
(a) x i(n) = u(n)
(b) jr2(n) = (-l)" « (n )
(c) jc3(n) = (coswon)u(n)
by evaluating their z-transforms on the unit circle.
Solution
(a) From Table 4.3 we find that
XiU) = r - t - j = - ^ r
ROC: |c| > 1
1 - z '1
z -1
Xi(z) has a pole, p\ = 1, on the unit circle, but convenges for |z| > 1.
268
Frequency Analysis of Signals and Systems
Chap. 4
If we evaluate A'l (:i on the unit circle, except at : = 1. we obtain
X \ ( w ) = ------ :------------ = - —:---------— e “
2i/ sin(w/2>
to =£ 2 r r k
2sm(«j/2)
k = 0 . J . . ..
At a) = 0 and multiples of 2,t , A’|( w ) contains impulses of area .t .
Hence the presence of a pole at ; = 1 (i.e.. at w = 0) creates a problem
only when we want to compute A'i(w) at w = 0 .because |A’](tu)i —*■ oc as
to —> 0. For any other value of to. X \ (to) is finite (i.e.. well behaved). Although,
at first glance one might expect the signal to have zero-frequency components
at all frequencies except at as = 0. this is not the case. This happens because
the signal .\|(n) is not a constant for all —oc < « < oc. Instead, it is turned
on at n = 0. This abrupt jump creates all frequency com ponents existing in
the range 0 < t o < tt. Generally, all signals which start at a finite time have
nonzero-frequency com ponents everywhere in the frequency axis from zero up
to the folding frequency.
(b ) From Table 3.3 we find that the .'-transform of a''uin) with a = —1 reduces to
1
XA:) =
---------- r =
1+ : !
—
—
;+ l
R O C :
|;i >
1
which has a pole at : = - 1 = c-n . The Fourier transform evaluated at frequen­
cies other than a> = tt and multiples of 2tt is
A'lfw) =
- — —------- —
2 cos (to/2)
a> *
2 , t (k r
0.
1. . . .
|)
k =
TT
^ = 0 . 1 . . . .
In this case the impulses occurs at w = tt + 2rrk.
Hence the magnitude is
I X-> ( t o ) | =
—
— — -------- --
2[ cos(to/2)
t!)
2 tZ k +
and the phase is
X; (to) =
^
if cos — < 0
Note that due to the presence of the pole at a = —1 (i.e.. at frequency w = tt),
the m agnitude of the Fourier transform becomes infinite. Now \X(w)\ —* oc as
w —>
■n. We observe that (—])nu(n) = (cosTrn)u(n), which is the fastest possible
oscillating signal in discrete time.
(c) From the discussion above, it follows that A?(w) is infinite at the frequency
com ponent w = too. Indeed, from Table 3.3. we find that
:
1 —C_1 COS to,)
1 - 2: ~‘ cos wo +
) = (coStonfi)tt(n) <— ► A-i(;) = --- -— ------------
R O C : |-| > 1
The Fourier transform is
1 - e ' 1"’ cos cuo
X 3 (to) - —--------- :------------------------------(1 -
„ .
to ^ ito o 4- 27tk
k = 0. 1 . . . .
Sec. 4.2
Frequency Analysis of Discrete-Time Signals
269
The magnitude of X3 M is given by
|1 —e~,<u cos tool
w
±wu -I- 2 n k
k = 0, 1 ....
Now if w = —cm or w = tuo. |X 3 (a>)| becomes infinite. For all other frequencies,
the Fourier transform is well behaved.
4.2.9 The Sampling Theorem Revisited
T o p rocess a con tinu ou s-tim e signal using digital signal processing techniques, it is
necessary to convert the signal into a seq u en ce o f num bers. A s w as discussed in
Section 1.4, this is usually d on e by sam pling the analog signal, say x a(t), periodically
every T secon ds to produce a discrete-tim e signal x ( n ) given by
— 00 < n < 00
x ( n ) = x a( n T )
(4.2.71)
T h e relationship (4.2.71) describes the sam pling process in the tim e dom ain.
A s discussed in C hapter 1, the sam pling frequency Fs = l / T m ust be selected large
enough such that the sam pling d oes not cause any loss o f spectral inform ation (no
aliasing). In d eed , if the spectrum of the analog signal can be recovered from the
spectrum o f the discrete- tim e signal, there is no loss of inform ation. C onsequently,
w e investigate the sam pling p rocess by finding the relationship b etw een the spectra
o f signals x a(t) and x ( n ).
If x a (t) is an aperiodic signal with finite energy, its (voltage) spectrum is given
by the F ourier transform relation
(4.2.72)
w hereas the signal x a (i) can be recovered from its spectrum by the inverse Fourier
transform
(4.2.73)
N o te that u tilization o f all frequency com p on en ts in the infinite frequency range
—00 < F < 00 is necessary to recover the signal x„(t) if the signal x„(t) is not
bandlim ited.
T h e spectrum o f a discrete-tim e signal x ( n ) , ob tain ed by sam pling x a (t), is
given by the Fourier transform relation
OO
(4.2.74)
or, equivalently,
OO
X (f) = 2 2 x ( n ) e - ^ n
dc
(4.2.75)
270
Frequency Analysis of Signals and Systems
Chap. 4
The sequ en ce x(n) can be recovered from its spectrum X(a>) or X ( / ) by the inverse
transform
-*{«) - —
f
2 tt J . x
X(co)eJU,ndco
(4.2.76)
= /
X { f ) e j2nJnd f
J-\/2
In order to determ ine the relationship b etw een the spectra of the discrete­
time signal and the analog signal, w e n ote that period ic sam pling im poses a rela­
tionship betw een the in d ep en dent variables t and n in the signals x a {t) and x(n),
respectively. That is,
r = nT = —
F,
(4.2.77)
This relationship in the tim e dom ain im plies a corresponding relationship betw een
the frequency variables F and / in Xa(F) and X( f ) . respectively.
Indeed, substitution o f (4.2.77) into (4.2.73) yields
x(n) = xa( " T ) =
f Xtl(F)ej2,TnF/F'd
F
(4.2.78)
If we com pare (4.2.76) with (4.2.78), we conclude that
r• 1/2
1/2
f oc
X ( f ) e j2nf " d f =
/
X u ( F ) e )2nnFlFd F
(4.2.79)
J -oc
\a
From the d evelop m en t in C hapter 1 w e know that periodic sam pling im poses a
relationship b etw een the frequency variables F and / of the corresponding analog
and discrete-tim e signals, respectively. That is,
/ = —
(4.2.80)
With the aid o f (4.2.80), w e can m ake a sim p le change in variable in (4.2,79), and
obtain the result
y
j
^
X ^ - 0 ej2”nF/F' d F =
J
X a ( F ) e j2,,nF/F- d F
(4.2.81)
W e now turn our atten tion to the integral on the right-hand side o f (4.2.81)T he integration range of this integral can be divided into an infinite num ber of
intervals o f width F,. Thus the integral over the infinite range can be expressed
as a sum o f integrals, that is,
/
oc
3C
X a ( F ) e J2,TnF/F' d F =
x
[{k+XflsF,
/
Jt= -o c J ( k - \ f l ) F s
X a ( F ) e J2jrnF/F’d F
(4.2.82)
Sec. 4.2
271
Frequency Analysis of Discrete-Time Signals
W e o b serve that X a {F) in the frequency interval (k — I ) F t to (k + ~)FS is identical
to X a( F — k F s) in the interval —Fs/ 2 to Fs/2. C onsequently.
oc
V
p{k+\l2)F,
/
x.
X a( F ) e i27TnF/F' d F =
Jk=-oc J ^ —\fl\Fs
Y2
* F\ :2
X a ( F - k F , ) e p^ nF!f d F
/
k=-?c
'FJ2
J2
X a( F - k F < )
■' ~"r ! d F
J-FJ2
(4.2.83)
where w e have used the p eriodicity of the exp on en tial, nam ely.
e j 2 n n { F + k F s )/Ft _ ^jTnnF/F,
Com paring (4.2.83). (4.2.82), and (4.2.81), w e conclude that
X ( t ) = F' T . X A F - k F . )
' *t
k=--XL
(4.2.84)
or, equivalently.
OC
X(f) = F
X a [ ( f - k ) F s]
(4.2.85)
This is the desired relationship betw een the spectrum X ( F / F , ) or X ( f ) o f the
discrete-tim e signal and the spectrum X a( F) o f the analog signal. The righl-hand
side o f (4.2.84) or (4.2.85) consists o f a periodic repetition of the sealed spectrum
Fs X a( F) with period F,. This periodicity is necessary because the spectrum X ( f )
or X ( F / F ,) o f the discrete-tim e signal is periodic with period f p = 1 or Fp = Fs .
For exam p le, su p p ose that the spectrum o f a band-lim ited analog signal is
as sh ow n in Fig. 4.18(a). T h e spectrum is zero for [FI > B. N ow . if the sam ­
pling frequency Fs is selected to be greater than 2 5 . the spectrum X ( F / F S) of the
discrete-tim e signal will appear as show n in Fig. 4.18(b). Thus, if the sam pling
frequency
is selected such that Fv > 2 B. where 2 B is the N yquist rate, then
= FsX a(F )
|f | < F J 2
(4.2.86)
In this case there is no aliasing and therefore, the spectrum o f the discrete-tim e
signal is identical (w ithin the scale factor F,) to the spectrum o f the analog signal,
within the fundam ental frequency range |F | < Fsf 2 or [ f \ <
O n the other hand, if the sam pling frequency Fs is selected such that Fs <
2 B, the periodic continuation o f X a( F ) results in spectral overlap, as illustrated
in Fig. 4.18(c) and (d). T h u s the spectrum X { F / F S) o f the discrete-tim e signal
contains aliased frequency com p onents o f the analog signal spectrum X a(F). The
end result is that the aliasing which occurs prevents us from recovering the original
signal x „(r) from the sam ples.
G iven the discrete-tim e signal x(n) with the spectrum X ( F / F S), as illustrated
in Fig. 4.18(b ), w ith no aliasing, it is n ow p ossible to reconstruct the original analog
Figure 4.18 Sampling of an analog bandlimited signal and aliasing of spectral
components.
Sec. 4.2
Frequency Analysis of Discrete-Time Signals
273
signal from the sam ples j ( n ) . Since in the absence of aliasing
*.<»-(£*(£)■
m
s
F
(4.2.87)
’r -
\ F \ > F 5/ 2
lo ,
and by the Fourier transform relationship (4.2.75).
f )- £
/
(4.2.88)
n= -o c
the inverse Fourier transform o f X a ( F ) is
xAD =
f F' p-
/
X a ( F ) e s~*F ' d F
(4.2.89)
Let us assum e that F s = 2 B . W ith the substitution o f (4.2.87) in to (4.2.89), we
have
Xa(t)
=
i
r
F, J - F J
y
x { n ) e ~ j2 * Fn/F‘
dF
(4.2.90)
s\n(n/T )(t - nT)
(ir/T W -n T )
where x (n ) = x a( n T ) and w here T = \ / F s — 1 /2 B is the sam pling interval. This
is the reconstruction form ula given by (1.4.24) in our discussion o f the sam pling
theorem .
T h e reconstruction form ula in (4.2.90) in volves the function
i KO =
s in (jr /7 );
s in 2 ^ B f
(n/T)t
2n B t
(4.2.91)
appropriately shifted by n T , n = 0 , ± 1 , ± 2 ........ and m ultiplied or w eigh ted by
the corresponding sam ples x „ (n T ) o f the signal. W e call (4.2.90) an in terp ola­
tion form ula for reconstructing x a U) from its sam ples, and g (f). given in (4.2.91),
is the interpolation function. W e note that at t = k T , the in terp olation function
g(t — n T ) is zero excep t at k = n. C onsequently. xa (t) evalu ated at t = k T is simply
the sam ple x 0{k T ). A t all other tim es the w eigh ted sum o f the tim e shifted versions
o f the interpolation function com b ine to yield exactly x a (t). T his com b ination is
illustrated in Fig. 4.19.
T he form ula in (4.2.90) for reconstructing the analog signal xa (t) from its
sam ples is called the ideal interpolation fo rm u la . It form s the basis for the sa m p lin g
theorem, which can b e stated as follow s.
Sam pling T h eo rem . A bandlim ited con tinu ou s-tim e signal, with highest fre­
quency (bandw idth) B H ertz, can be uniquely recovered from its sam ples provided
that the sam pling rate Fs > 2 B sam ples per second.
274
Frequency Analysis of Signals and Systems
Chap, 4
A ccordin g to the sam pling theorem and the reconstruction form ula in (4.2,90),
the recovery o f x a(t) from its sam ples ;c(«), requires an infinite num ber o f sam­
ples. H ow ever, in practice w e use a finite num ber o f sam ples o f the signal and
deal with finite-duration signals. A s a con seq u en ce, w e are con cern ed only with
reconstructing a finite-duration signal from a finite num ber o f sam ples.
W hen aliasing occurs due to to o low a sam pling rate, the effect can be de­
scribed by a m ultiple folding o f the frequency axis o f the frequency variable F for
the analog signal. Figure 4.20(a) show s the spectrum X a(F ) o f an analog signal.
A ccordin g to (4.2.84), sam pling of the signal with a sam pling frequency Fs results
in a periodic repetition o f X a( F ) with period Fs . If Fs < 2 B , the shifted replicas of
X g {F) overlap. T he overlap that occurs within the fundam ental frequency range
— F.J2 < F < Fs/2, is illustrated in Fig. 4.20(b ). T h e corresponding spectrum of
the discrete-tim e signal w ithin the fundam ental frequency range, is obtained by
adding all the shifted portions within the range | / | < j , to yield the spectrum
shown in Fig. 4.20(c).
A careful inspection of Fig. 4.20(a) and (b) reveals that the aliased spectrum
in Fig. 4.20(c) can be obtained by folding the original spectrum lik e an accordian
with pleats at every odd m ultiple o f Fs/2. C on sequ en tly, the frequency F J 2 is
called the f o l d i n g fre q u e n c y , as indicated in C hapter 1. Clearly, then, periodic
sam pling autom atically forces a folding o f the frequency axis o f an analog signal
at odd m ultiples o f Fs/2, and this results in the relationship F = f Fs b etw een the
frequencies for con tinu ou s-tim e signals and discrete-tim e signals. D u e to the fold­
ing o f the frequency axis, the relationship F — f F, is not truly linear, but piecew ise
linear, to accom m odate for the aliasing effect. T his relationship is illustrated in
Fig. 4.21.
If the analog signal is bandlim ited to B < Fs/2, the relationship betw een /
and F is linear and o n e-to-on e. In other words, there is no aliasing. In practice,
prefiltering with an antialiasing filter is usually em p loyed prior to sam pling. This
ensures that frequency com p onents o f the signal above F > B are sufficiently
attenuated so that, if aliased, they cause n egligib le distortion on th e desired signal.
T h e relationships am ong the tim e-d om ain and freq u en cy-d om ain functions
x a (t), x ( n ) , X „ ( F ), and X ( f ) are sum m arized in Fig, 4.22. T h e relationships for
Sec. 4.2
Frequency Analysis of Discrete-Time Signals
XJFi
0
(a)
o
T
T
(c)
Figure 4.20
Illustration of aliasing around the folding frequency.
/
Figure 4.21
R elationship betw een frequency variables F and / .
275
276
Frequency Analysis of Signals and Systems
Chap. 4
XJF)=
Fourier transform
pair
Xa(t)
\.
.
Xa(F)
xa(t) = j ~ Xa(F )ei -^' dF
a:0(F)
Reconstruction:
= £ tfrt)
F,
IFI < - j
sin 7r(r - nT)!T
7r(t - nT)!T
Sampling-,
x(n) - x a(nT)
x(n)
T
X (/)
Fourier transform
pair
x(n) = J
Figure 4.22
X(f)eill,f"df
Time-domain and frequency-domain relationships for sampled sig­
nals.
recovering the con tinu ou s-tim e functions, x 0{t) and X a{F ), from th e discrete-tim e
quantities x ( n ) and X ( f ) , assum e that the analog signal is bandlim ited and that it
is sam pled at the N yquist rate (or faster).
T h e follow in g exam p les serve to illustrate the problem o f the aliasing of
frequency com ponents.
Example 4.2.6 Aliasing in Sinusoidal Signals
The continuous-time signal
xa(r) ~ cos 2nF^t
LgjZxFai
has a discrete spectrum with spectral lines at F = ± F U> as shown in Fig. 4.23(a). The
process of sam pling this signal with a sam pling frequency Fs introduces replicas of the
spectrum about multiples of Fs. This is illustrated in Fig. 4.23(b) fo r Fs/2 < F0 < F,To reconstruct the continuous-time signal, we should select the frequency com­
ponents inside the fundam ental frequency range \F\ < Fs f l . The resulting spectrum
Sec. 4.2
277
Frequency Analysis of Discrete-Time Signals
Spectrum
_|
0
-Fo
F0
F
U)
Spectrum
' it
-F t
Fu - Fs 0 F,. - F0
(b)
Spcctrum
1
F, -<FS- /■(,) f-,~F0
^
2
(c)
Spectrum
1
2T\
- F q - Fs
—Fs F, - F0 0 F0 - F, F,
(d)
Spectrum
2
F,
-y
0
f3
T
(e)
Figure 4.23
1
F 0 + F,
A liasing of sinusoidal signals.
278
Frequency Analysis of Signals and Systems
Chap. 4
is shown in Fig. 4.23(c). The reconstructed signal is
* „ ( /) = c o s 2 j t ( F , -
F (t)i
Now. if Fv is selected such that Fs < F(l < 3F 5/2, the spectrum of the sampled
signal is shown in Fig. 4.23(d). The reconstructed signal, shown in Fig. 4.23(e). is
xu(t) = cos2jr(F (1 - Fs)i
In both cases, aliasing has occurred, so that the frequency of the reconstructed signal
is an aliased version of the frequency of the original signal.
E x a m . e 4.2.7
S am p lin g a N o n b a n d lim ited Signal
Consider the continuous-time signal
*„(/) = e - AI,i
A>0
whose spectrum is given hv
X J F ) = — ----- -------A- 4- (2;r F Y
Determ ine the spectrum of the sampled signal ,r(n) = xu{nT).
Solution
If we sample x„(t) with a sampling frequency F, = 1/7", we have
jr(n) — xuinT) — e~A1'"' = (e~A1
—oc < n < oc
The spectrum of x(n) can be found easily if we use a direct com putation of the Fourier
transform. We find that
F\
F, J
1 - e~1AT
1 - 2e~A1 cos I n FT + e~2AT
T =
1
Fs
Clearly, since c o s 2 x F T = c o s 2 t t ( F/ Fs ) is periodic with period Fs, so is X ( F / F S),
Since X a(F) is not bandlim ited. aliasing cannot be avoided. The spectrum of
the reconstructed signal i„(r) is
™ i - i .
\F\<!±
( t.
Xa(F) =
[o .
I F I > J
Figure 4.24(a) shows the original signal xa(t) and its spectrum X a( F ) for A = 1The sampled signal x(n) and its spectrum X ( F / F S) are shown in Fig. 4.24(b) for
= 1 Hz. The aliasing distortion is clearly noticeable in the frequency domain. The
reconstructed signal xa(r) is shown in Fig. 4.24(c). The distortion due to aliasing can
be reduced significantly by increasing the sampling rate. For example, Fig. 4.24(d)
illustrates the reconstructed signal corresponding to a sampling rate Fs = 20 Hz. It
is interesting to note that in every case xa( nT) = xa(nT), but x„{t) / xa(t) at other
values of time.
Sec. 4.2
279
Frequency Analysis of Discrete-Time Signals
x „(n :
.*,,(/) = e~A>11 .A = I
14
A - + (2 ttF) 2
(al
1.0
T
-2
!
-1
0
1 t
[
.
<b)
(c }
(d)
Figure 4.24 (a) A nalog signal xa(i ) and its spectrum Xa(Fy. (b) xin i = x a ( ti T)
and the spectrum of *(n) for A = 1 and F, = 1 Hz; (c) reconstructed signal x„(i)
for F, - 1 Hz: (dj reconstructed signal i aU) for Fs = 20 Hz.
4.2.10 Frequency-Domain Classification of Signals: The
Concept of Bandwidth
Just as w e have classified signals according to their tim e-dom ain characteristics, it
is also desirable to classify signals according to their frequency-dom ain character­
istics. It is com m on practice to classify signals in rather broad terms according to
their frequency content.
In particular, if a p ow er signal (or energy signal) has its p ow er density sp ec­
trum (or its energy density spectrum ) concentrated about zero frequency, such
a signal is called a low -freq ue ncy signal. Figure 4.25(a) illustrates the spectral
characteristics o f such a signal. O n the other hand, if the signal p ow er density
Frequency Analysis of Signals and Systems
280
Xa{F)
Chap. 4
X{a>)
fa)
X(w)
Xa(F)
(b)
XJF)
X(oj)
(c)
Figure 4.25 (a) Low-frequency, (b) high-frequency, and (c) medium-frequency
signals.
spectrum (or the energy density spectrum ) is concentrated at high frequencies,
the signal is called a h igh -fr eq uency signal. Such a signal spectrum is illustrated
in Fig. 4.25(b). A signal having a p ow er density spectrum (or an energy density
spectrum ) concentrated som ew h ere in the broad frequency range b etw een low fre­
q uencies and high frequencies is called a m e d i u m - f r e q u e n c y signal or a bandpass
signal. Figure 4.25(c) illustrates such a signal spectrum .
In addition to this relatively broad frequency-dom ain classification o f signals,
it is often desirable to express q uantitatively the range o f freq u en cies over which
the p ow er or energy density spectrum is concentrated. This q u an titative m easure
is called the b a n d w i d t h o f a signal. For exam p le, su p p ose that a con tinu ou s­
tim e signal has 95% o f its p ow er (or energy) density spectrum con cen trated in the
frequency range F\ < F < F 2. T hen the 95% bandw idth o f the signal is F2 — F 1 . In
a sim ilar m anner, w e may define the 75% or 90% or 99% bandw idth o f the signal.
Sec. 4.2
Frequency Analysis of Discrete-Time Signals
281
In the case o f a bandpass signal, the term n a r r o w b a n d is used to describe
the signal if its bandw idth Fz - F\ is m uch sm aller (say, by a factor o f 10 or m ore)
than th e m edian frequency {Fi + F \) f2. O therw ise, the signal is called wideband.
W e shall say that a signal is b a n d lim ite d if its spectrum is zero ou tsid e the
freq u en cy range
> B. For exam ple, a con tinu ou s-tim e finite-energy signal x (t)
is bandlim ited if its Fourier transform X ( F ) = 0 for m > B. A d iscrete-tim e
finite-en ergy signal x ( n ) is said to be (periodically) b a n d lim ited if
for a>o < M < x
jX(ti>)| = 0
Sim ilarly, a p eriod ic co n tinu ou s-tim e signal x p (r) is p eriodically bandlim ited if its
F ourier coefficien ts c* = 0 for |£| > M , w here M is som e p ositive integer. A
p erio d ic discrete-tim e signal w ith fundam ental period N is periodically bandlim ited
if the Fourier coefficients ck = 0 for kG < |£j < N . Figure 4.26 illustrates the four
types o f bandlim ited signals.
B y exploiting the duality b etw een the frequency dom ain and the tim e dom ain,
we can provide sim ilar m ean s for characterizing signals in the tim e dom ain. In
particular, a signal x ( t ) will be called tim e- lim ited if
x(r) = 0
|f|
> t
If the signal is p eriod ic with period Tn, it will be called periodic ally time- lim ited if
x p it)
=
0
r
<
\t\
<
T p f l
If w e have a d iscrete-tim e signal x( n ) o f finite duration, that is,
.r(n) = 0
In I > N
it is also called tim e-lim ited. W hen the signal is period ic with fundam ental period
it is said to be p eriod ically tim e-lim ited if
x(n} = 0
Figure 4 J 6
no < )n\ < N
Some exam ples of bandlim ited signals.
282
Frequency Analysis of Signals and Systems
Chap. 4
W e state, w ithout proof, that no sign al can be time-lim ited a n d ban dlimited
sim ultaneously. Furtherm ore, a reciprocal relationship exists b etw een the time
duration and the frequency duration o f a signal. T o elaborate, if w e have a shortduration rectangular pulse in the tim e dom ain, its spectrum has a width that is
inversely proportional to the duration o f the tim e- dom ain pulse. T he narrower
the pulse b eco m es in the tim e dom ain, the larger the bandwidth of the signal
b ecom es. C onsequently, the product o f the tim e duration and the bandwidth of
a signal cannot be m ade arbitrarily sm all. A short-duration signal has a large
bandwidth and a sm all bandwidth signal has a lon g duration. T hus, for any signal,
the tim e-b an dw id th product is fixed and cannot b e m ade arbitrarily small.
Finally, w e n o te that w e have discussed frequency analysis m eth od s for peri­
odic and aperiodic signals with finite energy. H o w ev er, there is a family o f deter­
m inistic aperiodic signals with finite pow er. T h ese signals consist o f a linear super­
p osition o f com plex exp on en tials with nonharm onically related frequencies, that is,
M
x{n) = Y l A ke ^ n
k= 1
w here &>i, a s , . . . , a >m are nonharm onically related. T h ese signals have discrete
spectra but the distances am ong the lin es are nonharm onically related. Signals
with discrete nonharm onic spectra are som etim es called quasi-periodic.
4.2.11 The Frequency Ranges of Some Natural Signals
T he frequency analysis tools that w e have d ev elo p ed in this chapter are usually
applied to a variety o f signals that are en cou n tered in practice (e.g., seism ic, b io lo g ­
ical, and electrom agn etic signals). In gen eral, the frequency analysis is perform ed
for the purpose o f extracting inform ation from the ob served signal. For exam ple,
in the case o f biological signals, such as an E C G signal, the analytical tools are
used to extract inform ation relevant for d iagn ostic purposes. In th e case o f seism ic
signals, w e may b e interested in detectin g th e p resen ce o f a nu clear exp losion or in
d eterm ining the characteristics and location o f an earthquake. A n electrom agnetic
signal, such as a radar signal reflected from an airplane, con tains inform ation on
the p osition o f the plane and its radial velocity. T h ese param eters can be estim ated
from observation o f the received radar signal.
In processing any signal for the p urpose o f m easuring param eters or ex ­
tracting other types o f inform ation, o n e m ust know approxim ately the range o f
freq u en cies contained by the signal. For referen ce, T ables 4.1, 4.2, and 4.3 give
approxim ate lim its in the frequency d om ain for b iological, seism ic, and electro­
m agnetic signals.
4.2.12 Physical and Mathematical Dualities
In the p revious section s o f the chapter w e have introduced several m eth od s for the
frequency analysis o f signals. Several m eth od s w ere n ecessary to accom m odate the
Sec. 4.2
Frequency Analysis of Discrete-Time Signals
TABLE 4.1
283
FR EQ U E N C Y R A NG ES OF SO M E BIO LOG ICAL
SIGN ALS
F requency R ange (Hz)
Type of Signal
0-20
0-20
E lectroretinogram 8
Electronystagtnogram h
P neum ogram '
Electrocardiogram (EC G )
E lectroencephalogram (E E G )
Electrom yogram d
Sphygm om anogram e
Speech
0-40
0-100
0-100
10-200
0-200
100-^000
"A graphic recording of retina characteristics.
’’A graphic recording of involuntary m ovem ent of the eyes.
CA graphic recording of respiratory activity.
dA graphic recording of m uscular action, such as m uscular contraction.
CA recording of blood pressure.
TABLE 4.2
FR E Q U E N C Y R A NG ES OF SO M E S E IS M IC SIGNALS
Type o f Signal
F requency R ange (Hz)
W ind noise
Seismic exploration signals
E arthquake and nuclear explosion signals
Seismic noise
TABLE 4.3
100-KXX)
10-11X1
(1.01-1(1
0.1-1
F R E Q U E N C Y RANGES O F E L E C T R O M A G N ETIC SIGN ALS
W avelength (m )
F requency R ange (Hz)
10M 02
io-’- i o - :
3 x lO4^ x JO6
3 x 1(^-3 x 10U1
Type of Signal
R adio broadcast
S hortw ave radio signals
R adar, satellite com m unications.
space com m unications.
com m on-carrier microwave
Infrared
Visible light
U ltraviolet
G am m a rays and x-rays
2
3.9
x
1-10
u rM o ^ 6
10_7-8.1
10~7
10 -7-]O - 8
u r M o - 10
X
3 x 10^-3 x 1010
3 x 10 11—3 x I0 i4
3.7 x 10w-7.7 x 1014
3 x 1015-3 x lO16
3 x 1017—3 x 101K
differen t types o f signals. T o sum m arize, the follow in g frequency analysis tools
have b een introduced:
1. T he F ourier series for con tinu ou s-tim e p eriod ic signals.
2. T he F ourier transform for con tinu ou s-tim e ap eriod ic signals.
3. T he F ourier series for discrete-tim e periodic signals.
4. T he F ourier transform for discrete-tim e aperiodic signals.
284
Frequency Analysis of Signals and Systems
Chap. 4
Figure 4.27 sum m arizes the analysis and synthesis form ulas for these types of
signals.
A s w e have already indicated several tim es, there are two tim e-dom ain char­
acteristics that determ ine the type of signal spectrum w e obtain. T h ese are whether
the tim e variable is con tinu ou s or discrete, and w h eth er the signal is periodic or
aperiodic. Let us briefly sum m arize the results o f the previous sections.
Continuous-time signals have aperiodic spectra. A close inspection of
the F ourier series and Fourier transform analysis form ulas for continuous-tim e
signals d o es not reveal any kind of periodicity in the spectral dom ain. This lack of
p eriodicity is a con sequ en ce of the fact that the com p lex ex p on en tial exp(y2jr Ft)
is a function o f the continuous variable t, and hence it is not p eriod ic in F . Thus
the frequency range o f continuous-tim e signals exten d s from F = 0 to F = 0 0 .
Discrete-time signals have periodic spectra. In d eed , both the Fourier
series and the Fourier transform for discrete-tim e signals are p eriod ic with period
co = 2n . A s a result o f this periodicity, the frequency range o f d iscrete-tim e signals
is finite and exten d s from co = —1r to w — tt radians, where w = n corresponds to
the highest p ossible rate o f oscillation.
Periodic signals have discrete spectra. A s we have ob served , periodic
signals are described by m eans o f Fourier series. T h e Fourier series coefficients
provide the “lines" that constitute the discrete spectrum . T h e line spacing A F
or A f is equal to the inverse o f the period Tp or N , resp ectiveiy, in the time
dom ain. That is. A F = 1/7,, for continuous-tim e period ic signals and A f = 1 / N
for discrete-tim e signals.
Aperiodic finite energy signals have continuous spectra. This prop­
erty is a direct con seq u en ce o f the fact that both X ( F ) and X ( w ) are functions
o f exp { j 2 n F t ) and exp {jeon), respectively, which are con tinu ou s functions o f the
variables F and co. The continuity in frequency is necessary to break the harmony
and thus create aperiodic signals.
In sum m ary, w e can conclude that periodic ity with “p e r i o d ” a in o n e domain
automatically im plies discretization with “s p a c i n g " o f 1 jot in the o th e r do m ain , and
vice versa.
If w e k eep in m ind that “p eriod ” in the frequency d om ain m eans the fre­
quency range, “spacing'' in the tim e dom ain is the sam pling p eriod T , line spacing
in the frequency dom ain is A F , then a = Tp im plies that 1 /a = \ j T p = A F , a = N
im plies that A f = \ / N , and a = Fs im plies that T = 1 /F S.
T h ese tim e-frequency dualities are apparent from observation o f Fig. 4.27.
W e stress, how ever, that the illustrations u sed in this figure do n ot correspond to
any actual transform pairs. Thus any com parison am ong them sh ou ld be a v o i d e d .
A careful in sp ection o f Fig. 4 .2 7 also reveals som e m athem atical s y m m e tr ie s
and dualities am ong the several frequency analysis relationships. In particular,
Figure 4.27
Summary of analysis and synthesis form ulas.
286
Frequency Analysis of Signals and Systems
Chap. 4
we observe that there are dualities b etw een the follow in g analysis and synthesis
equations:
1. T he analysis and synthesis eq u ation s o f the con tinu ou s-tim e Fourier trans­
form.
2. T he analysis and synthesis eq u ation s of the discrete-tim e F ourier series.
3. T he analysis eq u ation o f the con tinu ou s-tim e F ourier series and the synthesis
equation o f the discrete-tim e Fourier transform.
4. T he analysis equation o f the discrete-tim e F ourier transform and the synthesis
equation o f the continuous-tim e Fourier series.
N o te that all dual relations differ on ly in the sign of the exp on en t of the
corresponding com p lex exp on en tial. It is interesting to note that this change in
sign can be thought o f either as a folding of the signal or a folding o f the spectrum ,
since
e - j 2 n Fl _ e j 2n( - FU _
If w e turn our atten tion now to the spectral density o f signals, w e recall that
we have used the term ener gy density sp e ctru m for characterizing finite-energy
aperiodic signals and the term p o w e r density sp e ctru m for period ic signals. This
term inology is con sistent with the fact that periodic signals are p ow er signals and
aperiodic signals w ith finite energy are energy signals.
4.3 PROPERTIES OF THE FOURIER TRANSFORM FOR
DISCRETE-TIME SIGNALS
The Fourier transform for aperiodic finite-energy discrete-tim e signals described
in the preceding section p ossesses a num ber of properties that are very useful in
reducing the com p lexity o f frequency analysis problem s in m any practical appli­
cations. In this section w e d evelop the im portant p roperties o f the Fourier trans­
form. Similar p roperties hold for the Fourier transform of ap eriod ic finite-energy
continuous-tim e signals.
For co n v en ien ce, w e adopt the notation
OC
X(a>) = F {x{«)} =
x ( n ) e ~ Jcun
(4.3.1)
X{u>)ejmndoo
(4.3.2)
/J —— CC
for the direct transform (analysis eq u ation ) and
x ( n ) = F ~ 1{X(tv)] = —
[
2 * J 2n
for the inverse transform (syn thesis eq u ation ). W e also refer to x ( n ) and X(o)) as
a Fourier tra nsfo rm p a i r and d en ote this relationship with the n otation
Sec. 4.3
Properties of the Fourier Transform for Discrete-Time Signals
287
R ecall that X (a>) is periodic with period 2 n . C on sequ en tly, any interval
o f length 2 n is sufficient for the specification o f the spectrum . U sually, we plot
the spectrum in the fundam ental interval [—jr. tt J. W e em p hasize that all the
spectral inform ation contained in the fundam ental interval is necessary for the
co m p lete d escription or characterization o f the signal. For this reason, the range
o f integration in (4.3.2) is always 2jt, in d ep en dent o f the specific characteristics o f
the signal w ithin the fundam ental interval.
4.3,1 Symmetry Properties of the Fourier Transform
W hen a signal satisfies som e sym m etry p roperties in the tim e dom ain, these prop­
erties im pose so m e sym m etry conditions on its Fourier transform . E xp loitation
of any sym m etry characteristics leads to sim pler form ulas for both the direct and
inverse F ourier transform. A discussion o f various sym m etry properties and the
im plications o f these properties in the frequency dom ain is given here.
Suppose that both the signal x(n) and its transform X(co) are com plex-valued
functions. T hen they can be expressed in rectangular form as
jr(n) = x K(n) + jX f(n )
(4.3.4)
X'(co) = X k {(o ) + j X [ ( co)
(4.3.5)
By substituting (4.3.4) and e~>w — cosu> - / s in co into (4.3.1) and separating the
real and im aginary parts, we obtain
OC
Xk(lo) =
22
costu/i -f- xj{n) sin con]
(4.3.6)
ft = — OC
SC
X/(a>) = — 2 2 [*jfOO sin a>n—.*/(«) cos&inj
n=-oc
(4.3.7)
In a sim ilar m anner, by substituting (4.3.5) and eJm = cos co + j s m u > into (4.3.2),
w e obtain
x R(n) = - — f [X/?(a>) cosam — X/(co) sin con]dco
2 n J2n
(4.3.8)
X/(n) — —
(4.3.9)
f [X ft (co) sincon + X/ ( w) cos con]dco
"T J2k
N o w , let us in vestigate som e special cases.
Real signals. If x(n) is real, then x/t(n) = x(n) and xi(n) = 0.
(4.3.6) and (4.3.7) reduce to
H en ce
OC
X R(co) — 2 2 x ( n ) cos con
(4.3.10)
Frequency Analysis of Signals and System s
288
Chap. 4
and
X i { w) = —
* (« )s in o jrt
(4.3.11)
Since c o s (—con) = cos ton and sin (—con) = —sin cun, it follow s from (4.3.10) and
(4.3.11) that
X R( - w ) = X R(co)
(ev en )
(4.3.12)
X, ( -co) = -X /(o > )
(o d d )
(4.3.13)
If w e com bine (4.3.12) and (4.3.13) into a single eq u ation , w e have
X*(co) = X ( - w )
(4.3.14)
In this case we say that the spectrum o f a real signal has H e r m itia n sy m m etry.
W ith the aid o f Fig. 4.28, w e observe that the m agnitude and phase spectra
for real signals are
X(to)\ =
J X\{co) + Xj(co)
(4.3.15)
(4.3.16)
4 _X|oj| = tan
X r ( co)
A s a con sequ en ce o f (4.3.12) and (4.3.13), the m agnitude and p h ase spectra also
p ossess the sym m etry properties
|,Y(co)| = \ X { —co)\
(ev en )
(4.3.17)
liX (-co) = -& X (to )
(od d )
(4.3.18)
In the case o f the inverse transform o f a real-valued signal [i.e., Jt(n) = jcj?(n)],
(4.3.8) im plies that
(n) =
— f [X/f(a>) cosa>n — X /(a>)sinam ]dw
(4.3.19)
Jin
R(co)
Since both products X R
(to) cos ton and X /( oj) sin con are even fun ction s o f co, we
have
1 r
= — I [X^(cu) cos con — X/(co) sin con]dco
x Ja
Imaginary axis
functions.
(4.3.20)
Sec. 4.3
Properties of the Fourier Transform for Discrete-Time Signals
289
Real and even signals. If .*■<«) is real an d ev e n [i.e., x ( ~ n ) = * (« )], th en
jf(/i)cosw ?i is ev en a n d x ( n )s in w n is od d . H e n c e , fro m (4.3.10). (4.3.11). an d
(4.3.20) w e o b ta in
Xff(a>) = x (0 ) + 2
x(n)coswn
(e v e n )
(4.3.21)
X i ( w) = 0
(4.3.22)
jr(n) = j f * X R(o>) cos cun da>
(4.3.23)
T h u s real a n d ev en signals po ssess real-v a lu e d sp e c tra , w hich, in ad d itio n , a re even
fu n ctio n s o f th e freq u e n cy v ariab le a>.
Real and odd signals. If x(/i) is real an d o d d [i.e., x ( —n) = - x ( « ) ] , th e n
x ( n )c o s w n is o d d an d .vODsinwn is even. C o n se q u e n tly . (4.3.10). (4.3.11) and
(4.3.20) im ply th a t
A'k(oi) = 0
(4.3.24)
(o d d )
(4.3.25)
(4.3.26)
T h u s re a l-v a lu e d o d d signals possess p u rely im ag in arv -v alu ed sp e ctral c h a ra c te ris­
tics. w hich, in a d d itio n , a re o d d f unct i ons o f th e freq u e n cy v ariab le co.
Purely imaginary signals. In th is case x K(n) = 0 an d x ( n ) - j x f (n). T h u s
(4.3.6). (4.3.7), a n d (4.3.9) re d u c e to
(o d d )
(4.3.27)
(ev en )
(4.3.28)
(4.3.29)
If x / ( n ) is o d d [i.e., x / ( —n) = —x/ (n)], th e n
(o d d )
X/(a>) = 0
(4.3.30)
(4.3.31)
(4.3.32)
290
Frequency Analysis of Signals and Systems
Chap. 4
S im ilarly, if xj ( n) is ev en [i.e.. x / ( —n) — .v/(«)]. we h ave
X *M -
0
(4.3.33)
X
X [ ( oj) = x /(0 ) + 2 Y ^ x i ( n ) cos con
(ev en )
(4.3.34)
n= 1
1 r
x ; ( n) = — I
X Jo
X 1 (co) cos cun d w
(4.3.35)
A n a rb itra ry , possibly co m p lex -v alu ed signal x (n ) can be d e c o m p o s e d as
x ( n ) = Xfi(n) + j x i ( n ) = x R
e (n) + x ‘^ ( n) + j [ x ] ( n ) + *"(« )]
(4.3.36)
= Ar (/i) + Jr„(rt)
w h ere, by d efin itio n ,
x f (n) = x K
f ( n) - h j Xf ( n ) = j[at(«) + jr* (- n )]
x„(n) = x'x(n) + j x/ ( r i ) - |[ jr( n ) - x * ( - n ) ]
T h e su p e rscrip ts e a n d o d e n o te th e even an d o d d signal c o m p o n e n ts , respectively.
W e n o te th a t x e(n) — x e(—n) a n d x r,(—n) = - a :„(«). F ro m (4.3.36) a n d the F o u rie r
tra n sfo rm p ro p e rtie s e s tab lish ed ab o v e, w e o b ta in th e follow ing relatio n sh ip s:
x{n) = [*£(«) + j x l (n)\ + [.i*(/0+y'A'/(n)]
]
■
xito) = [x(R(co) + j x f u o ) 1 +
'
(4.3.37)
+ j x ) \ w) ]
T h e se sy m m e try p ro p e rtie s of th e F o u rie r tra n sfo rm are su m m a riz e d in T a ­
ble 4.4 an d in Fig. 4.29. T h ey a re o ften used to sim plify F o u rie r tra n sfo rm calcu ­
latio n s in p ractice.
Example 4.3.1
Determ ine and sketch X R(w), X , ( w ), l^ioOi. and ^X(co) for the Fourier transform
X(a>) =
1 ■■
- 1< a < 1
(4.3.38)
1 — a e ~ ,w
Solution By multiplying both the num erator and denom inator of (4 .3 .3 8 ) by the
complex conjugate of the denom inator, we obtain
1 —ae!w
1 — a co s co —j a sin co
X(w) - ----------------------------- = ------------------ ------—
(1 — a e ~ }U,) ( 1 - a e JU>)
1 — 2a c o s o j + a "
This expression can be subdivided into real and imaginary parts. Thus we obtain
1 —a cos co
X r (w ) —
1
—
2 a cos o j
+ a1
a sin w
X,(u>) = -■
1 — 2a cos co + a 2
Substitution of the last two equations into (4.3.15) and (4.3.16) yields the mag­
nitude and phase spectra as
|X(<u)i - ■■■■
1 - ■= =
v l — 2a cos co
a2
(4.3.39)
Sec. 4.3
Properties of the Fourier Transform for Discrete-Time Signals
TABLE 4.4
SYMMETRY PROPERTIES OF THE DISCRETE-TIME
FOURIER TRANSFORM
Sequence
DTFT
xin)
A' (w)
X’i-w)
X ’ (a>)
x ’ (n )
j 'l - n )
J«(n)
^
"I-
—^)]
A'/jlo)) =: ^[X(t^) — -V*(—aj)j
jx /[n )
A>(n) =
4 [ j r ( n } + a ' ( —n ) ]
X R\.u»
x„(n) =
- , v ’ (— n)]
j X, { t o)
Real Sienals
A n y real signal
x (n )
X (w ) = X ' ( - u i )
X r (w ) = X r { —id)
Xfiuj) — —
IX ( oj)| = ] X ( —<y)|
\ t.(n)
i (a (>>)
,r( —n )]
(r e a l an d e v e n )
x.An) —
i|.r(n) -
r l — n)]
(r e a l a n d o d d )
F igure 4.29
= -iX (-w )
An(to)
(real and even)
]X
(imaginary and odd)
Sum m ary of symmetry properties for the F o u rier transform .
291
Frequency Analysis of Signals and Systems
292
Chap, 4
and
XrX(w) =
- tan
(4.3.40)
1
1 — a cos w
Figures 4.30 and 4.31 show the graphical representation of these spectra foT
a = 0.8. The reader can easily verify that as expected, all symmetry properties for
the spectra of real signals apply to this case.
Example 4.3.2
D eterm ine the Fourier transform of the signal
| A.
—M < n < M
1 0,
elsewhere
(4.3.41)
Solution CleaTly, x { —n) = x{n). Thus *(«) is a T e a l and even signal. From (4.3.21)
we obtain
X(a>) = X K(a>) = A I 1 4- 2 ^ c o s
Xg((i>)
(a)
(b)
Figure 4 J 0
G raph of X r ( o>) and X / ( w ) for the transform in Exam ple 4,3.1.
Sec. 4.3
Properties of the Fourier Transform for Discrete-Time Signals
293
IXlcoM
(a)
^X{w)
Figure 4.31
Magnitude and phase spectra of the transform in Example 4.3.1.
If we use the identity given in Problem 4.13, we obtain the simpler form
X(u>) = A
sinfAf + l)oj
. - -J sin(tu/2)
Since X (a>) is real, the magnitude and phase spectra are given by
sin(Af +
IX M I =
A
sin(a)/2)
I
|
(4,3.42)
and
0,
jt,
Figure 4,32 shows the graphs for X(w).
if X{w) > 0
if X(a>) < 0
(4.3.43)
294
Frequency Analysis of Signals and Systems
Chap. 4
X(n)
- MOM
X(u>)
IXMI
ZX(w)
4.3.2 Fourier Transform Theorems and Properties
In th is se ctio n w e in tro d u c e se v era l F o u rie r tra n sfo rm th e o re m s a n d illu stra te th eir
u se in p ra c tic e b y ex am p les.
Linearity.
If
Sec. 4,3
295
Properties of the Fourier Transform for Discrete-Time Signais
an d
F
X2 (n) *— » X ;(a/)
th en
(4.3.44)
Sim ply sta te d , th e F o u rie r tra n sfo rm a tio n , view ed as an o p e ra tio n on a signal
jt(rt), is a iin e a r tra n sfo rm a tio n . T h u s th e F o u rie r tra n sfo rm o f a lin e a r c o m b in a tio n
o f tw o o r m o re signals is eq u al to th e sam e lin e a r co m b in a tio n o f th e F o u rie r
tra n sfo rm s o f th e in d iv id u al signals. T h is p ro p e rty is easily p ro v e d by using (4.3.1).
T h e lin earity p ro p e rty m a k e s th e F o u rie r tra n sfo rm su ita b le fo r th e stu d y o f lin ear
system s.
Example 4.3.3
Determ ine the Fourier transform of the signal
(4.3.45)
Beginning with the definition of the Fourier transform in (4.3.1), we have
Xi(a>) =
22 ■M'Oe- -""" = 22a
’'e
i=D
=
The summation is a geometric series that converges to
1 — ae~)w
provided that
\ae ""I = Icij • |e ""| = \a] < 1
which is a condition that is satisfied in this problem. Similarly, the Fourier transform
of X j i n ) is
296
Frequency Analysis of Signals and Systems
Chap, 4
By combining these two transforms, we obtain the Fourier transform of x(n) in the
form
X ((i>) =
l _ a:
1 -
(4.3.46)
2a co so j + a 2
Figure 4.33 illustrates *(n) and X(w) for the case in which a = 0.8.
Time shifting.
If
X (a))
x (n )
th e n
x ( n - k) «
e - ja,kX ( w )
(4.3.47)
T h e p ro o f o f th is p ro p e rty follow s im m e d ia te ly from (he F o u rie r tra n sfo rm of
x ( n — k) by m ak in g a change in th e su m m a tio n index. T h u s
F[x{n - k ) } = X(a>)e~J<uk
= \X(aj )\ej[^ Xuu]- <,,k]
X(a>)
Figure 4.33
Sequence x I n ) and its F ourier transform in Exam ple 4.3.3 with
Sec. 4.3
Properties of the Fourier Transform for Discrete-Time Signals
297
T h is re la tio n m ean s th a t if a signal is sh ifted in th e tim e d o m a in by k sam ­
p les, its m a g n itu d e sp e c tru m re m a in s u n c h a n g e d . H o w e v e r, th e p h ase sp e c tru m is
ch an g ed by an a m o u n t -cok. T his re su lt can easily b e e x p la in e d if w e recall th a t
th e fre q u e n c y c o n te n t o f a signal d e p e n d s only on its sh a p e. F ro m a m a th e m a tic a l
p o in t o f view , w e can say th a t shifting by k in th e tim e d o m a in , is e q u iv a le n t to
m u ltip ly in g th e sp e c tru m by e~i'ok in th e fre q u e n c y d o m ain .
T im e r e v e r s a f .
If
j:(«)
X(co)
x(-n)
X ( - a >)
th e n
(4.3.48)
T h is p ro p e rty can be e s tab lish ed by p e rfo rm in g th e F o u rie r tra n sfo rm a tio n
o f jt ( —n) an d m ak in g a sim ple ch an g e in th e su m m a tio n in d ex . T h u s
OC
F (x (-n )j =
22 x ( l '>e>Wi =
f= —OC
If x ( n ) is real, th e n from (4.3.17) an d (4.3.18) w e o b ta in
F ( * ( - n ) | = X{-a>) =
= \ X( w) \ e ~ / ^ XiM'
T h is m e a n s th a t if a signal is folded a b o u t th e origin in tim e, its m a g n itu d e sp e ctru m
re m a in s u n c h a n g e d , a n d th e p h a se sp e c tru m u n d e rg o e s a c h a n g e in sign (p h ase
re v e rsa l).
C o n v o lu tio n t h e o r e m .
If
F
x\ (n) 4— ►X ](w)
an d
x 2 (n)
X 2 {oj)
th e n
x (n ) = Xi (h) * X2 (n)
X(co) = X] (a))X 2 (&>)
(4.3.49)
T o p ro v e (4.3.49), w e recall th e co n v o lu tio n fo rm u la
OC
x ( n ) = ^ i(n ) * x 2 (n) = Y
x \ ( k ) x 2 (n - k)
k=—oc
B y m u ltip ly in g b o th sides o f this e q u a tio n by th e e x p o n e n tia l e x p ( —jeon) a n d
su m m in g o v e r all n, w e o b ta in
OC
X(a>)= ^
x ( n ) e ~ Jwn =
OC
^
" OC
^
x \ { k ) x 2(n — k)
e~]a>n
298
Frequency Analysis of Signals and Systems
Chap. 4
A fte r in te rc h a n g in g th e o rd e r o f th e su m m a tio n s a n d m ak in g a sim ple ch an g e in
th e su m m atio n in d ex , th e rig h t-h a n d side o f this e q u a tio n re d u c e s to th e p ro d u ct
Xi(a>)X 2 (io). T h u s (4.3.49) is estab lish ed .
T h e co n v o lu tio n th e o re m is o n e o f th e m o st p o w erfu l to o ls in lin e a r system s
analysis. T h a t is, if w e co nvolve tw o signals in th e tim e d o m a in , th e n this is
e q u iv a le n t to m u ltip ly in g th e ir sp e c tra in th e fre q u e n c y d o m ain . In la te r c h ap ters
we will see th a t th e co n v o lu tio n th e o re m p ro v id es an im p o rta n t c o m p u tatio n al
to o l fo r m an y d ig ital signal p ro cessin g ap p licatio n s.
Example 4.3.4
By use of (4.3.49). determ ine the convolution of the sequences
A'l(/1) = A:(h ) = {1. 1. 1)
t
Solution
By using (4.3.21). we obtain
X i ((o) = X 2(w ) = 1 + 2 cos w
Then
X (co) = A't (cijjASfa)) = (1 + 2coso>)‘
= 3 + 4 cos w -i- 2 cos ho
= 3 4- 2 (c"" + (■“ "") + ic' 2"' +
Hence the convolution of
with
is
.v (ii) = ( 1 2 3 2 1 )
T
Figure 4.34 illustrates the foregoing relationships.
The correlation theorem.
If
x i(n )
X \(a>)
A‘;(« )
Xzito))
an d
th en
ri,x2(m )
S „ ,; (w) = X \ ( c d) X 2 (—oj)
(4.3.50)
T h e p ro o f o f (4.3.50) is sim ilar to th e p ro o f o f (4.3.49). In th is case, w e have
X
r^xAn) =
^
x ] ( k ) x 2( k - n )
k=~-'X.
B y m u ltip ly in g b o th sides o f this e q u a tio n by th e e x p o n e n tia l ex p (—jeon) and
su m m in g o v er all n , w e o b ta in
OC
Sxll2(w) =
X
oc
x i ( k )x 2(k - n )
e ~ JmR
X2M
Figure 4 3 4
Graphical representation of the convolution property.
F in ally , w e in te rc h a n g e th e o rd e r o f th e su m m a tio n s an d m a k e a ch an g e in th e
su m m a tio n in d ex . T h u s w e find th a t th e rig h t-h a n d sid e o f th e e q u a tio n above
re d u c e s to X \ ( u ) ) X 2 ( —a>). T h e fu n ctio n SXlX2(a)) is called th e cross-energy density
s pe c t r um o f th e sig n a ls x\ (n) an d x 2 (n).
The W iener-Khintchine theorem.
rxz(l)
L e t jc(n) be a re a l signal. T h e n
s„(a)
(4-3.51)
T h a t is, th e e n e rg y sp e c tra l d en sity o f an en erg y signal is th e F o u rie r tra n sfo rm o f
its a u to c o rre la tio n se q u e n c e . T h is is a sp ecial case o f (4.3.50).
T h is is a v ery im p o rta n t resu lt. It m ean s th a t th e a u to c o rre la tio n se q u e n c e
o f a sig n al a n d its en e rg y sp e c tra l d e n sity c o n tain th e sam e in fo rm a tio n a b o u t th e
signal. S ince, n e ith e r o f th e s e c o n ta in s any p h ase in fo rm a tio n , it is im p o ssib le to
u n iq u e ly re c o n stru c t th e signal fro m th e a u to c o rre la tio n fu n ctio n o r th e en erg y
d e n sity sp e ctru m .
Example 4JJ
D eterm ine the energy density spectrum of the signal
x(n) = tf^ufn)
- 1<a < 1
300
Chap. 4
Frequency Analysts of Signals and Systems
Solution
signal is
From Example 2.6.2 we found that the autocorrelation function for this
f a il)
= --------- oc < / < oo
1 —a*
By using the result in (4.3.46) for the Fourier transform of a 1'1, derived in Exam­
ple 4.3.3. we have
= ----- — —----- - 1 —2a cos w + a-
1 —0“
Thus, according to the W iener-K hintchine theorem ,
1
1 —2a cos w + a 2
Sxx(w) =
Frequency shifting.
If
x( n) < F > X( u) )
th en
e i<onnx ( n ) «
X(cu-m )
(4.3.52)
T h is p ro p e rty is easily p ro v e d by d ire c t s u b s titu tio n in to th e analysis e q u a tio n
(4.3.1). A cco rd in g to this p ro p e rty , m u ltip lic a tio n o f a se q u e n c e x( n) by e iu>-)n is
eq u iv a le n t to a fre q u e n c y tra n sla tio n o f th e sp e c tru m X(co) by co<>. T h is freq u e n cy
tra n sla tio n is illu stra te d in Fig. 4.35. S ince th e s p e c tru m X(co) is p e rio d ic, th e shift
a\) ap p lies to th e sp e c tru m o f th e signal in every p erio d .
The modulation theorem.
If
x( n) < F > X (a>)
X(u>)
*
_2
1
2
0
w
(a)
X( uj- cjq)
- 2w
- 2* + wo
2i
2» + cjo
«
(b)
Figure 4 3 5
form.
Illustration of the frequency-shifting property of the Fourier trans­
Sec. 4.3
Properties o f the Fourier Transform for Discrete-Time Signals
301
th e n
+ cuo) + X(w — c^o)]
x ( n ) cos coqt i
(4.3.53)
T o p ro v e th e m o d u la tio n th e o re m , w e first ex p ress th e signal coswo'J as
+ e - J^'")
cos won =
U p o n m u ltip ly in g x (n ) by th e se tw o e x p o n e n tia ls a n d using th e freq u e n cy -sh iftin g
p ro p e rty d e sc rib e d in th e p re c e d in g sectio n , w e o b ta in th e d e sire d re su lt in (4.3.53).
A lth o u g h th e p ro p e rty given in (4.3.52) can also be v iew ed as (com plex)
m o d u la tio n , in p ractice w e p re fe r to use (4.3.53) b ecau se th e signal j (h ) c o s a ^ ?
is real. C learly , in th is case th e sy m m etry p ro p e rtie s (4.3.12) and (4.3.13) are
p re se rv e d .
T h e m o d u la tio n th e o re m is illu stra te d in Fig. 4.36, w hich c o n ta in s a plot of
th e sp e c tra o f th e signals jr(/;). vi(n) = ;c (n )c o s 0 .5 ;rn an d y ifn ) = .* (« )cos7n;.
2
(a)
2
2
(b)
2
2
(c)
Figure 4.36
G raphical representation of the m odulation theorem .
302
Frequency Analysis of Signals and Systems
Parseval’s theorem.
Chap. 4
If
jri(n)
an d
X 2 (a>)
X2 (n)
th en
oc
Y
1
/ ' ,7
xi(n)jL*(n) = — - /
X\((i>)X*
(4.3.54)
2jr J - *
‘
T o p ro v e this th e o re m , w e use (4.3.1) to elim in a te X\(<o) on th e rig h t-h an d
side o f (4.3.54). T h u s we h ave
h i
T ,
X](n)e'
X^i^dto
x
=
i
r
22
f
n= -3 c
*' 2n-
^
( w ) e ~ Jujnda> =
22
n —~ oc
In th e special case w h ere x 2 (n) = X](/i) = x ( n) , P a rse v a l’s re la tio n (4.3.54)
red u ces to
yx
1
k (« )| 2 = r —
/f \X (a > )\2dw
(4.3.55)
2 TT J 2,
W e o b se rv e th a t th e le ft-h a n d side o f (4.3.55) is sim ply th e e n e rg y E x o f th e signal
x(fj). It is also e q u a l to th e a u to c o rre la tio n o f * (« ), rxx(l), e v a lu a te d a t / = 0.
T h e in te g ra n d in th e rig h t-h a n d side o f (4.3.55) is e q u a l to th e en erg y density
sp e c tru m , so th e in te g ra l o v e r th e in terv al —it < a> < it y ields th e to ta l signal
en erg y . T h e re fo re , w e co n clu d e th a t
E x = r xA ® ) =
°°
y
i
\x ( n )\2 = —
2jt
r
l
—
J2n\X(a))\ 2d w = 2it
cn
Ss x (w)da>
Multiplication of two sequences (Windowing theorem).
(4.3.56)
If
x\ ( n) <— ►Xi(cu)
an d
x 2 (n) <— ►X 2 (co)
then
f
i
x j i n ) = x \ { n ) x 2 {n) ■*— ►X j ( w) = —
r
/ X] ( k ) X 2 (a) - k ) d k
J_n
(4.3.57)
Sec. 4.3
Properties of the Fourier Transform f o r Discrete-Time Signals
303
T h e in te g ra l on th e rig h t-h a n d side of (4.3.57) re p re se n ts th e co n v o lu tio n of the
F o u rie r tra n sfo rm s Xi(cl>) a n d X 2(w). T h is re la tio n is th e d u a l o f th e tim e-d o m ain
c o n v o lu tio n . In o th e r w ords, th e m u ltip lic atio n o f tw o tim e -d o m a in se q u en ces is
e q u iv a le n t to th e co n v o lu tio n of th e ir F o u rie r tran sfo rm s. O n th e o th e r h a n d , the
co n v o lu tio n o f tw o tim e -d o m a in se q u e n c e s is e q u iv a le n t to th e m u ltip lic a tio n o f
th e ir F o u rie r tran sfo rm s.
T o p ro v e (4.3.57) we b egin w ith th e F o u rie r tra n sfo rm o f
(n) = jci(rt)x 2(n)
an d u se th e fo rm u la fo r th e in v erse tra n sfo rm , nam ely,
T h u s, w e h av e
CC
X3(oj) =
CC
Y
x 3(n)e Jwn =
J
X\(k)
2 2 x ' ( n )x 2 (n )e
— k)d k
T h e co n v o lu tio n in teg ral in (4.3.57) is kn o w n as th e per i od i c convol ut i o n o f
Xi(o>) a n d X 2 (to) b e c a u se it is th e co n v o lu tio n o f tw o p e rio d ic fu n ctio n s h av in g th e
sam e p e rio d . W e n o te th a t th e lim its o f in te g ra tio n e x te n d o v e r a single p erio d .
F u rth e rm o re , w e n o te th a t d u e to th e p erio d icity o f th e F o u rie r tra n sfo rm fo r
d isc re te -tim e signals, th e re is no “p e rfe c t" d u ality b e tw e e n th e tim e an d freq u e n c y
d o m a in s w ith re sp e c t to th e c o n v o lu tio n o p e ra tio n , as in th e case o f c o n tin u o u s­
tim e signals. In d e e d , co n v o lu tio n in th e tim e d o m ain (a p e rio d ic su m m a tio n ) is
e q u iv a le n t to m u ltip lic a tio n o f co n tin u o u s p e rio d ic F o u rie r tra n sfo rm s. H o w ev er,
m u ltip lic a tio n o f a p e rio d ic se q u e n c e s is e q u iv a le n t to p e rio d ic co n v o lu tio n o f th e ir
F o u rie r tra n sfo rm s.
T h e F o u rie r tra n sfo rm p a ir in (4.3.57) will p ro v e u se fu l in o u r tr e a tm e n t of
F IR filter d esign b a se d on th e w indow te c h n iq u e .
Differentiation in the frequency domain.
If
F
x( n) «— ►X(a>)
then
f
nx(n)
. ii X( ( o)
da)
(4.3.58)
304
Frequency Analysis of Signals and System s
Chap. 4
T o p ro v e this p ro p e rty , we use th e d efin itio n o f the F o u rie r tra n sfo rm in
(4.3.1) an d d iffe re n tia te th e series te rm by te rm w ith resp ec t to w. T h u s we
o b ta in
d X ( u>)
d
doj
du>
7 2 n x ( n ) e Ju
N ow w e m u ltip ly b o th sides of th e e q u a tio n bv j to o b ta in th e d esired resu lt in
(4.3.58).
T h e p ro p e rtie s d eriv ed in this section are su m m a riz e d in T a b le 4.5, which
serv es as a co n v e n ie n t refe ren ce . T ab le 4.6 illu stra te s som e useful F o u rie r tran s­
form p a irs th a t will be e n c o u n te re d in la te r c h a p te rs.
TABLE 4.5
PR O P E R TIE S OF TH E FO U R IE R T R A N S F O R M FOR D IS C R E TE -TIM E
S IG N ALS
Property
Notation
Linearity
Time shifting
Time reversal
Convolution
C orrelation
W iener-Khm tchine
theorem
Frequency shifting
M odulation
Time Domain
Frequency Domain
,v(«)
A](ii)
x2(n)
a\X] (n) +
x(rt ~ k)
x(~n)
x \ { n ) * x 2 (n)
r llt,(/) = ^, (/) * j z( - / )
r„ (/)
A'(co)
X](w)
A'iUu)
a\X](a)) + 02 X 2 (01)
e_ ""l X(«)
X ( —a>)
Xi(u>)X;(oj)
5,,.,, iw) = X, ( w)X2(-to)
—- A j (ul>)X ■,(iu)
[if X2 (h) is real]
-S.t.r (w)
eja,0"x(n)
x(n) cosa>nr
X( w — o^j)
\ X ( w + tut,) +■ i X ( w - wn)
Multiplication
D ifferentiation in the
frequency domain
Conjugation
1 f”
— I X |(/.)X ’ (w —X)dk
2jr
d X (u»)
du>
J-n
nx(n)
x*{n)
X ’ ( - w)
X\ ( w) X*(io)doj
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
305
TABLE 4.6
SO M E U S EFU L F O U R IE R TR A N S FO R M PAIRS FO R D IS C R E TE -TIM E A P ER IO D IC
SIGNALS
4.4 FREQUENCY-DOMAIN CHARACTERISTICS OF LINEAR
TIME-INVARIANT SYSTEMS
In th is se ctio n w e d e v e lo p th e c h a ra c te riz a tio n o f lin e a r tim e -in v a ria n t system s in
th e fre q u e n c y d o m a in . T h e basic e x c ita tio n signals in th is d e v e lo p m e n t a re th e
co m p lex e x p o n e n tia ls a n d sin u so id al fu n ctio n s. T h e c h a ra c te ristic s o f th e system
a re d e s c rib e d by a fu n ctio n o f th e fre q u e n c y v a ria b le co called th e fre q u e n c y r e ­
sp o n se, w hich is th e F o u rie r tra n sfo rm o f th e im p u lse re sp o n s e h( n) o f th e system .
T h e fre q u e n c y re sp o n s e fu n ctio n c o m p le te ly c h a ra c te riz e s a lin e a r tim ein v a ria n t sy stem in th e fre q u e n c y d o m a in . T h is allow s us to d e te rm in e th e
306
Frequency Analysis of Signals and System s
Chap. 4
ste a d y -s ta te re sp o n se o f th e system to an y a rb itra ry w eig h ted lin e a r co m b in atio n
o f sin u so id s o r co m p lex e x p o n en tials. Since p e rio d ic se q u e n c e s, in p a rtic u la r, lend
th em se lv e s to a F o u rie r series d e c o m p o sitio n as a w eig h te d sum o f h a rm o n ically re­
lated co m p lex e x p o n e n tia ls, it b eco m es a sim ple m a tte r to d e te rm in e th e response
o f a lin e a r tim e -in v a ria n t system to this class o f signals. T his m e th o d o lo g y is also
a p p lied to a p e rio d ic signals since such signals can b e view ed as a su p e rp o s itio n of
infin itesim al size co m p lex ex p o n en tials.
4.4.1 Response to Complex Exponential and Sinusoidal
Signals: The Frequency Response Function
In C h a p te r 2, it w as d e m o n s tra te d th a t th e re sp o n se o f any re la x e d lin e a r tim ein v arian t sy stem to an a rb itra ry in p u t signal jr(n), is given by th e c o n v o lu tio n sum
fo rm u la
X
yin)-
ft(k)x(n-k)
(4.4.1)
£—
—
OC
In th is in p u t- o u tp u t re la tio n sh ip , th e sy stem is c h a ra c te riz e d in th e tim e dom ain
by its u n it sam p le re sp o n s e {h ( n ). - o c < n < oo}.
T o d ev e lo p a fre q u e n c y -d o m a in c h a ra c te riz a tio n o f th e sy stem , let us excite
th e sy stem w ith th e co m p lex e x p o n e n tia l
— 00 < n < cc
jr(n) = A e Jam
(4.4.2)
w h ere A is th e a m p litu d e and a> is an y a rb itra ry fre q u e n c y c o n fin ed to th e freq u en cy
in terv al [ - t t . j t ] . By su b stitu tin g (4.4.2) in to (4.4.1), w e o b ta in th e re sp o n se
v(n) =
22 h ( k ) [ A e Jb,in- k}]
X
k~ ~ X OC
= A
(4.4.3)
2 2 h ( k ) e ~ Juk
W e o b se rv e th a t th e te rm in b ra c k e ts in (4.4.3) is a fu n ctio n o f th e freq u en cy
v ariab le a>. In fact, th is te rm is th e F o u rie r tra n sfo rm o f th e u n it sa m p le resp o n se
h{k) o f th e system . H e n c e w e d e n o te th is fu n ctio n as
00
H(w)=
22 h ( k ) e ~ ja>k
(4.4.4)
C learly , th e fu n ctio n H(co) exists if th e sy stem is B IB O sta b le , th a t is, if
OC
y
|/i(rc)| <
00
n =-oc
W ith th e d efin itio n in (4.4.4), th e re sp o n se o f th e system to th e com plex
e x p o n e n tia l giv en in (4.4.2) is
y(n) = AH(oo)eJwn
(4.4.5)
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
307
W e n o te th a t th e re sp o n se is also in th e fo rm o f a co m p lex e x p o n e n tia l w ith the
sam e fre q u e n c y as th e in p u t, b u t a lte re d by th e m u ltip lic ativ e fa c to r H ( m ).
A s a re su lt o f th is c h a ra c te ristic b e h a v io r, th e e x p o n e n tia l signal in (4.4.2) is
called an ei genf unct i on o f th e system . In o th e r w o rd s, an eig e n fu n c tio n o f a system
is an in p u t signal th a t p ro d u c e s an o u tp u t th a t differs fro m th e in p u t by a c o n s ta n t
m u ltip lic ativ e facto r. T h e m u ltip lic ativ e fa c to r is called an ei genval ue o f th e system .
In th is case, a c o m p lex e x p o n e n tia l signal o f th e fo rm (4.4.2) is an e ig en fu n ctio n of
a lin e a r tim e -in v a ria n t system , a n d H(u>) e v a lu a te d at th e fre q u e n c y o f the in p u t
signal is th e c o rre sp o n d in g eig en v alu e.
Example 4.4.1
D eterm ine the output sequence of the system with impulse response
(4.4.6)
k{n) = (iY ‘u(/i)
when the input is the complex exponential sequence
xin) = Ac^"' -
— oo < n < oc
Solution First we evaluate the Fourier transform of the impulse response hin), and
then we use (4.4.5) to determ ine v(n). From Example 4.2.3 we recall that
(4.4.7 >
At ui = 7t/2, (4.4.7) yields
and therefore the output is
(4.4.8)
—oc < n < oc
T h is ex a m p le clearly illu stra te s th a t th e only effe ct o f th e system on th e in p u t
signal is to scale th e a m p litu d e by 2 /\/5 a n d shift th e p h ase by - 2 6 .6 “. T h u s th e
o u tp u t is also a co m p lex e x p o n e n tia l o f fre q u e n c y n / 2 , a m p litu d e 2 A / v /5. an d
p h a se - 2 6 .6 C.
If w e a lte r th e fre q u e n c y of th e in p u t signal, th e effe ct o f th e system on
th e in p u t also c h an g es a n d h en c e th e o u tp u t ch an g es. In p a rtic u la r, if th e in p u t
se q u e n c e is a c o m p lex e x p o n e n tia l o f fre q u e n c y tt. th a t is,
x(n) = A e ^ n
— oo < n < oc
then, at co = jr.
1
2
3
(4.4.9)
308
Frequency Analysts of Signals and Systems
Chap. 4
an d th e o u tp u t o f th e sy stem is
v(n) = 5 A e jnn
— 00 < n < 00
(4.4.10)
W e n o te th a t H ( n ) is p u re ly real [i.e., th e p h a se a s so ciated w ith H{u>) is zero at
co — t t ] . H e n c e , th e in p u t is scaled in a m p litu d e by th e fa c to r H ( n ) = =, b u t the
p h ase shift is zero .
In g e n e r a l H(co) is a co m p lex -v alu ed fu n ctio n o f th e fre q u e n c y variable w.
H e n c e it can b e e x p ressed in p o la r form as
H(to) = \H(a>)\eJH{w)
(4.4.11)
w h ere |//(a> )| is th e m ag n itu d e o f H(a>) and
@(w) — ^ H (c o)
w hich is th e p h a se shift im p a rte d on th e in p u t signal by the system at th e fre­
quen cy CO.
Since H( w) is th e F o u rie r tra n sfo rm o f i/i(£)}, it follow s th a t H(to) is a peri­
od ic fu n ctio n w ith p e rio d 2 n . F u rth e rm o re , w e can view (4.4.4) as th e ex ponential
F o u rie r series ex p an sio n for H(co), w ith h( k) as th e F o u rie r se rie s coefficients. C on­
se q u e n tly , th e u n it im p u lse h( k) is re la te d to H(co) th ro u g h th e in te g ra l expression
1
h( k) = —
H( t o) elwkdco
(4.4.12)
F o r a lin ear tim e -in v a ria n t system w ith a re a l-v a lu e d im p u lse resp o n se, the
m ag n itu d e an d p h a s e fu n ctio n s possess sy m m etry p ro p e rtie s w h ich are d eveloped
as follow s. F ro m th e d efin itio n o f H(co). w e have
H(co) =
h( k)e'
— 52
ju / k
h(k) coscok — j 5 2
k——Dc
h (jt)s in w £
(4 4 13)
k=—oc
= H R{io) -f j H 1 (to)
= y j H l i t o ) + H f ( c o ) e J tan- 1["/<")/"*<-)]
w h ere H K(to) an d Hi(co) d e n o te th e real a n d im ag in aary c o m p o n e n ts o f H(a>), d e­
fined as
DC
Hn(co) =
52
h ( k ) c os c ok
(4.4.14)
H/(co) = — 5 2 h( k) sin cok
k = -x
It is c lear from (4.4.12) th a t th e m a g n itu d e an d p h a s e o f H(co), ex p re sse d in term s
o f H r (co) an d Hi(co), are
\H(co)\ = J H 2r(co) + Hf(co)
(4.4.15)
©(w ) = tan
--------H R(to)
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
309
W e n o te th a t H R (a>) = H R( - t o ) a n d Hi {a>) = —/ / / ( —oj). so th a t H r ( c o ) is an
ev en fu n c tio n o f oj a n d Hi (co) is an o d d fu n ctio n o f w. A s a c o n s e q u e n c e , it follow s
th a t
is an ev en fu n ctio n of w an d 0(o>) is an o d d fu n c tio n o f a>. H e n c e ,
if w e k n o w j / f ( a > ) J a n d 0 ( a > ) fo r 0 < w < n . w e also k n o w th e s e fu n c tio n s for
—jr < w < 0.
Example 4.4.2
Moving Average Filter
D eterm ine the m agnitude and phase of H(w) for the three-point moving average
(MA) system
y ( n ) = ^fjc(n + 1) + x( n ) + x( n - 1)]
and plot these two functions for 0 < a> < n.
Solution
Since
h{n) = [ i i i }
t
it follows that
H(co) = ^
+
1 +
=
1 ( 1
2cos<w)
+
Hence
i 11 + 2 cos a)
IW M I
' 0.
B ( w )
=
(4.4.16)
0 < a> < 2?r/3
2 jt / 3
<
w
<
jt
Figure 4.37 illustrates the graphs of the m agnitude and phase of H(w). As indicated
previously, |W(w)| is an even function of frequency and C-)(a>) is an odd function of
3tt
4
Figure 4.37
Magnitude and phase
responses for the M A system in
Example 4.4.2.
310
Frequency Analysis of Signals and Systems
Chap. 4
frequency. It is apparent from the frequency response characteristic H(u>) that this
moving average filter smooths the input data, as we would expect from the inputoutput equation.
T h e sy m m etry p ro p e rtie s satisfied by th e m a g n itu d e a n d p h a se fu n ctio n s of
H ( u >), an d th e fact th a t a sinusoid can be e x p re sse d as a su m o r d ifferen ce of
tw o co m p lex -co n ju g ate e x p o n e n tia l fu n ctio n s, im ply th at th e re sp o n s e o f a linear
tim e -in v a ria n t sy stem to a sin u so id is sim ilar in fo rm to th e re sp o n se w hen the
in p u t is a co m p lex e x p o n e n tia l. In d e e d , if th e in p u t is
Jfi (n) — Ae-' w'’
th e o u tp u t is
vi(«) = A \ H( o j ) \ e j<rnw' eilM'
O n th e o th e r h a n d , if th e in p u t is
j::(« )
=
th e re sp o n se o f th e system is
y : ( / j) =
A\H{~
— A \ H (aj)\e~j H ""' e ~J'l>"
w h ere, in th e last ex p ressio n , w e h av e m a d e use of the sy m m etry p ro p e rtie s
\H(to)\ = \ H ( —cd)\ an d (~)(a;) = —(-)(—co). N ow , by ap p ly in g the su p e rp o sitio n
p ro p e rty o f th e lin e a r tim e -in v a ria n t system , w e find th at th e re sp o n se o f th e sys­
tem to th e in p u t
— i[jt] (n) + _V2 (/;)] = A cos con
is
}’(>0 = j[ v , (n) + \’2(/r)]
(4.4.17)
y i n ) = A \ H( c o) \ cosjw/? + (-)(o>)]
Sim ilarly, if th e in p u t is
x i n ) = —r[jf |(n ) —jt-*(/t) 1 = A sin ton
J2
th e resp o n se o f th e system is
v W = i [ v l( ,, )
(4 4 1 8 )
v(h) = A \ H { w ) \ sin[cu« + 0 (tn )]
I t is a p p a r e n t fro m th is discussion th a t H{co), o r e q u iv a le n tly . |//(c u )| and
0(cw), co m p le te ly c h a ra c te riz e the effect o f th e system on a sin u so id a l in p u t signal
o f any a rb itra ry freq u e n cy . In d e e d , w e n o te th a t \H(a>)\ d e te rm in e s th e am plifi­
catio n ( | / / M | > 1) o r a tte n u a tio n (\H(co)\ < 1) im p a rte d by th e system o n the
in p u t sin u so id . T h e p h a se @(co) d e te rm in e s th e a m o u n t o f p h a s e shift im p a rte d
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
311
by th e sy stem o n th e in p u t sinusoid. C o n se q u e n tly , by k n o w in g H{aj), w e are
ab le to d e te rm in e th e re sp o n s e o f the system to an y sin u so id a l in p u t signal. Since
H ( w ) specifies th e re sp o n s e o f th e sy stem in th e fre q u e n c y d o m a in , it is called th e
f r e q u e n c y response o f th e system . C o rre sp o n d in g ly , \H{to) \ is called th e ma g ni t ude
respons e a n d 0(o>) is c a lle d th e phas e response o f th e system .
I f th e in p u t to th e sy stem co n sists o f m o re th a n o n e sin u so id , th e s u p e rp o ­
sitio n p ro p e rty o f th e lin e a r system can b e u se d to d e te rm in e th e re sp o n se . T h e
fo llo w in g e x a m p le s illu stra te th e use o f th e su p e rp o s itio n p ro p e rty .
Exam ple 4.43
D eterm ine the response of the system in Example 4.4.1 to the input signal
;c(n) = 10 —5 sin —n + 2 0 co snn
2
Solution
— oc < n < oc
The frequency response of the system is given in (4.4.7) as
H(w) = ------*— 1-
The first term in the input signal is a fixed signal com ponent corresponding to w = 0.
Thus
H( 0) =
The second term in
response of the system is
=2
has a frequency jr/2.
At this frequency the frequency
Finally, the third term in x ( n ) has a frequency w = t t , At this frequency
H { tt) =
|
Hence the response of the system to *(n) is
10 . / jt
\
40
y (n ) = 2 0 — —
j= . sm ^ 2^ — 2 6 .6 J + — co sttw
— o c < n < oc
Example 4.4.4
A linear tim e-invariant system is described by the following difference equation:
y(n) = ay(n —1) + bx(n)
0 < a < 1
(a) D eterm ine the magnitude and phase of the frequency response H(w) of the
system,
(b ) Choose the param eter b so that the maximum value of \H{u>)\ is unity, and
sketch fH(<d)\ and 4 . H( w) for a = 0,9.
Frequency Analysis of Signals and Systems
312
Chap. 4
(c) D eterm ine the output of the system to the input signal
x (n )
Solution
=
+ 12 sin
5
—2 0 cos ( ^ z n + —^
The impulse response of the system is
h(n) = ba"u(n)
Since |a| < 1. the system is BIBO stable and hence H(co) exists,
(a) The frequency response is
oc
H (w) =
22, h ^
e ~iam
b
1 —ae~JW
Since
1 —ae~,‘“ = (1 —a cos co) 4- j a sin co
it follows that
[1 —ae J'“’| = ^ /(l —a cos co)2 4- (a sin co)2
= y/\ + a2 —2a cos co
and
4 (1 - ae J“’) = tan
osinct)
1 —a cos co
T herefore,
\b\
\H(co)\ =
V l 4- a2 — 2a cos 1u
^H(co) = &(co) = 4-b - tan 1
a sin co
1 —a cos (
(b) Since the param eter a is positive, the denom inator of \H(w)\ attains a minimum
at w = 0. Therefore, |//(a>)| attains its maximum value at a> = 0. A t this
frequency we have
1*1
|ff(0)| = r U - = ' l
1 —a
which implies that b = ±(1 —a). We choose b = 1 —a, so that
l —o
|W M I -
■J\ + a 2 —2a cos
and
&(w) = — tan"
,
asinoj
1 — a cos co
The frequency response plots for |tf(co)| and 0(o>) are illustrated in
Fig. 4.38. W e observe that this system attenuates high frequency signals.
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
313
Figure 4.38
Magnitude and
phase responses for the system in
Example 4.4.4 with a =
(c) The input signal consists of components of frequencies to = 0. tt/2, and n. For
co = 0. |//(0)| = 1 and 0 (0) = 0. For to = jr/2,
0.1
1 —a
¥ (-)l =
_________ ______ = 0.074
V2 / 1
y/\ -f a 2
-s/1.81
©
-- - tan-1 a = - 4 2
For co = tt,
1 -a
0.1
\H(n)\ = — — = — = 0.053
l+ o
1.9
0(7r) = 0
Therefore, the output of the system is
y(n ) = 5|ff(0)| + 1 2 | / / ( | ) | s i i i | j « + e ( ! ) ]
— 2 0 | / / ( j t ) | cos ^ 7 rn + — + 0 ( 7 r ) j
= 5 + 0.888sin
—42°^ —1.06 cos ^jrn + —^
—oc < n < oc
314
Frequency Analysis ol Signals and Systems
Chap. 4
In th e m o st g e n e ra l case, if th e in p u t to th e system co n sists o f an arbitrary
lin ear co m b in a tio n o f sinusoids o f th e form
L
x ( n ) = 5 2 A, cos(co,n 4• 4>i)
— oc < « < oc
1=1
w h ere (A, | an d {<£,} a re th e am p litu d e s a n d p h ases o f th e c o rre sp o n d in g sinusoidal
c o m p o n e n ts, th e n th e re sp o n s e of th e sy stem is sim ply
L
v{n) = 5 2 Aj \ H{ a),)| cos[oj,« 4- <p, 4- 0(ci>,)]
(4.4.19)
1= 1
w h ere \H(u>i)\ an d ©(oj,) are th e m a g n itu d e an d p h a se , re sp e c tiv e ly , im p a rte d by
th e system to th e in d iv id u al fre q u e n c y c o m p o n e n ts o f th e in p u t signal.
It is clear th a t d e p e n d in g on th e fre q u e n c y re sp o n se H ( uj) o f th e system , input
sinu soid s o f d iffe re n t fre q u e n c ie s will be affe c te d d ifferen tly by th e system . F o r ex­
am p le, so m e sin u so id s m ay be co m p letely su p p re sse d by th e sy stem if H ( w ) = 0 at
th e fre q u e n c ie s o f th e s e sinusoids. O th e r sin u so id s m ay receiv e n o a tte n u a tio n (or
p e rh a p s, som e am p lifica tio n ) by th e system . In effect, w e can view th e lin ear timein v arian t system fu n c tio n in g as a filter to sin u so id s o f d iffe re n t fre q u e n c ie s, passing
som e o f th e fre q u e n c y c o m p o n e n ts to th e o u tp u t a n d su p p re s sin g o r preventing
o th e r freq u e n cy c o m p o n e n ts fro m re ach in g th e o u tp u t. In fact, as discussed in
C h a p te r 8, th e basic dig ital filter design p ro b le m involves d e te rm in in g th e p a ra m e ­
ters o f a lin e a r tim e -in v a ria n t system to achieve a d e sire d fre q u e n c y re sp o n s e H (co).
4.4.2 Steady-State and Transient Response to Sinusoidal
Input Signals
In th e discu ssio n in th e p re ced in g sectio n , w e d e te rm in e d th e re s p o n s e o f a linear
tim e -in v a ria n t sy stem to e x p o n e n tia l a n d sin u so id al in p u t sig n als a p p lied to the
system at n = —oc. W e usually call such signals e te rn a l e x p o n e n tia ls o r etern al
sin u so id s, b ecau se th e y w ere a p p lied at n = - o c . In such a case, th e re sp o n se that
w e o b se rv e a t th e o u tp u t o f th e system is th e ste a d y -s ta te re s p o n s e . T h e re is no
tra n sie n t re sp o n se in th is case.
O n th e o th e r h a n d , if th e e x p o n e n tia l o r sin u so id al signal is a p p lie d a t som e
finite tim e in sta n t, say at n = 0, th e re sp o n se of th e sy stem co n sists of tw o term s, the
tra n sie n t re sp o n se a n d the ste a d y -sta te re sp o n se . T o d e m o n s tra te this b ehavior,
let us co n sid er, as an e x am p le, th e sy stem d e sc rib e d by th e firs t-o rd e r d ifferen ce
e q u a tio n
y( n) = av( n — 1 )4- j:(/i)
(4.4.20)
T his sy stem w as c o n s id e re d in S ectio n 2.4.2. Its re sp o n se to an y in p u t x ( n) applied
at n = 0 is given by (2.4.8) as
y( n) = a n+ly ( - l ) + J 2 a kx ( n - k)
*=0
w h ere y ( —1) is th e in itial co n d itio n .
n> 0
(4.4.21)
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
315
N o w , let us assu m e th a t th e in p u t to th e system is th e c o m p le x e x p o n e n tia l
x ( n ) - A e Jan
n> 0
(4.4.22)
a p p lied a t n = 0. W h e n w e s u b s titu te (4.4.22) in to (4.4.21), w e o b ta in
n
v(n) = a " +1y ( —1) +
= a
v ( —1) + A
k=0
1 _ gn+le ~juiUi +\)
= o "+1v ( - l ) + A ----- --------------------e jmn
1 -ae-J”
A/jn^
(4.4.23)
n > 0
A
= an+' v ( - l ) ---------- :
---eJua + ------ - e ia*
1 — a e ~ JW
1 ~ a e JW
n>0
W e recall th a t th e system in (4.4.20) is B IB O sta b le if |a | < 1 . In th is case
th e tw o te rm s in v o lv in g ^ " +1 in (4.4.23) d ecay to w a rd z e ro as n a p p ro a c h e s infinity.
C o n s e q u e n tly , w e a re left w ith th e ste a d y -s ta te re sp o n se
Vss(n) =
=
A
i„
hm v(«) = ------------ —e1
l-ae-J*
(4.4.24)
AH(a))eJwn
T h e first tw o te rm s in (4.4.23) c o n stitu te th e tra n s ie n t re s p o n s e o f th e svstem ,
th a t is,
Aan + '
I
Vlr(n) = a n+1 v ( - l ) ------- ;----------- :----- e JU,n
n > 0
(4.4.25)
1 - a e ~ JW
w hich d e c a y to w a rd z e ro as n a p p ro a c h e s infinity. T h e first te r m in th e tra n s ie n t
re sp o n se is th e z e ro -in p u t re sp o n se o f th e system an d th e se c o n d te rm is th e
tra n s ie n t p ro d u c e d by th e e x p o n e n tia l in p u t signal.
In g e n e ra l, all lin e a r tim e -in v a ria n t B IB O system s b e h a v e in a sim ilar fashion
w h e n e x c ite d by a co m p lex e x p o n e n tia l, o r by a sin u so id a t n = 0 o r at so m e o th e r
finite tim e in sta n t. T h a t is, th e tra n s ie n t re sp o n s e d ecay s to w a rd z e ro as n —►oo,
leav in g o n ly th e s te a d y -s ta te re sp o n s e th a t w e d e te rm in e d in th e p re c e d in g section.
In m an y p ractical a p p lic a tio n s, th e tra n s ie n t re sp o n se of th e sy ste m is u n im p o rta n t,
a n d th e r e fo r e it is u su ally ig n o re d in d e a lin g w ith th e re sp o n s e o f th e system to
sin u so id al in p u ts.
4.4.3 Steady-State Response to Periodic Input Signals
S u p p o se th a t th e in p u t to a sta b le lin e a r tim e -in v a ria n t system is a p e rio d ic signal
x ( n ) w ith fu n d a m e n ta l p e rio d N. S ince such a signal exists fro m —co < n < oc,
th e to ta l re sp o n s e o f th e sy stem a t an y tim e in s ta n t n, is sim p ly e q u a l to the
s te a d y -s ta te resp o n se.
316
Frequency Analysis of Signals and Systems
Chap. 4
T o d e te rm in e th e resp o n se v(n) o f th e system , we m a k e use o f th e Fourier
se ries re p re s e n ta tio n of th e p e rio d ic signal, w hich is
A’- l
jr(n) = 5 2 c ^ j27rkn/N
k=0
k = 0, 1........ N - 1
(4.4.26)
w h e re th e {c*} are th e F o u rie r se ries coefficients. N ow th e re sp o n s e of th e system
to th e co m p lex e x p o n e n tia l signal
Xk (n) -
ckej27rkn/N
k = 0 , 1 ............ N -
1
is
v* ( n ) = t \ H
k^j
v
fc = 0. 1.........A ' - l
/ Ink \
H ( — J = H ( w ) L = 2, i7A-
k = 0. 1 ........ A ' - l
(4.4.27)
w h ere
B y u sin g th e su p e rp o s itio n prin cip le fo r lin e a r system s, w e o b ta in th e resp o n se of
th e system to th e p erio d ic signal x( n) in (4.4.26) as
A—1
v(«) = 5 2
/ ?t k \
gjlxknlf*
— oc < 7; < OC
(4.4.28)
T his resu lt im p lies th a t th e re sp o n se of th e system to th e p e rio d ic in p u t signal
x ( n ) is also p e rio d ic w ith th e sam e p e rio d N . T h e F o u rie r se rie s coefficients for
y( n) a re
dk = ckH ( ^ ^ j
* = 0 , 1 .........A ' - l
(4.4.29)
H e n c e , th e lin e a r system can ch an g e th e sh a p e o f th e p e rio d ic input signal by
scaling th e a m p litu d e an d shifting th e p h ase o f th e F o u rie r se rie s c o m p o n en ts, but
it d o e s n o t affe ct th e p e rio d of the p e rio d ic in p u t signal.
4.4.4 Response to Aperiodic Input Signals
T h e c o n v o lu tio n th e o re m , given in (4.3.49). p ro v id e s th e d e s ire d freq u e n cy -d o m ain
re la tio n sh ip fo r d e te rm in in g the o u tp u t of an L T I system to an ap e rio d ic finiteen erg y signal. If {jt(n)} d e n o te s th e in p u t se q u e n c e , (v(n)} d e n o te s th e o u tp u t
se q u e n c e , an d (/?(«)} d e n o te s th e u n it sa m p le re sp o n se o f th e system , th e n from
th e co n v o lu tio n th e o re m , w e have
Y(w) = H(a>)X(w)
(4.4.30)
w h e re Y(to), X (o j), an d H(co) are th e c o rre sp o n d in g F o u rie r tra n sfo rm s o f {v(«)J.
(*(«)}, an d {h( n) ), resp ectiv ely . F ro m th is re la tio n sh ip w e o b se rv e th a t th e sp e c­
tru m o f th e o u tp u t signal is e q u a l to th e sp e c tru m o f th e in p u t signal m ultiplied
by th e fre q u e n c y resp o n se o f th e system .
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
317
If w e e x p re ss K(cu),
an d X(o>) in p o la r fo rm , th e m a g n itu d e a n d p h ase
o f th e o u tp u t signal can b e ex p re sse d as
|y M ! = I / / M I I X M 1
(4.4.31)
4-Y (o>) —
(4.4.32)
+ 4 - H ( co)
w h ere |//(a> )| a n d
a re th e m a g n itu d e an d p h ase re sp o n s e s o f th e system .
B y its v ery n a tu r e , a fin ite-en erg y a p e rio d ic signal c o n ta in s a c o n tin u u m of
fre q u e n c y c o m p o n e n ts . T h e lin e a r tim e -in v a ria n t system , th ro u g h its fre q u e n c y
re sp o n s e fu n c tio n , a tte n u a te s so m e fre q u e n c y c o m p o n e n ts of th e in p u t signal an d
am p lifies o th e r fre q u e n c y co m p o n e n ts. T h u s th e system acts as a filter to th e in p u t
signal. O b s e rv a tio n o f th e g ra p h o f \H(to)\ show s w hich fre q u e n c y c o m p o n e n ts
a re am p lified a n d w h ich a re a tte n u a te d . O n th e o th e r h a n d , th e angle o f H(a>)
d e te rm in e s th e p h a s e sh ift im p a rte d in th e c o n tin u u m o f fre q u e n c y c o m p o n e n ts of
th e in p u t signal as a fu n c tio n o f freq u e n cy . If th e in p u t signal s p e c tru m is ch an g ed
by th e sy stem in an u n d e s ira b le way, we say th a t th e system h a s cau sed mag ni t ude
a n d p h a s e distortion.
W e also o b se rv e th a t the o ut put o f a linear ti me-i nvariant sy s t em c annot c o n ­
tain f r e q u e n c y c o m p o n e n t s that are n o t cont ai ned in the input signal. It tak es e ith e r
a lin e a r tim e* v aria n t sy stem o r a n o n lin e a r system to c re a te fre q u e n c y c o m p o n e n ts
th a t a re n o t n ecessa rily c o n ta in e d in th e in p u t signal.
F ig u re 4.39 illu stra te s th e tim e-d o m ain an d fre q u e n c y -d o m a in relatio n sh ip s
th a t can b e u se d in th e an aly sis o f B IB O -s ta b le L T I system s. W e o b se rv e th a t
in tim e -d o m a in an aly sis, w e d eal w ith th e co n v o lu tio n o f th e in p u t signal w ith
th e im p u lse re sp o n s e o f th e system to o b ta in th e o u tp u t s e q u e n c e o f th e system .
O n th e o th e r h a n d , in fre q u e n c y -d o m a in analysis, w e deal w ith th e in p u t signal
sp e c tru m X(a)) a n d th e fre q u e n c y re sp o n s e H(co) o f th e sy stem , w hich a re re la te d
th ro u g h m u ltip lic a tio n , to yield th e sp e c tru m o f th e signal a t th e o u tp u t o f th e
system .
W e c a n u se th e re la tio n in (4.4.30) to d e te rm in e th e sp e c tru m Y( u>) o f th e
o u tp u t signal. T h e n th e o u tp u t se q u e n c e {v(«)} can b e d e te rm in e d fro m th e in v erse
F o u rie r tra n sfo rm
(4.4.33)
H o w e v e r, th is m e th o d is se ld o m used. In ste a d , th e z -tra n s fo rm in tro d u c e d in
C h a p te r 3 is a s im p le r m e th o d fo r solving th e p ro b le m o f d e te rm in in g th e o u tp u t
se q u e n c e {y(n)}.
X(n)
X( w)
Input
Linear
time-invariant
system
hin), H(a>)
Output
v(n) = h(n)+x(n)
KM = ma>)X(a>)
F,g“re 4 3 9
Tlm e' and
frequency-domain inpul-output
relationships in LTI systems.
318
Frequency Analysis of Signals and Systems
Chap. 4
L e t us r e tu rn to th e b asic in p u t-o u tp u t re la tio n in (4,4.30) a n d c o m p u te the
s q u a re d m a g n itu d e o f b o th sides. T h u s w e o b ta in
\Y(co)\2 = \H(oj)\ 2 \X(co )\2
(4.4.34)
Syy(u}) = \ H{ co)\2S xx {0>)
w h ere Sxx(oo) a n d S vv(cu) a re th e en erg y d en sity sp e c tra o f th e in p u t a n d o u tp u t
signals, resp ec tiv ely . By in te g ra tin g (4.4.34) o v e r th e fre q u e n c y ran g e ( - n , n ) , we
o b ta in th e e n e rg y o f th e o u tp u t signal as
r
i
(cu)da>
(4.4.35)
-
— r H ( co) |2 S j x ( co) d co
27r
J-y,
Example 4.4.5
A linear time-invariant system is characterized by its impulse response
h( n ) = (y)" u (n )
Determ ine the spectrum and the energy density spectrum of the output signal when
the system is excited by the signal
*(«) = ( j )"«(«)
Solution
The frequency response function of the system
OC
H(w)
=
1
1 - \e-^
Similarly, the input sequence U(n)) has a Fourier transform
1
X(w) =
1 — - e ~ JUI
Hence the spectrum of the signal at the output of the system is
Y (co) = H( w) X( c o )
1
(1 -
1-
The corresponding energy density spectrum is
= \Y(<o)\2 = \H(o>)\2\X(o>)\2
1
( j - coso))(j| - 1 cosa>)
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
319
4.4.5 Relationships Between the System Function and
the Frequency Response Function
F ro m th e d iscu ssio n in S ectio n 4.2.6 w e k now th a t if th e system fu n ctio n H( z )
co n v erg es on th e u n it circle, w e can o b ta in th e freq u e n c y re sp o n s e o f th e system
by e v a lu a tin g H ( z ) o n th e u n it circle. T h u s
(4.4.36)
In th e case w h e re H ( z ) is a ra tio n a l fu n ctio n of th e fo rm H{ z ) = B ( z ) / A ( z ) . w e have
(4.4.37)
M
(4.4.38)
]~~[U - pk? /<")
w h e re th e (fli) a n d {£>*} a re real, b u t {ct } and {pk} m ay be c o m p lcx -v alu cd .
It is so m e tim e s d e s ira b le to ex p ress th e m a g n itu d e s q u a re d o f H(a>) in term s
o f H( z ) . F irst, w e n o te th a t
|f f( a O r = H{ w) H*( u>)
F o r th e ra tio n a l sy stem fu n ctio n given by (4.4.38). we have
M
ft
(4.4.39)
It fo llo w s th a t H*( w) is o b ta in e d by e v a lu a tin g H*( 1/ z*) on th e unit circle, w h ere
fo r a ra tio n a l sy stem fu n ctio n .
M
(4.4.40)
*=i
H o w e v e r, w h e n {*(«)} is re a l or, e q u iv alen tly , th e coefficients {a*} and {bk} a re
real, c o m p lex -v alu ed p o le s a n d z e ro s o ccu r in c o m p le x -c o n ju g a te pairs. In this
320
Frequency Analysis of Signals and Systems
case. H*{ 1/;* ) = H ( z ~ l ). C o n se q u e n tly , H'(a>) =
Chap. 4
an d
| H(co )\2 = H{oj)H*{oj) = H ( w ) H ( - o > ) =
(4.4.41)
A cc o rd in g to th e c o rre la tio n th e o re m fo r th e z -tra n sfo rm (see T a b le 3.2), the
fu n ctio n H ( z ) H ( z ~ l ) is th e z -tra n sfo rm o f th e a u to c o rre la tio n se q u e n c e {rhh( m)}
o f th e u n it sa m p le re sp o n se (/i(n)J. T h e n it follow s fro m th e W ie n e r-K h in tc h in e
th e o re m th a t \ H { w )\2 is th e F o u rie r tra n sfo rm o f
Sim ilarly, if H ( z ) = B ( z ) / A ( z ) , th e tra n sfo rm s D( z ) = B tz J B tz - 1 ) an d C( z) =
/4 (;)A (c “ ') are th e z-tra n sfo rm s o f th e a u to c o rre la tio n s e q u e n c e s {c/} an d \di\,
w h ere
\rhh(m)}.
N - |J i
akak+i
Q =
—N < I < N
(4.4.42)
t=o
M - \t\
d, = 2 2 bkbk+i
~ M < / <M
(4.4.43)
Jk=0
Since th e system p a ra m e te rs (at) a n d {£*} a re re a l valu ed , it follow s th a t c, — c_/
an d di = d-i . B y using th is sy m m etry p ro p e rty , \ H( u >)\2 m ay be ex p ressed as
do + 2 2^2 dk cos kco
\H(co )\2 = ----------^ -------------(4.4.44)
ri
co 4- 2 ^ ' c'jt cos k.u)
i =1
F inally, w e n o te th a t c o s k w can be e x p re ss e d as a p o ly n o m ia l fu n ctio n of
cosoj. T h a t is,
t
cos kco = ^ / U c o s a , ) "
(4.4.45)
m=0
w h ere [j3m] a re th e coefficients in th e ex p an sio n . C o n s e q u e n tly , th e n u m e ra to r
an d d e n o m in a to r o f \H(u ))\2 can b e v iew ed as p o ly n o m ial fu n c tio n s o f coso;. T he
fo llow ing e x am p le illu stra te s th e fo reg o in g re la tio n sh ip s.
Example 4.4.6
Determ ine
for the system
y(n) = —0.1 v(n —1) + 0.2 y(n — 2 ) + x (n) + x(n — 1)
Solution
The system function is
1 4 :~l
H(Z) ~ 1 4 0 . 1 - “ - 0 . 2 : - 2
and its R O C is |z| > 0.5. Hence H{ od) exists. Now
1 + z“ ‘
1+ z
1 + 0 .1 Z '1 - 0 .2 z - 2
1 4- O.lz - 0.2z2
2 4- z + z 1
1.05 + 0.08(z 4- z_1) - 0.2(z~2 + z-2)
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
321
By evaluating H{z)H(z~' ) on the unit circle, we obtain
2 -j- 2 cos w
\H(u>)~ = --------------------------------------1.0^ -i- 0.16 cos w —0.4 cos 2w
However, cos2w = 2cos: w — 1. Consequently. | H(u>)\2 may be expressed as
2(1 cos u>)
1.45 4 0.16cosw - 0 .8 cos:
W e n o te th a t given H ( z ), it is stra ig h tfo rw a rd to d e te rm in e H ( z ~ l ) an d th e n
|//{ w ) |2. H o w e v e r, th e inverse p ro b lem o f d e te rm in in g H( z ) given 1/ / (cu)]2 or th e
c o rre sp o n d in g im p u lse resp o n se {/?(«))- is n o t stra ig h tfo rw a rd . Since |//(a > )l: d o es
n o t c o n tain th e p h ase in fo rm atio n in H(a>), it is n o t p o ssible to u n iq u ely d e te rm in e
H (:).
T o e la b o ra te on th e p oint, let us assum e th a t th e N p o les an d M ze ro s of
H{z) are [pk] an d {;.*). respectively. T h e c o rre sp o n d in g p o les and zero s o f H ( z ~ ])
are (1 j p^ j an d ll/c*. }, respectively. G iv en \ H( u >)\2 o r, eq u iv a le n tly . H ( z ) H ( z ~ l ). w e
can d e te rm in e d iffe re n t system fu n ctio n s H( z ) by assigning to H( z ) . a pole pt o r
its recip ro c al 1f p k . an d a z e ro zt o r its recip ro c al 1 fzk- F o r ex am p le, if A" = 2 a n d
M = 1. th e p o les an d z ero s of H l z ) H ( : _ l ) are [ p j. p z , 1/ p \ , l / p z ) an d (~ i, 1 /c i }. If
p i a n d pz are real, th e possible p o les for H(z.) a re {pi.
1. U /P i • 1 /P :K i / 'i • 1/ / ’:}.
a n d [/52. 1//>]} an d th e possible ze ro s are {~i) o r {1 / - 1 }, T h e re fo re , th e re are eight
p o ssib le ch o ices o f sy stem functions, all o f w hich re su lt in th e sam e j//(a> )|: . E ven
if we restrict th e p o les of H( z ) to be inside th e unit circle, th e re a re still tw o
d iffe re n t ch o ices for H( z ) . d e p e n d in g on w h e th e r w e pick th e zero j;i} o r { l/:.i).
T h e re fo re , we c a n n o t d e te rm in e H{z ) u n iq u ely given only th e m a g n itu d e resp o n se
\H(w)\.
4.4.6 Computation of the Frequency Response Function
In e v a lu a tin g th e m a g n itu d e re sp o n se an d th e p h a s e re sp o n se as fu n c tio n s o f f r e ­
q u en cy , it is c o n v e n ie n t to ex p ress H ( oj) in te rm s o f its p o le s an d zeros. H e n c e
we w rite H(co) in fa c to re d form as
M
n
H(co) =
a - z t e - i wk )
--------------------
(4.4.46)
f ] ( 1 - p ke->°*)
k=1
or, e q u iv alen tly , as
M
H M = b0eJU’^ - M) ------------------]"~[(eJ“ - p k)
(4.4.47)
Frequency Analysis of Signals and Systems
322
Chap. 4
L e t us ex p re ss th e c o m p lex -v alu ed facto rs in (4.4.47) in p o la r form as
ej a - Z k = Vk (co)ejB*M
(4.4.48)
e J0J - p k = Uk (a>)e] * k(m)
(4.4.49)
an d
w h ere
V*(w) s \eJW - z*|,
&k (co) = 4. (eJW - zk)
(4.4.50)
Uk(co) = \eJ“ - p kI,
= %.(eJW - p k)
(4.4.51)
an d
T h e m a g n itu d e o f H ( w ) is eq u al to th e p ro d u c t o f m a g n itu d e s of all te rm s in
(4.4.47). T h u s, u sin g (4.4.48) th ro u g h (4.4.51), w e o b ta in
Vi(£») ■• ■VM(co)
- ■
V\ (co)U2 ((o) ■■■Un(co)
|fr0l------ — -------
\H(co)\ =
(4.4.52)
’
since th e m a g n itu d e o f eJM(N~M) is 1.
T h e p h ase o f H(to) is th e sum o f th e p h a s e s o f th e n u m e r a to r facto rs, m i­
n us th e p h ases o f th e d e n o m in a to r facto rs. T h u s, by co m b in in g (4.4.48) th ro u g h
(4.4.51), we h av e
2^ H ( c o ) — 4-bo + co(N -
M
) +
0 i (co) + 0T(dti) +
• ■ ■ +
&m(oj)
(4.4.53)
— [<t>i(a>) + <I>2(^) + ■• ■+ $>(o>)]
T h e p h ase o f th e gain te rm
is z e ro o r jt, d e p e n d in g on w h e th e r bo is po sitiv e or
n eg ativ e. C learly , if w e k n o w th e z ero s an d th e p o les o f th e sy ste m fu n ctio n H(z),
we can ev a lu a te th e fre q u e n c y re sp o n se fro m (4.4.52) an d (4.4.53).
T h e re is a g e o m e tric in te r p re ta tio n of th e q u a n titie s a p p e a rin g in (4.4.52)
a n d (4.4.53). L e t us c o n sid e r a p o le p k a n d a z e ro z* lo c a te d a t p o in ts A a n d B
o f th e z-p lan e, as sh o w n in Fig. 4 .40(a). A ssu m e th a t w e w ish to c o m p u te H{io)
a t a specific v alu e o f fre q u e n c y co. T h e given v alu e o f co d e te rm in e s th e an g le of
ejw w ith th e p o sitiv e real axis. T h e tip o f th e v e c to r e
specifies a p o in t L o n the
u n it circle. T h e e v a lu a tio n o f th e F o u rie r tra n sfo rm fo r th e g iv en v alue of co is
e q u iv a le n t to e v a lu a tin g th e z -tra n sfo rm at th e p o in t L o f th e co m p lex p la n e . L et
us d raw th e v ecto rs A L a n d B L from th e p o le an d z e ro lo c a tio n s to th e p o in t L, at
w hich w e w ish to c o m p u te th e F o u rie r tra n sfo rm . F ro m Fig. 4 .4 0 (a) it follow s th at
CL = CA + A L
an d
CL = CB + BL
H o w ev er, C L = ej<u, C A = p k a n d C B = zk. T h u s
AL = ej* - p k
(4.4.54)
BL = e JW - z k
(4.4.55)
an d
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
323
ImU)
(a)
im(r)
Figure 4.40 Geom etric interpretation
of the contribution of a pole and a zero
to the Fourier transform (1) magnitude:
the factor V* / Vk, (2) phase; the factor
©* - <t>*B y c o m b in in g th e s e re la tio n s w ith (4.4.48) a n d (4.4.49), w e o b ta in
A L = eJ<u - p k = Uk {co)eJ* kM
(4.4.56)
BL = eJ* — Zk = Vk (aj)ejStUu)
(4.4.57)
T h u s Uk(a>) is th e len g th o f AL, th a t is, th e d istan c e o f th e p o le p k fro m th e p o in t
L c o rre sp o n d in g to e }w, w h e re a s Vk(cu) is th e d ista n c e o f th e z e ro z k fro m th e sam e
p o in t L. T h e p h a se s <!>*(&>) a n d Q*(a>) a re th e an g les o f th e v e c to rs A L an d BL
324
Frequency Analysis of Signals and Systems
Chap. 4
Im(c)
Pt = e>^‘
Zt =
mz)
Figure 4.41 A zero on the unit circle
causes |f/((u)| - 0 and w = 4-Zk- In
contrast, a pole on the unit circle results
in \H{w)\ = 0 0 at w = 4-Pt-
w ith th e p o sitiv e re a l axis, resp ec tiv ely . T h e s e g e o m e tric in te r p re ta tio n s a re show n
in Fig. 4.40(b).
G e o m e tric in te rp re ta tio n s are very useful in u n d e rs ta n d in g how th e location
o f p o le s an d ze ro s affects th e m ag n itu d e an d p h a s e o f th e F o u rie r tran sfo rm .
S u p p o se th a t a zero , say z*. an d a pole, say p k, a re o n th e u n it circle as sh o w n in
Fig. 4.41. W e n o te th a t at w = 4 z* , Vk (co) an d c o n s e q u e n tly | H(co)\ b eco m e zero.
S im ilarly, at to = 4-pk th e len g th Uk(w) b eco m es z e ro a n d h e n c e \H(co)\ b ecom es
infinite. C learly , th e e v a lu a tio n o f p h a s e in th e s e case s h a s n o m ean in g .
F ro m th is discu ssion we can easily see th a t th e p re se n c e o f a ze ro close to
th e u n it circle cau ses th e m a g n itu d e o f th e fre q u e n c y re sp o n s e , a t freq u e n cies
th a t c o rre sp o n d to p o in ts o f th e u n it circle close to th a t p o in t, to b e sm all. In
c o n tra st, th e p re se n c e o f a p o le close to th e u n it circle c au ses th e m a g n itu d e of
th e fre q u e n c y re sp o n s e to b e larg e a t fre q u e n c ie s close to th a t p o in t. T h u s poles
h av e th e o p p o site effe ct o f zero s. A lso , p lacin g a z e ro close to a p o le cancels
th e effe ct o f th e p o le, an d vice v ersa. T h is can b e also se en fro m (4.4.47), since
if Zk = Pk, th e te rm s eJW - z* an d ejb> — pk cancel. O b v io u sly , th e p re se n c e of
b o th p o les an d z e ro s in a tra n sfo rm resu lts in a g r e a te r v a rie ty o f sh a p e s for
\H(to)\ a n d ^ H( c o ) . T his o b se rv a tio n is very im p o rta n t in th e desig n o f digital
filters. W e co n clu d e o u r discussion w ith th e follow ing ex a m p le illu stra tin g th ese
co n cep ts.
Example 4.4.7
Evaluate the frequency response of the system described by the system function
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
Clearly. H(z) has a zero at ; = 0 and a pole at p = 0.8.
frequency response of the system is
S o lu tio n
H (w) =
325
Hence the
el- - 0.8
The m agnitude response is
| H (to) j =
1
\e-lw - 0.8|
Vl-64 — 1.6cose
and the phase response is
B(co) = ui — tan
sin oj
cos oj - 0.8
The magnitude and phase responses are illustrated in Fig. 4.42. Note that the peak
of the m agnitude response occurs at a s = 0. the point on the unit circle closest to the
pole located at 0.8.
If th e m a g n itu d e re sp o n se in (4.4.52) is ex p ressed in d ecib els,
M
\
\H(co)\lllt = 2 0 lo g ,(l |/j()| + 2 0 5 2 l°g.jci v/a M - 2 o 5 2 logui
A=1
i-l
(4.4.58)
T h u s th e m a g n itu d e resp o n se is ex p ressed as a sum o f th e m ag n itu d e facto rs in
| H i w )!.
4.4.7 Input-Output Correlation Functions and Spectra
In S ectio n 2.6.5 w e d e v e lo p e d several c o rre la tio n re la tio n sh ip s b etw een the input
an d th e o u tp u t se q u e n c e s o f an L TI system . S pecifically, w e d eriv ed th e follow ing
e q u a tio n s:
r vv(m) = rhh(m) * r xx(m)
(4.4.59)
ryx( m ) = h( m) * rxx(m)
(4.4.60)
w h ere rxx(m) is th e a u to c o rre la tio n se q u e n c e of th e in p u t signal {-*(/?)), ryv(m) is
th e a u to c o rre la tio n se q u e n c e o f th e o u tp u t {.y(n)K rhh(m ) is th e a u to c o rre la tio n se ­
q u e n c e o f th e im p u lse re sp o n se {Ai(/t)}. a n d ryx(m) is th e c ro ss c o rre la tio n se q u e n c e
b e tw e e n th e o u tp u t an d th e in p u t signals. Since (4,4.59) an d (4.4.60) involve th e
co n v o lu tio n o p e ra tio n , th e z-tra n sfo rm o f th e s e e q u a tio n s yields
Svvf;) = 5 m ( : ) 5 „ ( c )
(4.4.61)
= H ( z ) H ( z ~ 1 )Sxx(z)
S VJr(z) = H ( z ) S „ ( z )
(4.4.62)
If we su b s titu te z = e JW in (4,4.62), we o b tain
S y jc M
=
H(os)Sxx(oj)
(4.4.63)
= H(o>)\X(o>)\2
326
Frequency Analysis of Signals and Systems
- T
_ t
2
o
I
2
r
Chap. 4
Figure 4.42 Magnitude and phase of
system with H(z) = 1/(1 - 0 .8 ; - 1).
w h ere Syx(u>) is th e cro ss-en e rg y d en sity sp e c tru m o f [y(n)} a n d (jc(/7)}. Sim ilarly,
ev alu atin g Svv(z) o n th e u n it circle yields th e en erg y d e n sity s p e c tru m o f th e o u tp u t
signal as
S v v M = | / / ( w ) |2S „ M
(4.4.64)
w h ere Sxx(a>) is th e en erg y d en sity sp e c tru m o f th e in p u t signal.
S ince ryy(m) an d Syy(co) a re a F o u rie r tra n sfo rm p air, it fo llo w s th at
ryy(m) = ~
j
Syy(co)ejumdco
(4.4.65)
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-Invariant Systems
327
T h e to ta l e n erg y in th e o u tp u t signal is sim ply
Ey — --- f
Syy{CO)dlO = T,.v(0 >
~jT
i r
= — I \ H(to)\' S xx(cv)dcv
J-*
(4.4.66)
T h e re su lt in (4.4.66) m ay be used to easily p rove th a t £ v > 0.
F in ally , we n o te th a t if the in p u t signal has a flat sp e c tru m [i.e.. 5 v., (w) =
£ , = c o n s ta n t fo r n < co < — t t ] , (4.4.63) red u ces to
5 , v(oj) = H { w ) E x
(4.4.67)
w h ere £ , is th e c o n s ta n t v alu e o f the sp e ctru m . H e n c e
H ( w ) = — 5’vv(w)
(4.4.68)
h(n) — — r yxim)
(4.4.69)
E\
o r . e q u iv a le n tly .
T h e re la tio n in (4,4.69) im plies th at Inn) can be d e te rm in e d by exciting th e input
to th e system by a sp e ctrally flat signal ja («)), and c ro ssc o rre la tin g th e in p u t w ith
th e o u tp u t o f th e system . T his m e th o d is usef ul in m e a su rin g th e im pulse resp o n se
o f an u n k n o w n svstem .
4.4.8 Correlation Functions and Power Spectra for
Random Input Signals
T his d e v e lo p m e n t p a ra lle ls th e d e riv a tio n s in S ection 4.4.7, w ith th e e x cep tio n th a t
we no w d e a l w ith sta tistical m o m e n ts o f th e in p u t an d o u tp u t signals o f an L TI
system . T h e v ario u s sta tistical p a ra m e te rs are in tro d u c e d in A p p e n d ix A .
L et us c o n s id e r a d isc re te -tim e lin e a r tim e -in v a ria n t sy stem w ith u n it sam ple
re sp o n se (/z(n)} an d fre q u e n c y re sp o n se H ( f ) . F o r this d e v e lo p m e n t w e assum e
th a t {/?(«)} is real. L et _*•(«) be a sam ple fu n ctio n of a sta tio n a ry ra n d o m process
X ( n ) th a t ex cites th e system an d let y(/;) d e n o te th e re sp o n se o f th e system to xi n) .
F ro m th e c o n v o lu tio n su m m atio n th a t re la te s th e o u tp u t to th e in p u t w e have
OC
v(n) =
5 2 h ( k ) x ( n — k)
(4.4.70)
Jt = - oc
S ince jc(n) is a ra n d o m in p u t signal, th e o u tp u t is also a ra n d o m se q u en ce. In o th e r
w o rd s, fo r each sa m p le se q u e n c e x ( n ) o f th e p ro cess X ( n ), th e r e is a c o rre sp o n d in g
sa m p le se q u e n c e v(n) o f th e o u tp u t ra n d o m p ro c e ss Y(n) . W e w ish to relate
th e sta tistical c h a ra c te ristic s o f th e o u tp u t ra n d o m p ro cess Y( n) to th e statistical
c h a ra c te riz a tio n o f th e in p u t p ro c e ss a n d th e c h aracteristics o f th e system .
328
Frequency Analysis of Signals and Systems
Chap. 4
T h e ex p e c te d v alu e o f th e o u tp u t y(n) is
OC
m v s= £ [y (n )] = E[ 22 h(k)x{rt — fc)]
k ~ —rx.
oc
22 A ( * ) £ 0 ( n - * ) ]
=
(4.4.71)
A=^oc
cc
22 h( k)
my = mx
k ——oc
F ro m th e F o u rie r tra n sfo rm re la tio n sh ip
CC
H{u>)=
22 * (*)*"-'“*
(4.4.72)
k = - oc
we have
OC
22 h (*)
H ( 0) =
(4.4.73)
A — — DC
w hich is th e dc gain o f th e system . T h e re la tio n sh ip in (4.4.73) allow s us to express
th e m e a n v alu e in (4.4.71) as
m y = m xH{ 0)
(4.4.74)
T h e a u to c o rre la tio n se q u en ce fo r th e o u tp u t ra n d o m p ro c e ss is
yyy(m) — £ [ y * ( n ) y ( « + m ) ]
-- E
=
2 2 h( k)x*(n ~ k) Y 2
_k——OQ
OC
OC
22
V .
+ m - j)
(4.4.75)
h ( k ) h ( j ) E [ x * { n - k)x(rt -f- m - _/)]
£ = — C C _/ = —OC
OC
=
cc
22 Y2 h(k'>hU)Yxx(k - j + m)
k= —co
cc
T h is is th e g en eral fo rm fo r th e a u to c o rre la tio n o f th e o u tp u t in term s of the
a u to c o rre la tio n o f th e in p u t a n d th e im p u lse re sp o n s e o f th e system .
A sp ecial fo rm o f (4.4.75) is o b ta in e d w h en th e in p u t ra n d o m p ro c e ss is w hite,
th a t is, w h en m , = 0 an d
Yxx(m) = o’j S ( m )
(4.4.76)
w h ere o x = ^ j(O ) is th e in p u t signal p o w er. T h e n (4.4.75) re d u c e s to
OC
Yyy(m)
= <72 22 h{k) h{k + m)
(4.4.77)
t= -o c
U n d e r th is co n d itio n th e o u tp u t p ro cess h a s th e a v e ra g e p o w e r
MO) = ^
£
h ^ n) =
n=-e»
w h ere w e h av e a p p lie d P a rse v a l’s th e o re m .
(4-4.78)
J - 1/2
Sec. 4.4
Frequency-Domain Characteristics of Linear Tim e-invariant Systems
329
T h e re la tio n sh ip in (4.4.75) can b e tra n sfo rm e d in to th e fre q u e n c y d o m ain
by d e te rm in in g th e p o w e r d e n sity sp e c tru m of yvv(m ). W e h ave
r vvM
=
22 /vv
=
E
E
E
h ( k ) h ( l ) Y, A k - I + m)
k ^ —o o i =■—dc
(4.4.79)
E
E
V
h{k)h(l )
Yxx(k — I + m) e
k = — CC I = — DC
OC
= r r, ( / )
2 2 h{k) eJwk
£
h { l ) e - ]wl
^k= - c c
= tw (tu )|-r^ .v(w)
T h is is th e d e sire d re la tio n sh ip fo r th e p o w e r d en sity sp e c tru m o f th e o u tp u t p r o ­
cess, in te rm s o f th e p o w e r d en sity sp e c tru m o f th e in p u t p ro cess an d th e freq u e n cy
re sp o n s e o f th e system .
T h e e q u iv a le n t e x p ressio n fo r co n tin u o u s-tim e system s w ith ra n d o m in p u ts is
r vv( F ) = \ H{ F ) \ 2 r xA F )
(4.4.80)
w h ere th e p o w e r d e n sity s p e c tra T w ^ ) and T „ ( F ) are th e F o u rie r tra n sfo rm s
o f th e a u to c o rre la tio n fu n c tio n s y v v ( T ) a n d yJ t ( r ) , resp ec tiv ely , a n d w h e re H { F )
is th e fre q u e n c y re sp o n s e of th e system , w hich is re la te d to th e im p u lse re sp o n se
by th e F o u rie r tra n sfo rm , th a t is.
H(F)
- J-C
f h( t ) e
- j 2n Fi
dt
(4.4.81)
A s a final ex ercise , w e d e te rm in e th e c ro ssc o rre la tio n o f th e o u tp u t v(n) w ith
th e in p u t signal jcOi). If w e m u ltip ly b o th sides o f (4.4.70) by x*(n — m) a n d ta k e
th e e x p e c te d v a lu e , w e o b ta in
y
£ [ v (n )-**(« — wz)] = E
h( k)x*{n — m ) x ( n — k)
= —OC
y VJ(m) =
2 2 h( k) E[ x * ( n — m ) x ( n — k)]
(4.4.82)
= —CC
OC
=
22
h ( k ) Y x x ( m - k)
Since (4.4.82) h as th e fo rm o f a c o n v o lu tio n , th e fre q u e n c y -d o m a in e q u iv a le n t
e x p re ssio n is
r„(w ) = ff(<u)r„(<u)
(4.4.83)
In the special case where x ( n ) is white noise, (4.4.83) reduces to
r y, M
= a x2 H { a > )
(4.4.84)
330
Frequency Analysis of Signals and Systems
Chap. 4
w h ere a 2 is th e in p u t n o ise p o w er. T h is re su lt m e a n s th a t an u n k n o w n sy stem w ith
fre q u e n c y resp o n se H ( w ) can be id en tified by exciting th e in p u t w ith w h ite noise,
c ro ssco rrelatin g th e in p u t se q u e n c e w ith th e o u tp u t se q u e n c e to o b ta in y Vjr(m ), and
finally, c o m p u tin g th e F o u rie r tra n sfo rm o f yyx(m). T h e re su lt o f th e s e c o m p u ta ­
tio n s is p ro p o rtio n a l to H{co).
4.5 LINEAR TIME-INVARIANT SYSTEMS AS FREQUENCY-SELECTIVE
FILTERS
T h e te rm filter is co m m o n ly used to d escrib e a d ev ice th a t d isc rim in a te s, a c co rd ­
ing to so m e a ttr ib u te o f th e o b jects a p p lied at its in p u t, w h at passes th ro u g h it.
F o r ex am p le, an a ir filter allow s a ir to p ass th ro u g h it b u t p re v e n ts d u st p a r­
ticles th a t are p r e s e n t in th e a ir from passing th ro u g h . A n oil filter p erfo rm s
a sim ilar fu n ctio n , w ith th e e x cep tio n th a t oil is th e su b sta n c e allo w ed to pass
th ro u g h th e filter, w h ile p a rtic le s o f dirt are c o llected a t th e in p u t to th e filter
a n d p re v e n te d fro m p assin g th ro u g h . In p h o to g ra p h y , an u ltra v io le t filter is of­
ten u sed to p re v e n t u ltra v io le t light, w hich is p re s e n t in su n lig h t a n d w hich is not
a p a rt o f visible light, from passin g th ro u g h an d affe ctin g th e chem icals on the
film.
A s w e h av e o b se rv e d in th e p re c e d in g se ctio n , a lin e a r tim e -in v a ria n t system
also p e rfo rm s a ty p e o f d isc rim in atio n o r filtering am o n g th e v ario u s freq u e n cy
c o m p o n e n ts a t its in p u t. T h e n a tu re o f th is filtering actio n is d e te rm in e d by the
fre q u e n c y re sp o n se c h a ra c te ristic s H{co), w hich in tu r n d e p e n d s o n th e ch o ice of
th e sy stem p a ra m e te rs (e.g., th e coefficients (at ) a n d [bk ] in th e d iffe re n c e e q u a tio n
ch a ra c te riz a tio n o f th e sy stem ). T h u s, by p r o p e r selectio n o f th e coefficients, we
can d esig n freq u e n cy -selectiv e filters th a t p ass signals w ith fre q u e n c y co m p o n e n ts
in so m e b an d s w h ile th e y a tte n u a te signals co n ta in in g fre q u e n c y c o m p o n e n ts in
o th e r fre q u e n c y b an d s.
In g e n eral, a lin e a r tim e -in v a ria n t system m odifies th e in p u t signal sp e c­
tru m X((o) a cco rd in g to its fre q u e n c y re sp o n s e H(co) to yield an o u tp u t signal
w ith sp e c tru m Y(to) = H(o>)X(cl>). In a sense, H(co) acts as a wei ghti ng f u n c ­
tion o r a spectral s h a pi ng f un c t i o n to th e d iffe re n t fre q u e n c y c o m p o n e n ts in th e
in p u t signal. W h e n v iew ed in th is c o n te x t, any lin e a r tim e -in v a ria n t system can
b e c o n sid e re d to b e a fre q u e n c y -sh a p in g filter, ev en th o u g h it m ay n o t n ecessa r­
ily c o m p letely b lo ck a n y o r all fre q u e n c y c o m p o n e n ts. C o n s e q u e n tly , th e term s
“ lin e a r tim e -in v a ria n t sy stem ” a n d “filte r” a re sy n o n y m o u s a n d a re o fte n used
in terch an g eab ly .
W e use th e te r m filter to d escrib e a lin e a r tim e -in v a ria n t system u se d to
p e rfo rm sp e c tra l sh a p in g o r freq u e n cy -selectiv e filtering. F ilte rin g is u se d in dig­
ital sig n al p ro cessin g in a v a rie ty o f w ays. F o r e x am p le, re m o v a l o f u n d e sira b le
n o ise fro m d e s ire d signals, sp e c tra l sh a p in g such as e q u a liz a tio n o f c o m m u n icatio n
c h a n n e ls, signal d e te c tio n in r a d a r, so n a r, a n d c o m m u n ic a tio n s, a n d fo r p e rfo rm in g
sp e c tra l an aly sis o f signals, a n d so on.
Sec. 4.5
Linear Tim e-Invariant Systems as Frequency-Selective Filters
331
4.5.1 Ideal Filter Characteristics
F ilters a re usu ally classified acco rd in g to th e ir fre q u e n c y -d o m a in ch aracteristics
as low pass, h ig h p ass. b a n d p a ss, and b a n d s to p o r b a n d -e lim in a tio n filters. T he
id eal m a g n itu d e re sp o n s e ch a ra c te ristic s o f th ese ty p e s o f filters are illu strated
in Fig. 4.43. A s sh o w n , th ese ideal filters h ave a c o n sta n t-g a in (usually ta k e n as
u n itv -g a in ) p a ssb a n d c h a ra c te ristic an d z e ro gain in th e ir sto p b a n d .
Lowpass
IHu»)\
I
Hiphpa*
\HUo)I
Bandpass
-n
—
—a>0 — ai]
0
cu, iii(|Wi
n
t
Bandstop
~o>„
0
All-pass
Figure 4.43 Magnitude responses
for some ideal frequency-selective
discrete-time filters.
332
Frequency Analysis of Signals and Systems
Chap. 4
A n o th e r ch a ra c te ristic o f an id eal filter is a lin e a r p h ase re sp o n s e . T o dem on­
stra te this p o in t, let us assum e th a t a signal se q u e n c e (■*(/?)} w ith fre q u e n c y com­
p o n e n ts co n fin ed to th e fre q u e n c y ran g e coj < w < un is p a s se d th ro u g h a filter
w ith freq u e n cy resp o n se
H(co) = J
[ (J,
w h ere C an d
JWnv'
< co<c^
o th erw ise
(4_5
a re c o n stan ts. T h e signal at th e o u tp u t o f th e filte r h as a spectrum
Y( co) = X(co)H(co)
u>\ < co < co2
- CX(co)e }tL""’
(4.5.2)
By app ly in g th e scaling a n d tim e-sh iftin g p ro p e rtie s o f the F o u rie r tran sfo rm , we
o b ta in th e tim e -d o m a in o u tp u t
v(ri) = Cx ( n - h 0)
(4.5.3)
C o n se q u e n tly , th e filter o u tp u t is sim ply a d e la y e d an d a m p litu d e -s c a le d v ersio n of
th e in p u t signal. A p u re d elay is usually to le ra b le a n d is n o t c o n s id e re d a distortion
o f th e signal. N e ith e r is a m p litu d e scaling. T h e re fo re , ideal filters have a linear
p h ase c h a ra c te ristic w ithin th e ir p a ssb a n d , th a t is.
0(a>) = —cori(i
(4.5.4)
T h e d eriv ativ e o f th e p h a se w ith re sp e c t to fre q u e n c y has th e u n its o f delay.
H e n c e we can defin e th e signal d elay as a fu n ctio n o f fre q u e n c y as
1 ( o j) =
d&(a>)
--------- ----------
aco
.
(4.5.5)
Zg(co) is usually called th e e nvel ope del ay o r th e g r o u p del ay o f th e filter. W e
in te r p re t zg(co) as th e tim e d elay th a t a signal c o m p o n e n t o f fre q u e n c y co u n d erg o es
as it p asses fro m th e in p u t to th e o u tp u t o f the system . N o te th a t w h en 0(co) is
lin e a r as in (4.5.4), r s (a)) — no = c o n s ta n t. In th is case all fre q u e n c y co m p o n en ts
o f th e in p u t signal u n d e rg o th e sam e tim e delay.
In co n clu sio n , id eal filters h av e a c o n s ta n t m a g n itu d e c h a ra c te ristic an d a
lin ear p h ase c h a ra c te ristic w ith in th e ir p assb an d . In all cases, such filters a re not
p h ysically re a liz a b le b u t se rv e as a m a th e m a tic a l id e a liz a tio n o f p ra c tic a l filters.
F o r ex am p le, th e id eal low pass filter h a s an im p u lse re sp o n se
sin coc7Tn
hip(n) — -----------nn
— 00 < /7 < 00
(4.5.6)
W e n o te th a t th is filter is n o t causal an d it is n o t a b so lu te ly su m m a b le an d th e re fo re
it is also u n sta b le . C o n se q u e n tly , this id eal filter is physically u n re a liz a b le . N ev­
e rth e le ss, its fre q u e n c y re sp o n se c h a racteristics can be a p p ro x im a te d v ery closely
by p ractical, ph y sically re a liz a b le filters, as will be d e m o n s tra te d in C h a p te r 8.
In th e fo llo w in g d iscu ssio n , w e tr e a t th e design o f so m e sim p le d igital filters
by th e p la c e m e n t o f p o les a n d zero s in th e z-p lan e. W e h a v e a lre a d y d escrib ed
h o w th e lo catio n o f p o les a n d zero s affects th e fre q u e n c y re s p o n s e characteristics
Sec. 4.5
Linear Tim e-Invariant Systems as Frequency-Selective Filters
333
o f th e system . In p a rtic u la r, in S ection 4.4.6 w e p re s e n te d a g ra p h ic a l m e th o d for
c o m p u tin g th e fre q u e n c y re sp o n se c h a racteristics fro m th e p o le - z e r o p lo t. T his
sam e a p p ro a c h can be u sed to design a n u m b e r o f sim ple b u t im p o rta n t digital
filters w ith d e s ira b le freq u e n c y re sp o n se ch aracteristics.
T h e basic p rin cip le u n d e rly in g th e p o le - z e r o p la c e m e n t m e th o d is to lo cate
p o les n e a r p o in ts o f th e u n it circle c o rre sp o n d in g to fre q u e n c ie s to be em p h asized ,
a n d to p lace ze ro s n e a r th e fre q u e n c ie s to be d e e m p h a siz e d . F u rth e rm o re , the
fo llo w in g c o n s tra in ts m ust be im posed:
1. A ll p o le s sh o u ld be p laced inside th e unit circle in o r d e r fo r the filter to be
stab le. H o w e v e r, ze ro s can be p laced an y w h ere in th e z-p lan e.
2. A ll co m p lex zero s an d p o les m ust o ccu r in c o m p le x -c o n ju g a te p airs in o rd e r
fo r th e filter coefficients to be real.
F ro m o u r p re v io u s discussion w e recall th a t fo r a given p o le - z e r o p a tte rn ,
th e sy stem fu n ctio n H i : ) can be ex p ressed as
(4.5.7)
w h ere bi} is a jiain c o n sta n t selected to n o rm aliz e th e freq u e n cy re sp o n se at som e
sp ecified freq u e n cy . T h a t is, bo is se le c te d such th at
| H ( cl>o) | = 1
(4.5.8)
w h ere
is a fre q u e n c y in th e p a ssb a n d o f the filter. U sually, N is se le c te d to
eq u al o r ex cee d M , so th a t th e filter h as m o re n o n triv ial p o le s th a n zeros.
In th e n ex t se ctio n , we illu strate th e m e th o d o f p o le - z e r o p la c e m e n t in the
d esig n o f so m e sim p le low pass. highpass. an d b an d p a ss filters, d igital re so n a to rs,
a n d co m b filters. T h e d esign p ro c e d u re is facilita ted w h en c a rrie d o u t in te ra c tiv e ly
on a d ig ital c o m p u te r w ith a g raphics term in a l.
4.5.2 Lowpass, Highpass, and Bandpass Filters
In th e d esig n o f lo w pass digital filters, th e p oles sh o u ld be p la c e d n e a r th e unit
circle at p o in ts c o rre sp o n d in g to low fre q u e n c ie s (n e a r cm = 0) and ze ro s sh ould
b e p la c e d n e a r o r on th e u n it circle at p o in ts c o rre sp o n d in g to high freq u e n cies
( n e a r co = t t ) . T h e o p p o site h o ld s tru e fo r h ighpass filters.
F ig u re 4.44 illu strates th e p o le - z e r o p la c e m e n t o f th re e low pass a n d th re e
h ig h p ass filters. T h e m a g n itu d e an d p h a se re sp o n se s fo r th e sin g le-p o /e filter w ith
sy stem fu n ctio n
334
Frequency Analysis of Signals and System s
Chap. 4
Highpass
Figure 4.44
Pole-zero patterns for several lowpass and highpass fillers.
are illu stra te d in Fig. 4.45 fo r a = 0.9. T h e gain G w as se le c te d a s 1 — a, so th at
th e filter h as u n ity gain a t co = 0. T h e gain of this filter a t h ig h fre q u e n c ie s is
relativ ely sm all.
T h e a d d itio n o f a ze ro a t z = —1 f u rth e r a tte n u a te s th e re s p o n s e o f th e filter
a t high freq u e n cies. T h is lead s to a filter w ith a sy stem fu n ctio n
H 2 (z)
= 1- ~ ~ ~ ~l~~' .
2 1 — az~
(4.5.10)
an d a freq u e n cy re sp o n se c h a ra c te rstic th a t is also illu stra te d in Fig. 4.45. In this
case th e m ag n itu d e o f H 2 (co) g o es to z e ro a t co = n.
S im ilarly, w e can o b ta in sim p le h ig h p ass filters by reflec tin g (fo ld in g ) the
p o le -z e ro lo catio n s o f th e low pass filters a b o u t th e im ag in ary axis in th e z-plane.
T h u s w e o b tain th e sy stem fu n ctio n
H i ( z ) = -l ~ a
\
2 1 -f a z ~ [
(4.5.11)
w hich has th e freq u e n cy re sp o n s e c h a ra c te ristic s illu stra te d in Fig. 4.46 fo r a = 0.9.
Example 4.5.1
A two-pole lowpass filter has the system function
W(z) =
b0
(1 -
pz -')2
Sec. 4.5
Linear Tim e-invariant Systems as Frequency-Selective Filters
335
Figure 4.45 Magnitude and phase
response of (1) a singie-pole filter
and (2) a one-pole, one-zero
filter; Wi(;) = (1 - a ) /( 1 — a ; ' 1),
H j(z) = [(1 —a )/2 ][(l + r ~ * )/(l - a ; - 1 )]
and a = 0.9.
D eterm ine the values of h{>and p such that the frequency response H(w) satisfies the
conditions
H(0) = 1
and
I / K \ I2
1
r W I =2
Solution
A t o) = 0 we have
H( 0) =
(1 - p )2
Hence
bo = (1 - p Y
= 1
336
Chap. 4
20 log|0]HM|
Frequency Analysis of Signals and Systems
Figure 4.46 Magnitude and phase
response of a simple highpass fitter;
H(C) = [(1 - a )/2 ][(l - z - ' ) / ( l + a z - 1)]
with a = 0.9.
At w = tt/4.
(1 ) =
(1 ~ P?
v4 /
(1 —pe~i*/4)2
(1 ~ P )2
(1 - p cos(;r/4) + j p sin (tt/4))2
(1 - P ) 2
(1 - pj s / 2 + jp/ ^/ 2)2
Hence
(1 - p t
[(1 - p / - j2 ) 2 + p 212]2
1
2
Sec. 4.5
Unear Tim e-Invariant Systems as Frequency-Selective Filters
337
or, equivalently.
V ^d - p y = 1 + p1 - y'lp
The value of p = 0.32 satisfies this equation. Consequently, the system function for
the desired filter is
0.46
H( z)
(1 - 0.32c 1)=
T h e sam e p rin cip le s can be a p p lied fo r th e desig n of b a n d p a s s filters. B asi­
cally, th e b a n d p a s s filter sh o u ld c o n tain o n e o r m o re p airs o f co m p lex -co n ju g ate
p o les n e a r th e u n it circle, in the vicinity of th e freq u e n cy b a n d th a t c o n stitu te s the
p assb a n d o f th e filter. T h e follow ing ex am p le serves to illu strate th e basic ideas.
E x a m p le 4.5.2
Design a two-pole bandpass filler that has the center of its passband at w = n72.
zero in its frequency response characteristic at w = 0 and at = n . and its magnitude
response is 1 /V 2 at w = 4w /9 .
S o lu tio n
Clearly, the filter must have poles at
Pl : = r i b ­
and zeros at
— 1 and : = —1. Consequently, the system function is
/z o = a
(: - 1)<: + 1)
(: - j r)(: + j r)
= c-
- 1
The gain factor is determ ined by evaluating the frequency response H(w) of the filler
at u> = jr/2. Thus we have
" ( § ) - cr b - >
G _
1 - r2
The value of r is determ ined by evaluating H( w) at w = 4tt/9. Thus we have
4tt \
(1 - r1)2
9 /|
4
2 - 2 c o s ( 8 7 t/9 )
2r2 c o s ( 8 jt/9 )
1 4- r4 +
1
2
or. equivalently,
1.94(1
- r 1)1 = 1 - 1 .8 8 r2 + r 4
The value of r2 = 0.7 satisfies this equation. Therefore, the system function for the
desired filter is
"<:l = a,5IT5&
Its frequency response is illustrated in Fig. 4.47.
338
Frequency Analysis of Signals and Systems
_ T
x
~2
q
r
2
t
Chap. 4
Figure 4.47 Magnitude and
phase response of a simple
bandpass filter in Example 4.5.2;
H(z) = 0.15[(1 - z ~ 2) / a + 0 . 7 ; - 2)].
It sh o u ld b e e m p h a siz e d th a t th e m ain p u rp o se of th e fo re g o in g m eth o d o lo g y
fo r d esig n in g sim p le digital filters b y p o le - z e r o p la c e m e n t is to p ro v id e insight
in to th e effect th a t p o les a n d z e ro s have on th e fre q u e n c y re sp o n s e characteristic
o f system s. T h e m e th o d o lo g y is n o t in te n d e d as a goo d m e th o d fo r designing
d igital filters w ith w ell-specified p a s sb a n d an d sto p b a n d ch aracteristics. System atic
m e th o d s fo r th e d esig n of so p h is tic a te d d ig ital filters fo r p ra c tic a l ap p lic a tio n s are
d iscussed in C h a p te r 8.
A simple lowpass-to-highpass filter transformation. S u p p o se th a t we
h av e d esig n ed a p r o to ty p e low pass filter w ith im p u lse re sp o n se h\p(n). B y us-
Sec. 4.5
Linear Tim e-Invariant Systems as Frequency-Selective Filters
339
ing th e fre q u e n c y tra n sla tio n p ro p e rty o f th e F o u rie r tra n sfo rm , it is p o ssib le to
co n v ert th e p ro to ty p e filter to e ith e r a b a n d p a ss or a h ig h p ass filter. F re q u en cy
tra n sfo rm a tio n s fo r c o n v ertin g a p ro to ty p e low pass filter in to a filter of a n o th e r
ty p e are d e sc rib e d in d etail in S ection 8.3. In th is sectio n w e p re se n t a sim plefre q u e n c y tra n sfo rm a tio n fo r co n v e rtin g a low pass filter in to a h ig h p ass filter, and
vice v ersa.
If /i|p(n) d e n o te s th e im p u lse resp o n se of a low pass filter w ith freq u e n cy
re sp o n se H\r (co). a h ighpass filter can be o b ta in e d by tra n sla tin g H]P(a>) by t t rad ian s
(i.e., rep lacin g co by co — n ) . T h u s
^ h p ( ^ )
—
H\p(co
TT )
(4.5.12)
w h e re H hP(w) is th e fre q u e n c y re sp o n se o f th e h ig h p ass filter. Since a freq u e n cy
tra n sla tio n o f t t ra d ia n s is e q u iv a le n t to m u ltip lic atio n o f th e im pulse resp o n se
/iip(;i) by e 177", th e im p u lse resp o n se o f th e highpass filter is
/ihp(n) — (p 7't )''/7|p (/7) = ( —1 )"/ ii p(« )
(4.5.13)
T h e re fo re , th e im p u lse resp o n se of th e h ig h p ass filter is sim ply o b ta in e d from the
im pulse re sp o n se o f the low pass filter by ch an g in g th e signs o f th e o d d -n u m b e re d
sa m p le s in inp(n). C o n v ersely .
= ( - 1 )"/ihp(«)
(4.5.14)
If th e lo w pass filter is d e sc rib e d by th e d ifferen ce e q u a tio n
(4.5.15)
its fre q u e n c y re sp o n s e is
M
k—0
(4.5.16)
N ow , if w e re p la c e co by co — n , in (4.5.16). th en
M
tfhpM =
--------------
(4.5.17)
1 + £ ( - 1 )kake ~ iak
w hich c o rre sp o n d s to th e d iffe re n c e e q u a tio n
( 4 . 5 . 18 )
Frequency Analysis of Signals and Systems
340
Chap. 4
Example 4.5.3
Convert the lowpass filter described by the difference equation
v(n) = 0.9v(n — 1) + 0 .1 x 0 )
into a highpass filter.
Solution
The difference equation for the highpass filter, according to (4.5.18), is
y(n) = -0.9v(n - 1) + 0.1x(n)
and its frequency response is
0.1
- 1+ Q9e_ju
The reader may verify that H^((o) is indeed highpass.
4.5.3 Digital Resonators
A digital resonat or is a sp ecial tw o-pole b a n d p a ss filter w ith th e p a ir o f com plexc o n ju g ate p o les lo c a te d n e a r th e u n it circle as show n in Fig. 4 .4 8 (a). T h e m ag n itu d e
of th e fre q u e n c y re sp o n se of th e filter is show n in Fig. 4 .48(b). T h e n am e re so n a to r
re fe rs to th e fact th a t th e filter has a large m a g n itu d e re sp o n se (i.e., it re so n a te s ) in
th e v icinity o f th e p o le lo catio n . T h e a n g u lar p o sitio n of th e p o le d e te rm in e s th e
re so n a n t freq u e n c y o f th e filter. D igital re so n a to rs are useful in m a n y applications,
in clu d in g sim p le b a n d p a ss filterin g an d sp e ech g e n e ra tio n .
In th e d esig n o f a digital r e s o n a to r w ith a re so n a n t p e a k at o r n e a r to — o>o,
w e se lect th e c o m p lex -co n ju g ate p o les at
Pi .2 = r e ±ja*
0 < r < 1
In a d d itio n , we can select u p to tw o zeros. A lth o u g h th e re a re m an y possible
choices, tw o cases a re o f special in terest. O n e choice is to lo cate th e z ero s a t the
origin. T h e o th e r ch o ice is to locate a z e ro a t z = 1 an d a z e ro a t z = —1. T his
choice co m p letely e lim in a te s th e re sp o n s e o f th e filter a t fre q u e n c ie s co = 0 and
co = n , an d it is u se fu l in m an y p ractical a p p licatio n s.
T h e sy stem fu n ctio n o f th e digital r e s o n a to r w ith ze ro s a t th e origin is
H ( z ) = ----------- :------ r ~ -----------:-------r
(1 — r e ^ z
)(1 — re -'“ "z- 1 )
(4.5.19)
H ( z ) = ------- ----------(4.5.20)
1 — (2 r coscwo);-1 + r 2z ~2
S ince \H(co)\ h as its p e a k at o r n e a r co = coo, w e select th e gain bo so th a t
|W(too)I = 1- F ro m (4.5.19) w e o b ta in
b0
H{(Oo) =
---------:------- ----- ^-----------:------- :---(1 - r e j<*,e -J°*>)0. - r e - w e - w )
_______________
(1 — r ) ( l — r e - -'2'00)
an d h e n c e
bo
\H(coo)\ = ------;---I — •- === = 1
(1 — r)V 1 + r 2 — 2 r cos2a)o
(4 5 21)
Sec. 4.5
Linear Tim e-Invariant Systems as Frequency-Selective Filters
2
341
2
(b)
—tt
_ n
0
2
£r
2
(c)
Tt
Figure 4.48 (a) Pole-zero pattern and
(b) the corresponding magnitude and
phase response of a digital resonator
with (1) r — 0.8 and (2) r = 0.95.
342
Frequency Analysis of Signals and System s
Chap. 4
T h u s th e d esired n o rm a liz a tio n fa c to r is
i?o = (1 — r ) > /l + r 2 — 2r cos2a>o
(4.5.22)
T h e freq u e n cy re sp o n se o f th e r e s o n a to r in (4.5.19) can b e e x p re ss e d as
bo
\H(a>)\ = ----------------Ux(to)U2 ((o)
(4.5.23)
© (w ) = 2 to — <J>i (tfj) — <t>2 (co)
w h ere U\ (w) a n d 1/ 2 (01) are th e m a g n itu d e s of th e v e c to rs fro m p\ an d p 2 to the
p o in t w in th e u n it circle a n d <t>i(<w) an d <J>2 (£d) a re th e c o rre sp o n d in g angles of
th e se tw o v ecto rs. T h e m a g n itu d e s Ui(to) a n d U2 (co) m ay be e x p re ss e d as
Ui(co) = J \ + r 2 — 2 r cos(a>o — to)
(4.5.24)
U 2 (<o) = y / 1 + r 2 — 2 r c o s (w q + co)
F o r any v alu e o f r , U\(to) ta k e s its m in im u m v alu e (1 — r ) a t to = too- T he
p ro d u c t U\((o)U2 ({o) re a c h e s a m in im u m v alu e at th e fre q u e n c y
air = co s-1
coswo^j
(4.5.25)
w hich defin es p recisely th e r e s o n a n t fre q u e n c y o f th e filter. W e o b se rv e th a t w hen
r is very close to u n ity , tor
a>o, w hich is th e a n g u la r p o sitio n o f th e p o le. W e also
o b se rv e th a t as r a p p ro a c h e s unity, th e re so n a n c e p e a k b ec o m e s s h a rp e r becau se
U\(to) ch an g es m o re rap id ly in relativ e size in th e vicinity o f ojo- A q u a n tita tiv e
m e a su re o f th e sh a rp n e s s o f th e re so n a n c e is p ro v id e d by th e 3-dB b a n d w id th A w
o f th e filter. F o r v alu es o f r close to u nity.
Aw % 2(1 - r )
(4.5.26)
F ig u re 4.48 illu stra te s th e m a g n itu d e a n d p h a s e o f d igital re s o n a to rs w ith
coq = n / 3 , r = 0.8 an d too ~ *73, r = 0.95. W e n o te th a t th e p h a se resp o n se
u n d erg o es its g re a te st ra te o f c h a n g e n e a r th e re s o n a n t freq u e n cy .
If th e zero s o f th e digital r e s o n a to r a re p laced a t z = 1 an d z = — 1, th e
re s o n a to r has th e sy stem fu n ctio n
H(z) = G
= G
(1 - z - ' X l + z " 1)
(1 — re>aK,z ~ l )( 1 — r e - -'a*»2 -1 )
(4.5.27)
1 -z -2
1 — (2r coscdo)z-1 + r 2z ~2
an d a fre q u e n c y re sp o n s e c h a ra c te ristic
H{ w) = b° [ \ _
(«*-->][! _ r e - n » o +*>)J
(4.5.28)
W e o b se rv e th a t th e z e ro s a t z = ± 1 affe ct b o th th e m a g n itu d e a n d p h a se resp o n se
o f th e re so n a to r. F o r ex a m p le , th e m a g n itu d e re sp o n s e is
\H(to)\=b0
U\(to)U2(to)
( 4 -5 ’2 9 )
Sec. 4.5
Linear Tim e-Invariant Systems as Frequency-Selective Filters
343
Figure 4.49 Magnitude and phase
response of digital resonator with zeros
at 10 = 0 and
- t and ( I ) r = O.N and
(2) r = 0.95.
to=
w h ere N { oj) is d efin ed as
N ( w) — 7 2 (1 - cos2a>)
D u e to th e p re se n c e o f th e zero fa c to r, th e re s o n a n t fre q u e n c y is a lte re d from
th at given by th e ex p ressio n in (4.5.25). T h e b a n d w id th of th e filter is also a lte re d .
A lth o u g h ex act v alu es for th e s e tw o p a ra m e te rs are ra th e r te d io u s to d eriv e, we
can easily c o m p u te th e fre q u e n c y re sp o n se in (4.5.28) an d c o m p a re th e re su lt with
th e p re v io u s case in w hich th e zeros a re lo c a te d at th e origin.
F ig u re 4.49 illu strates th e m a g n itu d e an d p h a s e c h a ra c te ristic s fo r o>q — n / 3 .
r = 0.8 an d a>o — tt/ 3, r = 0.95. W e o b se rv e th a t th is filter h a s a slightly sm a ller
b a n d w id th th a n th e re s o n a to r, w hich h as zero s a t th e origin. In a d d itio n , th e re
a p p e a rs to be a v ery sm all shift in th e re so n a n t fre q u e n c y d u e to th e p re se n c e of
th e zero s.
4.5.4 Notch Filters
A n o tc h filter is a filter th a t c o n ta in s o n e o r m o re d e e p n o tc h e s o r, ideally, p e rfe c t
nulls in its fre q u e n c y re sp o n se ch a ra c te ristic . F ig u re 4.50 illu stra te s th e freq u e n cy
re sp o n se c h a ra c te ristic o f a n o tc h filter w ith nulls a t fre q u e n c ie s cl>o an d
. N otch
filters are useful in m an y a p p lic a tio n s w h e re specific fre q u e n c y c o m p o n e n ts m ust
be elim in a te d . F o r ex am p le, in stru m e n ta tio n a n d re c o rd in g sy stem s re q u ire th a t
th e p o w er-lin e fre q u e n c y o f 60 H z a n d its h a rm o n ic s b e e lim in ated .
344
Frequency Analysis of Signals and System s
Chap. 4
Figure 4,50 Frequency response
characteristic of a notch filter.
T o c re a te a null in th e fre q u e n c y re sp o n se o f a filter at a fre q u e n c y ojo, we
sim ply in tro d u c e a p a ir of c o m p lex -co n ju g ate ze ro s on th e u n it circle at an angle
too. T h a t is,
S,.2 = f ± j“°
T h u s th e system fu n ctio n fo r an F IR n o tch filter is sim ply
H( z ) = boQ - eJlo" z ~] Ml (4.5.30)
— bo (1 — 2 C O S tr>()c '
z ~)
A s an illu stratio n . Fig. 4.51 show s th e m a g n itu d e re sp o n se fo r a n o tc h filter having
a null a t a> = tt/4 .
T h e p ro b lem w ith th e F IR notch filter is th a t th e n o tch has a relativ ely large
b an d w id th , w hich m ean s th a t o th e r fre q u e n c y c o m p o n e n ts a ro u n d th e d esired null
are se v ere ly a tte n u a te d . T o re d u c e th e b an d w id th o f th e null, w e can re so rt to
a m o re so p h isticated , lo n g er F IR filter d esig n ed acco rd in g to c rite ria describ ed
in C h a p te r 8. A lte rn a tiv e ly , w e could, in an ad h o c m a n n e r, a tte m p t to im prove
on th e freq u e n cy re sp o n se ch a ra c te ristic s by in tro d u c in g p oies in th e system func­
tion.
S u p p o se th a t w e p lace a p a ir o f c o m p lex -co n ju g ate p o les at
Ph2 = r e ±i m
T h e effect o f th e p o les is to in tro d u c e a re so n a n c e in th e vicinity o f th e null and
th u s to red u ce th e b a n d w id th o f th e n o tch . T h e sy stem fu n ctio n fo r th e resulting
filter is
1 - 2 cos wo; 1 + z~
H ( c ) = bo1 - 2 r cosojoZ-1 + r 2z ~2
(4.5.31)
T h e m ag n itu d e re sp o n se \H(u>)\ o f th e filter in (4.5.31) is p lo tte d in Fig. 4.52 for
coo = t t /4 , r = 0.85, an d fo r ti>o = n / 4 , r = 0.95. W h en c o m p a re d w ith the
freq u e n cy resp o n se o f th e F IR filter in Fig. 4.51, w e n o te th a t th e effect o f the
p o les is to red u ce th e b an d w id th o f th e no tch .
In a d d itio n to red u cin g th e b a n d w id th o f th e n o tc h , th e in tro d u c tio n o f a
p o le in th e vicinity o f th e null m ay re su lt in a sm all rip p le in th e p a s sb a n d o f th e
filter d u e to th e re so n a n c e c re a te d by th e pole. T h e effe ct o f th e rip p le can be
re d u c e d by in tro d u c in g a d d itio n a l p o le s a n d /o r ze ro s in th e sy ste m fu n c tio n o f the
n o tch filter. T h e m a jo r p ro b le m w ith th is a p p ro a c h is th a t it is b a sic a lly an ad hoc,
tria l-a n d -e rro r m e th o d .
Sec. 4.5
Linear Tim e-Invariant Systems as Frequency-Selective Filters
345
Figure 4.51 Frequency response
characteristics of a notch filter with
a notch at co = n / 4 or / = 1/8;
H(z) = G[1 - 2cos£ooz-1 + z-2 ]-
4.5.5 Comb Filters
In its sim p lest fo rm , a c o m b filter can be v iew ed as a n o tc h filter in w h ich th e
n u lls o c c u r p e rio d ic a lly ac ro ss th e fre q u e n c y b a n d , h e n ce th e an a lo g y to an o rd i­
n a ry c o m b th a t h as p erio d ically sp a ced te e th . C o m b filters find ap p lic a tio n s in a
w ide ra n g e o f p ra c tic a l sy stem s such as in th e re je c tio n o f p o w e r-lin e h arm o n ics,
in th e se p a ra tio n o f so la r a n d lu n a r c o m p o n e n ts fro m io n o sp h e ric m e a s u re m e n ts
o f e le c tro n c o n c e n tra tio n , a n d in th e su p p re ssio n o f c lu tte r fro m fixed o b je c ts in
m o v in g -ta rg e t-in d ic a to r (M T I) ra d a rs.
346
Frequency Analysis of Signals and Systems
Chap. 4
Figure 4.52
Frequency response
characteristics of two notch filters with
poles at ( I ) r = 0,85 and 12) r — (1.95:
H(z) = />n|(l - - cos to u r'1
2r cosgju:-1 + r2z ~~)].
r " ")/(1 -
T o illu strate a sim ple form of a co m b filler, c o n sid er a m oving a v erag e (F IR )
filter d escrib ed by th e d ifferen ce e q u a tio n
xin - k )
v(«) = —
(4.5.32
T h e system fu n ctio n o f this F IR filter is
Hiz
+ * Jt=0
1
[ l - c - * ^ 11]
M + 1
( 1 - - - ’)
(4.5.33)
an d its freq u e n cy re sp o n se is
S]na)( « j d )
H(a>) =
(4.5.34)
M + 1
sin(w /2)
F ro m (4.5.33) we o b se rv e th a t th e filter has ze ro s on th e u n it circle at
k = 1 , 2 . 3 .........M
(4.5.35)
N o te th a t th e p o le a t ; = 1 is actu ally can ce le d by th e zero at ; = 1, so th a t in
effect th e F IR filter d o e s n ot c o n tain p o le s o u tsid e z = 0.
A p lo t o f th e m ag n itu d e c h a ra c te ristic o f (4.5.34) clearly illu stra te s th e ex­
isten ce o f th e p erio d ically sp a c e d zero s in fre q u e n c y at cot = 2 n k / ( M + 1) for
£ = 1 , 2 , -----M. F ig u re 4,53 sh o w s \ H( w) \ fo r M = 10.
Sec. 4.5
Linear Tim e-Invariant Systems as Frequency-Selective Filters
347
Figure 4.53 Magnitude response
characteristic for the comb filter given
by (5,4.32) with M = 10.
In m o re g e n eral te rm s, w e can c re a te a com b filter by ta k in g an F I R filter
w ith sy stem fu n ctio n
M
H( z ) = 2 2 h ( k ) z ~ k
(4.5.36)
k=0
an d rep la c in g z by z L, w h ere L is a po sitiv e in teg er. T h u s th e new F IR filter has
a sy stem fu n ctio n
H l (z ) = Y , h ( k ) z
(4.5.37)
If th e fre q u e n c y re sp o n se of th e original F IR filter is //(w ), th e freq u e n c y resp o n se
o f th e F IR in (4.5.37) is
M
H l (u>) = 2 2 h ( k ) e ~ jkLa
(4.5.38)
k=I)
= H(Lco)
C o n s e q u e n tly , th e fre q u e n c y re sp o n se c h a ra c te ristic H l {oj) is sim ply an L -o rd e r
re p e titio n o f H(co) in th e ra n g e 0 < co < 2 n . F ig u re 4.54 illu stra te s th e re la tio n sh ip
b e tw e e n H l ( c o) a n d H (w) fo r L — 5.
N o w , su p p o se th a t th e o rig in al F I R filter w ith system fu n ctio n H ( z ) h as a
sp e c tra l n ull (i-e -, a z e ro ), at so m e fre q u e n c y coo. T h e n th e filter w ith system
fu n ctio n H L(z) h a s p erio d ically sp a ced nulls a t av = coo + I n k j L , k = 0, 1 , 2 , . . . ,
L — 1. A s an illu stra tio n , F ig. 4.55 show s an F IR co m b filter w ith Af = 3 an d
L = 3. T h is F IR filter can b e v iew ed as an F IR filter o f le n g th 10, b u t only fo u r
o f th e 10 filter coefficients are n o n zero .
L e t us no w r e tu rn to th e m oving a v erag e filter w ith system fu n ctio n given by
(4.5.33). S u p p o se th a t we re p la c e z by z L. T h e n th e re su ltin g co m b filter h as th e
sy stem fu n ctio n
1
1_
1)
Hl(z) = T 7~~r
/
(4-5.39)
M + 1
l-z~ L
and a frequency response
H l ( co) =
1
sin[a>L(M + l ) /2 ]
M + 1
sin(coL/2)
(4.5.40)
348
Frequency Analysis of Signals and Systems
Chap. 4
H (u > )
(a)
Hl (co)
5
5
5
5
(b)
Figure 4.54
Comb fiJter with frequency response W/Joi) obtained from H(a>).
Figure 4.55
Realization of an FIR comb filter having M = 3 and L ~ 3.
T his filter has zero s on th e u n it circle at
Zk = e j 2* k / L l M+ h
(4.5.41)
fo r alt in te g e r v alu es o f k e x cep t k — 0, L, 2 L .........M L . F ig u re 4.56 illu strates
\ H l {(d )\ fo r L = 5 an d M = 10.
T h e co m b filter d esc rib e d by (4.5.39) finds a p p lic a tio n in th e s e p a ra tio n of
so la r a n d lu n a r sp e c tra l c o m p o n e n ts in io n o sp h e ric m e a s u re m e n ts o f e le c tro n c o n ­
c e n tra tio n as d e sc rib e d in th e p a p e r by B e rn h a rd t e t al. (1976). T h e so la r p e rio d
is Ts = 24 h o u rs an d resu lts in a so la r c o m p o n e n t o f o n e cycle p e r d ay a n d its
h arm o n ics. T h e lu n a r p e rio d is Tl = 24.84 h o u rs a n d p ro v id e s sp e c tra l lin es at
0.96618 cycle p e r d ay an d its h arm o n ics. F ig u re 4.57a show s a p lo t o f th e p o w e r
d en sity sp e c tru m o f th e u n filtered io n o sp h e ric m e a s u re m e n ts o f th e e le c tro n con-
Sec. 4.5
Linear Time-Invariant Systems as Frequency-Selective Filters
349
Figure 4.56 Magnitude response
characteristic for a comb filter given by
(4.5.40). with L — 3 anti M = II).
c e n tra tio n . N o te th a t th e w eak lu n a r sp e ctral c o m p o n e n ts are alm o st h id d en by
th e stro n g so la r sp e c tra l co m p o n en ts.
T h e tw o se ts o f sp e c tra l c o m p o n e n ts can be se p a ra te d by th e use o f com b
filters. If w e w ish to o b ta in th e so lar co m p o n e n ts, we can use a co m b filter w ith
a n a rro w p a s sb a n d a t m u ltip le s of o n e cycle p e r day. T h is can be ach iev ed by
selectin g L such th a t Fs/ L = 1 cycle p e r day. w h ere Fs is th e c o rre sp o n d in g
sa m p lin g freq u e n cv . T h e resu lt is a filter th a t has p e a k s in its freq u e n cy resp o n se
at m u ltip le s o f o n e cycle p e r day. By se lectin g M = 58. the filter will h ave nulls
at m u ltip le s o f ( F J L ) I ( M + 1) = 1/59 cycle p e r day. T h e se nulls a re very close
to th e lu n a r c o m p o n e n ts a n d result in good rejectio n . F ig u re 4.57(b) illu strates
Freq ue ncy (cycles/day)
<c)
Figure 4.57 (a) Spectrum of unfiltered electron content data; (b) spectrum of out­
put of solar filter; (c) spectrum of output of lunar filter. [From paper by Bernhardt
et al. (1976). Reprinted with permission of the American Geophysical Union.]
Frequency Analysis of Signals and Systems
350
Chap. 4
the p ow er spectral density of the output o f the com b filter that isolates the solar
com ponents. A com b filter that rejects the solar com p onents and passes the lunar
com ponents can be d esigned in a sim ilar m anner. Figure 4.57(c) illustrates the
pow er spectral density at the output o f such a lunar filter.
4.5.6 All-Pass Filters
An all-pass filter is defined as a system that has a constant m agnitude resp onse for
all frequencies, that is.
|ff{oj)l = l
0<o><7r
(4.5.42)
The sim plest exam ple o f an all-pass filter is a pure delay system with system func­
tion
H(z) = z~k
This system passes all signals w ithout m odification except for a d elay of k sam ples.
This is a trivial all-pass system that has a linear phase response characteristic.
A m ore interesting all-pass filter is described by the system function
a x + a /v '-i" ^ 1 + ■■■+ a \ Z
;
n(z) = -
1 + 0\Z
,.
=
' + • ■ ■ + Qn Z
N
n
(4.5.43)
.- N + k
----- IT
a° = 1
where all the filter coefficients \ak ) are real. If w e define the polyn om ial /\(~) as
A (;) = Y ^ a kz k
k=U
an — 1
then (4.5.43) can be expressed as
H(z) = z ~ n M ‘
]
(4.5.44)
A(z)
Since
\H(a>)\2 =
= 1
the system given by (4.5.44) is an all-pass system . Furtherm ore, if zo is a pole
o f H( z ) . then 1/zu is a zero o f H ( z ) (i.e., the p oles and zeros are reciprocals of
o n e another). Figure 4.58 illustrates typical p o le -z e r o patterns for a single-pole,
single-zero filter and a tw o-p ole, tw o-zero filter. A plot o f the p h ase characteristics
o f these filters is shown in Fig. 4.59 for a = 0.6 and r = 0.9, too = tt/4.
The m ost genera! form for the system function o f an all-pass system with real
coefficients, expressed in factored form in term s of p oles and zeros, is
Nr
H, f ^ - T T - _____
- 1J i -
n
Pk
l \ a - A z - ’ x i - P i z ~ x)
(4 5 45)
(4 ’5 '
where there are N R real p o les and zeros and N c com p lex-con ju gate pairs o f poles
and zeros. For causal and stable system s w e require that - 1 < a* < 1 and |/S*| < 1.
Sec. 4.5
Linear Time-Invariant Systems as Frequency-Selective Filters
351
(a)
Figure 4.58 Pole-zero patterns of (a) a
first-order and (b) a second-order
all-pass filter.
10
20 tog I H M I
0
-10
-20
-3 0
0(u)
-4 0
Figure 4.59 Frequency response
characteristics of an all-pass
filter with system functions
(1) H(z) = (0.6 + z - ') / ( l + 0 . 6 Z '1),
(2) H(z) = (r2 - 2tcoswqz~1 + Z ~ 2 ) /
(1 —2rcoscuoz-i + r2z~2), r = 0.9,
(jo — n/A.
352
Frequency Analysis of Signals and Systems
Chap. 4
E xp ression s for the phase response and group delay o f all-pass system s can
easily be obtained using the m ethod described in Section 4.4.6. For a single p o lesingle zero all-pass system w e have
Manioc) ‘—
Ju
H en ce
r sin(a) — 6}
C-)ap(w) = —cd — 2 tan'
— r COS(w - 6 )
1
and
di~).Ap(ct>)
1 - r2
ciaj
1 + i - — 2r cosiw — 0)
(4.5.46)
W e n ote that for a causal and stable system , r < 1 and hence rt, (w) > 0. Since the
group delay o f a higher-order p o le-zero system consists o f a sum o f positive terms
as in (4.5.46), the group delay will always be positive.
A ll-pass filters find application as phase equalizers. W hen placed in cascade
with a system that has an undesired phase response, a phase eq u alizer is designed
to com pensate for the poor phase characteristics of the system and therefore to
p roduce an overall linear-phase response.
4.5.7 Digital Sinusoidal Oscillators
A digital si nusoi dal oscillator can be view ed as a lim iting form o f a tw o-p ole res­
onator for which the com plex-conjugate p oles He on the unit circle. From our
previous discussion o f secon d-ord er system s, we recall that a system with system
function
H ( z ) = -----------------------1 + a j z " 1 + ai z
(4.5.47)
and param eters
a i = —2rcosa>o
and
a2 = r 2
(4.5.48)
has com plex-conjugate poles at p = r e ± m \ and a unit sam ple response
h(n) = —~ — sin(n + l)fDow(n)
sin cun
(4.5.49)
If the p oles are placed on the unit circle (r = 1) and bo is set to A sincuo, then
h(n) = A s i n ( n + l)u>ou(n)
(4.5.50)
Thus the im pulse response o f the second-order system with com plex-conjugate
poles on the unit circle is a sinusoid and the system is called a digital sinusoidal
oscillator or a digital si nusoi dal generator. A digital sinusoidal gen erator is a basic
com ponent o f a digital frequency synthesizer.
Sec. 4.5
Linear Time-Invariant Systems as Frequency-Selective Filters
353
T h e block diagram representation o f the system function given by (4.5.47) is
illustrated in Fig. 4.60. The corresponding difference equation for this system is
v ( n) = —a i y0? — 1) — y( n — 2) + boS(n)
(4.5.51)
w here the param eters are a\ = —2cosa>o and bo = Asincwo, and the initial con d i­
tions are y ( —1) = v (—2) = 0. By iterating the differen ce eq u ation in (4.5.51), we
obtain
y (0) = ^ sin w o
y ( l ) = 2cos<woy(0) - 2 A sin too cos tt>o = Asin2tL>o
y(2) — 2costL>oy(l) - y(0)
= 2 A cos a»o sin 2wo — A sin loq
- A {4 cos2 coq — 1) sin wq
— 3 A sin too ~ 4 sin 3 coq = A sin Ixoq
and so forth. W e n ote that the application of the im pulse at n = 0 serves the
purpose o f beginning the sinusoidal oscillation. T hereafter, th e oscillation is selfsustaining b ecau se the system has no dam ping (i.e., r = 1).
It is in terestin g to note that the sinusoidal oscillation ob tain ed from the sys­
tem in (4.5.51) can also be ob tain ed by setting the input to zero and setting the
initial co n d ition s to y ( - l ) = 0, v (—2) = -A sin a> o. Thus the zero-input response
to the secon d-ord er system described by the h om ogen eou s differen ce equation
y{n) = - a i y ( n - 1) - y( n - 2)
(4.5.52)
with initial con d ition s y ( - l ) — 0 and >■(—2) = —A sin two, is exactly the sam e as
the resp onse o f (4.5.51) to an im pulse excitation. In fact, the d ifferen ce equation
in (4.5.52) can b e ob tain ed directly from the trigonom etric identity
a +8
a - B
sin a + sin fi = 2 sin — - — cos — - —
(4.5.53)
w here, by definition, or = (n + !)&*>, 0 = (n — l)^ o , and y( n) = s i n( n + l)a>o-
354
Frequency Analysts of Signals and Systems
Chap. 4
In som e practical applications involving m odulation of tw o sinusoidal carrier
signals in phase quadrature, there is a need to generate the sinusoids A sin ojqn
and A cosaion. T hese signals can be gen erated from the so-called coupl ed- f orm
oscillator, which can be ob tain ed from the trigonom etric form ulas
co s(a + ft) = cos a cos ft — sin a sin ft
sin (a + ft) = sin a cos ft + cos a sin ft
where, by definition, a =
ft = u>o, and
yr (n) = COsnajoK(rc)
(4.5.54)
y , (n ) = sinna>t)U(fj)
(4.5.55)
Thus we obtain the tw o cou p led d ifference equations
)'(■(») = (COS a>o)}■<■(« - 1) - (s in a jo )y , (« - 1)
(4.5.56)
y.,(«) = (sin o>o)yt-(n - 1) + (cosw o)yr(n - 1)
(4.5.57)
which can also be expressed in matrix form as
yAn)
_vt(n) _
cos co{) sin too
sin £t?o cos lo{)
yt.(n - 1)
(4.5.58)
yAn ~
T h e structure for the realization of the coupled-form oscillator is illustrated in
Fig. 4.61. We note that this is a tw o-output system which is not driven by any input,
but which requires the initial con d ition s y<( —]) = /Icosaio and y.f ( —1) = -A sin a> o
in order to begin its self-sustaining oscillations.
Finally, it is interesting to note that (4.5.58) corresponds to vector rotation
in the tw o-dim ensional coordinate system with coordinates yc(n) and y.T(n). A s a
con sequ en ce, the coupled-form oscillator can also be im plem en ted by use o f the
so-called C O R D IC algorithm [see the b ook by K ung et al. (1985)].
Figure 4.61 Realization of the
coupled-form oscillator.
Sec. 4.6
inverse Systems and Deconvolution
355
4.6 INVERSE SYSTEMS AND DECONVOLUTION
A s we have seen , a linear tim e-invariant system takes an input signal v(/j) and
produces an output signal y (n ), which is the con volu tion of * (« ) with the unit
sam ple response h{n) of the system . In m any practical applications w e are given
an output signal from a system w hose characteristics are unknow n and we are
asked to determ ine the input signal. For exam p le, in the transm ission of digital
inform ation at high data rates over telep h on e channels, it is w ell known that the
channel distorts the signal and causes intersym bol interference am ong the data
sym bols. T h e intersym bol interference m ay cause errors when w e attem pt to re­
cover the data. In such a case the problem is to design a corrective system which,
w hen cascaded with the channel, produces an output that, in som e sense, corrects
for the distortion caused by the channel, and thus yields a replica of the desired
transm itted signal. In digital com m unications such a corrective system is called
an equalizer. In the general con text of linear system s theory, how ever, w e call
the corrective system an inverse s y s t e m , because the corrective system has a fre­
quency resp onse which is basically the reciprocal o f the frequency response o f
the system that caused the distortion. F urtherm ore, since the distortivc system
yields an output y i n ) that is the con volu tion of the input x ( n ) wiih the im pulse
response h(n). the inverse system operation that takes y(/i) and produces a ( / i ) is
called deconvol ut i on.
If the characteristics o f the distorlive system are unknow n, it is often nec­
essary, when p ossib le, to excite the system with a known signal, observe the
output, com pare it with the input, and in som e m anner, determ in e the charac­
teristics o f the system . For exam ple, in the digital com m unication problem just
described, w here the frequency response of the channel is unknow n, the m ea­
surem ent o f the channel frequency response can be accom p lish ed by transm itting
a set o f equal am plitude sinusoids, at different freq u en cies with a specified set
o f phases, within the frequency band o f the channel. The channel will atten ­
uate and phase shift each o f the sinusoids. By com paring the received signal
with the transm itted signal, the receiver obtains a m easu rem en t o f the channel
frequency resp onse which can be used to design the inverse system . The pro­
cess o f determ ining the characteristics of the unknow n system , either h( n) or
H(co), by a set o f m easurem ents perform ed on the system is called syst em identi­
fication.
T h e term “decon volu tion " is often used in seism ic signal p rocessing, and
m ore generally, in geophysics to describe the op eration of separating the input
signal from the characteristics o f the system which w e wish to m easure. T h e decon volu tion operation is actually in tend ed to identify the characteristics o f the
system , which in this case, is the earth, and can also be view ed as a system id en ­
tification problem . T he “inverse system ,” in this case, has a frequency response
that is the reciprocal o f the input signal spectrum that has b een used to excite the
system .
356
Frequency Analysis of Signals and Systems
Chap. 4
4.6.1 Invertibility of Linear Time-Invariant Systems
A system is said to be invertible if there is a o n e-to -o n e corresp on d en ce betw een
its input and output signals. This definition im plies that if w e know the output
sequ en ce y(n), —oc < n < oc, of an invertible system T , w e can un iqu eiv determ ine
its input A(n), —oc < n < oo. T he inverse syst em with input v(«) and output x{n)
is d en oted by T ~ l . Clearly, the cascade con n ection o f a system and its inverse is
equivalent to the identity system , since
w( n) = 7
1[>'(«)] = T ~ x [T[x{n)]) - x(n )
(4.6.1)
as illustrated in Fig. 4.62. For exam ple, the system s defined by the input-output
relations y{n) = ax ( n ) and y( n) = x ( n — 5) are invertible, w h ereas the input-output
relations y( n) = x 2(n) and y{n) = 0 represent noninvertible system s.
A s indicated above, inverse system s are im portant in m any practical appli­
cations. including geop h ysics and digital com m unications. Let us begin by con­
sidering the problem of determ ining the inverse o f a given system . W e limit our
discussion to the class o f linear tim e-invariant discrete-tim e system s.
N ow . suppose that the linear tim e-invariant system T has an im pulse response
h(n) and let h t (n) d en ote the im pulse response o f the inverse system T _ l . Then
(4.6.1) is equivalent to the con volu tion equation
U’(n) = h / ( n) * h( n) * x ( n ) = x( n)
(4.6.2)
h(n) * h/ ( n) = S(n)
(4.6.3)
But (4.6.2) im plies that
The convolution equation in (4.6.3) can be used to solve for h/ ( n) for a given
h(n). H ow ever, the solution o f (4.6.3) in the tim e dom ain is usually difficult. A
sim pler approach is to transform (4.6.3) into the z-dom ain and so lv e for T _1. Thus
in the z-transform dom ain, (4.6.3) becom es
H{z )H, (z) = 1
and therefore the system function for the inverse system is
*,(:)=^
(4.6.4)
If H ( z ) has a rational system function
H( z) =
(4.6.5)
/1(c)
Identity system
v( n )
T
T -i
Direct
system
Inverse
system
n '(n ) = x(n)
Figure 4.62 System T in cascade with
its inverse T _1.
Sec. 4.6
357
Inverse Systems and Deconvolution
then
(4.6.6)
Thus the zeros o f H ( z ) b ecom e the p oles of the inverse system , and vice versa.
Furtherm ore, if H{ z ) is an F IR system , then Hi ( z) is an all-p ole system , or if H{z )
is an all-p ole system , then H s {z) is an F IR system .
Example 4.6.1
Determine the inverse of the system with impulse response
h(n) = UyuOO
Solution
The system function corresponding to h(n) is
H(z) -
1-
This system is both causal and stable. Since H(z) is an all-pole system, its inverse is
FIR and is given by the system function
h ,{ z) =
i - h-1
Hence its impulse response is
= &(n) —
— 1)
Example 4.6.2
Determine the inverse of the system with impulse response
h(n) = <5(n) — j<5(h — 1)
Solution
This is an FIR system and its system function is
H(z) = 1 - k ~ '
ROC: |z| > 0
The inverse system has the system function
Thus H/ ( z) has a zero at the origin and a pole at z = j- In this case there are two
possible regions of convergence and hence two possible inverse systems, as illustrated
in Fig. 4.63. If we take the ROC of Hi(z) as |j| > j, the inverse transform yields
which is the impulse response of a causal and stable system. On the other hand, if
the ROC is assumed to be |j| < | , the inverse system has an impulse response
In this case the inverse system is anticausal and unstable.
358
Frequency Analysis of Signals and Systems
Chap. 4
Figure 4.63 Two possible regions of
convergence for H (z) = z/(z - Y>-
(b)
W e observe that (4.6.3) cannot be solved uniquely by using (4.6.6) unless we
specify the region of con vergen ce for the system function o f the inverse system.
In som e practical applications the im pulse response h(n) d o es not possess a
z-transform that can be expressed in closed form. A s an alternative w e may solve
(4.6.3) directly using a digital com puter. Since (4.6.3) d oes not. in general, possess
a unique solution, we assum e that the system and its inverse are causal. Then
(4.6.3) sim plifies to the equation
h(k)hi(rt — k) — S(/i)
^
(4.6.7)
By assum ption, h/ ( n) = 0 for n < 0. For n = 0 we obtain
h,(0 ) = l / h ( 0 )
(4.6.8)
T he values o f hi {n) for n > 1 can be ob tain ed recursively from the equation
^
h(k)h/(n-k)
= ~£
mo) —
,
n21
This recursive relation can easily be program m ed on a digital com puter.
( A -6
)
Sec. 4.6
359
Inverse Systems and Deconvolution
T here are two problem s associated with (4.6.9). First, the m ethod d oes not
work if h(0) = 0. H ow ever, this problem can easily be rem ed ied by introducing
an appropriate delay in the right-hand side o f (4.6.7), that is, by replacing <50) by
S(n — m) , where m = 1 if *(0) = 0 and /i( l) ^ 0, and so on. Second, the recursion
in (4.6.9) gives rise to rou n d -off errors which grow with n and, as a result, the
num erical accuracy o f h(n) d eteriorates for large n.
Example 4.6.3
D eterm ine the causal inverse of the FIR system with impulse response
h(n) = S(n) — aS(n — 1)
Solution
Since /i(0) = 1. h(1) = —a, and h(n) = 0 for n > a , we have
MO) = 1/MO) = 1
and
h/(n) = a M n _ 1 )
n —1
Consequently,
M l) = a .
M 2) = a 2..........
h,(n) =a"
which corresponds to a causal IIR system as expected,
4.6.2 Minimum-Phase, Maximum-Phase, and
Mixed-Phase Systems
T h e invertibility o f a linear tim e-invariant system is intim ately related to the char­
acteristics o f the phase spectral function o f the system . T o illustrate this point, let
us consider tw o FIR system s, characterized by the system functions
H\ {z) = 1 + j ; -1 = z_1(z + \ )
(4.6.10)
Hj i z ) =
(4,6.11)
— z ! ( |z + 1)
The system in (4.6.10) has a zero at z =
and an im pulse response fc(0) = 1,
/i(l) = 1/2. T h e system in (4.6.11) has a zero at z = —2 and an im pulse response
h(0) = 1/2, /i(1) = 1, which is the reverse of the system in (4.6.10). This is due to
the reciprocal relationship b etw een the zeros o f H\ ( z) and
In the frequency dom ain, the tw o system s are characterized by their fre­
quency response functions, which can be expressed as
|ffi(<y)| = |H 2(o))! = y | + cos to
(4.6.12)
,
sin w
-j------------| + cos OJ
(4.6.13)
_i
sin to
------------2 + cos to
(4.6.14)
and
©i((w) = —to + tan
© 2 (cl>) = —to + tan
The m agnitude characteristics for the tw o system s are identical becau se the zeros
o f Hi ( z ) and Hi ( z ) are reciprocals.
360
Frequency Analysis of Signals and Systems
Chap. 4
fr|I co)
94a.)
i
(b)
Figure 4.64 Phase response
charactcnslics for the systems in (4.6.10)
and (4.6.11).
The graphs o f 0 ] (co) and (~)2 (cv) are illustrated in Fig. 4.64. W e observe that
the phase characteristic 0i(cd ) for the first system begin s at zero phase at the fre­
quency w = 0 and term inates at zero phase at the frequency cv — 7t. H en ce the net
phase change, 0 i( ;r ) - 0i(O ) is zero. On the other hand, the p h ase characteristic
for the system with the zero outside the unit circle undergoes a n et phase change
0 ;(;r ) — ©2(0) = n radians. A s a con seq u en ce o f these different p h ase character­
istics, w e call the first system a m i n i mu m - p h a s e s y s t e m and the secon d system is
called a m a x i mu m - p h a s e system.
Th ese definitions are easily exten d ed to an F IR system o f arbitrary length.
T o be specific, an F IR system o f length M + 1 has M zeros. Its frequency response
can be expressed as
H(co) = bQ(\ - z \ e ~ iw)( 1 - z 2e ~ n • • ■ ( ! - z Me ~ J“)
(4.6.15)
where {;,} d en ote the zeros and bo is an arbitrary constant. W h en all the zeros
are inside the unit circle, each term in the product o f (4.6.15), corresponding to
a real-valued zero, will undergo a net phase change o f zero b etw een a> = 0 and
(*) = n . A lso , each pair of com p lex- conjugate factors in H(a>) will undergo a net
phase change o f zero. T herefore,
^ H ( n ) - ^H(O) = 0
(4.6.16)
and h en ce the system is called a m inim um -phase system . O n the o th er hand, when
all the zeros are outside the unit circle, a real-valued zero will con trib ute a net
Sec. 4.6
361
Inverse Systems and Deconvolution
phase change o f tt radians as the frequency varies from w = 0 to co — it. and each
pair o f com p lex-con ju gate zeros will contribute a net phase change of 2 tt radians
over the sam e range o f ai. T herefore.
iL H ( jt ) -
^ H ( O ) = M tt
(4.6.17}
which is the largest possible phase change for an FIR system with M zeros. H ence
the svstem is called m axim um phase. It follow s from the discussion above that
4- Hmax{7T) > 4 tf m,n(jr)
(4.6.18)
If the FIR system with M zeros has som e o f its zeros inside the unit circlc
and the rem aining zeros ou tsid e the unit circle, it is called a mi x e d - p h a s e system
or a n o n m i n i m u m - p h a s e syst em.
Since the derivative o f the phase characteristic o f the system is a m easure
of the tim e d elay that signal frequency com p onents undergo in passing through
the system , a m inim um -phase characteristic im plies a m inim um delay function,
while a m axim um -phase characteristic im plies that the delay characteristic is also
m axim um .
N ow suppose that we have an FIR system with real coefficients. .Then the
m agnitude square value of its frequency response is
|t f ( w ) |: =
)|r^ ,-
(4.6.19)
This relationship im plies that if we replace a zero
o f the system by its inverse
1 /zk- the m agnitude characteristic o f the system d oes not change. Thus if we re­
flect a zero zt that is inside the unit circle into a zero I/:* ou tsid e the unit circle,
we se e that the m agnitude characteristic o f the frequency response is invariant to
such a change.
It is apparent from this discussion that if \ H ( c o ) \ 2 is the m agnitude square
frequency resp onse o f an F IR system having M zeros, there are 2 M possible con ­
figurations for the M zeros, som e of which are inside the unit circle and the re­
m aining are ou tsid e the unit circle. Clearly, one configuration has all the zeros
inside the unit circle, which corresponds to the m inim um -phase system . A sec­
ond configuration has all the zeros outside the unit circle, which corresponds to
the m axim um -phase system . T h e rem aining 2 W — 2 configurations correspond to
m ixed-phase system s. H ow ever, not all 2 M - 2 m ixed-phase configurations n ec­
essarily correspond to F IR system s with real-valued coefficients. Specifically, any
pair o f com p lex-con ju gate zeros result in on ly two possib le configurations, w hereas
a pair o f real-valued zeros yield four possib le configurations.
Example 4.6.4
D eterm ine the zeros for the following FIR systems and indicate whether the system
is minimum phase, maximum phase, or mixed phase.
tf,(z) = 6 + z_1 - z~2
H 2{ z )
=
1 - r
1 - 6 ;" 2
362
Frequency Analysis of Signals and Systems
Chap. 4
H:Az) = 1 - ^c"1 - $z~z
HAZ) =
Solution
1+ f r ' -
By factoring the system functions we find the zeros for the four systems
are
H]{z) — *■ Ci.: = —*■ { — *■ minimum phase
H 2 (z )
-— » ci,: = - 2 , 3 — ► maximum phase
H:,(z) — *■ ci.: = - J. 3 — * mixed phase
Hi\z) — *■Ci.: = —2, t — *• mixed phase
Since the zeros of the fouT systems are reciprocals of one another, it follows that all
four systems have identical m agnitude frequency response characteristics but different
phase characteristics.
The m inim um -phase property o f F I R system s carries over to I I R system s that
have rational system functions. Specifically, an I I R system with system function
B(z)
H( z ) = —
(4.6.20)
A (c )
is called m i n i m u m p h a s e if all its poles and zeros are inside the unit circle. For a
stable and causal system [all roots of A (c) fall inside the unit circle] the system is
called m a x i m u m pha s e if all the zeros are outside the unit circle, and m i x e d phase
if som e, but not all. o f the zeros are ou tsid e the unit circle.
This discussion brings us to an im portant point that should be em phasized.
That is. a stable p o le-zero system that is m inim um phase has a stab le inverse which
is also minim um phase. The inverse system has the system function
= d(z)
(4 -6 -21)
H en ce the m inim um -phase property of H ( z ) ensures the stability o f the inverse
system H ~ l (z) and the stability o f H( z ) im plies the m inim um -phase property of
H ~ \ z ) . M ixed-phase system s and m axim um -phase system s result in unstable in­
verse system s.
D ecom p osition
of
n on m in im u m -p h ase
p o le -z e r o
sy stem s.
Any
nonm inim um -phase p o le -z e r o system can be expressed as
« (z) = /W z)tfa p (z)
(4.6.22)
w here
is a m inim um -phase system and / / ap(z) is an all-pass system . We
dem onstrate the validity o f this assertion for the class o f causal and stable systems
w-ith a rational system function H ( z ) = B ( z ) / A ( z ) . In general, if B(z) has one
or m ore roots ou tsid e the unit circle, w e factor B(z) in to the product B i( z ) # 2 (z)>
w here B i(z) has all its roots inside the unit circle and B 2(z) has all its roots outside
Sec. 4.6
Inverse Systems and Deconvolution
the unit circle. T h en
m inim um -phase system
363
has all its roots inside the unit circle. W e define the
B i i - J B j i z - 1)
Mz)
and the all-pass system
Hw (z) =
Biiz)
B 2(z - 1)
Thus H ( z ) — Fiminiz)Hap(z)■ N ote that / / ap(z) is a stable, all-pass, m axim um -phase
system .
Group delay of nonminimum-phase system. B ased on the d ecom p osi­
tion o f a nonm inim um -phase system given by (4.6.22), w e can express the group
delay o f H( z ) as
Tg(o>) = r™in(co) + ^apM
(4,6.23)
Since r “r (u>) > 0 for 0 < to < n , it follow s that 1 ^( 0;) > r™n(o;), 0 < a>< tt. From
(4.6.23) w e co n clu d e that am ong all p o le -z e r o system s having the sam e m agnitude
response, the m inim um -phase system has the sm allest group delay.
Partial energy o f nonminimum-phase system.
T he partial energy o f a
causal system with im pulse response h(n) is defined as
£Cn) = £ W * ) I 2
t=u
(4.6.24)
It can be show n that am ong all system s having the sam e m agnitude response and
the sam e total en ergy £ ( 0 0 ), the m inim um -phase system has the largest partial
energy [i.e., E min(n) > E( n) , where Emj„(n) is the partial energy o f the m inim um phase system ].
4.6.3 System Identification and Deconvolution
Suppose that w e excite an unknow n linear tim e-invariant system with an input se ­
quen ce x ( n ) and w e observe the output seq u en ce y( n). From the output sequ en ce
w e wish to determ ine the im pulse resp onse o f the unknow n system . This is a prob­
lem in s y s t em identification, which can be solved by deconvol ut i on. Thus w e have
y(n) = h( n) * *(n)
^
(4-6.25)
h( k ) x ( n — k)
=
k=—oo
A n analytical solution o f the d econ volu tion problem can b e obtained by
w orking with the z-transform o f (4.6.25). In the z-transform dom ain w e have
Y( z ) = H( z ) X( z )
364
Frequency Analysis of Signals and Systems
Chap, 4
an d h en ce
K(;)
HrJ = - —
* (;)
(4.6.26)
X u ) and
are th e ^ -tra n sfo rm s of the av ailab le in p u t signal x(/;) a n d the
o b se rv e d o u tp u t signal yi n) , resp ectiv ely . T his a p p ro a c h is a p p r o p ria te only w hen
th e re are clo sed -fo rm ex p ressio n s fo r X ( z ) an d F (c).
Example 4.6.5
A causal system produces the output sequence
n=0
n= 1
otherwise
f 1.
yin) = |
.
I 0.
when excited by the input sequence
n= 0
(1 .
x(n) = i
H i'
I 0,
otherwise
Determ ine its impulse response and its input-outpul equation.
Solution The system function is easily determ ined by taking the .--transforms of ,v(n)
and yin). Thus we have
no
H( z ) = —----- = — —
1 + 77,:"
1 - 17i-
-------:----- 7
+ i7>"~
1-r
(1 -
)(1 -
Since the system is causal, its ROC is |;| > i. The system is also stable since its poles
lie inside the unit circle.
The input-output difference equation for the system is
y i n ) = -jj;y(/i - 1 ) - ^ y i n - 2 ) + ,r(«) + y^.v(n - 1 )
Its impulse response is determined by performing a partial-fraction expansion of H(z)
and inverse transforming the result. This com putation yields
hin) = [4(i)" - 3({ )n]u(n)
W e observe that (4.6.26) d eterm in es the unknow n system uniquely if it is
known that the system is causal. H ow ever, the exam ple above is artificial, since
the system response {>(«)} is very likely to be infinite in duration. C onsequently,
this approach is usually im practical.
A s an alternative, we can deal directly with the tim e-dom ain expression given
by (4.6.25). If the system is causal, w e have
n
y( n) = ^ h{ k) x( n — k)
n> 0
*=o
Sec. 4.6
365
Inverse Systems and Deconvolution
and h ence
n-l
y { n )
-
Hn) =
(4.6.27)
-
y ~ ^ h ( k ) x ( n
k )
n
>
x(0)
1
This recursive solu tion requires that jr(0) ^ 0. H ow ever, we n ote again that w hen
(/i(n)) has infinite duration, this approach m ay not be practical unless w e truncate
the recursive solu tion at sam e stage [i.e., truncate {/z(«)}].
A n o th er m eth od for identifying an unknow n system is b ased on a crosscor­
relation technique. R ecall that the in p ut-ou tp u t crosscorrelation function derived
in Section 2.6.5 is given as
oc
r y x ( r 77)
=
^
h ( k ) r ( X ( m
-
=
k )
h ( n )
*
r x ! ( m )
(4.6.28)
t=0
where r y x ( m ) is the crosscorrelation seq u en ce o f the input {x(^)J to the system
with the output {>’(«)} o f the system , and r x x { m ) is the autocorrelation sequ en ce
o f the input signal. In the frequency dom ain, the corresponding relationship is
S vx((d) = J-!(io)S.fX (co) = H ( c o ) \X (u >)\2
H en ce
Svi(a>)
H ( w )
=
—
---------- =
SX i ( w )
5 Vj(w )
-
1 -
— r
\ X ( a >)\2
(4.6.29)
T h ese relation s suggest that the im pulse response (/?(k)1 or the frequency re­
sp on se o f an unknow n system can be determ ined (m easu red ) by crosscorrelating
the input seq u en ce {*(«)} with the output seq u en ce (y(n)}, and then solvin g the
d econ volu tion problem in (4.6.28) by m eans o f the recursive eq u ation in (4.6,27).
A ltern atively, w e cou ld sim ply com pute the Fourier transform o f (4.6.28) and d e ­
term ine the freq u en cy response given by (4.6.29). Furtherm ore, if w e select the
input seq u en ce (jc(n)} such that its autocorrelation seq u en ce { ^ ( n ) } , is a unit sam ­
ple seq u en ce, or eq u ivalen tly, that its spectrum is flat (con stan t) over the passband
o f H(a>), the valu es o f the im pulse resp onse {/?(«)} are sim ply equal to the values
o f the crosscorrelation seq u en ce {rVJ(«)}.
In general, the crosscorrelation m eth od described above is an effective and
practical m eth od for system identification. A n oth er practical approach based on
least-squares optim ization is described in C hapter 8.
4.6.4 Homomorphic Deconvolution
T h e com p lex cepstrum , introduced in Section 4.2.7, is a useful to o l for perform ing
d econ volu tion in so m e applications such as seism ic signal processing. T o describe
this m eth od , let us su p p ose that {>>(«)} is the output seq u en ce o f a linear timeinvariant system w hich is excited by the input seq u en ce (x(n )f. Then
Y( z) = X{ z ) H( z )
(4.6.30)
366
Frequency Analysis of Signals and Systems
Chap. 4
w here H( z ) is the system function. T h e logarithm o f Y ( z ) is
C A z ) = In Y (c)
= in X (c) + In H( z )
(4.6.31)
= C ,(c) + C>,U)
C onsequently, the com plex cepstrum o f the output sequence (y(n)} is expressed
as the sum o f the cepstrum o f |x(n )} and {/i(n)J, that is,
c y( n) = cx (n) + ch(n)
(4.6.32)
Thus w e observe that con volu tion of the tw o seq u en ces in the tim e dom ain corre­
sponds to the sum m ation o f the cepstrum seq u en ces in the cepstral dom ain. The
system for perform ing these transform ations is called a h o m o r m o r p h i c sy s t em and
is illustrated in Fig. 4.65.
In som e applications, such as seism ic signal processing and speech signal
processing, the characteristics of the cepstral sequ en ces (c,(/j)} and {c>(n)} are suf­
ficiently different so that they can be separated in the cepstral dom ain. Specifically,
suppose that {c* (/?)} has its main com p on en ts (m ain energy) in th e vicinity o f small
values o f n, w hereas |c r(n)} has its com p on en ts concentrated at large values of n.
W e m ay say that |c„(n)} is “lowpass" and {cx(«)l is “highpass.” W e can then sepa­
rate {cy,(n)} from {c,(r;)) using appropriate “low p ass” and “h igh pass” w indow s, as
illustrated in Fig. 4.66. Thus
ch(n) = c v(rt)wir (rr)
(4.6.33)
cx (n) = c_v(n )Whp(n)
(4.6.34)
and
z-Transform
logarithm
c j.)
;-iransform
Figure 4.65 Homomorphic system for obtaining the cepstrum (cv(n)} of the se­
quence (y(n)l-
F igure 4.66 S ep aratin g th e two
cepstral com ponents by “low pass" and
“ highpass” windows.
Sec. 4.7
367
Summary and References
w here
U ’) p ( J l)
=
1.
0.
< N,
otherw ise
(4.6.35)
0,
l«l < N\
\n\ >
(4.6.36)
U’hpO) = 1.1
O nce w e have separated the cepstrum sequ en ces ( o ,( « ) } and (c,-(n)} by w indow ing,
the seq u en ces {x(n)} and {/i(n)( are obtained b y p a ssin g (o ,(» )| and (c.v(n)) through
the inverse hom om orphic system , shown in Fig. 4.67.
In practice, a digital com puter w ould be used to com p ute the cepstrum o f the
seq u en ce {v(«)}. to perform the w indow ing functions, and to im plem ent the inverse
h om om orphic system shown in Fig. 4.67. In place o f the --transform and inverse
z-transform . we w ould substitute a special form of the Fourier transform and its
inverse. T his special form , called the discrete Fourier transform, is described in
C hapter 5.
C,<M)
C,(.v)
.--Transform
XO
Com plex
exponential
W(c)
| .v(n)
Inverse
■
"-rnmt l^nT> (
ir
i /iijn
.
Figure 4.67 Inverse homomorphic system for recovering the sequences {.kii )| mid
|/j(«)) from the corresponding cepstru.
4.7 SUMMARY AND REFERENCES
T he Fourier series and the Fourier transform are the m athem atical tools lor an­
alyzing the characteristics o f signals in the frequency dom ain, T he Fourier series
is appropriate for representing a periodic signal as a w eigh ted sum of harm oni­
cally related sinusoidal com p onents, w here the w eighting coefficients represent the
strengths o f each o f the harm onics, and the m agnitude squared of each w eighting
coefficient represents the pow er of the corresponding harm onic. A s we have in­
dicated, the Fourier series is on e o f m any possible orthogonal series expansions
for a p eriodic signal. Its im portance stem s from the characteristic behavior o f LTI
system s, as we shall see in Chapter 5.
T h e Fourier transform is appropriate for representing the spectral charac­
teristics o f aperiodic signals with finite energy. T he im portant properties of the
F ourier transform were also presented in this chapter.
There are m any excellen t texts on Fourier series and Fourier transforms.
F or reference, w e include the texts by B racew ell (1978), D avis (1963), Dvm and
M cK ean (1972). and P apoulis (1962).
In this chapter w e also considered the frequency-dom ain characteristics o f
LTI system s. W e show ed that an LTI system is characterized in the frequency
d om ain by its frequency response function H ( ai), w hich is the Fourier transform
368
Frequency Analysis of Signals and Systems
Chap. 4
o f the im pulse response o f the system . W e also observed that the frequency
response function d eterm ines the effect o f the system on any input signal. In fact,
by transform ing the input signal into the frequency dom ain, w e ob served that it is a
sim ple matter to determ ine the effect of the system on the signal and to determ ine
the system output. W hen view ed in the frequency dom ain, an LTI system performs
spectral shaping or spectral filtering on the input signal.
The design o f som e sim ple IIR filters was also considered in this chapter from
the view point o f p o le-zero placem ent. B y m eans o f this m eth od , we w ere able
to design sim ple digital resonators, notch filters, com b filters, ail-pass filters, and
digital sinusoidal generators. T h e design o f m ore com p lex IIR filters is treated in
detail in Chapter 8. which also includes several references. D igital sinusoidal gen­
erators find use in frequency synthesis applications. A com p reh en sive treatm ent of
frequency synthesis tech n iqu es is given in the text edited by G orski-P op iel (1975).
Finally, w e characterized LTI system s as either m inim um -phase, maximumphase, or m ixed-phase, d ep en d ing on the p osition o f their p o les and zeros in the
frequency dom ain. U sing these basic characteristics of LTI system s, w e considered
practical problem s in inverse filtering, d econ volu tion , and system identification.
W e concluded with the description o f a d econ volu tion m eth od based on cepstral
analysis o f the output signal from a linear system .
A vast am ount o f technical literature exists on the topics o f inverse filter­
ing. d econ volu tion , and system identification. In the context o f com m unications,
svstem identification, and inverse filtering as they relate to channel equalization
are treated in the book by Proakis (1995). D econ volu tion tech n iqu es are widely
used in seism ic signal processing. For reference, w e suggest the papers by W ood
and Treitel (1975), P eacock and T reitel (1969), and the b ook s by R obinson and
Treitel (1978, 1980). H om om orphic d econ volu tion and its ap p lication s to speech
processing is treated in the book by O p p en heim and Schafer (1989).
PROBLEMS
4.1 Consider the full-wave rectified sinusoid in Fig. P4.1.
(a ) Determine its spectrum X a(F).
(b) Compute the power of the signal.
Xa( ! )
Figure P4.1
Chap. 4
369
Problems
(c) Plot the power spectral density.
(d) Check the validity of Parseval's relation for this signal.
4.2 Com pute and sketch the m agnitude and phase spectra tor the following signals (a > 0).
Ae~a' ,
i > 0
la) V- , ' , = '0 .
,< 0
(b) xu(t) = Ae~“'r
4.3 Consider the signal
1 -
^
l r |h .
|r | < t
' 0.
elsewhere
(a) Determ ine and sketch its magnitude and phase spectra. |X „(F)| and 2^ X a(F),
respectively.
( b) Create a periodic signal x,.(t) with fundam ental period Tr > 2r. so that ,v(/) =
.v,,(n for |f| < T;,/2. What are the Fourier coefficients ci for the signal x;,u)?
(c) Using the results in p an s (a) and (b). show that a = (1/X;,)X „(k/Tr ).
4.4 Consider the following periodic signal:
xin) = { ..., I. (I. 1.2. 3.2. 1.0. I. . . ,|
t
(a) Sketch the signal vt/i) and its magnitude and phase spectra.
( b) Using the results in pari (a), verily Parseval's relation by computing the power
in the time and frequency domains.
4.5 Consider the signal
rr/i
nn
1
3nn
xin ) = 2 -f- 2 cos ------Hcos — h— c o s ----4
2
2
4
(a) Determ ine and sketch its power density spectrum.
( b) Evaluate the power of the signal.
4.6 Determ ine and sketch the magnitude and phase spectra of the following periodic
signals.
n(n - 2)
(a) ,v<(t)=4sin
3
In
. In
(b) ,v(n) = cos — n + sin — n
3
^
2n
. 2n
(c) x(n) = cos — n sin — n
(d) x( n) = { . . . , - 2 . - 1 . 0 . 1 . 2 . - 2 . - 1 . 0 . 1 . 2 . . . . )
t
( e) x i n ) = ( . . . . - 1 . 2 . 1. 2. - 1 . 0 . - 1 . 2 . 1 . 2 . . . . J
(f) x (n) = ( . . . . 0 , 0 . 1 . 1 . 0 . 0 . 0 . 1. 1 . 0 . 0 , . . . )
t
(g) x i n) = 1 . —oc < n < oc
( h) x(n) = ( - 1 ) " . - o c < n < oc
4.7 D eterm ine the periodic signals * 0 ), with fundam ental period N = 8. if their Fourier
coefficients are given by:
kn
3kn
370
Frequency Analysis of Signals and Systems
Chap. 4
kn
(b) c * = { Slny -
Q < k <6
k= 7
0,
(c) {c*} = { . . . . O . i . i . l . 2 . 1 . i , i . O .
t
4.8 Two DT signals. j*(«) and s{(n), are said to be orthogonal over an interval [N\, A^] if
k =t
Y2st ( n) s ?( n) = j 0 *'
k / I
If At = 1. the signals are called orthonorm al,
(a) Prove the relation
N '
ej2*kn/* =
' N,
0.
k = 0, ±jV, ±2N,
otherwise
(b) Illustrate the validity of the relation in part (a) by plotting for every value of
k ~ 1 ,2 ........ 6. the signals st (n) = eJI2,,/('}k" , n = 0, 1 ,___ 5. [Aro/e: For a given k,
n the signal
can be represented as a vector in the complex plane.]
(c) Show that the harmonically related signals
i*(n) = ejan,K'kB
are orthogonal over any interval of length N.
4.9 Compute the Fourier transform of the following signals.
(a) x(n) = u(n) —u(n —6)
(b) x(n) = 2"u{-n)
(c) x(n) = (j)"u(n + 4)
(d) x(n) = (a" sin a>on)u(n)
|a | < 1
(e) x(n) = laTsiniuiin
jar| < 1
(f) x(n) =
2 -U )n .
0,
|«| 5 4
elsewhere
(g) xin) = { - 2 . - 1 . 0. 1 . 2 )
t
\ n\ <M
^
.
U (2 A / + l - | n | ) .
(h) x(n) =
| 0.
. .
|n| > M
Sketch the magnitude and phase spectra for parts fa), (f), and (g).
4.10 D eterm ine the signals having the following Fourier transforms.
0,
0 < \a>| < <U()
(a) X(a>) = ,
I.
W() < \a>\ < 7T
(b) X (u>) = cos2 a>
(n\ Yt \ - \
O* - &i»/2 < M < O*) + to /2
W
1 0,
elsewhere
(d) The signal shown in Fig. P4.10.
4 .11 Consider the signal
x{n) = {1 , 0,- 1 ,2 .3 }
t
Chap. 4
371
Problems
Xitu)
3tt
jt
0
7r
3n
6 jt
7tt
tt
Figure P4.10
with Fourier transform Xtw) = X's (ct>) + j ( X t (a>)). D eterm ine and sketch the signal
y(n) with Fourier transform
Y(w) = X,(o>) + X R( ^) ei2,il
4.12 Determ ine the signal jc(h ) if its Fourier transform is as given in Fig. P4.12.
8k
10
0
(a)
X(ai)
(b)
X(u>)
(c )
Figure P4.12
8rr
971
10
To
7T
372
Frequency Analysis of Signals and Systems
Chap. 4
4.13 In Example 4.3.3. the Fourier transform of the signal
-M < n < M
otherwise
1,
0.
x (n) =
was shown to be
M
X (a>) = 1 + 2 2 2 cos am
f!=1
Show that the Fourier transform of
X|(n) =
1.
0.
0< n <M
otherwise
and
x 2(n) =
1.
0,
—M < n < —1
otherwise
are, respectively.
] _
I
Xi(a>) =
X 2{to) =
Thus prove that
X (to) -— X 1 (tt>) + X 2 (tt)}
sin(M + \) w
sintw /2)
and therefore.
coswn =
sin(M + \ ) w
sin(a»/2 j
4.14 Consider the signal
x(n) = ( - 1 ,2 , - 3 ,2 , -1}
t
with Fourier transform X((u). Compute the following quantities, without explicitly
computing X(a>):
(a) X(0)
(b) AX(co)
(c) f * n X(w) dw
(d) X (jt)
(e) f *JX( co ) \ 2 dco
4.15 The center of gravity of a signal x(w) is defined as
T , nx(n)
yx(n)
7I = —OC
and provides a measure of the “time delay” of the signal.
Chap. 4
373
Problems
Xui»
2
Figure P4.15
2
(a) Express c in terms of X{<o).
(b) Compute c for the signal x(n) whose Fourier transform is shown in Fig. P4.15.
4.16 Consider the Fourier transform pair
a"u(n)
|ti) < 1
-----------1 — at’
Use the differentiation in frequency theorem and induction to show that
< « + /-!> !
i
1
_v(/i) = --------------- a ‘u(n} -— * \ Uo) — ----------------/i!(f-l)!
(1 - f
4.17 Let vUil he an arbitrary signal, not necessarily real-valued, with Fourier transform
Express the Fourier transforms of the following signals in terms of X(a>).
(a) _v‘ i«)
(h) x ' ( - n )
(f) yin) = vf/1 ) - ,v(/; - 1 )
(d) v(fi I = ' y
xik i
(L‘) V|H)=.V(2«)
_
I x(n/2).
n even
(0 v(n) = L
,.
( 0.
n odd
4.18 Determ ine and sketch the Fourier transforms Xiiw), X 2(co). and Xi(a>) of the following
signals.
(a) -V!(fi) = (1. 1. 1. 1.1)
(b) x2(n) = ( 1. 0. 1 . 0. 1 , 0. 1 . 0, 1 )
(c) x:j n ) = (1 . 0. 0. 1 . 0. 0. 1 . 0. 0, 1 , 0. 0. 1 )
t
(d) Is there any relation between Xi(w). X:(g;). and X3(w)? What is its physical
meaning?
(e) Show that if
**(«) =
then
|
, ( r ).
if n j k integer
0.
otherwise
Xt(a>) = X (ka>)
4.19 Let x(n) be a signal with Fourier transform as shown in Fig. P4.19. D eterm ine and
sketch the Fourier transform s of the following signals.
Frequency Analysis of Signals and Systems
374
2
Chap. 4
Figure P4.19
2
(b ) A;(n) = x(n) sin(jrn/2)
(d) a4(/7) = x(n) cos nn
(a ) jri(fl) = jr(n) cos(;rn/4)
(c) xy(n) = x( n)cos(nn/2)
Note that these signal sequences are obtained by amplitude modulation of a carrier
co swrn or sinw,,/! by the sequence x(n).
4.20 Consider an aperiodic signal *(n) with Fourier transform X(u>). Show that the Fourier
series coefficients CA
' of the periodic signal
are eiven bv
c;
I .
N
— k
N
k = 0. 1........A '- l
4.21 Prove that
X v (w ) — ^ '
£—'
n=~ N
sin w,n
nn
may be expressed as
X v (o j )
1
= ^
r ° ‘ sinf(2A^ + 1 ){w — 8f lj \
db
- 0)/2\
.L ,
4.22 A signal jfn ) has the following Fourier transform:
1
X (oj) = ---------------
1 - ae~JW
Determine the Fourier transforms of the following signals:
(b) e*nf2x(n + 2 )
(b) x { —2n)
(d) x(n) cos(0.37rn)
(c) jt(«) * x(n —1 )
(0 x(n) * x (—n)
4.23 From a discrete-time signal x ( n ) with Fourier transform X(cv), shown in Fig. P4.23,
determine and sketch the Fourier transform of the following signals:
, „
1 x(n).
n even
( a )v i( n ) =
I 0,
n odd
(b) yiiri) - x(2n)
[ x( nj 2 ).
n even
(c) y,(«) = „
,,
I 0,
n odd
Note that vj (n) = x(n)s(n), where s(n) = {... 0, 1. 0, 1. 0. 1. 0. 1. ...}
t
(a) x(2n + \ )
Chap. 4
Problems
375
Xlw)
ai
Figure P4.23
4
4
4.24 The following inpul-output pairs have been observed during the operation of various
svstems:
(C) .t(H) — C‘ ‘ --- *• V(IJ) = _V' " '
■
a
s
,
(d) .v(u) = e n ■u(ii) — ►yin) — j r ' "
7*
(e) ,v(/i» = ,v(/i + Ar’i ) —— vin) = yin 4- N21
A',
A;. A't. N2 prime
Determ ine their frequency response if each of the above systems is LTI.
4.25 (a) D eterm ine and sketch the Fourier transform W/ffw) of the rectangular sequence
[I.
1 0.
(I < n < M
otherwise
(b) Consider the triangular sequence
( ) < / i < M/2
Mj2 < 2 < M
otherwise
Determ ine and sketch the Fourier transform Wr (a)) of u'70)) by expressing it as
the convolution of a rectangular sequence with itself.
(c) Consider the sequence
«■, in) = ^ (l -t- cos ^f-) u'R{n)
Determ ine and skctch Wricu) by using
4.26 Consider an LTI system with impulse response h i n ) =
u(n).
(a) Determ ine and skctch the magnitude and phase response \H(co)\ and
respectively.
(b) D eterm ine and sketch the magnitude and phase spectra for the input and output
signals for the following inputs:
(2) jr(n) = ( . . . . 1.0.0. 1. 1. 1.0. 1. 1. 1,0, 1. ...}
r
4.27 D eterm ine and sketch the magnitude and phase response of the following systems:
(a) y(n) = ^[.v(/i) + xin - 1 )]
(b) yfn) = 4[v(/i) - xin - 1 )]
(c) v(n) =
+ 1) - x(n - 1)]
376
Frequency Analysis of Signals and Systems
(d) y(n) =
Chap. 4
+ 1 ) + x(n - 1 )]
(e) v (n )= i[jr(rt) + jc(/t - 2 ) ]
(f) y(n) =
- x(n - 2 )]
(g) v(n) = j[jr(n) + x(n - 1 ) + x{n - 2 )]
(h) y(n) = x(n) — x(n — 8)
(i) y(n) = 2x(n —1 ) —x(n — 2 )
(j) v(n) =
+ x(n - 1) + x(n - 2) + x(« - 3)]
(k) y(«) =
+ 3x(n - 1) + 3x(n - 2) + x(n - 3)]
(I) y(n) = x(n - 4)
(m) y(n) = x(n + 4)
(n) y(n) = j[x(n) - 2x(n - 1 ) + xin - 2 )]
4.28 An FIR filter is described by the difference equation
y(n) = x(n) + xin - 10)
(a) Compute and sketch its magnitude and phase response.
(b) Determ ine its response to the inputs
„
TT
.
/ TT
IT '
(1) x(n) = cos — n + 3sm \ ^- n + — ^
(2 ) jr(n) = 10 + 5 cos ( ^ - n + y
—oc < n < oc
—OO <
< oc
4.29 D eterm ine the transient and steadv-statc responses of the FIR filler shown in Fig. P4.29
to the input signal A(n) = 10ejr"',~uin). Let b = 2 and v ( - l ) = v (-2 ) = v(—3) =
v(—4) = 0.
Figure P4.29
4.30 Consider the FIR filter
v(n) = x ( n ) + x ( n - 4)
(a) Compute and sketch its m agnitude and phase response.
(b) Compute its response to the input
x (n) = cos —n + cos —n
2
4
—oc < n < oc
(c) Explain the results obtained in part (b) in terms of the m agnitude and phase
responses obtained in part (a).
4.31 Determ ine the steady-state and transient responses of the system
y(n) = j [•*(«) — x ( n - 2)]
Chap. 4
377
Problems
to the input signal
x(n) = 5 + 3 cos
— oo < n < oc
+ 60“^
4.32 From our discussions it is apparent that an LTI system cannot produce frequencies
at its output that are different from those applied in its input. Thus, if a system
creates “new" frequencies, it must be nonlinear and/or time varying. D eterm ine the
frequency content of the outputs of the following systems to the input signal
4
(a) v (/i) = .v(2n)
(b) y(n) = x 2(n)
<c) y(n) = (cos nn)x(n)
4.33 D eterm ine and sketch the m agnitude and phase response of the systems shown in
Fig. P4.33(a) through (c).
(a)
(b)
8
—l
(c)
Figure P4J3
4.34 Determ ine the magnitude and phase response of the m ultipath channel
y( n) = x (n) + xfn —M)
A t what frequencies does H(co) = 0?
4.35 Consider the filter
v(«) = 0.9v(n - 1) + bx(n)
(a) D eterm ine b so that |H (0)| = 1.
(b) D eterm ine the frequency at which j/ / (cu)| = l/%/2.
378
Frequency Analysis of Signals and Systems
Chap. 4
(c) Is this filter lowpass, bandpass, or highpass?
(d) Repeat parts (b) and (c) for the filter v(n) = —0.9y(n - 1) + O.l.r(n).
4.36* Harmonic distortion in digital sinusoidal generators An ideal sinusoidal generator
produces the signal
x(n) = cos 2nf)n
— oc < n < cc
which is periodic with fundam ental period N if /<i = ki,/N and ky, N are relatively
prime numbers. The spectrum of such a “pure" sinusoid consist of two lines at k = ko
and k = N — ka (we limit ourselves in the fundam ental interval 0 < k < N - 1).
In practice, the approximations made in com puting the samples of a sinusoid of
relative frequency f t result in a certain am ount of power falling into other frequencies.
This spurious power results in distortion, which is referred to as harmonic distortion.
Harmonic distortion is usually m easured in terms of the total harmonic distortion
(TH D), which is defined as the ratio
THD =
spurious harmonic power
total power
(a) Show that
iQn t2
T H D == 1 - 2 —— P,
where
*-j
n = (I
(b) By using the Taylor approximation
COS 0 = 1 -------- ---------—
2!
4!
6!
compute one period of jr(n) for / 0 = 1/96, 1/32, 1/256 by increasing the number
of terms in the Taylor expansion from 2 to 8.
(c) Compute the THD and plot the power density spectrum for each sinusoid in
part (b) as well as for the sinusoids obtained using the com puter cosine function.
Comment on the results.
4.37* Measurement o f the total harmonic distortion in quantized sinusoids
Let x(n) be a
periodic sinusoidal signal with frequency fo = k / N , that is,
x(n) = sin 2jr/on
(a) Write a computer program that quantizes the signal x(n) into b bits or equivalently
into L = 2h levels by using rounding. The resulting signal is denoted by xq(n).
(b) For f a = 1/50 compute the THD of the quantized signals xq(n) obtained by using
b = 4, 6, 8, and 16 bits.
(c) Repeat part (b) for /« = 1/100.
(d) Comment on the results obtained in parts (b) and (c).
438* Consider the discrete-time system
y(n) = ay(n — 1 ) + (1 —a)x(n)
where a — 0.9 and y(—1) = 0.
n > 0
Chap. 4
379
Problems
(a) Compute and sketch the output y, (n) of the system to the input signals
.v ,(n ) = s in 2 jT /,n
0 £ ft < 100
where / , =
./? =
(b) Compute and sketch the magnitude and phase response of the system and use
these results to explain the response of the system to the signals given in part (a).
4.39* Consider an LTI system with impulse response h{n) = ( t )1"1
(a) Determ ine and sketch the magnitude and phase response Hi m) and
respectively.
(b) D eterm ine and sketch the magnitude and phase spectra for the input and output
signals for the following inputs:
3,t ii
( 1 ) .v(n ) = cos —
. —rc < n < cc
(2) .xin) = (....-1, 1. - 1 . 1 . - 1 . 1 . - 1 . 1 . - I . 1, - 1 . 1...-I
T
4.40* Time-domain sampling
(a)
(b)
(c)
(d)
(e)
Consider the continuous-time signal
Compute analytically the spectrum X„(F) of a „ U ) Compute analytically the spectrum of the signal a (/?> = x„{nT). T = 1 /F ,.
Plot the magnitude spectrum |X „(F)| for Ft, = K) Hz.
Plot the magnitude spectrum (X(F)I for F, = 10. 20, 40. and 100 Hz.
Explain the results obtained in part (d) in terms ol the aliasing effect.
4.41 Consider (he digital filter shown in Fig. P4.41.
(a) Determ ine the input-output relation and the impulse response h(n).
(b) Determ ine and sketch the magnitude ]W(w)| and the phase response 2^ H ( a » of
the filter and find which frequencies are completely blocked by the filter.
(c) When w,, = t / 2 , determ ine the output yin) to the input
.v(n) = 3 cos
+ 30 ^
1 ---- —( +
y(n>
-*——
u - - 2 cos aif,
— cc < n < oc
Figure P4.41
4.42 Consider the FIR filter
y(n) = xin) - xin - 4 )
(a) Compute and sketch its m agnitude and phase response.
(b) Com pute its response to the input
n
n
x ( n ) = cos —n + cos —n
— c c < n < oc
2
4
Frequency Analysts of Signals and Systems
380
Chap. 4
(c) Explain the results obtained in part (b) in terms of the answer given in part (a).
4.43 Determ ine the steady-state response of the system
y(n) = i[.v(n) - x(n - 2 )]
to the input signal
—oc < n < oc
4.44 Recall from Problem 4.32 that an LTI system cannot produce frequencies at its output
that are different from those applied in its input. Thus if a system creates “new”
frequencies, it must be nonlinear and/or time varying. Indicate w hether the following
systems are nonlinear and/or time varying and determ ine the output spectra when the
input spectrum is
(a) _v(n) = j:(2nt
(b) y ( « ) = x : (n)
(c) v(n) = (cos nn)xin)
4.45 Consider an LTI svstem with impulse response
(a) Determ ine its system function H(z).
(b) Is it possible to implement this system using a finite number of adders, multipliers,
and unit delays? If yes. how?
(c) Provide a rough sketch of \ H(w)| using the pole-zero plot.
(d) Determine the response of the system to the input
x(n) = (
4.46 An FIR filter is described by the difference equation
yin) = x(n) — xin — 10)
(a) Compute and sketch its m agnitude and phase response.
(b) Determ ine its response to the inputs
—oc < n < oc
— oc < n < oc4.47 The frequency response of an ideal bandpass filter is given by
Chap. 4
381
Problems
(a) D eterm ine its impulse response
(b) Show that this impulse response can be expressed as the product of cos(n7t / 4)
and the impulse response of a lowpass filter.
4.48 Consider the system described by the difference equation
v(n) = jy(n — I ) + .r(n) 4- |.r(n - 1 >
(a) D eterm ine its impulse response.
(b) D eterm ine its frequency response:
(1) From the impulse response
(2) From the difference equation
(c) D eterm ine its response to the input
I TT
x {n > = cos y — r +
.T \
j
- oc < n < oc
4.49 Sketch roughly the m agnitude |A"u<>)! of the Fourier transforms corresponding to the
pole-zero patterns given in Fig. P4.49.
Figure P4.49
4.50 Design an F IR filter that completely blocks the frequency a*, = rr/4 and then compute
its output if the input is
x(n)
=
^sin
j u(n)
for n = 0, 1 , 2, 3, 4. Does the filter fulfill your expectations? Explain.
382
Frequency Analysis of Signals and Systems
Chap. 4
4.51 A digital filter is characterized by the following properties:
(1) It is highpass and has one pole and one zero.
(2) The pole is at a distance r = 0.9 from the origin of the ; -plane.
(3) Constant signals do not pass through the system.
(a) Plot the pole-zero pattern of the filtei and determ ine its system function H(z).
(b) Compute the m agnitude response \H(cu}\ and the phase response ^ H( t n) of the
filter,
(c) Normalize the frequency response H( cd) so that
= 1.
(d) Determ ine the input-output relation (difference equation) of the filter in the time
domain.
(e) Compute the output of the system if the input is
— ex: < n < oc
(You can use either algebraic or geom etncal arguments.)
4.52 A causal first-order digital filter is described by the system function
(a) Sketch the direct form I and direct form II realizations of this filter and find the
corresponding difference equations.
(b) For a = 0.5 and b — —0.6, sketch the pole-zero pattern, is the system stable?
Why?
(c) For a = —0.5 and b = 0.5, determ ine bu. so that the maximum value of \H(w)\ is
equal to 1 .
(d) Sketch the m agnitude response \H{co)\ and the phase response 2t//(o>) of the
filter obtained in part (c).
(e) In a specific application it is known that a — 0.8. Does the resulting filter amplify
high frequencies or low frequencies in the input? Choose the value of b so as to
improve the characteristics of this filter (i.e., m ake it a better lowpass or a better
highpass filter).
4.53 Derive the expression for the resonant frequency of a two-pole filter with poles at
Pi = r e ’6 and p 2 = p*x. given by (4.5.25).
4.54 Determ ine and sketch the magnitude and phase responses of the Hanning filter char­
acterized by the (moving average) difference equation
y(n) ~ jx (n ) + \ x( n - 1 ) + j*(« - 2 )
4.55 A causal LTI system excited by the input
x(rt') = (j)"«(n) + u( —n - 1)
produces an output v(n) with r-transform
_ 3 „-l
(a) D eterm ine the system function H(z) and its ROC.
(b) Determ ine the output y(n) of the system.
(Hint: Pole cancellation increases the original ROC.)
Chap. 4
383
Problems
4.56 Determ ine the coefficients of a linear-phase FIR filter
v(n) = b^x(n) + btx(n —1) + thx(n — 2)
such that:
(a) It rejects completely a frequency com ponent at a* = 2jt/3.
(b) Its frequency response is normalized so that H{0) ~ 1.
(c) Compute and sketch the magnitude and phase response of the filter to check if
it satisfies the requirements.
4.57 D eterm ine the frequency response H(w) of the following moving average filters.
-v(") = 2F t t X
> - * )
1
1
J
(b) y(ff) = —r-x(n + Af) + —
Y
x(n - k) + ~r~rzx(n - M)
AM
2M k=- M + i
AM
Which filter provides better smoothing? Why?
4.58 The convolution *(r) of two continuous-time signals *((/) and x2(t), from which at
least one is nonperiodic, is defined by
x(t) = X] (/) * xj(t) — j
x\ {X)x2(t — k)dk
(a) Show that X(F) = X\ ( F) X2(F), where X, (F) and X 2(F) are the spectra of *1 (r)
and a:(/), respectively.
= ( I'
< X
p' ■
10.
elsewhere
(c) Determ ine the spectrum of x(t) using the results in part (a).
<b> Com pute .r(r) if jrt(r) =
4.59 Com pute the magnitude and phase response of a filter with system function
H(z) = l + c“ ‘
If the sampling frequency is F, = 1 kHz, determ ine the frequencies of the analog
sinusoids that cannot pass through the filter.
4.60 A second-order system has a double pole at p li2 = 0.5 and two zeros at
Z\.2 = e±s3*,A
Using geom etric arguments, choose the gain G of the filter so that |tf(0 )| = 1.
4.61 In this problem we consider the effect of a single zero on the frequency response of
a system. Let ; = re’9 be a zero inside the unit circle (r < 1). Then
H,(w) = 1 —re]0e~iu>
= 1 —r cos(a) —B) + j r sin(ti) —(?)
(a) Show that the m agnitude response is
\HZ((D)\ = [ 1 - 2 r cosito - 6 ) + r 2f n
or, equivalently,
201og10 |//:(cd)| = 101og10[l —2r cos{ct> —6) + r2]
384
Frequency Analysis of Signals and Systems
Chap. 4
(b) Show that the phase response is given as
0 : (ct>) = tan
-i
rsin(w - 8)
------------------1 - r cos(ct) -
(c) Show that the group delay is given as
r - rcos(w - 8)
1 + r1 — 2r COS(oj —■
(d) Plot the magnitude
(«)]dB. the phase ©( oj) and the group delay
and 0 = 0, n j 2, and n.
for r = 0.7
4.62 In this problem we consider the effect of a single pole on the frequency response of
a system. Hence, we let
«'•<»> =
' ■=1
Show that
[Wr (a>)ljB = ~ |W-(fi))|dD
4 //„ (u > ) = - Z - H . ( c o )
where H:(<x>) and
are defined in Problem 4.61.
4.63 In this problem we consider the effect of complex-conjugate pair of poles and zeros
on the frequency response of a system. Let
H.{u>) = (1 - reJ*'e~J'")(l - re~il'e~-"“)
(a) Show that the magnitude response in decibels is
| W - M | d B
—
1 0 1 o g 1()[ l
+
-
2 r
c o s ( o )
-
# ) ]
+ 101og,„[l + r 2 - 2r cos(a) + 6)]
(b) Show that the phase response is given as
®: (co) = tan '
rsinUo —8)
l-r c o s (tu -# )
_■
Asin(w + fi)
l- r c o s ( o > + 0)
(c) Show that the group delay is given as
r2 — r cos(o> — 8)
r2 — r c o s ( c j + 8)
= — — r— — — ------ — +
1 + r 2 - 2r cos(to — 9 )
1 4- r2 - 2r cos(a> +
(d) If Hp(w) = l / H : (a>), show that
&p(w) - -© .(oj)
Tg(a>) =
(e) Plot
®p(ci>) and
(w) for r = 0.9, and 8 — 0, jt/2.
Chap. 4
385
Problems
4.64 D e te rm in e th e 3-dB b a n d w id th o f the filte rs (0 < a < 1)
Which is a better lowpass filter?
4.65 Design a digital oscillator with adjustable phase, that is. a digital filter which produces
the signal
y(n) = cos(<W|,« + ft)u(n)
4.66 This problem provides another derivation of the structure for the coupled-form os­
cillator by considering the system
y i n ) = av(ri — 1 ) 4 xi / i )
for a — e' “".
Let xi/i) be real. Then yi n} is complex. Thus
yin) =
+
./'/(«)
(a) Determ ine the equations describing a system with one input x i n) and the two
outputs y t i U t ) and y f Ui ) .
(b) Determ ine a block diagram realization
(c) Show that if ,v(«) = bin), then
y^in) = (cosw()'i ii/tn I
y/ {ti) = (sin mull )u(n)
(d) Compute ynin), \/in). n = 0. 1....... 9 for &j(, = tt/6. Compare these with the true
values of the sine and cosine,
4.67 Consider a filter with system function
(1 -
1 -
H(Z) = h\ —--------:------j--- ;-------------- —r( 1 —reJU
I"z )(1 —re~,w"z )
(a) Sketch the pole-zero pattern.
<b) Using geom etric arguments, show that for r ^ 1, the system is a notch filter and
provide a rough sketch of its m agnitude response if ttx> = 60 .
(c) For (i>n = 60 , choose bo so that the maximum value of \Hia>)\ is 1.
(d) Draw a direct form II realization of the system
(e) D eterm ine the approximate 3-dB bandwidth of the system.
4.68* Design an FIR digital filter that will reject a very strong 60-Hz sinusoidal interference
contaminating a 200-Hz useful sinusoidal signal. Determ ine the gain of the filter so
that the useful signal does not change amplitude. The filter works at a sampling
frequency F, = 500 samples/s. Compute the output of the filter if the input is a 60-Hz
sinusoid or a 200-Hz sinusoid with unit amplitude. How does the perform ance of the
filter compare with your requirem ents?
4.69 Determ ine the gain bo for the digital resonator described by (4.5.28) so that
I//(«n) I = 1.
386
Frequency Analysis of Signals and Systems
Chap. 4
4.70 D em onstrate that the difference equation given in (4.5.52) can be obtained by apply­
ing the trigonometric identity
a + [i
a —
cos or + cos p = 2 cos —-— cos —-—
where a — (n-t-Dwu, ~ (n — l)a>o, and v(n) = coswon. Thus show that the sinusoidal
signal x(n) — A co sco^n can be generated from (4.5.52) by use of the initial conditions
v(—1) = A cos an) and y (—2) = j4cos2a>o.
4.71 Use the trigonometric identity in (4.5.53) with a = na>o and (i = (n —2)a^t to derive the
difference equation for generating the sinusoidal signal y(n) = A sin no*,. Determine
the corresponding initial conditions.
4.72 Using the --transform pairs 8 and 9 in Table 3.3. determine the difference equations
for the digital oscillators that have impulse responses h(n) = A cosna>it«(n) and h(n) =
A sin ncDf>u(n), respectively.
4.73 Determ ine the structure for the coupled-form oscillator by combining the structure
for the digital oscillators obtained in Problem 4.72.
4.74 Convert the highpass filter with system function
H(z) =
—az
a < 1
into a notch filter that rejects the frequency a*, = tt/ 4 and its harmonics.
(a) Determ ine the difference equation.
(b) Sketch the pole-zero pattern.
(c) Sketch the magnitude response for both filters.
4.75 Choose L and M for a lunar niter that must have narrow passbands at (k ± AF)
cycles/dav. where k = i. 2, 3 ,... and A F = 0.067726.
4.76 (a) Show that the systems corresponding to the pole-zero patterns of Fig. 4.58 are
all-pass.
(b) What is the number of delays and multipliers required for the efficient implemen­
tation of a second-order all-pass system?
4.77 A digital notch filter is required to remove an undesirable 60-Hz hum associated with
a power supply in an ECG recording application. The sampling frequency used is
Fs = 500 samples/s. (a) Design a second-order FIR notch filter and (b) a secondorder pole-zero notch filter for this purpose. In both cases choose the gain by so that
\H(w)\ = 1 for w = 0.
4.78 Determ ine the coefficients {/?(«)} of a highpass linear phase FIR filter of length M =
4 which has an antisymmetric unit sample response h{n) = —h(M - I - n) and a
frequency response that satisfies the condition
4.79 In an attem pt to design a four-pole bandpass digital filter with desired magnitude
response
0,
elsewhere
Chap. 4
Problems
387
we select the four poles at
/>,, = 0 .8 e-MT"
and four zeros at
(a) Determ ine the value of the gain so that
HDb
(b) Determ ine the system function H(:).
(c) Determ ine the magnitude of the frequency response H(a>) for 0 < a> < tt and
compare it with the desired response \Hd(w)[.
4.80 A discrete*time system with input
and output v ( n ) is described in the frequency
domain by the relation
V(oj) — v
dXiuj)
~ 1 X (as) -t- —
— •—
das
(a) Compute the response of the svstem to the input x i n ) =
(b ) C heck if the system is LTI and stable.
4.81 Consider an ideal lowpass filter with impulse response It in) and frequency response
I 1.
K"! 5 a),
I 0,
CIJ,
Hiw) = |
<
|< D | <
TT
What is the frequency response of the iiher defined by
X<n) =
h
f).
,
n ev e n
n odd
4.82 Consider the system shown in Fig. P4.S2. Determ ine its impulse response and its
frequency response if the system H(a>) is:
(a) Lowpass with cutoff frequency w,.
(b) Highpass with cutoff frequency o j , .
xi n )
Hiw)
I'
1
4.83 Frequency inverters have been used for many years for speech scrambling. Indeed,
a voice signal x(n) becomes unintelligible if we invert its spectrum as shown in
Fig. P4.S3.
(a) Determ ine how' frequency inversion can be perform ed in the time domain.
(b) Design an unscrambler. (Hint: The required operations are very simple and can
easily be done in real time.)
388
Frequency Analysis of Signals and Systems
Chap. 4
X(a>)
-x
n
0
(b)
Figure P4.83
(a) Original spectrum;
(b) frequency-inverted spectrum.
4.84 A lowpass filter is described by the difference equation
v(n) = 0.9v(/i - 1) + 0.1x(n)
(a) By performing a frequency translation of n f l . transform the filter into a bandpass
filter.
(b) What is the impulse response of the bandpass filter?
(c) What is the major problem with the frequency translation m ethod for transform­
ing a prototype lowpass filter into a bandpass filter?
4.85 Consider a system with a real-valued impulse response h(n) and frequency response
H(w) = \H (a>)\em -’'
The quantity
D =
2 2 ” 2^ ( n )
n = ~ oc
provides a m easure of the “effective duration” of h(n).
(a) Express D in terms of H(w).
(b) Show that D is minimized for 0(a>) = 0.
4.86 Consider the lowpass filter
v(n) = a y { n — 1) + fejr(n)
0 < a < 1
(a) D eterm ine b so that |//{0)| = 1.
(b) D eterm ine the 3-dB bandwidth an, for the normalized filter in part (a).
(c) How does the choice of the param eter a affect uyf!
<d) Repeat parts (a) through (c) for the highpass filter obtained by choosing - 1 <
a < 0.
4.87 Sketch the magnitude and phase response of the m ultipath channel
y (n ) = x ( n ) + a jr(n — M )
fo r a < < 1.
a > 0
Chap. 4
389
Problems
4.88 Determ ine the system functions and the pole-zero locations for the systems shown in
Fig. P4.88(a) through (c). and indicate w hether or not the systems are stable.
l
lc)
Figure P4.88
4.89 D eterm ine and sketch the impulse response and the magnitude and phase responses
of the FIR filter shown in Fig. P4.89 for b = 1 and b = -1 .
4.90 Consider the system
v(n) = x(n) - 0.95.r(fl - 6)
(a) Sketch its pole-zero pattern.
(b) Sketch its magnitude response using the pole-zero plot.
(c) D eterm ine the system function of its causal inverse system.
(d) Sketch the magnitude response of the inverse system using the pole-zero plot.
4.91 Determ ine the impulse response and the difference equation for all possible systems
specified by the system functions
390
Frequency Analysis of Signals and Systems
Chap. 4
(a) H( Z) = 1 _ rZ_ ' _ ,_2
0 » " W = ! _ eL z-4
0< « < 1
4.92 D eterm ine the impulse response of a causal LTI system which produces the response
v(n) = {1. - 1 .3 . - 1 ,6 )
t
when excited by the input signal
x(n) = (1,1,2}
t
4.93 The system
y(n) = i_v(n - 1) + x(n)
is excited with the input
x(n) = (j)"«(n)
Determ ine the sequences aj t (/), rhh(l), r,y(i). and rvy(l).
4.94 Determ ine if the following FIR systems are minimum phase.
(a) h(n) = {10,9. - 7 , - 8 , 0 , 5 . 3}
r
(b) h(n) = (5,4, - 3 . - 4 , 0 ,2 ,1 )
t
4.95 Can you determ ine the coefficients of the all-pole svstem
*=i
if you know its order N and the values /?(0), /i(l)........h ( L - l ) of its impulse response?
How? W hat happens if you do not know N1
4.96 Consider a system with impulse response
h(n) = baS(n) + bi$(n — D) + ^<5(n —2D)
(a ) Explain why the system generates echoes spaced D samples apart.
(b) Determ ine the m agnitude and phase response of the system.
(c) Show that for \b0 + b2\ < < |£>]|, the locations of maxima and minima of \H(a>)
are at
k
w = ± —n
k = 0. 1, 2. . . .
D
(d) Plot |//(w )| and
for b$ — 0.1, b\ = 1, and b2 = 0.05 and discuss the results.
4.97 Consider the pole-zero system
B(z)
1 + bz * v—'
W(z) = A(z) = T
1T
+ -----a z 1r = Y
“ l h^ z
(a) Determine >i(0), >i(l), h(2), and h(3) in terms of a and b.
(b) Let rhH(l) be the autocorrelation sequence of h(n). D eterm ine rkh{0), /-^(l), rw,(2),
and r/,h(3) in terms of a and b.
Chap. 4
391
Problems
4.98 Let
be a real-valued minimum-phase sequence. Modify xin I to obtain another
real-valued minimum-phase sequence y(/i) such that y(0) = x(0) and y(n) = |jr(n)|.
4.99 The frequency response of a stable LTI system is known to be real and even. Is the
inverse system stable?
4.100 Let h(n) be a real filter with nonzero linear or nonlinear phase response. Show that
the following operations are equivalent to filtering the signal x(n) with a zero-phase
filter.
(a) g(n) — Ji(n) * ,v(n)
/ (h ) = h (/ t ] * g t - i i )
yon = f( ~n)
(b)
g(n) — h(n) *
/ (r?) = h{n) * x ( —n)
y(nl = ain) + / ( - i i )
(Hint: D eterm ine the frequency response of the composite system _v(/i) = //[.v(«)].)
4.101 Cheek the validity of the following statements:
(a) The convolution of two minimum-phase sequences is always mimmum-phase se­
quence.
(h) The sum of two minimum-phase sequences is always minimum phase.
4.102 Determ ine the minimum-phase syslem whose squared magnitude response is given
by:
- cos w
^
(a) i H(</)}'- =
(b )
I H (u>)',: =
(i
a - ) - 2a cos w
4.103 Consider an FIR syslem with the following system function:
H(z) = (1 - 0 . 8 r '7V ' K l
- l..v " T,V 1HI - L S ^ ' V 1)
(a) Determ ine all systems that have the same magnitude response.
minimum-phase system?
(b) Deierm ine the impulse response of al) systems in part (a).
(c) Plot the partial energy
Which is the
i=t>
for every system and use il lo identify the minimum- and maximum-phase systems.
4.104 The causal system
//(.-) =
,v
392
Frequency Analysis of Signals and Systems
Chap, 4
(a) Show that by properly choosing A we can obtain a new stable system.
(b) What is the difference equation describing the new system?
4.105* Given a signal x(n), we can create echoes and reverberations by delaying and scaling
the signal as follows
>’(«) = 22 Skx (n ~ kD)
where D is positive integer and gk > g i —i > 0.
(a) Explain why the comb filter
H(z) =
1-az-°
can be used as a reverberator (i.e.. as a device to produce artificial reverberations).
(Hint: Determ ine and sketch its impulse response.)
(b) The all-pass comb filter
H(z) =
-a
1 - a z - 1’
is used in practice to build digital reverberators by cascading three to five such
filters and properly choosing the param eters a and D. Com pute and plot the
impulse response of two such reverberators each obtained by cascading three
sections with the following parameters.
UNIT 1
UNIT 2
Seclion
I)
a
Section
D
a
1
2
3
50
40
32
0.7
0.665
0.63175
1
2
3
50
17
6
0.7
0.77
0.847
(c) The difference between echo and reverberation is that with pure echo there are
clear repetitions of the signal, but with reverberations, there are not. How is this
reflected in the shape of the impulse response of the reverberator? Which unit
in part (b) is a better reverberator?
(d) If the delays D\, D2, Dj in a certain unit are prime numbers, the impulse response
of the unit is more “dense." Explain why.
(e) Plot the phase response of units 1 and 2 and comm ent on them.
(f) Plot h(n) for D |, D2, and
being nonprime. W hat do you notice?
More details about this application can be found in a paper by J. A. M oorer, “Signal
Processing Aspects of Com puter Music: A Survey," Proc, IEEE, vol. 65, No. 8, Aug.
1977, pp. 1108-1137.
4.106* By trial-and-error design a third-order lowpass filter with cutoff frequency at wc = Jt/9
radians/sample interval. Start your search with
(a) zi = Z2 = Z3 = 0, pi = r, p2J = re±,tu' . r = 0.8
(b) r = 0.9, zi = Z2 = Z3 = - 1
Chap. 4
393
Problems
4.107* A speech signal with bandwidth B = 10 kHz is sampled at F2 = 20 kHz. Suppose
that the signal is corrupted by four sinusoids with frequencies
F, = 10, 000 Hz.
F3 = 7778 Hz
F2 = 8889 Hz,
F4 = 6667 Hz
(a) Design a FIR filter that eliminates these frequency components.
(b) Choose the gain of the filter so that |H (0)| = 1 and then plot the log m agnitude
response and the phase response of the filter.
(c) Does the filter fulfill your objectives? Do you recom m end the use of this filter in
a practical application?
4.108* Com pute and sketch the frequency response of a digital resonator with co = t t / 6 and
r = 0.6, 0.9, 0.99. In each case, compute the bandwidth and the resonance frequency
from the graph, and check if they are in agreem ent with the theoretical results.
4.109* The system function of a communication channel is given by
H(z) = (1
- 0 . 9 f - 'u4'T;;-1)(l - 1.5^'afor; - 1)(l - 1.5<’- '" tez - 1)
D eterm ine the system function Ht.(;) of a causal and stable compensating system so
that the cascade interconnection of the two systems has a flat magnitude response.
Sketch the pole-zero plots and the m agnitude and phase responses of all systems in­
volved into the analysis process. [Hint: Use the decomposition H(z) = Wap(;) Wn,in(;)-]
The Discrete Fourier
Transform: Its Properties and
Applications
F requency analysis o f discrete-tim e signals is usually and most con ven ien tly per­
form ed on a digital signal processor, which may be a general-purpose digital com ­
puter or specially d esigned digital hardware. T o perform frequency analysis on a
d iscrete-tim e signal
w e convert the tim e-dom ain sequ en ce to an equivalent
frequencv-dom ain representation. We know that such a representation is given by
the Fourier transform X(cu) of the seq u en ce |a(>;)). H ow ever, A'u<j) is a contin­
uous function o f frequency and therefore, it is not a com p utationally convenient
representation o f the sequence {.v(/i)).
In this section we consider the representation o f a sequ en ce
by sam ples
o f its spectrum X ( uj). Such a frequency-dom ain representation lead s to the discrete
Fourier transform (D F T ), which is a pow erful com putational tool for perform ing
frequency analysis o f discrete-tim e signals.
5.1 FREQUENCY DOMAIN SAMPLING: THE DISCRETE FOURIER
TRANSFORM
B efore w e introduce the D F T , w e consider the sam pling of the F ourier transform of
an aperiodic discrete-tim e sequ en ce. Thus, w e establish the relationship betw een
the sam pled Fourier transform and the D FT.
5.1.1 Frequency-Domain Sampling and Reconstruction of
Discrete-Time Signals
W e recall that aperiodic finite-energy signals have con tinu ou s spectra. Let us
consider such an aperiodic discrete-tim e signal x ( n ) with Fourier transform
(5.1.1)
394
Sec. 5.1
Frequency Domain Sampling: The Discrete Fourier Transform
395
S uppose that w e sam ple X(a>) periodically in frequency at a spacing o f Sco radians
b etw een successive sam ples. Since X(a>) is period ic with period 2tt, only sam ples
in the fundam ental frequency range are necessary. For con ven ien ce, w e take N
equidistant sam ples in the interval 0 < w < 2tt with spacing Sco = 2 ttf N , as shown
in Fig. 5.1. First, w e consider the selection of N , the num ber o f sam ples in the
frequency dom ain.
If w e evaluate (5.1.1) at u> = 2 n k / N , w e obtain
(5.1.2)
T h e sum m ation in (5.1.2) can be subdivided in to an infinite n um ber of sum m ations,
where each sum con tains N term s. Thus
-i
oc
N-1
IN + N - 1
-)2nkn/\
}=-~yL f}=i A‘
If we change the index in the inner sum m ation from n to n — I N and interchange
the order o f the sum m ation, we obtain the result
(5.1.3)
for k = 0, 1, 2 .........N — 1.
T h e signal
OC
(5.1.4)
ob tain ed by the periodic repetition o f x{n) every N sam ples, is clearly periodic
with fundam ental period N. C onsequently, it can be expanded in a Fourier
0
F igure 5.1
kSa)
tt
Swh
Frequency-dom ain sam pling of the F ourier transform .
396
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
series as
A '-l
x p (n) = Y ^ c ke,2nkn/N
n = 0 , 1 .........N - 1
(5.1.5)
k=Q
with Fourier coefficients
l A '-l
k = 0 , 1 ........ N — 1
ck = - - ' £ x p ( n ) e - j2,rkn/N
N n=0
(5.1.6)
U p on com paring (5.1.3) with (5.1.6), we conclude that
1
f 2* \
ck = — X l — k )
N
\ N J
k = 0 . 1.........A ' - l
(5.1.7)
J
T herefore,
1 ^ ~ ^ / ^7T \
x.,(n) = — ' y * ( — * ) e llnkntN
\ N /
n = 0, 1........ A ' - l
(5.1.8)
The relationship in (5.1.8) provides the reconstruction of the periodic signal
x r (u) from the sam ples of the spectrum X ( oj). H ow ever, it d o es not imply that
we can recover X(u>) or x{n) from the sam ples. T o accom plish this, we need to
consider the relationship betw een x p (n) and j:(«).
Since x p(n) is the periodic exten sion of x{n) as given by (5.1.4). it is clear
that x (/ j) can be recovered from x p (n) if there is no aliasing in the tim e domain,
that is, if x( n) is tim e-lim ited to less than the period N of x p(n). This situation is
illustrated in Fig. 5.2, w here without loss of generality, we consider a finite-duration
rtn)
0
Hitt,....
L
xp(n)
N > L
ITIitttt
IITIt t t .. JTTTTttt
0
L
N
N< L
•
IT- I TNt t t O I TN T t
TTTt 1-
Figure 5.2 Aperiodic sequence x(n) of length L and its periodic extension for
N > L (no aliasing) and N < L (aliasing).
Sec. 5.1
Frequency Domain Sampling: The Discrete Fourier Transform
397
sequ en ce x( n) , which is n on zero in the interval 0 < n < L — 1. W e ob serve that
when N > L.
x(ti) = Xp(n)
0 < n < N —1
so that x ( n ) can be recovered from x r (n) w ithout am biguity. On the other hand,
if N < L, it is not possib le to recover
from its periodic exten sion due to timed o m a i n aliasing. Thus, w e conclude that the spectrum of an aperiodic discrete-tim e
signal with finite duration L . can be exactly recovered from its sam ples at frequen­
cies tot: = 2j rk/ N. if N > L. The procedure is to com pute x p (n). n = 0, 1.........N - 1
from (5.1.8); then
0 < n < N
elsew h ere
and finally, X(io) can be com p uted from (5.1.1).
A s in the case o f con tinu ou s-tim c signals, it is p ossible to express the spectrum
X ( w ) directly in term s o f its sam ples X Q n k / N ) , k = 0. 1........ N — 1. T o derive
such an interpolation formula for X{co), we assum e that N > L and begin with
(5.1.8). Since x( n) = x,,(n) for 0 < /; < A' - 1,
A
k ]
()<//< N - I
(5.1.1(1)
If w e use (5.1.1) and substitute for v(/;), we obtain
A'-I
X(a>) =
n = (l
(5.1.11)
A '-l
= 2>
The inner sum m ation term in the brackets o f (5.1.11) represents the basic
interpolation function shifted by 2 ttk / N in frequency. Indeed, if w e define
v/tuA-'
N 1 - e-J°‘
sin(o>Ar/2 )
N sin(w /2)
then (5.1.11) can be expressed as
jiuiN
(5.1.12)
398
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
The interpolation function P(co) is not the familiar ( si n 8 ) / 6 but instead, it
is a periodic counterpart of it, and it is due to the periodic nature of A’(co). The
phase shift in (5.1.12) reflects the fact that the signal j:(n) is a causal, finite-duration
seq u en ce o f length N. The function sln( a>N/ 2)/ ( N sin(a>/2)) is plotted in Fig. 5.3
for N = 5. We observe that the function P(to) has the property
0
k =
* = 1 .2 .........A ' - l
(5.1.14)
C onsequently, the interpolation form ula in (5.1,13) gives exactly the sam ple val­
ues X ( 2 j r k / N ) for oj = 2 i r k / N . A t all other freq u en cies, the form ula provides a
properly w eighted linear com bination o f the original spectral sam ples.
T h e follow ing exam ple illustrates the frequency-dom ain sam pling of a
discrete-tim e signal and the tim e-dom ain aliasing that results.
Example 5.1.1
Consider the sisinal
xin ) = a"uin)
0 < a < 1
The spectrum of this signal is sampled at frequencies a>t = l i r k / S . k — 0. 1....... A '- l.
Determ ine the reconstructed spectra for a = 0.8 when A' = 5 and N = 50.
Solution
The Fourier transform of the sequence x(n) is
Suppose thai we sample X(w) at N equidistant frequencies w*. = 2 x k / N , k = 0,
A' - 1. Thus we obtain the spectral samples
Xiw)
1.0
JV= 5
Figure S.3
Plot o f the function
[sin(ct>W/2)]/[jty sin(tu/2)].
Sec. 5.1
Frequency Domain Sampling: The Discrete Fourier Transform
399
The periodic sequence xp(n), corresponding to the frequency samples X ( 2 n k / N ) ,
k = 0, 1 , . . . . N — 1, can be obtained from either (5.1.4) or (5.1.8). Hence
0
oc
x p(n) = ^
jc(n - IN) = ^
a”~IN
0 <n < N - 1
where the factor 1/(1 - a N) represents the effect of aliasing. Since 0 < a < 1, the
aliasing error tends toward zero as N -+ oo.
For a = 0.8, the sequence x(n) and its spectrum X{w) are shown in Fig. 5,4a
and b, respectively. The aliased sequences xp(n) for N = 5 and N = 50 and the
corresponding spectral samples are shown in Fig. 5.4c and d, respectively. We note
that the aliasing effects are negligible for N = 50.
If we define the aliased finite-duration sequence x(n) as
xp{n),
0,
0 < n < N —1
otherwise
then its Fourier transform is
X(w) = y ^ i ( n ) e
1
1 —a N
1
1 —ae~,w
Note that although X(ou) ^ X(a>), the sample values at a>t = I n k f N are identical.
That is,
1
1- a N
1-a N
l - a e - ' 2”1"
5.1.2 The Discrete Fourier Transform (DFT)
The d ev elo p m en t in the preceding section is concerned with the frequency-dom ain
sam pling o f an aperiodic finite-energy sequ en ce j:(n). In general, the equally
spaced freq u en cy sam ples X (2n k / N ) , k = 0 , 1 ___ _ N — 1, do n ot uniquely represent
the original seq u en ce x ( n ) when x(n) has infinite duration. Instead, the frequency
sam ples X ( 2 n k / N ) , k = 0, 1........ N - I, correspond to a period ic seq u en ce x p(n)
o f period N , w here x p (n) is an aliased version o f *{«), as indicated by the relation
in (5.1.4), that is,
(5.1.15)
W hen the seq u en ce x ( n ) has a finite duration o f length L < N , then x p{n)
is sim ply a periodic repetition o f x ( n ) , w h ere x p (n) over a single period is
400
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
IAWH
1.0?
Tfrrmnx.
(a)
(b)
T
x(n)
1.0*
x \ f k)
1W
50
50
(1
fdi
Figure 5.4 (a) Plot of sequence xin) = (0.X)"h (h ): (b) its Fourier transform (magnitude
only): (c) effect of aliasing with A' = 5: (d) reduced effect of aliasing with A' = 50.
given as
x r (n) =
x( n) .
0.
0 < n < L —1
L < n < N —1
(5.1.16)
C onsequently, the frequency sam ples X ( 2 i r k / N ) , k = 0. 1.........N — 1, uniquely
represent the finite-duration seq u en ce *{/;)■ Since x ( n ) = x p (n) over a single pe­
riod (padded by N — L zeros), the original finite-duration seq u en ce x ( n ) can be
obtained from the frequency sam ples \ X ( 2 n k / N \ by m eans o f the formula (5.1.8)It is im portant to n ote that zero p a d d i n g d oes not provide any additional
inform ation about the spectrum X(a>) o f the seq u en ce \x(n)}. T he L equidis-
Sec. 5.1
Frequency Domain Sampling: The Discrete Fourier Transform
401
tant sam ples o f X(u>) are sufficient to reconstruct X(co) using the reconstruction
form ula (5.1.13). H o w ever, padding the seq u en ce {jc(« )} with N — L zeros and
com puting an A'-point D F T results in a “better display" of the Fourier transform
X(o>).
In sum m ary, a finite-duration seq u en ce x( n ) o f length L [i.e., x(n ) = 0 for
n < 0 and n > L\ has a Fourier transform
JL—1
X(u>) = ^ 2 x ( n ) e ~ Jwn
n=0
0 < a> < 2jt
(5.1.17)
where the upper and low er indices in the sum m ation reflect the fact that x ( n ) = 0
outside the range 0 < n < L — 1. W hen w e sam ple X{a>) at equally spaced
freq u en cies u>k = 2 n k / N . k ~ 0, 1, 2 ........ N — 1, where N > L. the resultant
sam ples are
X(k) = X
= J 2 x ( n ) e ^ j27,k"/N
N_ ) N 7
"=°
X(k) = y
x ( n ) e - p^ h,IN
(5.1.18)
k = 0, 1, 2, , . . , N - 1
n=<l
where for co n v en ien ce, the upper index in the sum has been increased from L — 1
to /V - 1 since x ( n ) = 0 for n > L.
T he relation in (5.1.18) is a form ula for transform ing a seq u en ce {jc(«)} of
length L < N in to a seq u en ce o f frequency sam ples ( X(£)) o f length N . Since
the frequency sam ples are ob tain ed by evaluating the Fourier transform X (a>)
at a set o f N (eq ually spaced) discrete frequencies, the relation in (5.1.18) is
called the discrete Fouri er t rans f or m (D F T ) o f jc(«). In turn, the relation given
by (5.1.10), which allow s us to recover the seq u en ce jr(n) from the frequency
sam ples
1 A'-l
x ( n ) = - £ X f c ) e ,23tknlN
n = 0 . 1 .........N ~ 1
(5.1.19)
is called the i nverse D F T (ID F T ). Clearly, when x ( n ) has length L < N , the Ap­
point ID F T yield s x (n ) = 0 for L < n < N — 1. T o sum m arize, the form ulas for
the D F T and ID F T are
DFT
A'-l
X ( k ) = J ^ x ( n ) e - j2nkn/N
^7—0
k = 0 , 1, 2,
(5.1.18)
IDFT
, A'-l
x( n) = — Y * X ( k ) e JZnkn/N
N
*=o
n = 0 , 1 , 2 .........N - 1
(5.1.19)
402
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
Example 5.1.2
A finite-duration sequence of length L is given as
x(n) — I
1 0,
0 < n <L -l
otherwise
Determine the /V-point DFT of this sequence for N > L.
Solution
The Fourier transform of this sequence is
£-1
L-l
The magnitude and phase of X (w) are illustrated in Fig. 5.5 for L = 10. The Appoint
DFT of x (n) is simply X(w) evaluated at the set of N equally spaced frequencies
wt = 2it k / N , k = 0, 1........N — 1. Hence
p-jkrkLiN
sm(*kL/N)
, - j x k l L - 1 I/A/
sm(i t k/ N)
IX(w)l
1"
2
7C
—
2
2n
W
|H 0
0(a>)
TT
ID
Figure SS Magnitude and phase
characteristics of the Fourier transform
for signal in Exam ple 5.1.2.
Sec. 5.1
Frequency Domain Sampling: The Discrete Fourier Transform
403
If N is selected such that N = L, then the D FT becomes
L,
* (* ) = 10.
k =0
* = 1 .2 ........L - l
Thus there is only one nonzero value in the DFT. This is apparent from obser­
vation of X(w), since X(a>) = 0 at the frequencies an = I n k f L , k ^ 0. The
reader should verify that x(n) can be recovered from X(k) by perform ing an Z.-point
IDFT.
Although the L-point D FT is sufficient to uniquely represent the sequence x{ n)
in the frequency domain, it is apparent that it does not provide sufficient detail to yield
a good picture of the spectral characteristics of x(n). If we wish to have better picture,
we must evaluate (interpolate) X(a>) at more closely spaced frequencies, say wt, =
2n k / N , where N > L. In effect, we can view this com putation as expanding the size
of the sequence from L points to N points by appending A '- L zeros to the sequence
x ( n ). that is, zero padding. Then the A*-point DFT provides finer interpolation than
the L-point DFT.
Figure 5.6 provides a plot of the Appoint DFT, in magnitude and phase, for
L = 10, N = 50, and N = 100. Now the spectral characteristics of the sequence
are more clearly evident, as one will conclude by comparing these spectra with the
continuous spectrum XUo).
5.1.3 The DFT as a Linear Transformation
T he form ulas for the D F T and ID F T given by (5.1.18) and (5.1.19) m ay be ex ­
pressed as
N -1
X ( k ) = J ^ j r O i) ^ "
FT—(I
] JV-1
x( n) = — J ] x a ) ^ in
^ k=o
A- = 0 , 1 , . . . , A/ — 1
n = 0 , 1 .........N - 1
(5.1.20)
(5.1.21)
w here, by definition,
W N = e ~ ^ IN
(5.1.22)
which is an N th root o f unity.
W e n o te that the com putation o f each point o f th e D F T can be accom plished
by N com p lex m ultiplications and ( N ~ 1) com plex additions. H en ce the W-point
D F T valu es can b e com p uted in a total o f N 2 com p lex m ultiplications and N ( N —1)
com p lex additions.
It is instructive to view the D F T and ID F T as linear transform ations on
seq u en ces {jc(«)} and {X(fc)}, respectively. Let us d efine an Af-point vector %N o f
th e signal se q u en ce x( n) , n — 0, 1 , . . . , N — 1, an N -p oin t vector X N o f frequency
404
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
Figure 5.6 Magnitude and phase of an N-poini DFT in Example 6.4.2; (a) L =
N = 50; (b) L = 10, N = 100.
sam ples, and an N x N m atrix
as
- *(0)
X* =
■
1
x(l)
X,v =
,
1
1. ..
1
1
w*
K
•••
w*
■■■
W%N~ 0
•••
.1
<
_1
-I
- X ( N - 1 ).
'1
1!
3
*
.x(N-l).
x(0)
X ( l)
(5.1.23)
N
y y 2 ( N — 1)
(* -!)< A '-l)
Sec. 5.1
Frequency Domain Sampling: The Discrete Fourier Transform
405
W ith th ese definitions, the W-point D F T m ay b e exp ressed in matrix form as
X* =
(5.1.24)
w h ere
is the m atrix o f the linear transform ation. W e ob serve that Wjv is a
sym m etric matrix. If w e assum e that the inverse o f W * exists, then (5.1.24) can
be inverted by prem ultiplying both sid es by W ^1. T hus w e ob tain
x/v =
(5.1.25)
B u t this is just an exp ression for the ID F T .
In fact, th e ID F T as given by (5.1.21), can b e exp ressed in m atrix form as
** =
N
(5.1.26)
The Discrete Fourier Transform: Its Properties and Applications
406
where
d en o tes the com p lex conjugate o f the m atrix W A.
(5.1.26) with (5.1.25) leads us to con clu d e that
Chap. 5
C om parison of
W *1 = -W * w
(5.1.27)
W „W ;, = N l N
(5.1.28)
w hich, in turn, im plies that
w here I * is an N x N identity matrix. T h erefore, the m atrix
in the trans­
form ation is an orthogonal (unitary) m atrix. F urtherm ore, its inverse exists and
is given as W *n / N . O f course, the existen ce o f the inverse o f W,v w as established
p reviously from our derivation o f the ID FT .
Example 5.L3
Compute the D FT of the four-point sequence
1
x(n) = (0
2
3)
Solution The first step is to determ ine the matrix W4. By exploiting the periodicity
property of W4 and the symmetry property
= -v v ‘
the matrix W4 may he expressed as
~w"
w"
W< =
w" < 1
w1
1.
w44 w
9
w; w
Wl
-W«
-1
1
1
.1
1
-j
-1
j
1
- 1
1
'1
1
1
J
1
w4‘
W;
w*
1
M'i
W?
I"
W4-
1“
j
-1
-j
-
Then
T
X4 = W 4X4 =
6
-2 + 2;
-2
L-2-2JJ
The IDFT of X4 may be determ ined by conjugating the elements in W 4 to obtain WJ
and then applying the formula (5.1.26).
T h e D F T and ID F T are com putational to o ls that play a very im portant role
in m any digital signal processing applications, such as frequency analysis (spectrum
analysis) o f signals, p ow er spectrum estim ation , and lineaT filtering. T h e im por­
tance o f the D F T and ID F T in such practical ap p lication s is du e to a large extent
on the ex isten ce o f com putationally efficient algorithm s, know n co llectiv ely as fast
Sec. 5.1
Frequency Domain Sampling: The Discrete Fourier Transform
407
F ourier transform (F F T ) algorithm s, for com puting the D F T and ID F T . T his class
o f algorithm s is d escrib ed in Chapter 6.
5.1.4 Relationship of the DFT to Other Transforms
In this d iscu ssion w e h ave indicated that the D F T is an im portant com putational
to o l for perform ing frequency analysis o f signals on digital signal p rocessors. In
v iew o f the other frequency analysis to o ls and transform s that w e have d e v e l­
o p ed , it is im portant to establish the relationships b etw een the D F T to th ese other
transform s.
Relationship to the Fourier series coefficients of a periodic sequence.
A p eriod ic se q u en ce
F ourier series o f th e form
with fundam ental period N can b e represented in a
iV~l
x p (n) — ^ £kei2l,nk,N
t=o
— oo < n < oo
(5.1.29)
w h ere the F ourier series coefficients are given by the expression
1 A'-l
Ct = ~ J 2 XP(/1 )r~j2”"k/N
^ n=(l
Jt = 0 , 1 .........AT - 1
(5.1.30)
If w e com p are (5.1.29) and (5.1.30) with (5.1.18) and (5.1.19), w e ob serve that the
form ula for the F ourier series coefficients has the form o f a D F T . In fact, if we
d efine a se q u en ce x(rt) = x p(n), 0 < n < N - 1 , the D F T o f this se q u en ce is sim ply
X(k) = Nc k
(5.1.31)
F urtherm ore, (5.1.29) has the form o f an ID F T . T hus th e N -p oin t D F T provid es
the exact line spectrum o f a p eriod ic seq u en ce with fundam ental period N.
Relationship to the Fourier transform of an aperiodic sequence. W e
have already sh ow n that if -t(n) is an aperiodic finite en ergy seq u en ce w ith Fourier
transform X(a>), w h ich is sam pled at N eq u ally spaced freq u en cies
= 2n k / N ,
k = 0 , 1 , . . . , N — 1, the spectral com p onents
OO
X(k) =
=
£
x ( n ) e - j2* nk,N
k = 0,1,..., N - 1
(5.1.32)
n « —oo
are th e D F T coefficien ts o f the period ic seq u en ce o f period N , given by
OO
J;p (n) = ^
x(n-lN )
(5.1.33)
/* —OO
Thus x p (n) is d eterm in ed by aliasing {jc(n)J o ver th e interval 0 < n < N - 1. T h e
finite-duration seq u en ce
408
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
bears no resem blance to the original seq u en ce {x(n )), u nless ;c(n) is o f finite dura­
tion and length L < N, in which case
x( n) = x{n)
0 < n < N —1
(5.1.35)
O nly in this case will the ID F T o f {X(jfc)} yield the original seq u en ce {*(«)}.
Relationship to the z*transform.
Let us consider a seq u en ce x( n) having
the ^-transform
(5.1.36)
with a R O C that includes the unit circle. If X ( z ) is sam pled at the N equally
spaced points on the unit circle zt = e i2*k/ N, 0, 1, 2 , . . . , N — 1, w e obtain
X( k ) = X(z)U=, i»«t »
k = 0, 1 , . . . , N - 1
(5.1.37)
=
£
x ( n ) e ~ j2nnk/N
T he expression in (5.1.37) is identical to the Fourier transform X(io) evalu ated at
the N equally spaced freq u en cies <ot = 2 n k / N , k = 0, 1 , ___ N ~ 1, which is the
topic treated in Section 5.1.1.
If the seq u en ce x( n) has a finite duration o f length N or less, the seq u en ce can
be recovered from its /V-point D F T . H en ce its z-transform is un iqu ely determ ined
by its N -p oin t D F T . C onsequently, X ( z ) can be expressed as a function o f the
D F T fX (k)} as follow s
N~ 1
* (Z ) = ^ * ( « ) 2 “ n
n=0
/V-l
X(z) = £
n=0 L
N
*=0
-N N- 1
X ( Z) =
N
(5.1.38)
X( k)
ej2*k/Nz - i
W hen evaluated on the unit circle, (5.1.38) yield s the F ourier transform o f the
finite-duration seq u en ce in term s o f its D F T , in th e form
k=0 1
T his expression for th e Fourier transform is a p olyn om ial (L agrange) interpolation
form ula for X ( w ) exp ressed in term s o f the valu es {X (Jt)) o f th e polyn om ial at a
set o f equally spaced d iscrete freq u en cies <ok = 2 n k / N , k = 0, 1.........N - 1. With
Sec. 5.2
Properties of the DFT
409
so m e algebraic m anipulations, it is p ossib le to reduce (5.1.39) to the interpolation
form ula given p reviously in (5.1.13).
Relationship to the Fourier series coefficients of a continuous-time
signal. S uppose that xa (t) is a con tinu ou s-tim e periodic signal with fundam ental
p eriod Tp = 1 /F 0. T he signal can be expressed in a Fourier series
OC
xaU) =
CteJ2,TkF"
t = —3C
(5-1-40)
w here { q ) are the Fourier coefficients. If we sam ple xc,(t) at a uniform rate
Fs = N / T p = 1 / T , w e obtain the discrete-tim e sequ en ce
x ( n ) = x a( nT) =
y
Cl;ej2,rkF"’,T =
ckej2nt,l/N
k=—'\,
k=—cc
N - 1
J 2 x k n /N
- E
E
l~~CX.
It is clear that (5.1.41) is in the form o f an ID F T form ula, where
rv
X{ k) = N j 2 Ck-iN = N c t
/=--v
and
•V
Q = y
t'k-iN
/=-ac
(5.1.42)
(5.1.43)
T hus the {q } seq u en ce is an aliased version o f the seq u en ce (cA}.
5.2 PROPERTIES OF THE DFT
In S ection 5.1.2 w e introduced the D F T as a set o f N sam p les {X(Jt)} of the
F ourier transform X(a>) for a finite-duration seq u en ce |jr(n)} o f length L < N.
T h e sam pling o f X (to) occurs at the N equally spaced freq u en cies cd* = 2 n k / N ,
k — 0, 1, 2 .........N — 1. W e dem onstrated that the N sam p les (X(A)} uniquely
represent the seq u en ce (;c(n)} in the frequency dom ain. R ecall that the D F T and
inverse D F T (ID F T ) for an //-p o in t seq u en ce {*(«)} are given as
N- 1
DFT: X( k ) = J ^ x ^n ) W N
n=0
* - 0 , 1 .........A ' - l
(5.2.1)
1 N- 1
ID FT: jc(n) = — £ X ( k ) W ~ kn
i=0
n = 0 , 1 .........N - 1
(5.2.2)
where Wn is defined as
WN = e~ j2n,N
(5.2.3)
410
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
In this section w e present the im portant p rop erties o f the D F T . In view o f the
relationships estab lish ed in Section 5.1.4 b etw een the D F T and Fourier series,
and Fourier transform s and --transform s o f d iscrete-tim e signals, w e exp ect the
p roperties o f the D F T to resem ble the properties o f these o th er transform s and
series. H ow ever, som e im portant d ifferen ces exist, o n e o f w hich is the circular
con v o lu tio n property derived in the follow in g section . A good understanding of
these properties is extrem ely helpful in the application o f the D F T to practical
p roblem s.
T h e notation used b elow to d en o te the N -point D F T pair x ( n ) and AT(Jt) is
*(„) S
N
x(k)
5.2.1 Periodicity, Linearity, and Symmetry Properties
Periodicity.
If x(n) and X (k ) are an Af-point D F T pair, then
x(n + N) = x( n)
for all n
(5.2.4)
X( k + N) = X( k )
for all k
(5.2.5)
T h ese periodicities in .r(n) and A' (A:) follow im m ed iately from form ulas (5.2.1) and
(5.2.2) for the D F T and ID FT , respectively.
W e previously illustrated the periodicity property in the seq u en ce x (n) for a
given D FT. H ow ever, we had not p reviously view ed the D F T X( k ) as a periodic
seq u en ce. In so m e applications it is ad van tageou s to do this.
Linearity.
If
DFT
* ,(„ ) «—
X l (k)
N
and
x 2(n) K
N
X 2(k)
then for any real-valued or com p lex-valu ed con stan ts a\ and a2,
DFT
a\ X] ( n) + a 2x 2(n) <— + a {X\ ( k) + a2X 2(k)
(5.2.6)
This property follow s im m ediately from the definition o f the D F T given by (5.2.1).
Circular Symmetries of a Sequence. A s w e h ave seen , the Appoint D FT
o f a finite duration seq u en ce, x(n) o f length L < N is eq u ivalen t to the W-point
D F T o f a periodic seq u en ce xp (n), o f period N, w hich is ob tain ed by periodically
exten d ing jc(n), that is,
OO
xp (n) =
£
x( n - I N)
/=—OO
(5.2.7)
Sec. 52.
Properties of the DFT
411
N o w su p p ose that w e shift the periodic seq u en ce x p (n) by k units to the right.
Thus w e obtain a n oth er period ic sequ en ce
OC
x'p (n) = x p (n - k) =
^ x( n - k - IN)
/=-OC
T h e finite-duration seq u en ce
1 0,
otherw ise
(5.2.8)
(5.2.9)
is related to the original seq u en ce x ( n ) by a circular shift. T h is relationship is
illustrated in Fig. 5.7 for N = 4.
In gen eral, th e circular shift of the seq u en ce can b e rep resen ted as the index
m o d u lo N . T h u s w e can w rite
x'(n) = x(n — k, m odu lo N )
(5.2.10)
s x((n ~ k) ) N
For exam p le, if k = 2 and N = 4, w e have
x'(n) = * ((« - 2))4
which im plies that
jc'(0) = x ( ( - 2 ) ) 4 = x ( 2 )
x' { l ) =
jc ( ( - 1))4 =
;c ( 3 )
* '( 2 ) = jt<(0))4 = j (0)
x '( 3 )
= j t ( ( 1))4 = j t (1 )
H en ce x'(n) is sim ply x (n) shifted circularly by tw o units in tim e, w here the cou n ­
terclock w ise direction has b een arbitrarily selected as the p ositive direction. Thus
w e con clu d e that a circular shift o f an
-point seq u en ce is eq u ivalen t to a linear
shift o f its p eriod ic ex ten sion , and vice versa.
T h e in h eren t p eriod icity resulting from the arrangem ent o f the Af-point se ­
q u en ce o n th e circum ference o f a circle dictates a differen t definition o f even and
od d sym m etry, and tim e reversal o f a sequ en ce.
A n Af-point seq u en ce is called circularly even if it is sym m etric ab ou t the
p oin t zero on th e circle. T h is im plies that
x ( N —n ) = x ( n )
1 < n < N —1
(5.2.11)
A n W -point seq u en ce is called circularly o d d if it is antisym m etric ab ou t the point
zero o n the circle. T h is im plies that
x ( N — n) = —x( n)
1 < n < N —1
(5.2.12)
T h e tim e reversal o f an Af-point seq u en ce is attained by reversing its sam ples
about the p o in t ze r o on the circle. T h u s th e seq u en ce jc((—n ) ) # is sim ply given as
* ((-« ))* = x ( N - n )
0 < n < N - l
(5.2.13)
T h is tim e reversal is eq u ivalen t to plottin g j:(/i ) in a clock w ise direction on a circle.
412
The Discrete Fourier Transform: Its Properties and Applications
4<
j(n)
3
2t
'±
<a)
4'
xM
41
4
3
‘ T2J
-4
r l l '
. . ■t’
-3 - 2 - I
0
I
2
3
4
6
5
7
(b)
Jr,,(n - 2)
4<
3 <
4'
4'
■- t6 ’- 5I - 4 - 3 ,- -2 t- I" l0 1i m2 ’ i3 ’I4 5
(O
4
3
■TI
*(0)
*'(2)
<e)
Figure 5.7
Circular shift o f a sequence.
Chap. 5
Sec. 5.2
Properties of the DFT
413
A n eq u ivalen t definition o f even and od d seq u en ces for th e associated peri­
odic seq u en ce x p (n) is given as follow s
even:
x„(n) =
= x p( N - n)
odd:
x p (n) = - x p( —n) - - x p ( N - n)
(5.2.14)
If the periodic seq u en ce is com p lex-valu ed , w e have
conjugate even:
x „(n) = x U N — n)
P
x p (n) = —x * ( N — n )
conjugate odd:
(5.2.15)
T h ese relationships suggest that w e d ecom p ose the seq u en ce x p (n) as
xp (n) = xP'(n) + x ^ i n )
(5.2.16)
xPAn) = \ [ x p (n) + x p( N - n)]
,
P
x p„{n) = \ [ x P(n) - x * ( N - n ) ]
(5.2.17)
w here
Symmetry properties of the DFT. T h e sym m etry properties for the D F T
can be o b tain ed by applying the m eth od ology p reviously used for the Fourier
transform . Let us assum e that the N -p oin t seq u en ce jc(n) and its D F T are both
com plex valued. T h en the seq u en ces can be exp ressed as
jr(n) = XR(n) + j x / ( n )
0 < n < N —1
(5.2.18)
X( k ) = Xg ( k ) + j X , ( k )
0 < k < N - 1
(5.2.19)
B y substituting (5.2.18) in to the expression for the D F T given by (5.2.1), w e obtain
* * (* ) = £
j*J?(") cos
X,(k) = “ E
+ x , ( n ) sin
j
(5.2.20)
[•**(" )sin ~ J p ' — -*■/(«) cos
n=0 L
(5-2.21)
J
Sim ilarly, by substituting (5.2.19) into the expression for the ID F T given by (5.2.2),
w e obtain
x K(n) = i
1
* /(n ) = —
£ [”* * ( * ) cos
*=0 L
f
Real-valued sequences.
.
- X ,( fc ) s in ^ ^ j
J
. 2nkn
sin —^ — h
2n kn "I
cos —^ —J
(5.2.22)
(5.2.23)
If the seq u en ce jc(n) is real, it follow s directly
from (5.2.1) that
X ( N - k ) = X m(k) = X(-Jfc)
(5.2.24)
414
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
C onsequently, \ X ( N ~ jfc)| = |AT(A:)| and I X ( N - k) = —l X ( k ) . Furtherm ore,
xi (n) = 0 and therefore ;r(rj) can b e d eterm in ed from (5.2.22). which is another
form for the ID F T .
Real and even sequences.
If x (n) is real and even , that is,
x(n ) — x ( N — n)
0 < n < N —1
then (5.2.21) yield s X / ( k ) = 0. H en ce the D F T reduces to
Cy
2nkn
X ( k ) = y x ( n ) c o s -------
^
0 < k < N —\
(5.2.25)
N
which is itself real-valued and even .
reduces to
Furtherm ore, since X / ( k ) = 0, the ID FT
jc («) — — ^ X (k ) c o s -------*=o
Real and odd sequences.
0 < n < N —1
(5.2.26)
If x ( n ) is real and odd, that is,
x ( n ) = —x ( N — n)
0 < n < N ~ 1
then (5.2.20) yields X R(k) = 0. H en ce
X( k ) — —j Y x ( n ) sin
t z
o < k < N —1
(5.2.27)
N
which is purely im aginary and odd. Since X K(k) = 0, the ID F T reduces to
* (« ) = j ~ y
X (k ) sin
N
0 < n < N - 1
(5.2.28)
N
Purely imaginary sequences.
In this case, x( n) = j x f (n). C onsequently,
(5.2.20) and (5.2.21) reduce to
**< *) -
E
n=0
x >(k) -
JC,(n)sin
(5.2.29)
N
£ * / ( " ) 0 0 8 —77n=0
(5.2.30)
W e ob serve that Xn( k) is odd and X } (k) is even.
If x / ( n ) is odd, then X / ( k ) = 0 and h en ce X(Jt) is purely real. O n th e other
hand, if x r (n ) is ev en , then X * (Jt) = 0 and h en ce X(i fc) is purely im aginary.
Sec. 5.2
Properties of the DFT
TABLE 5.1
415
SYMMETRY PROPERTIES OF THE DFT
yV-Point Sequence x(n).
0< n < N - 1
/V-Point DFT
X(k)
X ’ ( N —k)
X' ( k)
x(n)
x*(n)
x*( N - n )
xx(n)
X t,
*)]
X,-,„<*) = HX( k ) - X ' ( N - *)]
X K(k)
JXj (n)
xcf(n) = i[jr(n) + x*(N - «)]
xcJ n ) = |[T (n) - x*( N - /j)]
j X/ ( k )
Real Signals
X(k) = X*( N - k )
X K(k) = X „ W - k )
X, (k) = - X , i N - k )
\X(k)\ = \ X( N - k ) \
I X( k) = - I X { N - k)
X K{k)
jX/tt)
A ny real signal
jr{n)
x, An) = j[* (n ) + x ( N - n)]
x„.(n) = 5(a-(») - x( N - »)]
T h e sym m etry properties given ab ove may be sum m arized as follow s:
Mu) - xjf(n) + x 'kOi ) + j x j‘{n) + jx'/in)
I
’
\
x(k) = x H
‘ (k) + x ;‘ (k) + j x ;t (k) + j x l<,(k)
(5-2-3D
AH the sym m etry properties o f the D F T can easily be d ed u ced from (5.2.31). For
exam p le, the D F T o f the seq u en ce
x pr(n) = j ^ f n ) + x * ( N - n)]
is
* * ( * ) = X'K(k) + X°K(k)
T h e sym m etry p rop erties of the D F T are sum m arized in T able 5.1. E x­
p loitation o f th e se properties for the efficient com putation o f the D F T o f special
se q u en ces is con sid ered in so m e o f the problem s at the end o f the chapter.
5.2.2 M ultiplication o f Two DFTs and Circular C on volu tion
S u p pose that w e have tw o finite-duration seq u en ces o f length N, Jti(n) and
T h eir resp ective Appoint D F T s are
JV-1
X, (k ) = Y L jc, (n)e~i2* HklN
iV-1
X 2(k) = Y x 2( n ) e - j2*nk/N
n=0
k = 0,1,..., N —1
(5.2.32)
k = 0,1,..., N - 1
(5.2.33)
The Discrete Fourier Transform: Its Properties and Applications
416
Chap. 5
If w e m ultiply the tw o D F T s togeth er, the result is a D F T , say
o f a se­
quen ce jf3(n) o f length N. L et us determ in e the relationship b etw een x3(n) and
the sequ en ces X] ( n) and jr2(n).
W e have
Xi ( k) = X t ( k ) X2(k)
k = 0 , 1 .........N - 1
(5.2.34)
T h e ID F T o f {X3(Jt)} is
1 N-1
x 3 (m) = ~
X j ( k ) e i2* km/N
*=o
E
(5.2.35)
A'-l
Suppose that we substitute for X\ ( k ) and X 2(k) in (5.2.35) using the D F T s given
in (5.2.32) and (5.2.33). T hus w e obtain
—
j27rkn/N
k=0
-j2nkl/N
„j2nkm/N
/=0
(5.2.36)
Af-l
T he inner sum in the brackets in (5.2.36) has the form
« -i
f N.
a = 1
(5.2.37)
y > A= l l - a "
*-«
0751
w here a is defined as
Q _ e j2n(m-n-l)/N
W e observe that a = 1 w hen m — n — I is a m ultiple o f N. O n the other hand,
a N = 1 for any value o f a =£ 0. C on sequ en tly, (5.2.37) reduces to
/V- 1
E
ks=0
t
a
,
I N,
1 0,
1
I = m - n + p N = ((m - n ) ) N,
otherw ise
p an in teger
2 381
If w e substitute the result in (5.2.38) in to (5.2.36), w e obtain the desired expression
for ^ ( m ) in the form
xi ( m) = Y
x \ ( n) x 2((m - n ) ) N
m = 0,1,..., N - 1
(5.2.39)
T h e expression in (5.2.39) has th e form o f a con volu tion sum . H ow ever, it is
not the ordinary linear con volu tion that w as introduced in C hapter 2, which relates
the output seq u en ce y( n) o f a linear system to th e input seq u en ce x( n) and the
im pulse response h(n). Instead, the con volu tion sum in (5.2.39) in volves the index
Sec. 5.2
417
Properties of the DFT
((m —fl))w and is called circular co nvolution. Thus w e con clu d e that m ultiplication
o f the D F T s o f tw o seq u en ces is eq u ivalen t to the circular con volu tion o f the tw o
seq u en ces in the tim e dom ain.
T h e fo llow in g exam p le illustrates the op eration s in volved in circular con vo­
lution.
Example Si.1
Perform the circular convolution of the following two sequences:
*!(«) = { 2,1,2( 1}
t
x 2(n) = {1,2,3,4}
t
Solution Each sequence consists of four nonzero points. For the purposes of illus­
trating the operations involved in circular convolution, it is desirable to graph each
sequence as points on a circle. Thus the sequences x\{n) and x 2(n) are graphed as
illustrated in Fig. 5.8(a). We note that the sequences are graphed in a counterclock­
wise direction on a circle. This establishes the reference direction in rotating one of
the sequences relative to the other.
Now, xi(m) is obtained by circularly convolving jci<«) with J 2(«) as specified by
(5.2.39). Beginning with m = 0 we have
jrj(O ) = y ^ j r i ( n ) j r 2( ( - n ) ) y
n=U
jc2((—«)>4 is simply the sequence x 2(n) folded and graphed on a circle as illustrated in
Fig. 5.8(b). In other words, the folded sequence is simply xz(n) graphed in a clockwise
direction.
The product sequence is obtained by multiplying jci(h) with * j( ( - n ) ) 4, point by
point. This sequence is also illustrated in Fig. 5.8(b). Finally, we sum the values in
the product sequence to obtain
*3(0) = 14
For m = 1 we have
3
*3(1) = ^ x l ( n ) x 2( ( l - « » 4
«*0
It is easily verified that *2((1 —n ))4 is simply the sequence * 2((~ n ))4 rotated coun­
terclockwise by one unit in time as illustrated in Fig. 5.8(c). This rotated sequence
multiplies x\ (n) to yield the product sequence, also illustrated in Fig. 5.8(c). Finally,
we sum the values in the product sequence to obtain *3(1). Thus
* 3d) = 16
F or m = 2 we have
3
*3(2) = E
Xl (n )*2((2 - n ))«
n«0
Now *2 ((2 —n ))4 is the folded sequence in Fig. 5.8(b) rotated two units of time in
the counterclockwise direction. The resultant sequence is illustrated in Fig. 5.8(d)
*|0) = 1
jr2( l) = 2
* ,(0) = 2
jri(2> - 2
■*2(2) = 3
*2( 0 ) =1
(a)
jc2 ( 2 )
2
* 2( 0 ) = 1
=3
jc2(I) = 2
Folded sequence
Product sequence
(b)
x2(0) = 1
jr2(3 )= 4
4
x2(l) = 2
jc2(2> = 3
Folded sequence rotated by one unit in time
(c)
x 2( l ) = 2
*2(2)= 3
*2(0) = 1
6
*2(3) = 4
Folded sequence rotated by two units in time
(d)
*j(2) = 3
*2(3) = 4
*j(0)=l
Folded sequence rotated by three units in time
Figure 5.8
Product sequence
(e)
Circular oonvolutioa o f tw o sequences.
Sec. 5.2
Properties of the DFT
419
along with the product sequence x l (n)xi((2 - n))A. By summing the four term s in the
product sequence, we obtain
* j ( 2 ) = 14
For m = 3 we have
3
X30 ) = y^xi(n)x;((3 - n ) ) 4
i,=Q
The folded sequence X2((—n))4 is now rotated by three units in time to yield jc2(<3—n))4
and the resultant sequence is multiplied by *j(n) to yield the product sequence as
illustrated in Fig. 5.8(e). The sum of the values in the product sequence is
x 3( 3) = 16
We observe that if the com putation above is continued beyond m = 3, we
simply repeat the sequence of four values obtained above. Therefore, the circular
convolution of the two sequences x \ ( n ) and x2(n) yields the sequence
xi(n) = {14,16,14,16)
t
From this exam p le, w e observe that circular con volu tion involves basically
the sam e four step s as the ordinary linear co n vo lu tio n introduced in C hapter 2:
fo l d i n g (tim e reversing) on e seq u en ce, shifting the fold ed seq u en ce, m u ltip ly in g the
tw o seq u en ces to obtain a product seq u en ce, and finally, s u m m i n g the valu es o f the
product seq u en ce. T h e basic d ifference b etw een th ese tw o types o f con volu tion
is that, in circular con v olu tion , the foldin g and shifting (rotatin g) op eration s are
p erform ed in a circular fashion by com puting the index o f o n e o f the seq u en ces
m o d u lo N . In linear co n volu tion , there is n o m odu lo N op eration .
T h e reader can easily sh ow from our previous d evelop m en t that eith er on e
o f the tw o seq u en ces m ay b e fold ed and rotated w ithou t changing the result o f the
circular con volu tion . Thus
N- 1
x$(m) = Y ' X 2 ( n )x i ( ( m — n ) ) s
m = 0 , 1 , ... t N — 1
(5.2.40)
n=0
T h e fo llo w in g exam p le serves to illustrate the com p utation o f x 3 (n) by m eans
o f the D F T and ID F T .
Exam ple SJJ1
By m eans of the D FT and IDFT, determ ine the sequence xj(n) corresponding to the
circular convolution of the sequences x\ (n) and X2 (n) given in Exam ple 5.2.1.
Solution
xi(n) is
First we com pute the DFTs of xi(n) and * 2(0 ). T he four-point D FT of
3
Xi<*) = Y Xl
ft-0
k = 0 ,1 ,2 ,3
420
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
Thus
*1(0) = 6
JCid) = 0
Jlf,(2) = 2
A'1(3) = 0
The D FT o f Jt2(n) is
3
X 2(.k) = Y
fl«0
k = 0 ,1 ,2 ,3
= 1 + 2 e - iltkri + 3e~>*k + 4e~J^ kri
Thus
X2(0) = 10
X 2a ) = - 2 + j 2
X2( 2 ) = - 2
X 2Q) = - 2 - j 2
When we multiply the two DFTs, we obtain the product
X3(*) = X t (k)X 2(k)
or, equivalently,
X3(0) = 60
* 3(1) = 0
X3( 2 ) = - 4
Xj (3) = 0
Now, the ID FT of X 3(k) is
jr3(n) = Y X3(Jt)cj7,rn*/4
£*41
n = 0, 1, 2, 3
= j(6 0 - 4eJ”n)
Thus
j 3(0> = 14
jc3(1) =
16
x3(2) = 14
;c3(3) = 16
which is the result obtained in Exam ple 5.2.1 from circular convolution.
W e conclude this section by form ally stating this im portant property o f the
DFT.
Circular co n v o lu tio n .
If
* i(n )
DFT
x m
and
then
DFT
*1 (H) (g> * 2 (n) ^
(*)X 2(Jt)
(5.2.41)
w here x\ (n) (N) x2(n) d en o te s th e circular con volu tion o f the se q u en ce x i(n ) and
x%(n).
Sec. 5.2
Properties of the DFT
421
42)
*6)
*(6)
jt(2)
Figure 5.9 Tim e reversal of a sequence.
5.2.3 Additional DFT Properties
Time reversal of a sequence.
If
DFT
th e n
x((-n))v = x ( N - n )
X ( ( - k ) ) N = X ( N - k)
(5.2.42)
H e n c e re v e rsin g th e /V-point se q u e n c e in tim e is e q u iv a le n t to rev ersin g th e D F T
values. T im e rev ersal of a s e q u e n c e x(rt) is illu stra te d in Fig. 5.9.
Proof . F ro m th e d efin itio n of th e D F T in (5.2.1) w e h a v e
JV-I
D F T { * W - «)} = £ x ( N ~ n ) e ~ j2nkn/N
n=0
I f w e c h a n g e th e in d ex fro m n to m = N - n, th e n
N -l
D F T [ x ( N - n ) } = £ x ( m ) < T j7,r*(* - m)/,v
=o
A '-l
= Y , x ( m ) e > 2*kmIN
m=0
= Y x ( m ) e - i2*m(N- k)/N = X ( N — k)
m=0
W e n o te th a t X ( N - k) = X { ( - k ) ) N, 0 < k < N - l .
Circular time shift of a sequence.
If
422
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
th e n
* « n - l))N «
X ( k ) e ~ j2* k!/N
(5.2.43)
Proof. From the definition o f the D F T w e have
N- 1
D F T { * ((# i- / ) ) * } = £ * ( ( n - / ) ) „ e - ' ' 2ir*"/N
n=0
= £ > ( ( « - l ) ) Ne ~ * * k”/N
n=0
+ Y x ( n - l ) e - j ”kn/N
n=l
B ut x( (n - 1)) h = x ( N - l + n). C onsequently,
~ 1 + n ) e ' j2jrkn/N
£ x ( ( n - l ) ) Ne~j2*kn/N = ] T
n—0
rr=s()
=
£
x ( m ) e ' ^ k^
m=N-l
N
F urtherm ore,
y x { n - l ) e - J2,Tkn/N =
/»=/
Y
x ( m ) e - j2”kim+IWN
T herefore,
N~ 1
DFT{jc((« - / ) ) } = ^ x{m)e~-'2xk(m+,)/N
m=0
= X ( k ) e ' ^ u/N
Circular freq u en cy sh ift.
If
DFT
x( n) «
X( k)
then
x ( n ) e j2*In/N £ 5 - X ( ( k - / » *
(5.2.44)
H en ce, the m ultiplication o f the seq u en ce jc(/i) w ith the com p lex exp on en tial se­
quence eP***1* is eq u ivalen t to th e circular shift o f the D F T by I u n its in frequency.
This is th e dual to the circular tim e-shifting p roperty and its p r o o f is sim ilar to the
latter.
Sec. 5.2
Properties of the DFT
423
C o m p lex -co n ju g a te p roperties.
If
X(k)
jr(n) K
th e n
DFT
x\n) «
* * ( ( - * ) ) * = * * ( * - k)
N
(5.2.45)
T h e p r o o f o f th is p r o p e rty is left as an ex ercise fo r th e re a d e r. T h e ID F T o f X m(k)
is
— Y
X * ( k ) e J2nkn/N =
— £ x ( J k ) e ' 2**(A'- " )' A'
^ *=o
T h e re fo re ,
x * ( ( - n ) ) N = x+( N - b) «
Circular correlation.
A/
X'(k)
(5.2.46)
In g e n e ra l, fo r c o m p le x -v a lu e d se q u e n c e s x ( n ) and
V(n), if
DF!'
x(n) «
/v
X(A-)
an d
v(«)
N
y<*)
th e n
FJV(/) «
n
R „ ( k ) = X ( k ) Y r (k)
(5.2.47)
w h ere r ,,.(/) is th e (u n n o rm a liz e d ) c irc u la r c ro ssc o rre la tio n se q u e n c e , d efin e d as
N- 1
^ v (0 = ^ * ( n ) / ( ( n - l))N
F r o o / W e c a n w rite f*v(/) as th e circ u la r co n v o lu tio n o f x ( n ) w ith y*(—n),
th a t is,
T h e n , w ith th e a id o f th e p r o p e rtie s in (5.2.41) a n d (5.2.46), th e W -point D F T o f
rxy(l) is
R i y (k) = X (k)F *(k)
In th e sp e c ia l case w h e re y (n ) = x ( n ) , w e h a v e th e c o rre sp o n d in g e x p ressio n
fo r th e c irc u la r a u to c o r re la tio n o f x ( n ) ,
424
The Discrete Fourier Transform: Its Properties and Applications
Multiplication of two sequences.
Chap. 5
If
jr,(n ) «
* ,( * )
x 2(n) «
X 2(k)
and
then
x A n ) x 2(n) K
^ X ,( J t ) ( N ) X 2(*)
(5.2.49)
This property is the dual o f (5.2.41). Its p ro o f follow s sim ply by interchanging
the roles o f tim e and frequency in the exp ression for the circular con volu tion of
tw o sequ en ces.
Parseval’s theorem.
For com p lex-valu ed seq u en ces * (n ) and _y(n), in gen­
eral, if
and
then
A N- 1
/V — 1
/V — 1
£ Jt(n)y*(n) = - £ X ( J t ) r (/:)
N *=0
n=0
(5-2 ‘50)
Proof. T h e property follow s im m ed iately from the circular correlation prop­
erty in (5.2.47). W e have
N-1
^ x ( n ) y * ( n ) = r JV(0)
and
k~0
H en ce (5.2.50) fo llo w s by evalu atin g the ID F T at I = 0.
T he exp ression in (5.2.50) is th e gen eral form o f P arseval’s theorem . In th e
sp ecial case w h ere ;y(n) = x ( n ) , (5.2.50) red u ces to
£
/11=0
\x{n)\2 = i
£
JtseO
|X ( * ) | 2
(5.2.51)
Sec. 5.3
Linear Filtering Methods Based on the DFT
TABLE 5.2
425
PROPERTIES OF THE DFT
Time Dom ain
Frequency Domain
Notation
Periodicity
Linearity
Time reversal
Circular time shift
Circular frequency shift
Complex conjugate
Circular convolution
x(n), yOi)
x (n) =s x(rt + iV)
a]Xi(n) + a2x 2(n)
x ( N —n)
*((" - D) n
x(n)ei2jr,n/N
x "(n)
xi (n )@ jr2(n)
*<*), Y(k)
X( k) = X( k + N)
0\ Xi ( k) + a2X 2(k)
X(N - k )
X(k)e~J2*kl/N
X ((k-l))N
X*(N — k)
xm xiik)
Circular correlation
x (n )® y * (-n )
X(k) Y' ( k)
Multiplication of two sequences
jri(n)x2(n)
/V—1
Property
Parseval’s theorem
n=(!
*3=0
w hich ex p resses the energy in the finite-duration seq u en ce x ( n) in term s o f the
frequency co m p o n en ts {X(£)l T h e properties o f the D F T given ab ove are sum m arized in T able 5.2.
5.3 LINEAR FILTERING METHODS BASED ON THE DFT
Since the D F T provides a discrete frequency rep resen tation o f a finite-duration
seq u en ce in the frequency dom ain, it is in terestin g to exp lore its use as a com ­
p utational to o l for linear system analysis and, especially, for linear filtering. W e
have already estab lish ed that a system with freq u en cy resp on se H { w ) y w hen e x ­
cited with an input signal that has a spectrum X(a>), p o ssesses an output spectrum
Y(a>) = X ( oj) H ( w ). T he output seq u en ce y ( n) is d eterm in ed from its spectrum via
the inverse F ourier transform . C om putationally, the p rob lem with this frequencydom ain approach is that X(a>), H(a>), and Y(a>) are fun ction s o f the continuous
variable o>. A s a con seq u en ce, the com p utations can n ot be d o n e on a d igital com ­
puter, since th e com puter can on ly store and perform com p utations on quantities
at discrete frequencies.
O n the other hand, th e D F T d o es len d itself to com p utation on a digital
com puter. In th e discussion that follow s, w e d escrib e h ow th e D F T can b e used
to perform lin ear filtering in the frequency dom ain. In particular, w e p resent
a com p utational p rocedure that serves as an alternative to tim e-d om ain co n v o ­
lution. In fact, the frequency-dom ain approach b ased o n the D F T , is com pu­
tationally m ore efficien t than tim e-dom ain con volu tion d u e to the existen ce o f
efficient algorithm s for com p utin g the D F T . T h ese algorithm s, w hich are d e ­
scribed in C hapter 6, are collectively called fast F ourier transform (FFT ) algo­
rithms.
426
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
5.3.1 Use of the DFT in Linear Filtering
In the p receding section it w as d em onstrated that the product o f tw o D F T s is
equivalent to th e circular con volu tion o f the corresponding tim e-d om ain sequ en ces.
U nfortunately, circular con volu tion is o f no use to us if our ob jectiv e is to deter­
m ine th e output o f a linear filter to a given input seq u en ce. In this case w e seek
a frequency-dom ain m eth od ology eq u ivalen t to lin ear con volu tion .
S u ppose that w e have a finite-duration seq u en ce x( n) o f length L which
excites an F IR filter o f length Af. W ithout loss o f generality, let
jc(n) = 0,
n < 0 and n > L
h(n) = 0,
n < 0 and n > M
where h(n) is the im pulse resp onse o f the F IR filter.
The output seq u en ce y ( n ) o f the FIR filter can be exp ressed in the tim e
dom ain as the con volu tion o f x( n) and h(n), that is
y(n) =
M- 1
h(k) x( n — k )
(5.3.1)
Since h(n) and jc(n) are finite-duration seq u en ces, their con volu tion is also finite
in duration. In fact, the duration o f y( n) is L + M — 1.
T h e frequency-dom ain equivalent to (5.3.1) is
Y(co) = X( w) H( ( o )
(5.3.2)
If the seq u en ce y (n ) is to be represented u n iqu ely in the freq u en cy dom ain by
sam ples o f its spectrum Y (to) at a set o f d iscrete freq u en cies, th e n um ber o f distinct
sam ples m ust equal or exceed L + M - 1. T h erefore, a D F T o f size N > L + M —I,
is required to represent [y(n)\ in the frequency dom ain.
N ow if
Y(k) = Y{a>)\m 2 ,tkfN
= X M H M U h t /N
k = 0 , 1 ,. . . , N — 1
it = 0 ,1 ....... N - 1
then
Y(k) = X ( k ) H ( k )
Jfc = 0 , l , . . . , t f - 1
(5.3.3)
w h ere {X(Jfc)} and {f/(Jt)} are the N -p oin t D F T s o f th e corresp on d ing sequences
x( n) and h (n), resp ectively. Since the seq u en ces x ( n) and h ( n ) have a duration
less than N , w e sim ply pad th ese seq u en ces w ith zeros to in crease their length to
N. T his increase in th e size o f the seq u en ces d o e s n ot alter their spectra X(o>) and
H(a>), which are con tin u ou s spectra, sin ce the seq u en ces are ap eriod ic. H ow ever,
by sam pling their spectra at N equally sp aced p oin ts in freq u en cy (com p u ting the
JV-point D F T s), w e have increased the num ber o f sam p les that represent these
seq u en ces in the frequency dom ain b eyon d the m inim um n u m b er (L or M, re­
sp ectively).
Sec. 5.3
427
Linear Filtering Methods Based on the DFT
Since the N = L + M — 1-point D F T o f the output seq u en ce y ( n ) is sufficient
to represent y ( n ) in the frequency dom ain, it follow s that the m ultiplication o f the
N -point D F T s X( k ) and H{ k) , according to (5.3.3), follow ed by the com putation
o f the Appoint ID F T , m ust yield the seq u en ce {,v(n)J. In turn, this im plies that
the Appoint circular con volu tion o f x( n) with h(n) m ust b e eq u ivalen t to th e linear
con v o lu tio n o f x ( n ) with h(n). In other words, by increasing th e length o f the
seq u en ces x( n) and h(n) to N points (by appending zeros), and then circularly
con volvin g the resulting seq u en ces, w e obtain the sam e result as w ou ld have b een
ob tain ed with linear con volu tion . Thus with zero padding, the D F T can b e used
to perform linear filtering.
T h e follow in g exam p le illustrates the m eth od ology in the u se o f the D F T in
linear filtering.
Example 5.3.1
By m eans of the D FT and IDFT, determine the response of the FIR filter with impulse
response
h(n) = 11.2.3}
t
to the input sequence
x(n) = |1. 2, 2.1)
t
Solution The input sequence has length L = 4 and the impulse response has length
M = 3. Linear convolution of these two sequences produces a sequence of length
N = 6. Consequently, the size of the DFTs must be at least six.
For simplicity we com pute eight-point DFTs. We should also m ention that the
efficient com putation of the DFT via the fast Fourier transform (FFT) algorithm is
usually perform ed for a length N that is a power of 2. Hence the eight-point D FT of
jr(n) is
7
Jkn/8
= 1 + 2e~W
+ le->*k’2 -I- e~‘**k,A
This com putation yields
X (4) = 0
X to -i+ J
XC7
+
k = 0 .1 ........7
The Discrete Fourier Transform: Its Properties and Applications
428
Chap. 5
The eight-point DFT of h(n) is
7
H(k) =
= 1 + 2e~J*k/* + 3>e- jnkr2
Hence
H(0) = 6,
H (l) = l + V 2 - > (3 + V 5 ) ,
H( 2 ) = - 2 - j 2
HO) = 1 - ^ 2 + > (3 - V 2 ) ,
H( 4) = 2
H(5) = 1 - 7 2 - y (3 - J 2 ) ,
H( 6) = ~2 + >2
W(7) = 1 + V2 + j ( 3 + J 2 )
The product of these two DFTs yields Y{k), which is
Y (0) = 36,
y (l) = - 1 4 .0 7 -> 1 7 .4 8
y(4) = 0,
V(5) = 0.07 —y 0.515
Y (2) = >4
Y( 6) = ~ j 4
X(3) = 0.07 + ,0.515
Y(7) = -14.07 + >17.48
Finally, the eight-point ID FT is
7
v(n) = Y
y(k)ej2nk"/H
n = 0, 1........7
*=■0
This computation yields the Tesult
>(«) = (1 ,4 ,9 ,1 1 .8 ,3 ,0 ,0 }
t
We observe that the first six values of y(rt) constitute the set of desired output
values. The last two values are zero because we used an eight-point D FT and IDFT,
when, in fact, the minimum number of points required is six.
A lthough the m ultiplication o f tw o D F T s corresp on d s to circular convolution
in the tim e dom ain, w e have ob served that padding the seq u en ces x ( n ) and h(n)
with a sufficient num ber o f zeros forces the circular con volu tion to yield the sam e
output sequ en ce as linear con volu tion . In the case o f the F IR filtering problem
in E xam p le 5.3.1, it is a sim ple m atter to d em onstrate that the six-point circular
convolution o f the sequ en ces
h( n) = {1, 2 , 3 , 0 , 0, 0}
t
(5.3.4)
x ( n ) = {1 ,2 , 2 , 1 , 0 , 0}
t
(5.3.5)
results in the output sequ en ce
y ( n) = {1, 4, 9 , 1 1 , 8 , 3}
t
w hich is the sam e seq u en ce ob tain ed from linear con volu tion .
(5.3.6)
Sec. 5.3
Linear Filtering Methods Based on the DFT
429
I t is im p o rta n t f o r u s to u n d e rs ta n d th e aliasing th a t re su lts in th e tim e d o m a in
w h e n th e size o f th e D F T s is sm a lle r th a n L + M —I. T h e fo llo w in g e x am p le focuses
o n th e aliasin g p ro b le m .
Exam ple 5.3.2
D eterm ine the sequence v(n) that results from the use of four point DFTs in Exam­
ple 5.3.1.
Solution
The four-point D F T of h(n) is
t*)jre.-jink*!*
H(k) = ^ A (fi n\
'
ff-0
H(k) = \ + 2 e - ink/1 + 3 e - ikn
k = 0 ,1 ,2 , 3
Hence
H(0) = 6,
H( \ ) = - 2 - j 2 ,
H( 2) = 2,
H ( 3 ) = - 2 + j2
The four-point DFT of x («} is
X(k) = \ + 2c ~jnk/2 + 2e~,7,k + 3e ~J%7,t/2
k = 0, 1, 2, 3
Hencc
*(()) = 6,
X (l) = - l - j ,
X(2) = 0,
Jf(3) = —1 + y
The product of these two four-point DFTs is
K(0) = 36,
f ( l ) = ;4 ,
Y( 2) = 0.
K(3) = ~ j 4
The four-point ID FT yields
y(n) = \ Y ^ k^eJ2xk”,A
*=o
« = 0 , 1 ,2 ,3
= 1(36 + j 4e Jn"^ - j 4e J1*na)
Therefore,
v(n) = {9,7, 9,11}
t
The reader can verify that the four-point circular convolution of h(n) with x(n)
yields the same sequence y(n).
I f w e c o m p a re th e re su lt y (n ), o b ta in e d fro m fo u r-p o in t D F T s w ith th e s e ­
q u e n c e y (n ) o b ta in e d fro m th e use o f e ig h t-p o in t (o r six -p o in t) D F T s, th e tim ed o m a in aliasin g effe cts d e riv e d in S ectio n 5.2.2 a re clearly e v id e n t. In p a rtic u la r,
>(4) is a liased in to y (0) to yield
y (0) = y((» + y ( 4) = 9
S im ilarly, _y(5) is a liased in to _y(l) to yield
?(1) = ? (!) + ?(5) = 7
430
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
A ll other aliasing has n o effect since y(n ) = 0 for n > 6 . C on sequ en tly, w e have
y ( 2) = y ( 2) = 9
y(3) = y(3) = 1 1
T h erefore, on ly the first tw o p oints o f y ( n ) are corrupted by th e effect o f aliasing
[i.e., y(0) 7^ y(0) and v( l ) ^ y (l)]. T his observation has im portan t ram ifications
in the discussion o f the follow in g section , in which w e treat th e filtering o f long
sequ en ces.
5.3.2 Filtering of Long Data Sequences
In practical applications involving linear filtering o f signals, the input sequ en ce
x ( n ) is often a very long sequ en ce. T his is especially true in so m e real-tim e signal
processing applications concerned with signal m onitorin g and analysis.
Since linear filtering perform ed via the D F T in volves op eration s on a block
o f data, w hich by necessity m ust be lim ited in size due to lim ited m em ory o f a
digital com puter, a long input signal seq u en ce m ust be se g m en ted to fixed-size
blocks prior to processing. Since the filtering is linear, su ccessive blocks can be
p rocessed on e at a tim e via the D F T and the output blocks are fitted togeth er to
form the overall output signal sequ en ce.
W e now d escribe tw o m eth od s for linear F IR filtering a lon g seq u en ce on a
block-by-block basis using the D F T . T h e input seq u en ce is segm en ted in to blocks
and each block is p rocessed via the D F T and ID F T to prod u ce a block o f output
data. T h e output blocks are fitted togeth er to form an overall output sequ en ce
which is identical to the seq u en ce ob tain ed if the lon g block had b een processed
via tim e-dom ain con volu tion .
T h e tw o m eth od s are called the overlap-save m e t h o d and the o verlap-a dd
m eth o d . For b oth m eth od s w e assum e that the F IR filter has duration M . The
input data seq u en ce is segm en ted in to blocks o f L points, w h ere, by assum ption,
L »
M w ithout lo ss o f generality.
Overlap-save method. In this m eth od the size o f the in p ut data blocks is
N — L 4- M — 1 and th e size o f the D F T s and ID F T are o f len gth N . E ach data
block consists o f the last M - 1 data points o f the p reviou s data b lock follow ed by
L new data p oin ts to form a data seq u en ce o f len gth N = L 4- M — 1. A n N -point
D F T is com p uted for each data block. T h e im pulse resp onse o f the F IR filter is
increased in length by appending L - l zero s and an Appoint D F T o f the sequ en ce
is com p uted on ce and stored. T h e m ultiplication o f the tw o Af-point D F T s {//(Jt)}
and {Xm(Jt)} for the m th b lock o f data yields
Ym(k) = H ( k ) X m(k)
k = 0 , 1 .........N - 1
(5.3.7)
T hen the Appoint ID F T yield s the result
L ( n ) = { ^ (0 )y ffl(l) • • ■ym(M - 1)ym{M ) • ■■ym(N - 1)}
(5.3.8)
Sec. 5.3
Linear Filtering Methods Based on the DFT
431
Sin ce the data record is o f length N, the first Af — 1 p oin ts o f >m(n) are corrupted
by aliasing and m ust be discarded. T h e last L p oints o f y„( n) are exactly the sam e
as the result from linear con volu tion and, as a con seq u en ce,
>«(n) = y»( n) , n = M, M + 1.........N - 1
(5.3.9)
T o avoid lo ss o f data du e to aliasing, the last Af —1 p oin ts o f each data record
are saved and th ese p oin ts b ecom e the first Af — 1 data p oin ts o f the subsequent
record, as indicated above. T o begin the processing, the first Af — 1 p oints o f the
first record are set to zero. Thus the blocks o f data seq u en ces are
jci(n) = (0, 0 ........ 0, x(0), x ( l ) .......... x ( L - 1)}
(5.3.10)
M —1 points
x2(n) = {x ( L - M + 1).........x ( L — 1 ) , x ( L ) , . . . , x ( 2 L - l) j
M - 1 data points
from i|(n)
x3(rt) = {x ( 2 L - M + l ) .........x ( 2 L — 1), x ( 2 L ) .......... x ( 3 L - l } )
M - 1 data points
from
(5.3.11)
L new data points
(5.3.12)
I new data points
and so forth. T h e resulting data sequ en ces from the ID F T are given by (5.3.8),
w here the first M — 1 points are discarded due to aliasing and the rem aining L
p oin ts con stitute the d esired result from linear con volu tion . T h is segm en tation o f
th e input data and the fitting o f the output data blocks togeth er to form the output
seq u en ce are graphically illustrated in Fig. 5.10.
Overlap-add method. In this m eth od the size o f the input data block is L
p oin ts and the size o f th e D F T s and ID F T is N ~ L + Af - 1. T o each data block
w e append Af — 1 zeros and com p ute the N -point D F T . T h u s the data b lock s may
be rep resen ted as
jti(n) = {jc(0), jc(1 )......... jc( £ - 1 ) , 0 , 0 _____0}
M -l
(5.3.13)
zeros
X2(n) = [ x( L) , x ( L + 1), • . . , X(2L - 1), 0, 0 , . . . , 0}
(5.3.14)
M-1 zeros
jt3(n) = { x ( 2 L ) .........x( 3 L - 1), 0 , 0 , . . . , 0)
M -\
(5.3.15)
zeros
and so on . T h e tw o Appoint D F T s are m ultiplied togeth er to form
Ym(k) = H ( k ) X m(k)
* = 0 , 1 .........N - 1
(5.3.16)
T h e ID F T y ield s data blocks o f length N that are free o f aliasing sin ce the size o f
th e D F T s and ID F T isW = Z, + Af —1 and the seq u en ces are increased to N -p oin ts
b y ap p en d in g zero s to each block.
432
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
Output signal
points
/
Discard
P°ints
Figure 5.10 L in ear F IR filtering by the
overlap-save m ethod.
Since each data block is term inated with M — 1 zeros, the last M — 1 points
from each output block m ust be overlap p ed and added to the first M - 1 p oin ts of
the succeeding block . H en ce this m eth od is called th e overlap-add m ethod. This
overlapping and adding yields the output seq u en ce
y (« ) = {y i(0 ), ^ i ( l ) , . . . , y \ ( L - l ) , y i ( L ) + y2(0 ), ^ ( L + 1) +
(5.3.17)
>■2( 1 ) .........y \ ( N - 1) + y i ( M — 1), y 2 ( M ) , . . . }
T he segm en tation o f the input data in to b lock s and th e fitting o f th e output data
blocks to form the output seq u en ce are graphically illustrated in F ig. 5.11.
A t this poin t, it m ay appear to the reader that th e use o f th e D F T in linear
F IR filtering is n ot o n ly an indirect m eth od o f com puting the o u tp u t o f an FIR
filter, b ut it m ay a lso be m ore exp en sive com p utationally since th e input data must
first b e converted to the frequency d om ain via th e D I T , m ultip lied by th e D F T
o f th e F IR filter, and finally, converted back to th e tim e d om ain via the ID FT.
O n th e contrary, h ow ever, by using the fast F ourier transform algorithm , as will
b e show n in C hapter 6, th e D F T s and ID F T require few er com p utations to com ­
pu te th e output seq u en ce than th e direct realization o f the F IR filter in the time
Sec. 5.4
Frequency Analysis of Signals Using the DFT
433
Input data
\
MA
Output data
M-1 point.*;/ PZJ
add
— V/y
together
M-l points^
add
—
together
Figure 5.11 Linear FIR filtering by the
overlap-add method.
dom ain. This com putational efficiency is the basic advantage o f using the D F T to
com p ute the output o f an F IR filter.
5.4 FREQUENCY ANALYSIS OF SIGNALS USING THE DFT
T o com p ute the spectrum o f either a con tinu ou s-tim e or d iscrete-tim e signal, the
valu es o f the signal for all tim e are required. H ow ever, in practice, w e observe
signals for on ly a finite duration. C on sequ en tly, the spectrum o f a signal can
on ly b e approxim ated from a finite data record. In this section w e exam ine the
im plications o f a finite data record in frequency analysis using the D F T .
If the signal to b e analyzed is an analog signal, w e w ou ld first pass it through
an an tialiasing filter and th en sam ple it at a rate F, > 2 5 , w h ere B is the band­
width o f the filtered signal. T h u s th e highest frequency that is contained in the
sam pled signal is Fs f l . F inally, for practical purposes, w e lim it the duration o f
the signal to th e tim e interval To — L T , w h ere L is the num ber o f sam ples and T
The Discrete Fourier Transform: Its Properties and Applications
434
Chap. 5
is the sam ple interval. A s w e shall ob serve in the follow in g d iscu ssion , the finite
observation interval for the signal places a limit on the freq u en cy resolution; that
is, it lim its our ability to distinguish tw o frequency com p on en ts that are separated
by less than 1 /To = 1/Z-7" in frequency.
L et {*(«)} d en o te the seq u en ce to b e analyzed. L im iting th e duration o f the
seq u en ce to L sam ples, in the interval 0 < n < L — 1, is eq u ivalen t to m ultiplying
{jc(/i)} by a rectangular w indow w ( n ) o f length L. That is,
x (n) = x ( n ) w ( n )
(5.4.1)
w h ere
u;w
= {J ;
, v,,
0
< n < L - 1
otherw ise
(5.4.2)
N ow su p p ose that the sequ en ce x ( n ) consists o f a single sin u soid , that is,
x( n ) = coscoon
(5.4.3)
Then the Fourier transform o f the finite-duration seq u en ce x ( n ) can be expressed
X(£u) = $[W '(cu-<u0) + W 'to + wo)]
(5.4.4)
where W(a>) is the Fourier transform o f the w indow seq u en ce, w h ich is (for the
rectangular w indow )
W{0}) =
D/2
sin(df/ 2 )
(5.4.5)
T o com pute X(a>) w e use the D F T . B y padding the seq u en ce x ( n ) w ith N — L zeros,
we ca n c o m p u te the N -point D F T o f the truncated (L p oin ts) seq u en ce {Jt(n)J.
T he m agnitude spectrum |A W I = |X(£t»*) | for o* = 2 n k / N , k = 0, l , . . . , A f , is
illustrated in Fig. 5.12 for L = 25 and N = 2048. W e n ote that the w indow ed
spectrum X((o) is n ot localized to a single frequency, but instead it is spread out
over the w h ole frequency range. Thus the p ow er o f the origin al signal sequence
{jr(n)} that was concentrated at a single frequency has b een spread by the window
into the entire freq u en cy range. W e say that the p ow er has “lea k ed o u t” in to the
entire frequency range. C onsequently, this p h en om en on , which is a characteristic
o f w indow ing the signal, is called leakage.
Frequency
Figure 5.12 Magnitude spectrum for
I = 25 and n = 2048, illustrating the
occurrence of Leakage.
Sec. 5.4
435
Frequency Analysis of Signals Using the DFT
W in d o w in g n o t o n ly d isto rts th e sp e ctral e stim a te d u e to th e le a k a g e effects,
it also re d u c e s s p e c tra l re so lu tio n . T o illu stra te th is p ro b le m , let us c o n s id e r a
signal s e q u e n c e co n sistin g o f tw o fre q u e n c y co m p o n e n ts,
x ( n ) = cos co\ n + c o s a ^ n
(5.4.6)
W h e n th is se q u e n c e is tr u n c a te d to L sam ples in th e ra n g e 0 < n < L — 1, th e
w in d o w e d s p e c tru m is
X ( w ) — \ [ W (to — u>i) + W (a> — a>2 ) + W((o + ati) + W(a> + a>2)]
(5.4.7)
T h e s p e c tru m W (to) o f th e re c ta n g u la r w indow se q u e n c e h as its first z e ro crossing
a t co = 2 n / L . N o w if |<yi — a ^ l < 2n / L , th e tw o w in d o w fu n c tio n s W(co — a>\) a n d
V/(o) — an) o v e rla p a n d , as a co n se q u e n c e , th e tw o sp e c tra l lin es in x ( n ) a re n o t
d istin g u ish a b le . O n ly if (a>\ — a>i) > 2 n / L will w e se e tw o s e p a ra te lo b es in th e
sp e c tru m X(a>). T h u s o u r a b ility to reso lv e sp e c tra l lines o f d iffe re n t fre q u e n c ie s
is lim ite d b y th e w in d o w m ain lo b e w idth. F ig u re 5.13 illu s tra te s th e m a g n itu d e
sp e c tru m |X (a»)|, c o m p u te d via th e D F T , fo r th e se q u e n c e
+ cosa^n
(5.4.8)
Magnitude
x ( n ) = costuo n + cos
Frequency
Frequency
(a)
<b)
(c)
Figure 5.13 Magnitude spectrum for the signal given by (5.4.8), as observed through a
rectangular window.
436
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
w here ato = 0.2n ,
= 0. 22n, and coi = 0.6jt. T h e w indow len gth s selected are
L = 25, 50, and 100. N o te that o>o and
are not resolvab le for L = 25 and 50,
but th ey are resolvab le for L = 100.
T o reduce leak age, w e can select a data w indow w( n) that h as low er sid elob es
in the frequency d om ain com pared with the rectangular w in d ow . H ow ever, as we
d escribe in m ore detail in Chapter 8, a reduction o f the sid elo b es in a w indow
W( w) is obtained at the ex p en se o f an increase in the width o f th e m ain lo b e o f
W(a>) and h en ce a loss in resolution. T o illustrate this point, let us consider the
H anning w indow , w hich is specified as
w( n ) =
^(1 - cos jTTj-n),
0 < n < i. — 1
0,
otherw ise
(5.4.9)
Figure 5.14 show s |A"(<y)| for the w indow o f (5.4.9). Its sid elo b es are significantly
sm aller than th o se o f the rectangular w in d ow , but its m ain lob e is approxim ately
tw ice as w ide. Figure 5.15 sh ow s the spectrum o f the signal in (5.4.8), after it is
w indow ed by the H anning w indow , for L = 50, 75, and 100. T h e reduction o f
the sid elo b es and th e d ecrease in the resolu tion , com pared with the rectangular
w indow , is clearly evident.
For a general signal seq u en ce ljr(n)}, the frequency-dom ain relationship be­
tw een the w indow ed seq u en ce i ( n ) and the original seq u en ce x ( n ) is given by the
con volu tion form ula
(5.4.10)
T h e D F T o f the w in d ow ed seq u en ce x( n) is the sam pled version o f the spectrum
X(a>). T hus w e have
X( k ) = X ( » ) U W
(5.4.11)
k = 0,1,..., N - 1
Just as in the case o f th e sinusoidal seq u en ce , if the spectrum o f th e w indow is
relatively narrow in w idth com pared to the spectrum X (to) o f th e signal, the win­
dow function has o n ly a sm all (sm ooth in g) effect on the spectrum X (w). O n the
other hand, if the w in d ow function has a w ide spectrum com pared to the w idth of
6
25
0
—
r
r
2
0
Frequency
r
2
Figure 5.14 Magnitude spectrum of tbe
Hanning window.
437
Frequency Analysis of Signals Using the DFT
Magnitude
Sec. 5.4
9
8
L = 100
7
«; 6
-r
- T
0
2
I
2
*
Frequency
(O
Figure 5.15 Magnitude spectrum of the signal in (5.4.8) as observed through a Hanning
window.
X(a>), as w ould b e the case w hen the num ber o f sam ples L is sm all, the w indow
spectrum m asks the signal spectrum and, con sequ en tly, the D F T o f the data re­
flects th e spectral characteristics o f the w in d ow function. O f course, this situation
should b e avoided.
Example 5.4.1
The exponential signal
*..(0
H r
t >o
t <0
is sampled at the rate F, = 20 samples per second, and a block of 100 samples is used
to estimate its spectrum. Determine the spectral characteristics of the signal x„(t) by
computing the DFT of the finite-duration sequence. Compare the spectrum of the
truncated discrete-time signal to the spectrum of the analog signal.
Solution
The spectrum of the analog signal is
Xa(F) =
1
1 + ;2 jtF
The exponential analog signal sampled at the rate of 20 samples per second yields
438
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
the sequence
x(n) = <r"r = c - '/ 20,
n > 0
Now, let
0 < n < 99
(0.95)",
0,
otherwise
The A/-point D F T of die L = 100 point sequence is
99
X(k) = Y
i<-n '>e~i2*k/N
* = 0 ,1 ........A ' - l
To obtain sufficient detail in the spectrum we choose N = 200. This is equivalent to
padding the sequence x ( n) with 100 zeros.
The graph of the analog signal xa(t) and its magnitude spectrum |Xa(F )| are
illustrated in Fig. 5.16(a) and (b), respectively. The truncated sequence xin) and its
N = 200 point D FT (m agnitude) are illustrated in Fig. 5.16(c) and (d), respectively.
1.0
0.8
0.6
0.4
0.2
0
0
2
3
5
4
(a)
_■
-5 0
■
-4 0
■
-3 0
■
-2 0
-1 0
JL
0
10
20
i
■
30
40
■ F
50
(b)
Figure 5.1ti Effect o f windowing (truncating) the sampled version o f the analog
signal in Example 5.4.1.
Sec. 5.4
Frequency Analysis of Signals Using the DFT
439
(d)
(*)
Figure 5.16 Continued
In this case the D FT {X (Jt)} bears a close resem blance to the spectrum of the analog
signal. The effect of the window function is relatively small.
O n the other hand, suppose that a window function of length L = 20 is selected.
T hen the truncated sequence x(n) is now given as
.£(„) _ [ ( ° - 95)"1 0,
0 < n < 19
otherwise
440
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
Its N = 200 point DFT is illustrated in Fig. 5.16(e). Now the effect of the wider
spectral window function is dearly evident. First, the main peak is very wide as a
result of the wide spectral window. Second, the sinusoidal envelope variations in the
spectrum away from the main peak are due to the large sidelobes of the rectangular
window spectrum. Consequently, the DFT is no longer a good approximation of the
analog signal spectrum.
5.5 SUMMARY AND REFERENCES
T h e m a jo r focus o f th is c h a p te r w as o n th e d isc re te F o u rie r tra n s fo rm , its p ro p e rtie s
an d its ap p licatio n s. W e d e v e lo p e d th e D F T by sa m p lin g th e sp e c tru m X (&>) o f
th e se q u en ce x(n').
F re q u e n c y -d o m a in sa m p lin g o f th e sp e c tru m o f a d is c re te -tim e signal is p a r­
ticu larly im p o rta n t in th e p ro cessin g o f d ig ital signals. O f p a r tic u la r significance
is th e D F T , w hich w as show n to u n iq u e ly re p re s e n t a fin ite -d u ra tio n se q u e n c e in
th e fre q u e n c y d o m a in . T h e ex iste n ce o f c o m p u ta tio n a lly efficien t a lg o rith m s fo r
th e D F T , w hich a re d e sc rib e d in C h a p te r 6, m a k e it p o ssib le to d ig itally p ro cess
sig n als in th e fre q u e n c y d o m a in m u ch fa ste r th a n in th e tim e d o m a in . T h e p ro ­
cessin g m e th o d s in w hich th e D F T is esp ecially su ita b le in clu d e lin e a r filtering as
d e sc rib e d in th is c h a p te r a n d c o rre la tio n , an d sp e c tru m an aly sis, w h ich a re tre a te d
in C h a p te rs 6 a n d 12. A p a rtic u la rly lucid an d co n cise tre a tm e n t o f th e D F T and
its ap p licatio n to fre q u e n c y an aly sis is given in th e b o o k by B rig h a m (1988).
PROBLEMS
5.1 The first five points of the eight-point DFT of a real-valued sequence are (0.25,
0.125 - j 0.3018, 0, 0.125 - y0.0518, 0}. Determine the remaining three points.
5.2 Compute the eight-point circular convolution for the following sequences.
(a) *,(«) = (1,1,1,1,0,0,0,0}
. 3jt
x2(n) = sin — -n
8
„
_
0 < n < 7
0 <n <7
3n
.
_
*2(n) = cos — n
0< n < 7
O
(c) Compute the DFT of the two circular convolution sequences using the DFTs of
*i(n) and X2 (n).
(b) *,(») = (i)"
S 3 Let X (Jt), 0 < Jt < N —1, be the Appoint DFT of the sequence *(n), 0 < n < N —1.
We define
y /ia _ f
0 < k < k c, N - k c < k < N - 1
{)
1 0,
kc < k < N —ke
and we compute the inverse //-point DFT of X(k), 0 < k < N - 1. What is the effect
of this process on the sequence x («)? Explain.
Chap. 5
441
Problems
5.4 F or the sequences
jci(n) = cos ^ - n
N
jc2(«) = sin
N
n
0 <n <N - 1
determ ine the N -point:
(a) Circular convolution jc^n) (N)x 2(n)
(b) Circular correlation of *i(n) and x 2(n)
(c) Circular autocorrelation of xi(n)
(d) Circular autocorrelation of x 2(n)
5.5 Com pute the quantity
N-l
y ^ x ](n)x 2(n)
»=0
for the following pairs of sequences.
(a) JC](n) = x 2 (n) = cos ~ n
N
0 < n < N —1
x 2(n) = sin — n
0 < n < N —1
N
N
(c) *](n) = 6(n) + 5(n - 8)
*2(«) = « ( « ) - u(n - N)
(b) Xi (n) = cos — n
5.6 D eterm ine the A/-point D FT of the Blackman window
if(n) = 0.42 —0.5 cos —— - -I- 0.08c o s ------ Ar — 1
A '- l
0 < n < N —1
5.7 If X (Jt) is the D FT of the sequence *(n), determ ine the Af-point D FTs of the sequences
2nkn
xc(n) — x(n) c o s ------N
. . .
0< n < N - 1
. 2nkn
x,(n) = x ( n ) sin ——
Ar
0 < n < N —1
and
in terms of X (Jt).
5.8 Determine the circular convolution of the sequences
*i(n) = {1,2,3,1}
t
x2(n) = {4,3,2,2}
t
using the time-domain formula in (5.2.39).
5.9 Use the four-point DFT and IDFT to determine the sequence
X3(n) = *i(n)(§)*2(*)
where xi(n) and x 2(n) are the sequence given in Problem 5.8.
5.10 Compute the energy of the N -point sequence
442
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
5.11 Given the eight-point D FT of the sequence
*(«) =
1.
0,
0 <n < 3
4 <n < 7
compute the D FT of the sequences:
1,
n= 0
(a) x\(n) = 0,
1< n < 4
1.
5 <n < 7
0,
0 <n < 1
(b) jr2(n) = 1 ,
2 <n < 5
0,
6< n < 7
5.12 Consider a finite-duration sequence
jc(n) = {0 ,1 .2 ,3 .4 )
t
(a) Sketch the sequence .t(n) wilh six-poini DFT
£(*) = Wj’ XOfc)
k = 0 .1 ........6
(b) D eterm ine the sequence y(n) with six-point D FT K(jt) = Re |X(A)|.
(c) Determ ine the sequence u(w) with six-point D FT V(k) = Im |X(A-)|.
5.13 Let x p(n) be a periodic sequence with fundamental period N. Consider the following
DFTs:
DFT
x p(n) *— *• *,(*)
N
x („)
D FT
3^
X ,(k)
(a) W hat is the relationship between Xi(^) and XiOt)?
(b) Verify the result in part (a) using the sequence
xp(n) = {■■• 1. 2 ,1 ,2 ,1 ,2 ,1 .2 - ■]
t
5.14 Consider the sequences
X] (n) = {0,1, 2,3,4}
t
x 2(n) = {0 ,1 ,0 ,0 .0 ]
t
s(n) = {1. 0, 0 .0 ,0 )
t
and their 5-point DFTs.
(a) Determine a sequence y( n ) so that Y(k) = A-!
fJt).
(b) Is there a sequence x3(n) such that S(k) = X^(k)X^(k)'!
5.15 Consider a causal LTI system with system function
The output y(n) of the system is known for 0 < n < 63. Assuming that H(z) is
available, can you develop a 64-point DFT method to recover the sequence x{n),
0 < n < 63? Can you recover all values of x(n) in this interval?
5.16* The impulse response of an LTI system is given by h(n) = S(n) - ji(n - /to). To
determine the impulse response g(n) of the inverse system, an engineer computes the
Af-point D FT H(k), N = 4ko, of h(n) and then defines g(n) as the inverse DFT of
Chap. 5
443
Problems
G (k) = 1j H ( k ), k = 0, 1, 2 , . . . , A' - l . Determine g(n) and the convolution h(n)*g(n),
and comment on whether the system with impulse response g(n) is the inverse of the
system with impulse response h(n).
5.17* Determine the eight-point DFT of the signal
*(«) = [ 1 , 1 , 1 . 1 . 1 , 1 , 0 , 0 }
and sketch its magnitude and phase.
5.18 A linear time-invariant system with frequency response H(u>) is excited with the
periodic input
OO
jc(n) = ^ S(n —kN)
*=-oo
Suppose that we compute the Af-point DFT Y(k) of the samples >■(«), 0 < it < N - 1
of the output sequence. How is ^(Jt) related to //(w)?
5.19 D F T o f real sequences with special symmetries
(a) Using the symmetry properties of Section 5.2 (especially the decomposition prop­
erties), explain how we can compute the DFT of two real symmetric (even) and
two real antisymmetric (odd) sequences simultaneously using an W-point DFT
only.
(b) Suppose now that we are given four real sequences *,(«), i = 1, 2, 3, 4, that are
all symmetric [i.e., X j ( n ) = x j ( N — n ), 0 < n < N — 1], Show that the sequences
j,-(n) = X j ( n + 1) — X j ( n — 1)
are antisymmetric [i.e., s, (n) = —s;(N - n) and 5, (0 ) = 0 ].
(c) Form a sequence *(/i) using Jti(n), JC2 (n), ^(n), and ^(n) and show how to compute
the DFT Xj(k) of x,(n), i = 1, 2, 3, 4 from the N-point DFT X(k) of x(n).
(d) Are there any frequency samples of X, (k) that cannot be recovered from X(k)?
Explain.
5.20 D F T o f real sequences with o dd harmonics only Let x(n) be an A'-point real sequence
with Af-point DFT X(k) (N even). In addition, x(n) satisfies the following symmetry
property:
/
N \
x ( n + y j = -x (n )
N
n = 0 , 1 ........y - 1
that is, the upper half of the sequence is the negative of the lower half.
(a) Show that
X (k) = 0
k even
that is, the sequence has a spectrum with odd harmonics.
(b) Show that the values of this odd-harmonic spectrum can be computed by evaluat­
ing the A”/2-point DFT of a complex modulated version of the original sequence
x(n).
5.21 Let x„(t) be an analog signal with bandwidth B = 3 kHz. We wish to use a N = 2"point DFT to compute the spectrum of the signal with a resolution less than or equal
to 50 Hz. Determine (a) the minimum sampling rate, (b) the minimum number of
required samples, and (c) the minimum length of the analog signal record.
444
The Discrete Fourier Transform: Its Properties and Applications
Chap. 5
5.22 Consider the periodic sequence
2x
— oo < n < oo
x p(n) = COS — n
with frequency /o = ^ and fundamental period N = 10. Determine the 10-point
DFT of the sequence x ( n ) = x p(n), 0 < n < Af - 1.
5.23 Compute the W-point DFTs of the signals
(a) ■>:(") = ^(«)
(b) x ( n) = S(n — no)
(c) Jt(n) = a "
(d) x(n)
0 < fiq < N
0 < n < N — 1
1,
, Q
0 < n < N/ 2 - l ( N even)
N p.< n< N
(e) x(n) = e’a*IN)k<> 0 < n < N - 1
2x
(I) x(n) = cos — kon 0 < n < N —1
N
(g) j:(/i) = sin -j^kort
0< n < N - 1
i
I
n even
) ^
10,
« odd 0 < n < Af —1
5.24 Consider the finite-duration signal
x(n) = {1, 2, 3, 1}
(a) Compute its four-point DFT by solving explicitly the 4-by-4 system of linear
equations defined by the inverse DFT formula.
(b) Check the answer in part (a) by computing the four-point DFT, using its defini­
tion.
5.25 (a) Determine the Fourier transform X (tu) of the signal
x(n) =
{ 1 ,2 ,3 ,2 ,1 ,0 1
t
(b) Compute the 6-point DFT V (k) of the signal
i»(n) = { 3 ,2 ,1 ,0 ,1 ,2 }
(c) Is there any relation between X(w) and V(Jt)? Explain.
5.26 Prove the identity
Y & ( n + l N ) = ± Y eJQ*/,'*K
J—OC
kmC
(Hint: Find the DFT of the periodic signal in the left-hand side.)
5 J 7 Computation o f the even and odd harmonics using the DFT
sequence with an Appoint DFT X(k) ( N even)
(a) Consider the time-aliased sequence
x ( n + I M ),
Let x(n) be an Appoint
0 < n < M —1
fv-00
0,
elsewhere
What is the relationship between the Af-point DFT K(Jt) of y(/i) and the Fourier
transform X(w) of x(n)?
Chap. 5
Problems
445
(b) Let
0 < n < N —1
— J x (n ) + x ( n + ~ l '
(-*?)■
I 0,
elsewhere
and
y(n)
DFT
K(Jt)
N/l
Show that AT(Jt) = Y(k/2), k = 2, 4 , . . . , N - 2.
(c) Use the results in parts (a) and (b) to develop a procedure that computes the
odd harmonics of X(k) using an jV/2-point DFT.
5•28"' Frequency-domain sampling
Consider the following discrete-time signal
C( n ) = h ' " '
} 0,
|n| > L
where a = 0.95 and L = 10
(a) Com pute and plot the signal x(n).
(b) Show that
x(n)e
X (w )=
= x(0) + 2 ^ ^ x(n) cos wn
Plot X{i») by computing it at w = jri/1 0 0 , k = 0, 1........100.
(c) Com pute
N
\N
)
for N = 30.
(d) D eterm ine and plot the signal
x(n) =
*=o
W hat is the relation between the signals x(n) and x ( n)l Explain.
(e) Com pute and plot the signal ii(n ) =
X x(n - IN), - L < n < L for N = 30.
Com pare the signals x(rt) and X](n).
(f) R epeat parts (c) to (e) for N = 15.
5.29* Frequency-domain sampling The signal x(n) = a 1"1, - 1 < a < 1 has a Fourier
transform
X(w) =
1—
1 —2a cos <o + a2
(a) Plot X(w) for 0 < w < 2jt, a = 0.8.
Reconstruct and plot X(w) from its samples X Qj r k / N) , 0 < k < N - 1 for:
(b) N = 20
(c) N = 100
(d) Com pare the spectra obtained in parts (b) and (c) with the original spectrum
X(a>) and explain the differences.
(e) Illustrate the time-domain aliasing when N = 20.
446
The Discrete Fourier Transform: Its Properties and Applications
5 J0 * Frequency analysis o f amplitude-modulated discrete-time signal
Chap. 5
T he discrete-time
(a) Sketch the signals jr(n), xc(/i), and *»m(n), 0 < n < 255.
(b) Com pute and sketch the 128-point D FT of the signal
0 < n < 127.
(c) Compute and sketch the 128-point D FT of the signal xtm(n), 0 < n < 99.
(d) Com pute and sketch the 256-point D FT of the signal jcani{n), 0 < n < 179.
(e) Explain the results obtained in parts (b) through (d), by deriving the spectrum of
the am plitude-m odulated signal and comparing it with the experim ental results.
5.31* The sawtooth waveform in Fig. P5.31 can be expressed in the form of a Fourier series
as
)
(a) D eterm ine the Fourier series coefficients c*.
(b) Use an N -point subroutine to generate samples of this signal in the time domain
using the first six term s of the expansion for N « 64 and N = 128. Plot the signal
x(f) and the samples generated, and com m ent on the results.
Figure P531
5 3 2 Recall that the Fourier transform of x (r) = eJ0* is X ( j i 2) = 2jtS(£2 - i2o) and the
Fourier transform of
0 < t < To
otherwise
is
e-jOT<i/l
(a) D eterm ine the Fourier transform Y ( j n ) of
y(r) = p(t)eja°'
and roughly sketch |K0 '£2)| versus £2.
Chap. 5
Problems
447
(b) Now consider the exponential sequence
jr(n) =
where <uo is some arbitrary frequency in the range 0 < ojo < tt radians. Give the
most general condition that a>o must satisfy in order for x(n) to be periodic with
period P (P is a positive integer).
(c) Let y(n) be the finite-duration sequence
v (n ) = x ( n ) w N ( n ) = e i w ° " u i s ( n )
where w^( n) is a finite-duration rectangular sequence of length N and where
x(n) is not necessarily periodic. D eterm ine Y(a)) and roughly sketch \Y(a>)\ for
0 < to < 2n. W hat effect does N have in | y ( o j ) | ? Briefly com ment on the
similarities and differences between jY(a>)\ and ]y(j'S2)|.
(d> Suppose that
*(n) = e>{1' ! p)n
p a positive integer
and
y(n) = w N(n)x(n)
where N = IP, I a positive integer. Determ ine and sketch the N-point D FT of
y(n). R elate your answer to the characteristics of |K{w)|.
Is the frequency sampling for the D FT in part (d) adequate for obtaining a rough
approxim ation of |K(w)| directly from the magnitude of the D FT sequence |K(/t)|?
If not. explain briefly how the sampling can be increased so that it will be possible
to obtain a rough sketch of |K(£o)| from an appropriate sequence |y (Jt) | .
Efficient Computation of the
DFT: Fast Fourier Transform
Algorithms
A s w e h av e o b se rv e d in th e p reced in g c h a p te r, th e D isc re te F o u rie r T ra n sfo rm
(D F T ) play s an im p o rta n t ro le in m an y a p p lic a tio n s o f digital signal processing,
in clu d in g lin e a r filterin g , c o rre la tio n an aly sis, a n d sp e c tru m analysis. A m ajor
re a so n fo r its im p o rta n c e is th e ex isten ce o f efficient a lg o rith m s fo r c o m p u tin g the
DFT.
T h e m ain to p ic o f this c h a p te r is th e d e sc rip tio n o f c o m p u ta tio n a lly efficient
a lg o rith m s fo r e v a lu a tin g th e D F T . T w o d iffe re n t a p p ro a c h e s a re d e sc rib e d . O n e is
a d iv id e -a n d -c o n q u e r a p p ro a c h in w hich a D F T o f size N , w h e re jV is a c o m p o site
n u m b e r, is re d u c e d to th e c o m p u ta tio n o f sm a lle r D F T s fro m w hich th e larg er
D F T is co m p u te d . In p a rtic u la r, w e p r e s e n t im p o rta n t c o m p u ta tio n a l alg o rith m s,
called fast F o u rie r tra n sfo rm (F F T ) a lg o rith m s, fo r c o m p u tin g th e D F T w h en the
size N is a p o w e r o f 2 a n d w h e n it is a p o w e r o f 4.
T h e se co n d a p p ro a c h is b a s e d o n th e fo rm u la tio n o f th e D F T as a lin ear
filterin g o p e ra tio n o n th e d a ta . T h is a p p ro a c h le a d s to tw o a lg o rith m s, th e G o ertzel
alg o rith m an d th e ch irp -z tra n sfo rm a lg o rith m fo r co m p u tin g th e D F T via linear
filterin g o f th e d a ta se q u e n c e .
6.1 EFFICIENT COMPUTATION OF THE DFT: FFT ALGORITHMS
In th is sectio n w e p re s e n t se v e ra l m e th o d s fo r c o m p u tin g th e D F T efficiently.
In view o f th e im p o rta n c e o f th e D F T in v a rio u s d ig ital sig n a l p ro cessin g ap ­
p licatio n s, such as lin e a r filtering, c o rre la tio n an aly sis, a n d s p e c tru m analysis, its
efficien t c o m p u ta tio n is a to p ic th a t h a s re c e iv e d c o n s id e ra b le a tte n tio n by m any
m a th e m a tic ia n s, e n g in e e rs, a n d a p p lie d scien tists.
B asically , th e c o m p u ta tio n a l p ro b le m fo r th e D F T is t o c o m p u te th e se q u en ce
{X(*)} o f N c o m p lex -v alu ed n u m b e rs g iv en a n o th e r se q u e n c e o f d a ta (x(n)} of
448
Sec. 6.1
Efficient Computation of the DFT: FFT Algorithms
449
le n g th N , a c c o rd in g to th e fo rm u la
N- 1
* (* ) = V
Jt(n) W N
kn
0 < k < N -l
(6.1.1)
w h ere
WN = e - }7* ,N
(6.1.2)
In g e n e ra l, th e d a ta se q u e n c e x ( n ) is also a s su m e d to b e c o m p lex v alu ed .
S im ilarly , th e I D F T b ec o m e s
1 n -i
jr(n ) = — ^ X ( J k ) W ^ " *
^ *=o
0 < n < N —I
(6.1.3)
Since th e D F T a n d ID F T involve b asically th e sa m e ty p e o f c o m p u ta tio n s, o u r
d iscu ssio n o f efficien t c o m p u ta tio n a l a lg o rith m s fo r th e D F T a p p lie s as w ell to th e
efficien t c o m p u ta tio n o f th e ID F T .
W e o b se rv e th a t fo r each v alu e o f k , d ire c t c o m p u ta tio n o f X ( k ) involves
N co m p lex m u ltip lic a tio n s ( 4 N real m u ltip lic a tio n s) a n d N — 1 co m p lex a d d itio n s
(4 JV -2 re a l a d d itio n s). C o n se q u e n tly , to c o m p u te all N v alu es o f th e D F T re q u ire s
jV2 c o m p lex m u ltip lic atio n s a n d N 2 — N c o m p lex ad d itio n s.
D ire c t c o m p u ta tio n o f th e D F T is b asically in efficien t p rim a rily b e c a u se it
d o e s n o t e x p lo it th e sy m m etry a n d p e rio d ic ity p ro p e rtie s o f th e p h a s e fa c to r WV
In p a rtic u la r, th e s e tw o p r o p e rtie s are:
S y m m etry p ro p e rty :
Wk
N+N/2 = —W N
L
(6.1.4)
P e rio d ic ity p ro p e rty :
W#+N = W N
k
(6.1.5)
T h e c o m p u ta tio n a lly efficient a lg o rith m s d e s c rib e d in th is se c tio n , k n o w n collec­
tiv ely as fa st F o u rie r tra n sfo rm (F F T ) a lg o rith m s, e x p lo it th e s e tw o b asic p ro p e rtie s
o f th e p h a s e facto r.
6.1.1 Direct Computation of the DFT
F o r a co m p le x -v a lu e d se q u e n c e x ( n ) o f N p o in ts, th e D F T m a y b e ex p re sse d as
X R{k) = Y
|* * ( n ) c o s ^ ~ p - + x / ( n ) s i n ^ ^ - j
Xi(k) = - Y
|x ,f ( n ) s in
- x , ( n ) co s
T h e d ire c t c o m p u ta tio n o f (6.1.6) a n d (6.1.7) re q u ire s :
L 2 N 2 e v a lu a tio n s o f trig o n o m e tric fu n ctio n s.
2. 4 N 2 re a l m u ltip lic atio n s.
(6.1.6)
(6.1.7)
450
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
3. AN ( N — 1) real additions.
4. A num ber o f indexing and addressing op eration s.
T hese operations are typical o f D F T com putational algorithm s. T he operations
in item s 2 and 3 result in the D F T values X K(k) and X i ( k ) . T h e indexing and
addressing op eration s are necessary to fetch the data x(n'), 0 < « < Ar — 1, and
the phase factors and to store th e results. T h e variety o f D F T algorithm s optim ize
each o f these com p utational p rocesses in a different way.
6.1.2 Divide-and-Conquer Approach to Computation of
the DFT
T h e d evelop m en t o f com p utationally efficient algorithm s for the D F T is m ade pos­
sible if w e adopt a divide-and-conquer approach. T his approach is based on the
d ecom p osition o f an Af-point D F T in to su ccessively sm aller D F T s. T his basic ap­
proach leads to a fam ily o f com putationally efficient algorithm s k n ow n collectively
as FFT algorithm s.
T o illustrate the basic n otion s, let us consider the com p utation o f an Appoint
D F T , where N can be factored as a product o f tw o integers, that is,
N = LM
(6.1.8)
T h e assum ption that N is not a prim e num ber is n ot restrictive, sin ce w e can pad
any sequ en ce with zeros to ensure a factorization o f the form ( 6 . 1 .8 ).
N o w the seq u en ce j ( n ) , 0 < n < N — 1, can be stored in either a one­
dim ensional array in d exed by n or as a tw o-d im en sion al array in d exed by I and
m, w here 0 < / < L — 1 and 0 < m < A / - l a s illustrated in Fig. 6.1. N o te that / is
the row index and m is the colum n index. T hus, the se q u en ce x (n ) can b e stored
in a rectangular array in a variety o f w ays, each o f which d ep en d s on the mapping
o f index n to the in d exes (/, m).
For exam p le, su p p ose that w e select the m apping
n = Ml + m
(6.1.9)
T his leads to an arrangem ent in which the first row consists o f th e first M elem ents
o f x ( n ) , the secon d row consists o f the n ext M elem en ts o f x ( n ) , and so on, as
illustrated in Fig. 6.2 (a ). O n the other hand, the m apping
n — l + mL
(6.1.10)
stores the first L elem en ts o f x ( n ) in the first colu m n, the n ext L elem en ts in the
second colum n, and so on , as illustrated in Fig. 6.2(b ).
A sim ilar arrangem ent can be u sed to store th e com p u ted D F T valu es. In
particular, the m apping is from the in d ex it to a pair o f in d ices (p , q), where
0 < p < L — 1 and 0 < q < M — 1. If w e se lect the m apping
k — Mp + q
(6 .1 .1 1 )
Sec. 6.1
Efficient Computation of the DFT: FFT Algorithms
n ----------
0
1
m
x(\)
...
M2)
451
N -
1
JcCW-1)
(a)
column index
row index
K
0
1
0
*(0,0)
x(0,1)
1
* 1 .0 )
* 1 ,1 )
2
*(2,0)
*2,1)
Af-1
L- 1
(b)
Figure 6.1
N -l.
T w o dim ensional data array for storing the sequence x ( n ) . 0 < n £
the D F T is stored on a row -w ise basis, w here the first row contains the first M
elem en ts o f the D F T X ( k ) , the second row contains the next set of M elem en ts,
and so on . O n the other hand, th e m apping
(6 . 1. 12)
k = qL + p
results in a colu m n-w ise storage o f X (Jt), w here the first L elem en ts are stored in
the first colu m n, th e secon d se t o f L elem en ts are stored in the secon d colum n,
and so on.
N o w su p p ose that x ( n) is m apped in to the rectangular array x ( l , m ) and X( k )
is m app ed in to a corresp on d ing rectangular array X ( p , q). T h en the D F T can be
exp ressed as a d o u b le sum o ver th e elem en ts o f the rectangular array m ultiplied
b y th e corresp on d ing p h ase factors. T o b e specific, let us ad op t a colum n-w ise
m apping fo r x ( n ) g iven by (6.1.10) and th e row -w ise m apping for the D F T given
by (6.1.11). T hen
(6.1.13)
X ( p , q) = Y Y x{1' m ) W <“ p+qHmL+t)
rrtacO 1=0
But
p+ q){m L+ i)
_ ^M Lrnp^m Lq
H o w ev er, W%mp = 1, W%*L = W $ L = W**, and W * pl = W?'
(6.1.14)
= W[l
452
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Row-wise
Chap.
n = Ml + m
M- 1
0
*0)
1
MM)
MM
2
M 2M )
M 2M +
*2)
M M -
M M + 2)
M2M
M D
+ 1)
1)
M 2M
+ 2)
L - 1 x( (L- IW) M(L - 1 )Af + 1) x «L -l)M + 2)
1)
- 1)
M 'iM -
1)
x{L M -
1)
(a)
Column-wise
M- 1
0
MO )
ML)
1
jt(1)
M L+
2
M2)
M L + 2)
L -l
x(L -
I)
M U -
x ( ( M — 1) L )
M 2L)
1)
1)
x(2L
+ 1)
M 2 L + 2)
M IL
- 1)
x((M -
I )L + 1)
M (M -l)L + 2 )
MLM -
1)
(b)
Figure <L2 Two arrangements for the data arrays.
W ith th ese sim p lification s, (6.1.13) can be exp ressed as
L- l
X( p, q) = Y
/■*0
(6.1.15)
^|R«0
The expression in (6.1.15) in volves the com p utation o f D F T s o f length M and
length L. T o elab orate, let us subdivide th e com p utation into th ree steps:
L First, w e com p ute th e M -point D F T s
M-l
F(l,q) = £ x ( / ,m ) W ^ \
for each of the rows I = 0 ,1 ........ L - l .
0 < q < M - 1
( 6 .1 .1 6 )
Sec. 6.1
Efficient Computation of the DFT: FFT Algorithms
453
2 . S econ d , w e com p ute a new rectangular array G ( l , q ) defined as
(6.1.17)
3. F inally, w e com p ute the L -point D F T s
L-l
X ( p , q ) = J 2 G ^l ' ^ WL
(6.1.18)
for each colum n q = 0 , 1 , . . . , M — 1, o f the array G (l, q).
O n the surface it m ay appear that the com putational procedure outlined
above is m ore com p lex than the direct com p utation o f the D F T . H ow ever, let
us evalu ate the com putational com p lexity o f (6.1.15). T h e first step in volves the
com p utation o f L D F T s, each o f M points. H en ce this step requires L M 2 com ­
plex m ultiplications and L M { M — 1) com p lex additions. T h e secon d step requires
L M com p lex m ultiplications. Finally, the third step in the com p utation requires
M L 2 com p lex m ultiplications and M L ( L — 1) com p lex additions. T h erefore, the
com p utational com plexity is
C om plex m ultiplications:
N ( M + L + 1)
C om plex additions:
N ( M + L — 2)
(6.1.19)
w here N = M L . T hus the num ber o f m ultiplications has b een reduced from N 2
to N ( M + L + 1 ) and the num ber o f additions has b een reduced from N ( N — 1) to
N ( M + L — 2).
For exam p le, suppose that N = 1000 and w e select L = 2 and M = 500.
T h en , instead o f having to perform 106 com p lex m ultiplications via direct com pu­
tation o f the D F T , this approach leads to 503,000 com p lex m ultiplications. This
rep resents a reduction by approxim ately a factor o f 2. T h e num ber o f additions is
also red u ced by ab ou t a factor o f 2.
W h en N is a highly com p osite num ber, that is, N can be factored in to a
product o f prim e num bers o f th e form
N = r\ r 2 ■■- r v
(6.1.20)
then the d ecom p osition ab ove can b e rep eated (v - 1 ) m ore tim es. T his procedure
results in sm aller D F T s, w hich, in turn, lead s to a m ore efficient com putational
algorithm .
In effect, th e first segm en tation o f th e seq u en ce x ( n ) in to a rectangular array
o f M colu m ns w ith L elem en ts in each colu m n resu lted in D F T s o f sizes L and M .
Further d eco m p o sitio n o f th e data in effect in volves th e segm en tation o f each row
(or colu m n ) into sm aller rectangular arrays w hich result in sm aller D F T s. This
p roced u re term inates w h en N is factored in to its prim e factors.
Example 6.1.1
To illustrate this computational procedure, let us consider the computation of an
N = 15 point DFT. Since N = 5 x 3 = 15, we select L = 5 and M = 3. In other
454
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
words, we store the 15-point sequence *(n) column-wise as follows:
Row 1:
Row 2:
Row 3:
Row 4:
Row 5:
*(0, 0) = *(0)
*(1,0) = * (1 )
*(2,0) = * ( 2 )
*(3,0) = * (3 )
* (4,0) = * ( 4 )
*(0.1) = * ( 5 )
*(1, 1) = jc(6)
x ( 2 , 1) = x(7)
*(3,1) = * ( 8 )
*(4,1) = * ( 9 )
*(0, 2) = *(10)
*(1,2) = *(11)
x(2,2)=x(12)
*(3,2) = .r{13)
*(4,2) = *(14)
Now. we com pute the three-point DFTs for each of the five rows. This leads
to the following 5 x 3 array:
F(0, 0)
F a , o)
F( 2. 0)
F(3, 0)
F(4. 0)
F (O .l)
F O .l)
F( 2,1)
FO. 1)
F (4.1)
F( 0.2)
F (1.2)
F( 2.2)
FO- 2)
F( 4.2)
The next step is to multiply each of the terms F(l, q) by the phase factors
= M/jj. 0 < / < 4 and 0 < q < 2. This computation results in the 5 x 3 array:
Column 1
Column 2
Column 3
G (0.0)
G (1,0)
G(2, 0)
G (3.0)
G (4 .0)
C(0. 1)
C ( l.l)
C(2. 1)
G (3 ,1)
G (4 .1)
C (0.2)
G (l. 2)
C ( 2 .2)
G (3 .2)
G (4.2)
The final step is to com pute the five-point DFTs for each of the three columns.
This com putation yields the desired values of the D FT in the form
X (0.0) = X(0)
X (1,0) = X(3)
X (2.0) = X (6)
X (3,0) = X (9)
X (4,0) = X (12)
X (0 ,1) = X (l)
X (1.1) = X (4)
X (2 .1) = X(7)
X (3,1) = X(10)
X (4,1) = X (13)
X (0,2) = X(2)
X (1,2) = X(5)
X (2,2) = X(8>
X (3 ,2) = X ( ll)
X (4 ,2) = X(14)
Figure 6.3 illustrates the steps in the computation.
It is interesting to view the segm ented data sequence and the resulting D FT in
term s of one-dim ensional arrays. When the input sequence x(n) and the output DFT
X(jfc) in the two-dimensional arrays are read across from row 1 through row 5, we
obtain the following sequences:
IN PU T A R R A Y
*(0) x{5) *(10) *(1) *(6) *(11) x(2) *(7) *(12) *(3) *(8) *(13) x(4) *{9) *(14)
O U T PU T A R R A Y
X(0) X (l) X(2) X(3) X(4) X(5) X(6) X(7) X(8) X(9) X(10) X ( ll) X(12) X(13) X(14)
W e observe that the input data sequence is shuffled from the norm al order
in the com putation of the D FT. On the other hand, the output sequence occurs in
norm al order. In this case the rearrangem ent of the input data array is due to the
Sec. 6.1
455
Efficient Computation of the DFT: FFT Algorithms
Figure 63
DFTs.
Computation of N = 15-point DFT by means of 3-point and 5-point
segm entation of the one-dim ensional array into a rectangular array and the order in
which the D FTs are com puted. This shuffling of either the input data sequence or
the output D FT sequence is a characteristic of most FFT algorithms.
T o sum m arize, the algorithm that w e have introduced in volves the follow in g
com putations:
Algorithm 1
1. S tore the signal colu m n-w ise.
2. C om pute the Af-point D F T o f each row.
3. M ultiply the resulting array by the p h ase factors
4. C om p ute the L -point D F T o f each colum n
5. R ea d the resulting array row -w ise.
A n additional algorithm with a sim ilar com putational structure can b e o b ­
tained if th e input signal is stored row -w ise and the resulting transform ation is.
colu m n-w ise. In this case w e select as
n = Ml + m
k
(6.1.21)
= qL + p
This ch o ice o f in d ices lead s to the form ula for the D F T in the form
X {p ,q )
= £ £ * (/,
msO 1*0
(6. 1.22)
M- l
= E > c
Thus w e obtain a secon d algorithm .
urmp
WN
456
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
A lg o rith m 2
1. S to re th e sig n al row -w ise.
2. C o m p u te th e L -p o in t D F T a t each co lu m n .
3. M u ltip ly th e re su ltin g a rra y by th e fa c to rs W%m.
4. C o m p u te th e A f-point D F T o f e a c h row .
5. R e a d th e re su ltin g a rra y colum n-w ise.
T h e tw o a lg o rith m s given a b o v e h a v e th e sa m e c o m p lex ity . H o w e v e r, they
d iffer in th e a rra n g e m e n t o f th e c o m p u ta tio n s. In th e follow ing se c tio n s w e exploit
th e d iv id e -a n d -c o n q u e r a p p ro a c h to d e riv e fast a lg o rith m s w h e n th e size o f the
D F T is re stric te d to b e a p o w e r o f 2 o r a p o w e r o f 4.
6.1.3 Radix-2 FFT Algorithms
In th e p re c e d in g se ctio n w e d e sc rib e d fo u r a lg o rith m s fo r efficien t c o m p u ta tio n of
th e D F T b ased o n th e d iv id e -a n d -c o n q u e r a p p ro a c h . S uch an a p p ro a c h is applica­
b le w h en th e n u m b e r N o f d a ta p o in ts is n o t a p rim e . In p a rtic u la r, th e a p p ro ach
is v ery efficien t w h en N is highly c o m p o s ite , th a t is, w h en N can b e fa c to re d as
N = r\r2ry • ■■rv, w h e re th e {r,} are prim e.
O f p a rtic u la r im p o rta n c e as th e case in w hich r i = r2 — ■■• = r v = r , so th at
N = r ' \ In such a case th e D F T s a re o f size r , so th a t th e c o m p u ta tio n o f the
N -p o in t D F T h a s a re g u la r p a tte rn . T h e n u m b e r r is called th e rad ix o f th e F F T
alg o rith m .
In th is se c tio n w e d escrib e rad ix -2 a lg o rith m s, w hich a re by far th e m ost
w idely u sed F F T alg o rith m s. R ad ix -4 a lg o rith m s a re d e sc rib e d in th e follow ing
sectio n .
L e t us c o n s id e r th e c o m p u ta tio n o f th e N — 2 V p o in t D F T by th e dividea n d -c o n q u e r a p p ro a c h specified by (6.1.16) th ro u g h (6.1.18). W e select M = N / 2
a n d L = 2. T h is se lectio n re su lts in a sp lit o f th e N -p o in t d a ta se q u e n c e in to two
// /2 - p o in t d a ta se q u e n c e s f \ ( n ) a n d f 2(ri), c o rre sp o n d in g to th e ev en -n u m b ered
a n d o d d -n u m b e re d sa m p le s o f x ( n ), resp ec tiv ely , th a t is,
/ i ( n ) = x ( 2 n)
f i ( n ) = x (2n + 1),
N
n = 0 ,1 ,..., — ~ 1
(6.1.23)
T h u s f i ( n ) a n d f j ( n ) a re o b ta in e d by d e c im a tin g x ( n) b y a f a c to r o f 2, a n d hence
th e re su ltin g F F T a lg o rith m is called a d e c im a tio n -in -tim e a lg o rith m .
N o w th e Af-point D F T c a n b e e x p re ss e d in te rm s o f th e D F T s o f th e deci­
m a te d se q u e n c e s as follow s:
A'-l
X(k) = Y x ^
n*0
wn
* = 0 ,1 ,..., A f-1
Sec. 6.1
Efficient Computation of the DFT: FFT Algorithms
=
£
x (n )H # + 5 3
n even
(6.1.24)
n odd
<W/2)-l
(Af/2)-l
J ! x ( 2 m ) W ] f k 4- 5 3 *(2m + l)W * (2'n+1>
m=0
m=0
=
B ut
457
= W ^/ 2- W ith this su b stitu tio n , (6.1.24) c a n b e e x p re ss e d as
(N/Zi-i
x(k)=
5 3
(N/2)—l
M m )w *ra + K
= fi(Jfc) + W * f 2(*)
£
it = 0 , 1 , . . . , W - l
w h e re F i(it) a n d F2 (k) a re th e N /2 -p o in t D F T s o f th e se q u e n c e s f \ ( m ) an d f 2 (m),
resp ec tiv ely .
S ince F\ ( k) a n d F 2 (k) a re p e rio d ic, w ith p e rio d N f l , w e h a v e F](A -f N / 2) =
F i(* ) a n d /^(jfc + N f l ) = / i t * ) - 1° a d d itio n , th e fa c to r W ^ Nfl = —Wfa. H e n c e
(6.1.25) can b e ex p re sse d as
X ( k ) = f i(J t) + W* F2(fc)
* = 0 , 1 .........~ — 1
(6.1.26)
= F , ( * ) - < F 2(/:)
* = 0 , 1 ....... y - 1
(6.1.27)
+
W e o b se rv e th a t th e d ire c t c o m p u ta tio n o f F\ ( k) re q u ire s ( N /2 )2 com plex
m u ltip lic a tio n s. T h e sa m e a p p lie s to th e c o m p u ta tio n o f F2 (k). F u rth e rm o r e , th e re
a re N f l a d d itio n a l co m p lex m u ltip lic a tio n s re q u ire d to c o m p u te W kN F 2 (k). H en ce
th e c o m p u ta tio n o f X ( k ) re q u ire s 2 ( N f l )1 4- N f l = N 2/ 2 + N f l c o m p lex m u ltip li­
c a tio n s. T h is first s te p resu lts in a re d u c tio n o f th e n u m b e r o f m u ltip lic a tio n s from
N 2 to N 1 f l + N f l , w h ich is a b o u t a fa c to r o f 2 fo r N large.
T o b e c o n s iste n t w ith o u r p rev io u s n o ta tio n , w e m ay define
</!(*) = F l(* )
* = 0 ,l,...,y - l
G 2 (k) = W N
l F2 (k)
* = 0 , l , . . . , y —1
T h e n th e D F T X (it) m ay b e ex p re sse d as
X ( k ) = G x(k) + G 2 (k)
it = 0 , 1 ____ y - 1
(6.1.28)
X(k + j )
= G x( k ) - G 2 (k)
* = 0 , 1 .........y - 1
T h is c o m p u ta tio n is illu stra te d in Fig. 6.4.
H a v in g p e rfo rm e d th e d e c im a tio n -in -tim e o n ce, w e c a n r e p e a t th e p ro cess
fo r e a c h o f th e se q u e n c e s f \ ( n ) a n d f 2 (n). T h u s f \ ( n ) w o u ld re s u lt in th e tw o
458
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
* 0 ) x{2) x(4)
Chap. 6
JrW -2)
Figure 6.4 F irst step in the decim ation-in-tim e algorithm .
/V /4-point se q u e n c e s
v n (n ) = / ] ( 2n)
N
« = 0 , 1 ................... 1
4
v\ 2 (n) = f \ Q n + 1)
n = 0, 1.........j
(6.1.29)
- 1
an d f 2 (n) w o u ld yield
V2\(n) = f 2 (2 n)
N
n = 0, 1.........— - 1
4
V22(n) = /2(2n + 1)
N
n = 0 , 1.........— - 1
(6.1.30)
B y co m p u tin g jV /4-point D F T s , w e w o u ld o b ta in th e ///2 - p o in t D F T s Fi(Jfc) and
F2 (k) fro m th e re la tio n s
FiOfc) = V„(Jfc) + W k
Nf2 Vn (k)
k = 0 ,1 ,
1
(6.1.31)
Fx (* + t ) = Vl1{k) - KpynW
F 2 (k) = V21(*) + W N
k / 2 V22(k)
k=
1..... 7 - 1
k = 0 ,1 ,..., J
- 1
(6.1.32)
F2 ( * + j )
= V2i (*) -
N
k = 0, . , , , j - l
where the (Vi; (jt)} are the ///4 -p o in t D F T s o f th e seq u en ces {u,;(n)}.
Sec. 6.1
Efficient Computation of the DFT: FFT Algorithms
459
COMPARISON OF COMPUTATIONAL COMPLEXITY FOR THE
DIRECT COMPUTATION OF THE DFT VERSUS THE FFT ALGORITHM
TABLE 6.1
Number of
Points,
N
Complex Multiplications
in Direct Computation,
N2
Complex Multiplications
in FFT Algorithm,
(JV/2) log2 N
Speed
Improvement
Factor
4
8
16
32
64
128
256
512
1,024
16
64
256
1,024
4,096
16,384
65,536
262.144
1,048,576
4
12
32
80
192
448
1,024
2,304
5,120
4.0
5.3
8.0
12.8
21.3
36.6
64.0
113.8
204.8
W e o b se rv e th a t th e c o m p u ta tio n o f {V(J(*)} re q u ire s 4{W /4)2 m u ltip lic a tio n s
a n d h e n c e th e c o m p u ta tio n o f F\ ( k) a n d F 2(Jt) can b e a c c o m p lish ed w ith N 2/ 4 +
N f l c o m p lex m u ltip lic a tio n s. A n a d d itio n a l N f l co m p lex m u ltip lic a tio n s a re re ­
q u ire d to c o m p u te X ( k ) fro m F i(it) a n d Fi{k), C o n se q u e n tly , th e to ta l n u m b e r o f
m u ltip lic a tio n s is re d u c e d a p p ro x im a te ly by a fa c to r o f 2 ag ain to N 2/ 4 + N.
T h e d e c im a tio n o f th e d a ta se q u e n c e can b e re p e a te d ag ain a n d ag ain until
th e re su ltin g se q u e n c e s a re re d u c e d to o n e -p o in t se q u en ces. F o r N = 2 V, this
d e c im a tio n can b e p e rfo rm e d v = log2 N tim es. T h u s th e to ta l n u m b e r o f co m p lex
m u ltip lic a tio n s is re d u c e d to { N f l ) log2 N . T h e n u m b e r o f co m p lex a d d itio n s is
N log2 N . T a b le 6.1 p re s e n ts a c o m p a riso n o f th e n u m b e r o f co m p lex m u ltip lic a­
tio n s in th e F F T a n d in th e d ire c t c o m p u ta tio n o f th e D F T .
F o r illu stra tiv e p u rp o se s , Fig. 6.5 d e p ic ts th e c o m p u ta tio n o f an N = 8 p o in t
D F T . W e o b se rv e th a t th e c o m p u ta tio n is p e rfo rm e d in th r e e stag es, b eg in n in g
w ith th e c o m p u ta tio n s o f fo u r tw o -p o in t D F T s, th e n tw o fo u r-p o in t D F T s , an d
Figure <L5 Three stages in the computation o f an N = 8-point DFT.
460
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Stage
Stage 2
Chap. 6
Stage 3
X (0 )
X(l)
X(2)
X(3)
X(4)
X{5)
*(6)
X p )
finally, o n e eigh t-p oin t D F T . T he com bination o f the sm aller D F T s to form the
larger D F T is illustrated in Fig. 6. 6 for N = 8 .
O bserve that the basic com putation perform ed at every stage, as illustrated
in Fig. 6 .6 , is to take tw o com p lex num bers, say th e pair (a, b), m ultiply b by W N
r,
and then add and subtract the product from a to form tw o new com p lex numbers
(A, B ). This basic com putation, which is show n in Fig. 6.7, is called a butterfly
b ecau se the flow graph resem bles a butterfly.
In general, each butterfly involves on e com p lex m ultiplication and tw o com ­
plex additions. F or N = 2 V, there are N f l butterflies per stage o f th e com putation
p rocess and log 2 N stages. T h erefore, as previously indicated th e total num ber of
com plex m ultiplications is ( N f l ) log 2 N and com p lex additions is Arlog 2 N .
O nce a butterfly operation is perform ed on a pair o f com p lex num bers (a, b)
to p roduce ( A , B ) , there is no n eed to 'sa v e the input pair ( a , b ) . H en ce w e can
>A = a + W^b
B=a-Wt/b
F igure 6.7 Basic butterfly com putation
in th e decim ation-in-tim e F FT
algorithm .
Sec. 6.1
Efficient Computation of the DFT: FFT Algorithms
461
store th e result (A , B ) in the sam e location s as ( a , b ) . C on sequ en tly, w e require
a fixed am ount o f storage, n am ely, 2 N storage registers, in order to store the
results ( N com p lex num bers) o f the com p u tation s at each stage. Since th e sam e
2 N storage loca tio n s are used throughout the com p utation o f the JV-point D F T ,
w e say that the c o m p u ta tio n s are d o n e in place.
A secon d im portant observation is con cern ed with the ord er o f the input
data seq u en ce after it is d ecim ated (v - 1 ) tim es. For exam p le, if w e consider
the case w h ere N = 8 , w e know that th e first d ecim ation yield s the seq u en ce
jc(0), x ( 2 ) , x (4 ), * ( 6 ), * (1 ), Jt(3), jr(5), jc(7), and the secon d d ecim ation results in
the seq u en ce jc(0), x (4 ), x (2 ), x ( 6 ), jt(1), x (5 ), jc(3), jc(7). T h is sh u fflin g o f the
input data seq u en ce has a w ell-d efin ed order as can b e ascertained from observing
Fig. 6 .8 , w hich illustrates the d ecim ation o f th e eigh t-p oin t seq u en ce. B y expressing
the in d ex n, in the seq u en ce x ( n ) , in binary form , w e n o te that th e order o f the
d ecim ated d ata seq u en ce is easily ob tain ed by reading the binary representation
o f th e index n in reverse order. T hus the data p oin t jt(3) = *(011) is placed in
position m = 110 or m = 6 in the d ecim ated array. Thus w e say that the data x ( n )
after d ecim ation is stored in bit-reversed order.
W ith th e input data seq u en ce stored in bit-reversed order and the butterfly
com p utations perform ed in p lace, the resulting D F T seq u en ce X ( k ) is ob tain ed
in natural order (i.e., k = 0 , 1 , . . . , N — 1). O n the oth er hand, w e should indi­
ca te that it is p ossib le to arrange the F F T algorithm such that the input is left
in natural order and the resulting output D F T will occur in bit-reversed order.
Furtherm ore, w e can im pose the restriction that b oth the input data x ( n ) and the
output D F T X ( k ) be in natural order, and d erive an FFT algorithm in which the
com p utations are not d on e in place. H en ce such an algorithm requires additional
storage.
A n o th e r im portant radix-2 FFT algorithm , called th e decim ation -in-freq u en cy
algorithm , is o b tain ed by using the divide-and-conquer approach described in S ec­
tion 6.1.2 w ith th e ch oice o f M = 2 and L = N f l . T his ch oice o f param eters
im plies a colu m n-w ise storage o f the input data seq u en ce. T o d erive the algo­
rithm, w e begin by splitting the D F T form ula in to tw o sum m ations, o n e o f which
in v o lv es the sum over the first N t 2 data p oin ts and th e secon d sum in volves the
last N I 2 data points. Thus w e obtain
W 2 > -1
X (k) =
£
N- 1
x( n)W *? + Y
x(n)W %
(6.1.33)
Since W„N/2 = (—1)*, the exp ression (6.1.33) can b e rew ritten as
(6.1.34)
462
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
Data
decimation 1
Memory
Memory address
(decimal) (binary)
0
000
(ninino)
-
(/To"2« ()
—
(nnn in ;)
(0 0 0)
(0 0 1)
(0 1 0)
(0 1 1)
(1 0 0)
(1 0 !)
(1 1 0)
(1 1 1)
-»
—*■
-►
-►
-*>
-*>
—4
(0 0 0 )
(1 0 0 )
(0 0 1)
(1 0 1 )
(0 10)
(1 1 0 )
(0 1 1)
(1 1 1 )
-►
-►
-►
”*
-►
-►
(0 0 0 )
(1 0 0)
(0 10)
(t 10)
(0 0 1)
(1 0 1)
(0 1 1)
(1 1 ))
(b)
Figure 6Jt
Shuffling of the data and bit reversal.
N ow , let us split (d e c im a te ) X ( k ) in to th e ev en - a n d o d d - n u m b e re d sam p les. Thus
w e o b ta in
(W /2)-l r
x(n) + x
v tr k n
K
N/2
)
k = Q, 1 , . . . , y - 1
(6.1.35)
an d
(A 72)-l
X{2k + \) =
£
, r
/
JV \"1
+
"“ °
w h ere w e h av e u se d th e fact th a t Wf, = 'Wsp.-
1
N
* = 0 , 1 ......y - 1
(6.1.36)
Sec. 6.1
Efficient Computation of the DFT: FFT Algorithms
463
If w e d e fin e th e N /2 -p o in t se q u e n c e s gi ( n) a n d gz(n) as
g i( n ) = * ( « ) + *
(6.1.37)
g 2 (n) = |* ( n ) - x
+ y^)
n = 0 , 1 , 2 .........J
K
- 1
th e n
(N/2)-l
x ( 2 k) =
Y
s m K )2
n=0
(6.1.38)
(AT/2)—1
& W WN/2
X (2* + l ) =
n=0
T h e c o m p u ta tio n o f th e s e q u e n c e s g i(n ) a n d g 2 (n) a c co rd in g to (6.1.37) a n d th e
su b s e q u e n t u se o f th e s e se q u e n c e s to c o m p u te th e N /2 -p o in t D F T s a re d e p ic te d in
Fig. 6.9. W e o b se rv e th a t th e b asic c o m p u ta tio n in th is figure in v o lv es th e b u tte rfly
o p e r a tio n illu stra te d in Fig. 6.10.
T h is c o m p u ta tio n a l p ro c e d u re can b e re p e a te d th ro u g h d e c im a tio n o f th e
N /2 -p o in t D F T s , X ( 2 k ) a n d X ( 2 k + 1). T h e e n tire p ro cess in v o lv es v = log2 N
464
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
A = a + b
wi,
B = ( a — b)W f/
-
Figure 6.10 Basic butterfly com putation
in the decim ation-in-frequency F FT
algorithm .
1
stages o f decim ation , w here each stage in volves N t l butterflies o f the type shown in
Fig. 6.10. C onsequently, the com putation o f the Af-point D F T via the decim ationin-frequency FFT algorithm , requires ( N / 2) iog 2 N com p lex m ultiplications and
N lo g 2 N com p lex additions, just as in the d ecim ation -in-tim e algorithm . For il­
lustrative purposes, the eight-point d ecim ation-in-frequency algorithm is given in
Fig. 6.11.
W e observe from Fig. 6.11, that the input data x ( n ) occurs in natural order,
but the output D F T occurs in bit-reversed order. W e also n ote that the com puta­
tions are perform ed in place. H ow ever, it is p ossib le to reconfigure the decim ationin-frequency algorithm so that the input seq u en ce occurs in bit-reversed order
w hile the output D F T occurs in norm al order. F urtherm ore, if w e abandon the
requirem ent that the com putations b e d on e in place, it is also p ossib le to have
both the input data and the output D F T in norm al order.
Figure 6.11
N = 8-point decimation-in-frequency FFT algorithmn.
Sec. 6.1
Efficient Computation of the DFT: FFT Algorithms
465
6.1.4 Radix-4 FFT Algorithms
W h e n th e n u m b e r o f d a ta p o in ts N in th e D F T is a p o w e r o f 4 (i.e., N = 4 l ), w e
can , o f c o u rse, alw ays use a radix-2 a lg o rith m fo r th e c o m p u ta tio n . H o w e v e r, fo r
th is case, it is m o re efficien t c o m p u ta tio n a lly to em p lo y a rad ix -4 F F T alg o rith m .
L e t u s b eg in by d escrib in g a rad ix -4 d e c im a tio n -in -tim e F F T a lg o rith m , w hich
is o b ta in e d by se lectin g L = 4 an d M = N / 4 in th e d iv id e -a n d -c o n q u e r a p p ro a c h
d e s c rib e d in S ectio n 6.1.2. F o r this ch o ice o f L a n d M , w e h av e /, p — 0 ,1 , 2, 3: m,
q = 0, 1.........N J4 - 1; n = 4m + /; a n d k = ( N / 4) p + q. T h u s w e sp lit o r d e c im a te
th e W -point in p u t se q u e n c e in to f o u r su b s e q u e n c e s, x ( 4 n ), jc(4n + 1), x( 4n + 2),
x ( 4 n -f 3), n = 0, 1.........N / 4 — 1By a p p ly in g (6.1.15) w e o b ta in
3
* ( /> .« ) = £
[w ^F il'q ^W ?
0 ,1 .2 .3
(6.1.39)
w h ere F ( I . q ) is giv en by (6.1.16), th a t is.
(iV/4 |—!
F(l.q)=
£
mq
x ( l - n i ) W N/A
I = 0 .1 , 2. 3.
N
.........4 - 1
« =
(6.1.40)
an d
x(l . m ) = x ( 4 m -j- /)
(6.1.41)
(N
X(p.q) = X / — p + q
(6.1.42)
T h u s, th e fo u r ///4 -p o in t D F T s o b ta in e d fro m (6.1.40) a re c o m b in e d a cco rd in g
to (6.1.39) to yield th e W -point D F T . T h e ex p re ssio n in (6.1.39) fo r co m b in in g
th e ///4 -p o in t D F T s d efin es a rad ix -4 d e c im a tio n -in -tim e b u tterfly , w hich can be
e x p re ss e d in m atrix fo rm as
~X( Q , q ) ‘
X(\,q)
X (2,q)
-X(3,q)J
- 1 1 1 1
1
' j
-1
W °F(0,<7)
j
1 - 1 1 - 1
Li j -i - j
W«F(Lq)
W % F(2 ,q)
(6.1.43)
w l qF { X q )
T h e ra d ix -4 b u tte rfly is d e p ic te d in Fig. 6 .1 2 (a) a n d in a m o re co m p a c t fo rm
in Fig. 6 .1 2 (b ). N o te th a t since W® = 1, e a c h b u tte rfly in v o lv es th r e e co m p lex
m u ltip lic a tio n s , a n d 12 c o m p lex a d d itio n s.
T h is d e c im a tio n -in -tim e p ro c e d u re can b e re p e a te d recu rsiv ely v tim es. H e n c e
th e re su ltin g F F T alg o rith m co n sists o f v sta g es, w h e re e a c h sta g e c o n ta in s A74
b u tte rflie s . C o n s e q u e n tly , th e c o m p u ta tio n a l b u r d e n fo r th e a lg o rith m is 3 v N / 4 =
(3jV /8) lo g ; N c o m p lex m u ltip lic a tio n s a n d O N f l ) log2 N co m p lex a d d itio n s. W e
n o te th a t th e n u m b e r o f m u ltip lic a tio n s is re d u c e d by 2 5 % , b u t th e n u m b e r o f
a d d itio n s h a s in c re a se d b y 50% fro m N log2 N to O N f l ) log2 N .
466
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
Figure 6.12 Basic butterfly computation in o radix-4 FFT algorithm.
It is in te re stin g to n o te , h o w ev er, th a t by p e rfo rm in g th e a d d itio n s in tw o
step s, it is p o ssib le to re d u c e th e n u m b e r o f a d d itio n s p e r b u tte rfly fro m 12 to 8.
T h is can b e acco m p lish ed by e x p ressin g th e m atrix o f th e lin e a r tra n s fo rm a tio n in
(6.1.43) as a p ro d u c t o f tw o m atric es as follow s:
• m ? n
X ( l,9 )
X ( 2,q)
-1
0
1
.0
0
1
0
1
1
0
-1
0
0 -j
0
j
-
‘1
1
0
.0
0
0
1
1
1
-1
0
0
0
0
1
-1
w jjm g )
WqF(\,q)
W % F ( 2 .q)
(6.1.44)
lW *F(3,q).
N ow ea c h m atrix m u ltip lic a tio n involves fo u r a d d itio n s fo r a to ta l o f e ig h t ad d i­
tio n s. T h u s th e to ta l n u m b e r o f co m p lex a d d itio n s is re d u c e d to N log2 N , w hich
is id en tical to th e ra d ix -2 F F T a lg o rith m . T h e c o m p u ta tio n a l sa v in g s re su lts from
th e 25% re d u c tio n in th e n u m b e r o f co m p lex m u ltip lic atio n s.
A n illu stra tio n o f a rad ix -4 d e c im a tio n -in -tim e F F T a lg o rith m is sh o w n in
Fig. 6.13 fo r N = 16. N o te th a t in th is a lg o rith m , th e in p u t se q u e n c e is in norm al
o r d e r w h ile th e o u tp u t D F T is shuffled. In th e rad ix -4 F F T a lg o rith m , w here
th e d e c im a tio n is b y a f a c to r o f 4, th e o r d e r o f th e d e c im a te d se q u e n c e can be
d e te rm in e d b y re v e rsin g th e o r d e r o f th e n u m b e r th a t re p re s e n ts th e in d ex n
in a q u a te rn a ry n u m b e r sy stem (i.e., th e n u m b e r sy stem b a s e d o n th e digits 0,
1, 2, 3).
A rad ix -4 d e c im a tio n -in -fre q u e n c y F F T a lg o rith m can b e o b ta in e d by se lect­
ing L = N / 4 , M = 4; /, p = 0, 1.........N / 4 - 1; m, q = 0, 1, 2, 3; n = {N / 4 ) m + /;
a n d k = 4 p + q. W ith th is ch o ice o f p a ra m e te rs , th e g e n e ra l e q u a tio n g iven by
Efficient Computation of the DFT: FFT Algorithms
467
Figure 6.13 Sixteen-point radix-4 decimation-in-time algorithm with input in nor­
mal order and output in digit-reversed order.
(6.1.15) can be exp ressed as
(A y 4 )- l
X(p,q) =
Ip
C ( l , q) W' NfA
l
£
(6.1.45)
1=0
w here
q = 0 , 1 , 2, 3
G ( l , q ) = w ‘ hF (l, q)
(6.1.46)
i - 0 . 1 .........£ - 1
4
and
q =0,1,2,3
F(l,q) = Y x ( l , m ) W ?
N
/ = 0 , 1 , 2 , 3 .........- - 1
4
(6.1.47)
W e n o te that X ( p , q ) = X ( 4 p + q ), q = 0, 1, 2, 3. C on sequ en tly, the Af-point
D F T is d ecim a ted into four N /4 -p o in t D F T s and h en ce w e have a decim ationin -frequency F F T algorithm . T h e com putations in (6.1.46) and (6.1.47) define
468
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
Figure 6.14 Sixteen-point, radix-4 decimation-in-frequency algorithm with input
in normal order and output in digit-reversed order.
the basic radix-4 butterfly for the d ecim ation-in-frequency algorithm . N o te that
the m ultiplications by the factors W% occur after the com b ination o f the data
p oin ts x (/, m), just as in the case o f th e radix-2 decim ation -in-freq u en cy algo­
rithm.
A 16-point radix-4 decim ation -in-freq u en cy F F T algorithm is show n in
Fig. 6.14. Its input is in norm al order and its output is in digit-reversed order.
It has exactly the sam e com putational com p lexity as the d ecim ation -in-tim e radix4 F F T algorithm .
For illustrative purposes, let us rederive the radix-4 decim ation-in-frequency
algorithm by breaking the jV-point D F T form ula in to four sm aller D F T s. We
have
N- 1
X( k) = T x ( n ) W N
kn
n=0
=
JV/4-1
N/Z-1
3N/4-1
fif-l
£
x { n ) W kNn + £
x{n)W%' + £
* (* )< " + £
x(n)W N
kn
n=0
n=N/4
n=Nfi
n=JN/4
Sec. 6.1
469
Efficient Computation of the DFT: FFT Algorithms
/A» 74/ * ♦ - 1J
=
h= (I
«=o
A'/4 —1
+ <
E*("
A’// 4t -- iI
A
£ ,< „ ,< ■ + < «
V/2 E
/
/
A/ N
+ t
'
\i \
* (« + y ) <
N / A —\
+ < A'/4 E
/
'l \ ! \
A (" + t )
(6.1.48)
F ro m th e d efin itio n o f th e tw id d le facto rs, w e h av e
lNk/4
w\N
(6.1.49)
(jf
A fte r su b s titu tio n o f (6.1.49) in to (6.1.48). we o b ta in
N/ 4-1
xa)=
E
x(») + (
(6.1.50)
N
+ { -1 f x [ n + - J + ( ;)
W\
T h e re la tio n in (6.1.50) is n o t an N /4 -p o in t D F T b e c a u se th e tw id d le facto r
d e p e n d s o n N a n d n o t on N / 4 . T o c o n v ert it in to an A '/4-point D F T , wc su b d iv id e
th e D F T se q u e n c e in to four /V /4-point su b se q u e n c e s, X ( 4 k ) . X ( 4 k + !), X (4£ + 2),
an d X { 4 k + 3), k — 0, 1........ N / 4 — 1. T h u s we o b ta in th e rad ix -4 d ecim atio n -in fre q u e n c y D F T as
x ( n ) + .v
(6.1.51)
■ +? )
+a(', + i ) +j:(" + t )
X (4k + l ) =
0 urkn
w"w
N r r N /4
(6.1.52)
■*(” ) ~ i x ( ” + " j }
E
IV " w kn
N w yv/4
X ( 4 k + 2) =
T ; 1r
(
n \
E
* (« )“ * ( « + j J
- K
r
X ( 4 k + 3) =
E
M
(6.1.53)
kn
' + t ) W 2" ww Nf4
/
x (n '>+ J x ( " +
n \
(6.1.54)
J
~ x ( n ~h j ) - Jx (b+ t ) ] w^
kn
S/4
470
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
w h ere w e h av e u se d th e p ro p e rty W^ kn = W^"4. N o te th a t th e in p u t to ea c h N/ 4p o in t D F T is a lin e a r co m b in a tio n o f fo u r signal sa m p le s sc aled by a tw id d le factor.
T h is p ro c e d u re is r e p e a te d v tim es, w h ere v = log,, N.
6.1.5 Split-Radix FFT Algorithms
A n in sp e ctio n o f th e radix-2 d e c im a tio n -in -fre q u e n c y flo w g rap h show n in Fig. 6.11
in d icates th a t th e e v e n -n u m b e re d p o in ts o f th e D F T can b e c o m p u te d in d e p e n ­
d en tly o f th e o d d -n u m b e re d p o in ts. T h is suggests th e p o ssib ility o f using d ifferen t
c o m p u ta tio n a l m e th o d s fo r in d e p e n d e n t p a rts o f th e a lg o rith m w ith th e ob jectiv e
o f re d u c in g th e n u m b e r o f c o m p u ta tio n s. T h e sp lit-ra d ix F F T (S R F F T ) alg o rith m s
ex p lo it th is id ea by u sing b o th a radix-2 a n d a radix-4 d e c o m p o s itio n in th e sam e
F I T alg o rith m .
W e illu stra te th is a p p ro a c h w ith a d e c im a tio n -in -fre q u e n c y S R F F T alg o rith m
d u e to D u h a m e l (1986). F irst, w e recall th a t in th e rad ix -2 d ec im a tio n -in -fre q u e n c y
F F T alg o rith m , th e e v e n -n u m b e re d sa m p le s o f th e /V -point D F T a re given as
N o te th a t th ese D F T p o in ts can b e o b ta in e d fro m an N /2 -p o in t D F T w ith o u t any
a d d itio n a l m u ltip lic atio n s. C o n se q u e n tly , a radix-2 suffices fo r th is c o m p u ta tio n .
T h e o d d -n u m b e re d sa m p le s {X{2k + 1)) o f th e D F T re q u ire th e p re m u ltip li­
catio n o f th e in p u t se q u e n c e w ith th e tw id d le fa c to rs W N
n . F o r th e s e sa m p les a
rad ix -4 d eco m p o sitio n p ro d u c e s so m e c o m p u ta tio n a l efficiency b e c a u se th e fourp o in t D F T h as th e larg est m u ltip lic a tio n -fre e b u tterfly . I n d e e d , it can b e show n
th a t usin g a rad ix g r e a te r th a n 4, d o e s n o t re su lt in a significant re d u c tio n in com ­
p u ta tio n a l co m p lex ity .
I f we u se a rad ix -4 d e c im a tio n -in -fre q u e n c y F F T a lg o rith m fo r th e oddn u m b e re d sa m p le s o f th e /V -point D F T , w e o b ta in th e fo llo w in g N /4 -p o in t D FTs:
N/4-1
(6.1.56)
- j [ x ( n + N / 4 ) - x( n + 3 N / 4 )]}
A74-1
(6.1.57)
+ j [ x ( n + N / 4 ) - x{ n + 3 N / 4 ) ] } W ^
T h u s th e N -p o in t D F T is d e c o m p o se d in to o n e N /2 -p o in t D F T w ith o u t ad d itio n al
tw id d le facto rs a n d tw o N /4 -p o in t D F T s w ith tw id d le facto rs. T h e /V-p o in t D F T
is o b ta in e d by su ccessiv e u se o f th e s e d e c o m p o s itio n s u p to th e la st stag e. T hus
w e o b ta in a d e c im a tio n -in -fre q u e n c y S R F F T alg o rith m .
F ig u re 6.15 sh o w s th e flow g ra p h fo r a n in -p la ce 3 2 -p o in t d ecim atio n in -freq u en cy S R F F T a lg o rith m . A t stag e A of th e c o m p u ta tio n fo r N = 32, the
Sec. 6.1
471
Efficient Computation of the DFT: FFT Algorithms
A
B
Figure 6,15 L ength 32 split-radix F F T algorithm s from p ap er by D uham el (1986); rep rin ted
w ith perm ission from the IE E E .
to p 16 p o in ts c o n s titu te th e se q u e n c e
go(«) = x ( n ) + x ( n + N / 2)
0 < n < 15
(6.1.58)
T h is is th e s e q u e n c e re q u ire d fo r th e c o m p u ta tio n o f X ( 2 k ) . T h e n e x t 8 p o in ts
c o n s titu te th e se q u e n c e
gi(n) = x(n) - x(n + N/ 2)
0<n<7
(6.1.59)
472
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
T h e b o tto m e ig h t p o in ts c o n s titu te th e se q u e n c e j g 2 (71). w h ere
82 (n) = x ( n + N / 4 ) — x ( n + 3 N / 4 )
0 < « < 7
(6.1.60)
T h e se q u e n c e s gj ( n) a n d gi ( n) a re u sed in th e c o m p u ta tio n o f X ( 4 k 4 - 1) and
A'( 4 * + 3). T h u s, a t stag e A w e h ave c o m p le te d th e first d e c im a tio n fo r th e radix-2
c o m p o n e n t o f th e alg o rith m . A t sta g e B , th e b o tto m eig h t p o in ts c o n stitu te the
c o m p u ta tio n o f [# i(n ) + 7 ( ” )]^32 ->0 < /i < 7, w hich is u sed to c o m p u te X (4A -f 3),
0 < k < 7. T h e n ex t eig h t p o in ts fro m th e b o tto m c o n s titu te th e c o m p u ta tio n of
[tfi(n) — j g 2(h)] VVjj, 0 < n < 7, w hich is u se d to c o m p u te X ( 4 k 4 -1 ), 0 < k < 7.
T h u s a t stag e B , w e h av e c o m p le te d th e first d e c im a tio n fo r th e ra d ix -4 alg o rith m ,
w hich resu lts in tw o 8 -p o in t se q u e n c e s. H e n c e th e b asic b u tte rfly c o m p u ta tio n fo r
th e S R F F T a lg o rith m h a s th e “L -s h a p e d ” form illu stra te d in Fig. 6.16.
N ow w e r e p e a t th e ste p s in th e c o m p u ta tio n a b o v e. B e g in n in g w ith th e to p
16 p o in ts a t stag e A , w e r e p e a t th e d e c o m p o s itio n f o r th e 1 6 -p o in t D F T . In o th e r
w o rd s, w e d e c o m p o se th e c o m p u ta tio n in to an e ig h t-p o in t, rad ix -2 D F T an d tw o
fo u r-p o in t, rad ix -4 D F T s. T h u s a t sta g e B , th e to p eig h t p o in ts c o n s titu te the
se q u e n c e (w ith N = 16)
g'o(*) = 8o(n) + go(n 4- N / 2)
0 <n < 7
(6.1.61)
an d th e n ex t eig h t p o in ts c o n s titu te th e tw o fo u r-p o in t se q u e n c e s g[(n) a n d jg'2(n),
w h ere
g[ (n) = go(n) ~ go(n + N f l )
0 < n < 3
(6.1.62)
82 («) = 8o(n + N / 4 ) - g0(n + 3 N / 4 )
0 < n < 3
T h e b o tto m 16 p o in ts o f sta g e B a re in th e fo rm o f tw o e ig h t-p o in t D F T s. H en ce
ea c h e ig h t-p o in t D F T is d e c o m p o s e d in to a fo u r-p o in t, rad ix -2 D F T a n d a fourp o in t, rad ix -4 D F T . In th e final stag e, th e c o m p u ta tio n s in v o lv e th e co m b in atio n
o f tw o -p o in t se q u en ces.
T a b le 6.2 p r e s e n ts a c o m p a riso n o f th e n u m b e r o f nont ri vi al re a l m u ltip li­
ca tio n s a n d a d d itio n s re q u ire d to p e rfo rm a n jY -point D F T w ith co m p lex -v alu ed
Sec. 6 .1
473
Efficient Computation of the DFT: FFT Algorithms
TABLE 6.2 NUMBER OF NONTRIVIAL REAL MULTIPLICATIONS AND
ADDITIONS TO COMPUTE AN N-POINT COMPLEX DFT
Real M ultiplications
Radix
R adix
4
24
88
264
712
1,800
4.360
10.248
20
N
16
32
64
128
256
512
1,024
208
Radix
8
204
1.392
3.204
7,856
Real A dditions
Split
Radix
Radix
2
Radix
4
20
68
196
516
1.284
3.076
7,172
152
408
1.032
2.504
5,896
13.566
30.728
148
976
R adix
8
972
5,488
12,420
28.336
Split
Radix
148
388
964
2308
5.380
12.292
27,652
Source: E xtracted from D uham el (1986).
d a ta , using a rad ix -2 , ra d ix -4, radix-8, a n d a sp lit-ra d ix F F T . N o te th a t th e S R F F T
alg o rith m re q u ire s th e lo w est n u m b e r o f m u ltip lic a tio n a n d a d d itio n s. F o r this
re a so n , it is p re fe ra b le in m an y p ra c tic a l a p p licatio n s.
A n o th e r ty p e o f S R F F T a lg o rith m has b e e n d e v e lo p e d by P rice (1990). Its
re la tio n to D u h a m e l’s a lg o rith m d e sc rib e d p rev io u sly can b e seen by n o tin g th a t
th e rad ix -4 D F T te rm s X ( 4 k 4- 1) an d X ( 4 k + 3) involve th e N /4 -p o in t D F T s o f th e
se q u e n c e s [g i(n ) a n d [ # i( '0 + ,/£ 2(« )]W $ \ resp ec tiv ely . In effect, th e
se q u e n c e s g i(/i) a n d g 2 (n) are m u ltip lie d by th e fa c to r (v e c to r) (1, —j ) = (1, H ^ )
an d by WJJ fo r th e c o m p u ta tio n o f X ( 4 k + 1), w hile th e c o m p u ta tio n o f X (4k + 3)
in v o lv es th e fa c to r (1 , j ) = (1, W{2*) an d W j f .
In s te a d , o n e can r e a rra n g e th e
c o m p u ta tio n so th a t th e fa c to r fo r X ( 4 k + 3) is ( —j , —1) = —(W £ 8, 1). A s a resu lt
o f th is p h a s e ro ta tio n , th e tw id d le fa c to rs in th e c o m p u ta tio n o f X (4k -f 3) b eco m e
ex actly th e sam e as th o se fo r X ( 4 k + 1), e x cep t th a t th e y o c c u r in m irr o r im age
o rd e r. F o r e x am p le, at sta g e B o f Fig. 6.15, th e tw id d le fa c to rs W21, W 18, . . . , W3
a re re p la c e d by (V1, W 2, . . . , W 1, resp ec tiv ely . T h is m irro r-im a g e sy m m etry occurs
a t ev ery s u b s e q u e n t sta g e o f th e alg o rith m . A s a c o n s e q u e n c e , th e n u m b e r o f
tw id d le facto rs th a t m u st b e c o m p u te d a n d s to re d is r e d u c e d by a fa c to r o f 2 in
c o m p a riso n to D u h a m e l’s a lg o rith m . T h e re su ltin g a lg o rith m is called th e “m irr o r ”
F F T (M F F T ) alg o rith m .
A n a d d itio n a l facto r-o f-2 savings in sto ra g e o f tw id d le fa c to rs can be o b ta in e d
b y in tro d u c in g a 90° p h a s e o ffset a t th e m id p o in t o f ea c h tw id d le a rra y , w hich can
b e re m o v e d if n ecessa ry a t th e o u tp u t o f th e S R F F T c o m p u ta tio n . T h e in co r­
p o r a tio n o f th is im p ro v e m e n t in to th e S R F F T ( o r th e M F F T ) re su lts in a n o th e r
a lg o rith m , also d u e to P ric e (1990), c a lle d th e “p h a s e ” F F T (P F F T ) alg o rith m .
6.1.6 Implementation of FFT Algorithms
N o w th a t w e h av e d e s c rib e d th e b asic rad ix -2 a n d rad ix -4 F F T a lg o rith m s, let
us c o n s id e r so m e o f th e im p le m e n ta tio n issues. O u r r e m a rk s ap p ly d ire c tly to
474
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
rad ix -2 a lg o rith m s, a lth o u g h sim ilar c o m m e n ts m ay b e m a d e a b o u t rad ix -4 a n d
h ig h er-rad ix alg o rith m s.
B asically, th e rad ix -2 F F T a lg o rith m co n sists o f ta k in g tw o d a ta p o in ts a t a
tim e fro m m e m o ry , p e rfo rm in g th e b u tte rfly c o m p u ta tio n s a n d r e tu rn in g th e r e ­
su ltin g n u m b e rs to m em o ry . T h is p ro c e d u re is r e p e a te d m an y tim e s ( ( N log2 N) [ 2
tim es) in th e c o m p u ta tio n o f an JV-point D F T .
T h e b u tterfly c o m p u ta tio n s re q u ire th e tw id d le fa c to rs {W^} a t v a rio u s stages
in e ith e r n a tu ra l o r b it-re v e rse d o rd e r. In an efficien t im p le m e n ta tio n o f th e algo­
rith m , th e p h ase fa c to rs a re c o m p u te d o n ce a n d s to re d in a ta b le , e ith e r in n o rm a l
o r d e r o r in b it-re v e rse d o rd e r, d e p e n d in g o n th e specific im p le m e n ta tio n o f th e
alg o rith m .
M e m o ry re q u ire m e n t is a n o th e r fa c to r th a t m u st b e c o n s id e re d . If th e c o m ­
p u ta tio n s a re p e rfo rm e d in p lace, th e n u m b e r o f m e m o ry lo c a tio n s re q u ire d is 2 N
since th e n u m b e rs a re com plex. H o w e v e r, w e can in ste a d d o u b le th e m e m o ry to
4N , th u s sim p lifying th e in d ex in g a n d c o n tro l o p e r a tio n s in th e F F T a lg o rith m s. In
th is case w e sim p ly a lte rn a te in th e use o f th e tw o se ts o f m e m o ry lo c a tio n s from
o n e sta g e o f th e F F T a lg o rith m to th e o th e r. D o u b lin g o f th e m e m o ry also allow s
us to h av e b o th th e in p u t se q u e n c e a n d th e o u tp u t se q u e n c e in n o rm a l o rd e r.
T h e re are a n u m b e r o f o th e r im p le m e n ta tio n issues re g a rd in g ind ex in g , bit
rev ersal, an d th e d e g re e o f p arallelism in th e c o m p u ta tio n s. T o a larg e ex ten t,
th e se issues a re a fu n ctio n o f th e specific a lg o rith m a n d th e ty p e o f im p le m e n ta ­
tio n , n am ely , a h a rd w a re o r so ftw are im p le m e n ta tio n . In im p le m e n ta tio n s b ased
o n a fix ed -p o in t a rith m e tic , o r flo atin g -p o in t a rith m e tic o n sm a ll m a ch in es, th e re
is also th e issue o f ro u n d -o ff e rro rs in th e c o m p u ta tio n . T h is to p ic is co n sid e re d
in S ectio n 6.4.
A lth o u g h th e F F T a lg o rith m s d e sc rib e d p re v io u sly w e re p re s e n te d in th e
c o n te x t o f co m p u tin g th e D F T efficiently, th e y can also b e u s e d to c o m p u te th e
ID F T , w hich is
j
A '- l
* (") = 7 T ] C * (
*=0
(6' ll63)
T h e o n ly d iffe re n c e b e tw e e n th e tw o tra n sfo rm s is th e n o rm a liz a tio n fa c to r l / N
a n d th e sign o f th e p h a se fa c to r WN. C o n s e q u e n tly , a n F F T a lg o rith m fo r co m ­
p u tin g th e D F T , c a n b e c o n v e rte d to an F F T a lg o rith m fo r c o m p u tin g th e ID F T
by ch an g in g th e sign o n all th e p h a se fa c to rs a n d d iv id in g th e final o u tp u t o f th e
alg o rith m by N.
In fact, if w e ta k e th e d e c im a tio n -in -tim e a lg o rith m t h a t w e d e sc rib e d in
S ectio n 6.1.3, re v e rse th e d ire c tio n o f th e flow g ra p h , c h a n g e th e sign o n th e p h ase
facto rs, in te rc h a n g e th e o u tp u t a n d in p u t, a n d finally, d iv id e th e o u tp u t by N , w e
o b ta in a d e c im a tio n -in -fre q u e n c y F F T a lg o rith m fo r c o m p u tin g th e ID F T . O n th e
o th e r h a n d , if w e b eg in w ith th e d e c im a tio n -in -fre q u e n c y F F T a lg o rith m d escrib ed
in S ectio n 6.1.3 a n d r e p e a t th e ch an g es d e s c rib e d a b o v e , w e d b ta in a d ecim atio n in -tim e F F T a lg o rith m fo r c o m p u tin g th e ID F T . T h u s it is a sim p le m a tte r to devise
F F T a lg o rith m s fo r co m p u tin g th e ID F T .
Sec. 6.2
Applications of FFT Algorithms
475
F in ally , w e n o te th a t th e em p h asis in o u r discussion o f F F T a lg o rith m s w as
o n rad ix -2 , rad ix -4 , a n d sp lit-ra d ix a lg o rith m s. T h e se are by fa r th e m o st w idely
u se d in p ra c tic e . W h e n th e n u m b e r o f d a ta p o in ts is n o t a p o w e r o f 2 o r 4. it is a
sim p le m a tte r to p a d th e se q u e n c e x ( n ) w ith zero s such th a t /V = 2 1’ o r N = 4 '.
T h e m e a s u re o f co m p lex ity fo r F F T alg o rith m s th a t w e h av e e m p h a siz e d
is th e r e q u ire d n u m b e r o f a rith m e tic o p e ra tio n s (m u ltip lic a tio n s an d a d d itio n s).
A lth o u g h this is a v ery im p o rta n t b e n c h m a rk fo r c o m p u ta tio n a l co m p lex ity , th e re
a re o th e r issues to b e c o n s id e re d in p ractical im p le m e n ta tio n o f F F T a lg o rith m s.
T h e se in clu d e th e a rc h ite c tu re o f th e p ro cesso r, th e av ailab le in stru c tio n set, th e
d a ta s tru c tu re s fo r s to rin g tw id d le facto rs, an d o th e r c o n sid e ra tio n s.
F o r g e n e ra l-p u rp o s e c o m p u te rs, w h e re th e cost o f th e n u m e ric a l o p e ra tio n s
d o m in a te , rad ix -2 , rad ix-4, an d sp lit-ra d ix F F T a lg o rith m s a re go o d c a n d id a te s.
H o w e v e r, in th e case o f sp e c ia l-p u rp o se digital signal p ro c e ss o rs , fe a tu rin g sin g le­
cycle m u ltip ly -a n d -a c c u m u la te o p e ra tio n , b it-re v e rse d ad d re ssin g , a n d a high d e ­
g ree o f in stru c tio n p a ra lle lism , th e stru c tu ra l re g u la rity o f th e a lg o rith m is eq u ally
im p o rta n t as a rith m e tic co m p lex ity . H e n c e fo r D S P p ro c e sso rs, rad ix -2 o r radix4 d e c im a tio n -in -fre q u e n c y F F T a lg o rith m s are p re fe ra b le in te rm s o f sp e e d an d
a ccu racy . T h e irre g u la r stru c tu re o f th e S R F F T m ay r e n d e r it less su ita b le fo r
im p le m e n ta tio n o n d ig ital signal pro cesso rs. S tru c tu ra l re g u la rity is also im p o rta n t
in th e im p le m e n ta tio n o f F F T a lg o rith m s on v e c to r p ro c e sso rs, m u ltip ro c e sso rs,
a n d in V L SI. I n te rp ro c e s s o r co m m u n icatio n is an im p o rta n t c o n s id e ra tio n in such
im p le m e n ta tio n s o n p a rallel pro cesso rs.
In co n clu sio n , we h av e p re s e n te d several im p o rta n t c o n s id e ra tio n s in th e
im p le m e n ta tio n o f F F T a lg o rith m s. A d v an ce s in digital signal p ro cessin g te c h n o l­
ogy, in h a rd w a re a n d so ftw are, will c o n tin u e to influence th e ch o ice a m o n g F F T
a lg o rith m s fo r v a rio u s p ractical ap p licatio n s.
6.2 APPLICATIONS OF FFT ALGORITHMS
T h e F F T a lg o rith m s d e sc rib e d in th e p re c e d in g se ctio n find a p p lic a tio n in a v ariety
o f a re a s , in clu d in g lin e a r filtering, c o rre la tio n , a n d s p e c tru m analysis. B asically,
th e F F T a lg o rith m is u sed as a n efficient m e a n s to c o m p u te th e D F T a n d th e ID F T .
In th is se c tio n w e c o n s id e r th e u se o f th e F F T a lg o rith m in lin e a r filterin g
a n d in th e c o m p u ta tio n o f th e c ro ssco rrelatio n o f tw o se q u e n c e s. T h e use o f th e
F F T in s p e c tru m an aly sis is c o n sid e re d in C h a p te r 12. In a d d itio n w e illu strate
h o w to e n h a n c e th e efficiency o f th e F F T a lg o rith m by fo rm in g co m p le x -v a lu e d
se q u e n c e s fro m re a l-v a lu e d se q u e n c e s p rio r to th e c o m p u ta tio n o f th e D F T .
6.2.1 Efficient Computation of the DFT of Two Real
Sequences
T h e F F T a lg o rith m is d esig n e d to p e rfo rm co m p lex m u ltip lic a tio n s a n d a d d itio n s,
ev en th o u g h th e in p u t d a ta m ay b e re a l v alued. T h e b asic re a s o n fo r th is s itu a tio n is
476
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
th a t th e p h a se fa c to rs a re co m p lex a n d h e n c e , a fte r th e first sta g e o f th e alg o rith m ,
all v a riab les a re b asically co m p lex -v alu ed .
In view o f th e fact th a t th e a lg o rith m can h a n d le c o m p le x -v a lu e d in p u t se­
q u en ces, w e c a n ex p lo it th is cap ab ility in th e c o m p u ta tio n o f th e D F T o f tw o
re a l-v a lu e d se q u e n c e s.
S u p p o se th a t * i(n ) a n d x 2(n) are tw o re a l-v a lu e d se q u e n c e s o f len g th N , and
let x( n ) b e a co m p le x -v a lu e d se q u e n c e d efin ed as
x ( n ) = X] (n) + j x 2 (n)
0 < n < N —1
(6.2.1)
T h e D F T o p e ra tio n is lin e a r a n d h en ce th e D F T o f x ( n ) can be ex p re sse d as
X ( k ) = X ](k) + j X 2(k)
(6.2.2)
T h e se q u e n c e s ^ i(« ) an d J 2 OO can be ex p re sse d in te rm s o f x ( n ) as follow s:
, 4
ac(H) + J:*(n)
* i(« ) = -------- 2--------
(6.2.3)
x(n)-x*(n)
•*:(«) = ------- tt.--------
(6.2.4)
H e n c e th e D F T s o f jri(n ) an d x 2(n) are
-] \ D F T [ x { n ) } + D F T [ x \ n ) } )
(6.2.5)
X 2(k) = j - \ D F T [ x ( n )] - DF T [ x * ( n ) ] )
(6.2.6)
* ,( * ) =
R ecall th a t th e D F T o f x*( n) is X * ( N — k). T h e re fo re ,
X] (k) = i[X (* r) + X * ( N - jt)]
(6.2.7)
X 2(k) = -^ [X (* > - X * ( N - *)]
;2
(6.2.8)
T h u s, by p e rfo rm in g a single D F T o n th e co m p le x -v a lu e d se q u e n c e x ( n ), we
h av e o b ta in e d th e D F T o f th e tw o re a l se q u e n c e s w ith only a sm all a m o u n t of
a d d itio n a l c o m p u ta tio n th a t is involved in co m p u tin g Xi (Jt) a n d X 2 (k) fro m X(k)
by u se o f (6.2.7) a n d (6.2.8).
6.2.2 Efficient Computation of the DFT of a 2/V-Point
Real Sequence
S u p p o se th a t g( n) is a re a l-v a lu e d se q u e n c e o f 2 N p o in ts. W e n o w d e m o n s tra te
h o w to o b ta in th e 2 N -p o in t D F T o f g( n) fro m c o m p u ta tio n o f o n e A ppoint D F T
involving c o m p le x -v a lu e d d a ta . F irst, w e define
* i(n )
=
g(2 n)
(6.2.9)
*2(n) = g ( 2 n + 1)
Sec. 6.2
477
Applications of FFT Algorithms
T h u s w e h a v e su b d iv id e d th e 2 N -p o in t re a l se q u e n c e in to tw o W -point real se ­
q u e n c e s. N o w w e can ap p ly th e m e th o d d escrib ed in th e p re c e d in g sectio n .
L et jc(n) b e th e A7-p o in t c o m p lex -v alu ed se q u e n c e
A-(n) = * i ( n ) + j x i i n )
(6 .2 .10)
F ro m th e re su lts o f th e p re c e d in g se ctio n , w e h av e
x m
= ^ [* (* ) + * * ( * - * ) ]
j
(6.2.11)
X 2(k) = — [ X( k) - X * ( N - k)]
F inally, w e m u st ex p re ss th e 2/V -point D F T in te rm s o f th e tw o /V -point D F T s,
Xi(A) a n d X 2(k). T o acco m p lish this, w e p ro c e e d as in th e d e c im a tio n -in -tim e F F T
a lg o rith m , n am ely ,
N -1
N-1
C( k ) = £ s < 2 h ) H $ * + J 2 s ( 2 n + ^ W7 N ^ k
n=tl
n=0
N- l
N-1
«=()
n=()
C o n s e q u e n tly ,
G( k ) = X t (k) + W i N X 2(k)
k = 0 . 1 ..........N - 1
( 6 . 2 . 12 )
G( k + N ) = X i ( k ) - W%N X 2(k)
k = Q . \ ..........N - l
T h u s w e h av e c o m p u te d th e D F T o f a 2/V -point real se q u e n c e from o n e jV-point
D F T an d so m e a d d itio n a l c o m p u ta tio n as in d icated by (6.2.11) an d (6.2.12).
6.2.3 Use of the FFT Algorithm in Linear Filtering and
Correlation
A n im p o rta n t ap p lic a tio n o f th e F F T a lg o rith m is in F IR lin e a r filterin g o f lo n g
d a ta se q u e n c e s. In C h a p te r 5 w e d e sc rib e d tw o m e th o d s, th e o v e rla p -a d d an d th e
o v e rla p -sa v e m e th o d s fo r filterin g a lo n g d a ta se q u e n c e w ith an F I R filter, b a s e d
o n th e u se o f th e D F T . In th is se ctio n w e c o n sid e r th e u se o f th e s e tw o m e th o d s
in c o n ju n c tio n w ith th e F F T a lg o rith m fo r co m p u tin g th e D F T an d th e ID F T .
L e t h( n), 0 < n < M - 1 , b e th e u n it sa m p le re sp o n s e o f th e F IR filter an d let
x ( n ) d e n o te th e in p u t d a ta se q u e n c e . T h e block size o f th e F F T alg o rith m is N ,
w h e re N = L + M — 1 an d L is th e n u m b e r o f new d a ta sa m p le s b e in g p ro cessed
by th e filter. W e a ssu m e th a t fo r a n y given v alu e o f Af, th e n u m b e r L o f d a ta
sa m p le s is se le c te d so th a t N is a p o w e r o f 2. F o r p u rp o se s o f th is discussion, w e
c o n s id e r o n ly rad ix -2 F F T alg o rith m s.
T h e /V -point D F T o f h(n), w hich is p a d d e d b y L — 1 z e ro s, is d e n o te d as H( k ) .
T h is c o m p u ta tio n is p e rfo rm e d o n c e via th e F F T an d th e re su ltin g N co m p lex
n u m b e rs a r e sto re d . T o be specific w e a ssu m e th a t th e d e c im a tio n -in -fre q u e n c y
478
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
F F T a lg o rith m is u se d to c o m p u te H( k ) . T h is y ields H ( k ) in b it-re v e rse d o rd e r,
w hich is th e w ay it is s to re d in m em ory.
In th e o v e rlap -sav e m e th o d , th e first M —1 d a ta p o in ts o f e a c h d a ta b lo ck are
th e last M — 1 d a ta p o in ts o f th e p rev io u s d a ta b lo ck . E a c h d a ta b lo c k c o n ta in s L
new d a ta p o in ts, su ch th a t N = L + M — 1. T h e N -p o in t D F T o f ea c h d a ta block
is p e rfo rm e d by th e F F T alg o rith m . If th e d e c im a tio n -in -fre q u e n c y alg o rith m is
e m p lo y ed , th e in p u t d a ta b lo ck re q u ire s n o shuffling a n d th e v a lu e s o f th e D F T
o ccu r in b it-re v e rse d o rd e r. S ince th is is ex actly th e o r d e r o f H ( k ) . w e can m ultiply
th e D F T o f th e d a ta , say Xm(fc), w ith //(Jt) a n d th u s th e re su lt
Ym(k) = H ( k ) X m(k)
is also in b it-re v e rse d o rd e r.
T h e in v erse D F T (ID F T ) can b e c o m p u te d by use o f an F F T alg o rith m th a t
ta k e s th e in p u t in b it-re v e rse d o r d e r a n d p ro d u c e s an o u tp u t in n o rm al o rd er.
T h u s th e r e is n o n e e d to shuffle any b lo ck o f d a ta e ith e r in c o m p u tin g th e D F T
o r th e ID F T .
If th e o v e rla p -a d d m e th o d is used to p e rfo rm th e lin e a r filterin g , th e co m p u ­
ta tio n a l m e th o d u sin g th e F F T a lg o rith m is basically th e sa m e. T h e only differen ce
is th a t th e N -p o in t d a ta b lo ck s consist o f L new d a ta p o in ts a n d M — 1 a d d itio n a l
zero s. A fte r th e I D F T is c o m p u te d fo r ea c h d a ta b lo ck , th e W -point filtered blocks
a re o v e rla p p e d as in d ic a te d in S ectio n 5.3.2, a n d th e M - 1 o v e rla p p in g d a ta p o in ts
b e tw e e n successive o u tp u t re c o rd s a re a d d e d to g e th e r.
L et u s assess th e c o m p u ta tio n a l co m p lex ity o f th e F F T m e th o d fo r lin e a r fil­
terin g . F o r th is p u rp o se , th e o n e -tim e c o m p u ta tio n o f H ( k ) is in sig n ifican t an d can
b e ig n o red . E ach F F T re q u ire s ( N / 2) log2 N co m p lex m u ltip lic a tio n s an d N Iog2 N
a d d itio n s. Since th e F F T is p e rfo rm e d tw ice, o n ce fo r th e D F T a n d o n ce fo r th e
ID F T , th e c o m p u ta tio n a l b u rd e n is N log2 N co m p lex m u ltip lic a tio n s an d 2 N log2 N
a d d itio n s. T h e re a re also N co m p lex m u ltip lic a tio n s a n d N — 1 a d d itio n s re q u ire d
to c o m p u te ym(Jfc). T h e re fo re , w e h av e ( N \ o g 2 2 N ) / L co m p lex m u ltip lic a tio n s p er
o u tp u t d a ta p o in t a n d a p p ro x im a te ly ( 2 N \ o g 2 2 N ) / L a d d itio n s p e r o u tp u t d ata
p o in t. T h e o v e rla p -a d d m e th o d re q u ire s an in c re m e n ta l in c re a se o f ( M — \ ) / L in
th e n u m b e r o f ad d itio n s.
B y w ay o f c o m p a riso n , a d ire c t fo rm re a liz a tio n o f th e F I R filter involves M
real m u ltip lic atio n s p e r o u tp u t p o in t if th e filter is n o t lin e a r p h a s e , a n d M / 2 if it
is lin e a r p h ase (sy m m etric ). A lso , th e n u m b e r o f a d d itio n s is M - 1 p e r o u tp u t
p o in t (see Sec. 8.2).
I t is in te re stin g to co m p a re th e efficiency o f th e F F T a lg o rith m w ith th e direct
fo rm re a liz a tio n o f th e F IR filter. L e t us focus o n th e n u m b e r o f m ultip lic atio n s,
w h ich a re m o re tim e co n su m in g th a n a d d itio n s. S u p p o se th a t M = 128 = 27 an d
N = 2 V. T h e n th e n u m b e r o f co m p lex m u ltip lic a tio n s p e r o u tp u t p o in t fo r an F F T
size o f N = 2 V is
Sec. 6.3
A Linear Filtering Approach to Computation of the DFT
TABLE 6.3
479
COMPUTATIONAL COMPLEXITY
Size of FFT
i) —log2 N
f(v)
Number of Complex Multiplications
per Output Point
9
10
11
12
14
13.3
12.6
12.8
13.4
15.1
T h e v alu es o f c( v) fo r d iffe re n t v alu es o f i> are given in T a b le 6.3. W e o b se rv e
th a t th e re is an o p tim u m v a lu e o f i< w h ich m in im iz es c(u ). F o r th e F IR filter of
size M = 128, th e o p tim u m o ccu rs at d = 10.
W e sh o u ld e m p h asize th a t c ( f ) r e p re s e n ts th e n u m b e r o f co m p lex m u ltip lic a­
tio n s fo r th e F F T -b a se d m e th o d . T h e n u m b e r o f re a l m u ltip lic a tio n s is fo u r tim es
th is n u m b e r. H o w e v e r, ev en if th e F IR filter has lin e a r p h a s e (see Sec. 8.2), th e
n u m b e r o f c o m p u ta tio n s p e r o u tp u t p o in t is still less w ith th e F F T -b a se d m eth o d .
F u rth e rm o r e , th e efficiency o f th e F F T m e th o d can be im p ro v e d by c o m p u tin g
th e D F T o f tw o successive d a ta b lo ck s sim u lta n e o u sly , ac c o rd in g to th e m eth o d
ju st d e sc rib e d . C o n s e q u e n tly , th e F F T -b a se d m e th o d is in d e e d su p e rio r from a
c o m p u ta tio n a l p o in t o f view w h en th e filter len g th is re lativ ely large.
T h e c o m p u ta tio n o f th e cross c o rre la tio n b e tw e e n tw o se q u e n c e s by m e a n s o f
th e F F T a lg o rith m is sim ilar to th e lin e a r F IR filtering p ro b le m ju st d esc rib e d . In
p ractical a p p lic a tio n s involving c ro ssc o rre la tio n , a t least o n e o f th e se q u e n c e s has
finite d u ra tio n an d is a k in to th e im p u lse re sp o n s e o f th e F IR filter. T h e seco n d
s e q u e n c e m ay be a lo n g se q u e n c e w hich c o n ta in s th e d e s ire d se q u e n c e c o rru p te d
b y a d d itiv e n o ise. H e n c e th e se co n d se q u e n c e is a k in to th e in p u t to th e F I R filter.
B y tim e rev e rsin g th e first se q u e n c e a n d co m p u tin g its D F T , w e h av e r e d u c e d th e
cro ss c o rre la tio n to an e q u iv a le n t co n v o lu tio n p ro b le m (i.e.. a lin e a r F I R filtering
p ro b le m ). T h e re fo re , th e m e th o d o lo g y w e d e v e lo p e d fo r lin e a r F IR filterin g by
u se o f th e F F T a p p lie s directly.
6.3 A LINEAR FILTERING APPROACH TO COMPUTATION OF THE
DFT
T h e F F T alg o rith m ta k e s N p o in ts o f in p u t d a ta a n d p ro d u c e s an o u tp u t se q u e n c e
o f N p o in ts c o rre sp o n d in g to th e D F T o f th e in p u t d a ta . A s w e h a v e show n,
th e rad ix -2 F F T a lg o rith m p e rfo rm s th e c o m p u ta tio n of th e D F T in ( N f l ) log2 N
m u ltip lic a tio n s a n d N log2 N a d d itio n s fo r a n N -p o in t se q u e n c e .
T h e re a re so m e a p p lic a tio n s w h e re o n ly a se le c te d n u m b e r o f valu es o f
th e D F T a re d e s ire d , b u t th e e n tire D F T is n o t re q u ire d . In such a case, th e
F F T a lg o rith m m ay n o lo n g e r be m o r e efficien t th a n a d ire c t c o m p u ta tio n o f
th e d e s ire d v alu es o f th e D F T . In fact, w h e n th e d e s ire d n u m b e r o f valu es o f
480
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
th e D F T is less th a n log2 N , a d ire c t c o m p u ta tio n o f th e d e s ire d v alu es is m o re
efficient.
T h e d irect c o m p u ta tio n o f th e D F T can b e fo rm u la te d as a lin e a r filtering
o p e ra tio n o n th e in p u t d a ta se q u en ce. A s w e will d e m o n s tra te , th e lin e a r filter
ta k e s th e fo rm o f a p a ra lle l b a n k o f re so n a to rs w h e re ea c h r e s o n a to r se lects o n e
o f th e fre q u e n c ie s a>k = 2 n k / N , k = 0, 1 , . . . , N — 1, c o rre sp o n d in g to th e N
fre q u e n c ie s in th e D F T .
T h e re a re o th e r a p p lic a tio n s in w hich w e re q u ire th e e v a lu a tio n o f th e ztra n sfo rm o f a fin ite -d u ra tio n se q u e n c e a t p o in ts o th e r th a n th e u n it circle. If
th e se t o f d e sire d p o in ts in th e z-p lan e po ssesses so m e re g u la rity , it is possible
to also ex p ress th e c o m p u ta tio n o f th e z -tra n s fo rm a s a lin e a r filte rin g o p e ra tio n .
In th is c o n n e c tio n , w e in tro d u c e a n o th e r alg o rith m , called th e c h irp -z tra n sfo rm
alg o rith m , w hich is su ita b le fo r e v a lu a tin g th e z -tra n s fo rm o f a se t o f d a ta o n a
v ariety o f c o n to u rs in th e z-p lan e. T h is alg o rith m is also fo rm u la te d as a lin ear
filtering o f a set o f in p u t d a ta . A s a co n se q u e n c e , th e F F T a lg o rith m can b e used
to c o m p u te th e ch irp -z tra n sfo rm a n d th u s to e v a lu a te th e z -tra n s fo rm at various
c o n to u rs in th e z -p la n e , in clu d in g th e u n it circle.
6.3.1 The Goertzel Algorithm
T h e G o e rtz e l a lg o rith m ex p lo its th e p erio d icity o f th e p h ase fa c to rs {W£} an d
allow s us to ex p re ss th e c o m p u ta tio n o f th e D F T as a lin e a r filterin g o p e ra tio n .
Since W # kN = 1, w e can m u ltip ly th e D F T by th is fa c to r. T h u s
(6.3.1)
W e n o te th a t (6.3.1) is in th e fo rm of a c o n v o lu tio n .
se q u e n c e yk(n) as
In d e e d , if w e d efin e the
(6.3.2)
m=0
th e n it is c le a r th a t » ( n ) is th e co n v o lu tio n o f th e fin ite -d u ra tio n in p u t se q u en ce
x( n ) o f len g th N w ith a filter th a t h as an im pulse re sp o n s e
h k(n) = W ~ knu ( n )
(6.3.3)
T h e o u tp u t o f th is filter a t n = N y ields th e v alu e o f th e D F T a t th e freq u e n cy
an = h r k / N . T h a t is,
X ( k ) = >*(n)|n=JV
(6.3.4)
as can b e verified b y c o m p a rin g (6.3.1) w ith (6.3.2).
T h e filter w ith im p u lse re s p o n s e h k (n) h a s th e sy stem fu n c tio n
(6.3.5)
Sec. 6.3
A Linear Filtering Approach to Computation of the DFT
481
T h is filter h as a p o le o n th e u n it circle a t th e fre q u e n c y cd* = 2n k / N . T h u s, the
e n tire D F T can b e c o m p u te d by passin g th e block o f in p u t d a ta in to a p a ra l­
lel b a n k o f N sin g le-p o le filters (re s o n a to rs), w h ere each filter h as a p o le at the
c o rre sp o n d in g fre q u e n c y o f th e D F T .
I n s te a d o f p e rfo rm in g th e c o m p u ta tio n o f th e D F T as in (6.3.2), via co n v o lu ­
tio n , w e can use th e d iffe re n c e e q u a tio n c o rre sp o n d in g to th e filter given by (6.3.5)
to c o m p u te y k(ir) recu rsiv ely . T h u s we h av e
y t (n) = W ^ kyt ( n - 1) + x i n )
V i-(-l) = 0
(6.3.6)
T h e d e sire d o u tp u t is X ( k ) = y k( N) , fo r k = 0, 1 , . . . , N — 1. T o p e rfo rm this
c o m p u ta tio n , w e can c o m p u te o n ce a n d sto re th e p h a s e facto rs W # k.
T h e co m p lex m u ltip lic a tio n s an d a d d itio n s in h e re n t in (6.3.6) can be av o id ed
by co m b in in g th e p airs o f re so n a to rs p o ssessin g c o m p le x -c o n ju g a te p oles. T his
lead s to tw o -p o le filters w ith system fu n c tio n s o f th e form
] _ iy*
HkL ) ~ 1 - 2 c o s ( 2 t i k / N ) : ~ l + C ' 2
(6'3 '7)
T h e d irect form II re a liz a tio n o f th e system illu stra te d in Fig. 6.17 is d e sc rib e d by
th e d iffe re n c e e q u a tio n
2:rk
v k(n) = 2 cos —
N
v*.(zi — 1) - vk(n - 2) + x( i t )
Vi(h) = vk in) - W N
k vk (n - 1)
(6.3.8)
(6.3.9)
w ith in itial c o n d itio n s iv - ( - l) = vk{ - 2 ) = 0.
T h e recu rsiv e re la tio n in (6.3.8) is ite ra te d for n = 0, 1.........N , b u t th e e q u a ­
tio n in (6.3.9) is c o m p u te d o n ly o n ce a t tim e n = N. E ach ite ra tio n re q u ire s o n e
real m u ltip lic a tio n a n d tw o a d d itio n s. C o n se q u e n tly , fo r a re a l in p u t se q u e n c e
x ( n) . th is a lg o rith m re q u ire s N + 1 re a l m u ltip lic a tio n s to yield n o t o n ly X ( k ) b ut
also, d u e to sy m m etry , th e v a lu e o f X ( N — k).
T h e G o e rtz e l alg o rith m is p a rtic u la rly a ttra c tiv e w h en th e D F T is to b e c o m ­
p u te d at a re lativ ely sm all n u m b e r M o f values, w h e re M < Iog2 N . O th erw ise,
th e F F T a lg o rith m is a m o re efficient m e th o d .
Figure 6.17 Direct form It realization
of two-pole resonator for computing the
DFT.
482
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
6.3.2 The Chirp-z Transform Algorithm
T he D F T o f an W -point data seq u en ce x(n ) has b een view ed as the z-transform
o f x i n ) evaluated at N equally spaced p oin ts on the unit circle. It has also been
view ed as N equally spaced sam ples o f the Fourier transform o f th e data sequ en ce
x (n ). In this section w e consider the evaluation o f X ( z ) on other contours in the
z-plane, including th e unit circle.
S u ppose that w e wish to com p ute the values o f the z-transform o f jc(n) at a
set o f p oints {z*}. T hen,
A '- l
X i z k) = J 2 x ( n ) z r
* = 0 , 1 .........L - 1
(6.3.10)
n=0
For exam ple, if the contour is a circle o f radius r and the z* are N equally spaced
points, then
Zk = r e j 2”*"/"
2=1
X ( z k) = J 2 i x M r ~n}e
n=0
,
k = 0 1,2
..... N - 1
(6.3.11)
n/N
k = 0 , 1 , 2 .........N - 1
In this case the FFT algorithm can be applied on the m odified seq u en ce x { n ) r ~ n.
M ore generally, suppose that the p oin ts z* in the z-plane fall on an arc which
begins at som e point
Zo =
r0eJlk’
and spirals either in toward the origin or out away from the origin such that the
points
are defined as
zk = rQe je°(Roei *‘)i
k = 0 ,1 ,..., L - 1
(6.3.12)
N o te that if R0 < 1, the points fall on a con tour that spirals tow ard th e origin and if
R0 > 1, the contour spirals away from the origin. If Ro — 1, the con tou r is a circular
arc o f radius ro. If r0 = 1 and Ro = l , the con tour is an arc o f th e unit circle. The
latter contour w ould allow us to com p ute the frequency con ten t o f the sequence
x ( n ) at a dense set o f L freq u en cies in the range covered by the arc w ithout having
to com pute a large D F T , that is, a D F T o f the seq u en ce x ( n ) pad d ed with many
zeros to obtain the desired resolution in frequency. Finally, if r0 = Ro = 1,
= 0,
0o = 2n / N , and L = N , the contour is the entire unit circle and the frequencies
are those o f the D F T . T h e various contours are illustrated in Fig. 6.18.
W hen points {z*J in (6.3.12) are substituted in to the exp ression for the ztransform, w e obtain
* ( z t ) = X ! -* ( ” > z r i
n=0
n=°
N-1
= j > ( n ) ( r 0e j * ) ~ ”V
(6.3.13)
Sec. 6.3
A Linear Filtering Approach to Computation of the DFT
lm(r)
ImU)
lm (;l
Im(;)
483
n=0
Figure 6.18 Some examples of contours on which we may evaluate the ztransform.
w here, by definition.
V = R veJ^
(6.3.14)
W e can exp ress (6.3.13) in the form o f a con volu tion , by n oting that
nk = j[n 2 + k 2 — (k — n) 2]
(6.3.15)
Substitution o f (6.3.15) into (6.3.13) yield s
N- 1
X(C*) = V - l ' Z /2 J 2 [ x ( n ) ( r 0eJlk' ) - nV - n2f2] V (k- n)2fZ
(6.3.16)
Let us define a n ew seq u en ce g ( n ) as
g (n) = x ( n )( r ^ e j<hr n V - n^
(6.3.17)
T h en (6.3.16) can b e exp ressed as
X ( z k) = V - k2/2y g ( n ) V {k-',)1/2
(6.3.18)
484
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
T he sum m ation in (6.3.18) can be interpreted as the con volu tion o f the sequ en ce
g (n) with the im pulse resp onse h (n) o f a filter, where
h(n) = V n2/2
(6.3.19)
C onsequently, (6.3.18) m ay b e expressed as
X(zt) = V -*Vy(k)
(6.3.20)
w here y(Jt) is the ou tp u t o f the filter
s —1
y ( k ) = Y ' g ( n ) h ( k — n)
n=0
k = 0. 1.........L — 1
(6.3.21)
W e observe that b oth h (n) and g(n) are com p lex-valu ed seq u en ces.
T he sequ en ce h (n) with R 0 = 1 has the form o f a com plex exp on en tial with
argum ent (on = n 2<(>o /2 = (n 0 o /2 )n. T h e quantity
rep resents the frequency
o f the com plex exp on en tial signal, which increases linearly with tim e. Such signals
are used in radar system s and are called chi rp signals. H en ce th e z-transform
evaluated as in (6.3.18) is called the chi rp-z t ransf orm.
T h e linear con volu tion in (6.3.21) is m ost efficien tly d on e by use o f the FFT
algorithm . T he seq u en ce g( n) is o f length N . H ow ever, h{n) has infinite du­
ration. Fortunately, only a portion h{n) is required to co m p u te the L values
o f X (z).
Since w e will com p ute the con volu tion in (6.3.1) via the F FT, let us consider
the circular con volu tion o f the W-point seq u en ce g{n) with an M -p oint section of
/i(n), w here M > N . In such a case, w e k n ow that the first N — 1 p oin ts contain
aliasing and that the rem aining M — N + 1 p oints are identical to the result that
would b e obtained from a linear con volu tion o f h( n) with g(n). In view o f this, we
should select a D F T o f size
M - L + N - 1
which would yield L valid p oin ts and N - 1 points corrupted by aliasing.
T he section o f h(n) that is n eed ed for this com putation corresp on d s to the
values o f h{ri) for —( N - 1) < n < (L — 1), which is o f length M = L + N — 1, as
observed from (6.3.21). Let us define the seq u en ce h \{n ) o f length M as
/ii(n ) = h(n — N -f 1)
n — 0 , 1 .........M — 1
(6.3.22)
and com p ute its Af-poin t D F T via the FFT algorithm to obtain H \ ( k ) . F rom x (n )
w e com p ute g ( n ) as specified by (6.3.17), pad g(n ) w ith L — 1 zeros, and com ­
pute its Af-point D F T to yield G(Jfc). T h e ID F T o f th e product y i(* ) = G ( k ) H \( k )
yields the Af-point seq u en ce > i(n ), n = 0, 1 , . . . , Af — 1. T h e first N — 1 p oints of
y i(« ) are corrupted by aliasing and are discarded. T h e desired valu es are yi(n)
f o r N — 1 < n < M — 1, w hich correspond to the range 0 < n < L — l i n (6.3.21),
Sec. 6.3
A Linear Filtering Approach to Computation of the DFT
485
that is,
y(n) = y t ( n + N — 1)
n = 0, 1.........L — 1
(6.3.23)
A ltern atively, w e can define a seq u en ce ft 2 (n) as
h 2(n) =
h (n),
h ( n ~ N - L + l),
0 < n < L —1
L < ti < M — 1
(6.3.24)
The A f-point D F T o f h 2{n) yields H2(k), which w hen m ultiplied by G( k ) yields
Y2(k) = G( k ) Hz ( k ) . T he ID F T o f Y2(k) yield s the seq u en ce y2( n) for 0 < n < A f - 1 .
N o w the desired valu es o f >’2 (") are in the range 0 < n < L — 1, that is,
y ( n ) = y 2(n)
n = 0, 1 , . . . , L — 1
(6.3.25)
Finally, the com p lex valu es X(Zi) are com p uted by dividing y( k) by h ( k ),
k = 0, 1.........L — 1, as specified by (6.3.20).
In gen eral, the com p utational com p lexity o f the chirp-z transform algorithm
described ab ove is o f the order of Af log 2 M com plex m ultiplications, where M =
N + L ~ 1. T h is num ber should be com pared with the product, N ■L, the num ber
o f com p utations required by direct evaluation o f the z-transform . Clearly, if L is
sm all, direct com p utation is m ore efficient. H ow ever, if L is large, then the chirp-z
transform algorithm is m ore efficient.
T h e chirp-z transform m eth od has b een im plem ented in hardware to com pute
the D F T o f signals. For the com putation of the D FT, w e select ro = /?(i = 1, 6\j = 0,
</>o = 2n / N , and L = N. In this case
y-ir/2 _
e -jjin -/N
nn2
. Tin2
= c o s --------- j s i n ------N
N
<6 '3 '26 >
T he chirp filter with im pulse response
h( n) = V nlfl
t2
. nn2
= c o s -------i n ----— (- j/ ssin
—
N
N
(6.3.27)
= h r(n) + jh , { n )
has b een im p lem en ted as a pair o f F IR filters w ith coefficients h r (n) and A,(n),
resp ectively. B o th su rface acous tic w av e (SAW ) d evices and charge co u p l e d d e ­
vices (C C D ) h ave b een u sed in practice for the F IR filters. T h e cosine and sine
seq u en ces given in (6.3.26) n eed ed for the prem ultiplications and postm ultiplica­
tion s are usually stored in a read-only m em ory (R O M ). Furtherm ore, w e n ote that
if o n ly the m agnitu d e o f the D F T is desired, the postm ultiplications are u n n eces­
sary. In this case,
|X (z*)l = \y(k)\
k = 0 ,1 ,..., n - 1
(6.3.28)
as illustrated in Fig. 6.19. T hus the linear F IR filtering approach using th e chirp-z
transform has b een im p lem en ted for the com putation o f the D F T .
486
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
Chirp Fillers
Figure 6.19 Block diagram illustrating the implementation of the chirp-z transform for com­
puting the DFT (magnitude only).
6.4 QUANTIZATION EFFECTS IN THE COMPUTATION OF THE DFT*
A s w e have ob served in our p reviou s discussions, the D F T plays an im portant role
in m any digital signal p rocessing applications, including F IR filtering, the com pu­
tation o f the correlation betw een signals, and spectral analysis. For this reason
it is im portant for us to kn ow th e effect o f quantization errors in its com puta­
tion. In particular, w e shall consider the effect o f rou n d -off errors due to the
m ultiplications perform ed in the D F T with fixed-point arithm etic.
T h e m odel that w e shall adopt for characterizing rou n d -off errors in m ulti­
plication is the additive w hite n oise m o d el that w e use in the statistical analysis
o f Tound-off errors in IIR and F IR filters (see Fig. 7.34). A lth ou gh the statistical
*It is recommended that the reader review Section 7.5 prior to reading this section.
Sec. 6.4
487
Quantization Effects in the Computation of the DFT
analysis is perform ed for rounding, the analysis can be easily m odified to apply to
truncation in tw o's-com p lem en t arithm etic (see Sec. 7.5.3).
O f particular interest is the analysis o f rou n d -off errors in the com putation
o f the D F T via the FFT algorithm . H ow ever, w e shall first establish a benchm ark
by determ ining the round-off errors in the direct com p utation o f the D F T .
6.4.1 Quantization Errors in the Direct Computation of
the DFT
G iven a finite-duration seq u en ce (jt(n)], 0 < n < N — 1, the D F T o f {jc(h)1 is
defined as
A/-1
* (* ) = Y l x ( n ) w "'
j,=0
£ = 0 , 1 ........ N - 1
(6.4.1)
w here IVyv = c ~ )2r,/N. W e assum e that in general, {*(«)] is a com p lex-valu ed se ­
quence. W e also assum e that the real and im aginary com p on en ts o f {a (h)I and
{VV^"] are represented by b bits. C onsequently, the com putation o f the product
requires four real m ultiplications. Each real m ultiplication is rounded
from 2b bits to b bits, and hence there are four quantization errors for each
com p lex-valu ed m ultiplication.
In the direct com putation o f the D F T , there are N com p lex-valu ed m ultiplica­
tions for each point in the D FT. T herefore, the total num ber o f real m ultiplications
in the com putation o f a single point in the D F T is 4 N. C on sequ en tly, there are
4 N quantization errors.
Let us evaluate the variance o f the quantization errors in a fixed-point com ­
putation o f the D F T . First, w e m ake the follow in g assum ptions about the statistical
properties o f the quantization errors.
1. T h e quantization errors due to rounding are uniform ly distributed random
variables in the range (—A /2 , A /2 ) where A = 2~ b.
2. T h e 4 N quantization errors are m utually uncorrelated.
3. T h e 4 N quantization errors are uncorrelated with the seq u en ce |jc{«}}.
Since each o f the quantization errors has a variance
A 2
7~2b
" ' = 1 2 = 1 2
<6A2>
the variance o f the quantization errors from the 4 N m ultiplications is
<r2 = 4 N o ]
(6A3)
3
H en ce the variance o f the quantization error is p roportional to the size o f D FT.
N o te that w hen Af is a p ow er o f 2 (i.e., N = 2 1’), the variance can be expressed
488
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
2 —2(h—1*/2»
a ] = -------------
Chap. 6
(6.4.4)
This expression im plies that every fourfold increase in the size N o f the D F T
requires an additional bit in com putational precision to offset the additional quan­
tization errors.
T o prevent overflow , the input seq u en ce to the D F T requires scaling. Clearly,
an upper bound on | X (A:) | is
A '- l
[* (£)! < Y
|*(n )|
(6.4.5)
n=0
If the dynam ic range in addition is ( - 1 , 1 ) , then |X (/:)| < 1 requires that
A '-l
Y
|jr(/i)| < 1
(6.4.6)
n=0
If U (/i)| is initially scaled such that |a (/j)| < 1 for all n, then each point in the
seq u en ce can be divided by N to ensure that (6.4.6) is satisfied.
T h e scaling im plied by (6.4.6) is extrem ely severe. For exam p le, su p p ose
that the signal seq u en ce {*(«)} is white and. after scaling, each valu e |.r(n)l o f the
seq u en ce is uniform ly distributed in the range (-1 /7 V , I/ N ) . T h en the variance of
the signal sequ en ce is
<2 / * > 2 = —1 o ,2 = ----------
3N1
12
tzA-n
(6.4.7)
>
and the variance o f the output D F T coefficients |Jf(/t)l is
al = N a2
1
(6.4.8)
~ 3 ~N
Thus the signal-to-noise p ow er ratio is
(6.4.9)
W e observe that the scaling is responsible for reducing th e S N R by N and
the com bination o f scaling and quantization errors result in a total reduction that
is proportional to N 2. H en ce scaling the input seq u en ce (j(n )} to satisfy (6.4.6)
im poses a severe p en alty on the signal-to-noise ratio in the D F T .
Exam ple 6.4.1
Use (6.4.9) to determ ine the num ber of bits required to com pute the D FT of a 1024point sequence with a SNR of 30 dB.
Solution
The size of the sequence is N = 210. Hence the SNR is
Sec. 6.4
Quantization Effects in the Computation of the DFT
489
For an SNR o f 30 dB, we have
3(2* - 20) = 30
b = 15 bits
N ote that the 15 bits is the precision for both multiplication and addition.
Instead o f scaling the input sequ en ce {Jt(n)}, suppose w e sim ply require that
|x(n)l < 1. T h en w e m ust provide a sufficiently large dynam ic range for addition
such that |* ( * ) l < N . In such a case, the variance o f the seq u en ce {|jc(n)|) is
a 2 = 5 , and h en ce th e variance o f |X (* )| is
(6.4.10)
C on sequ en tly, the S N R is
(6.4.11)
If w e repeat the com putation in E xam ple 6.4,1, w e find that the num ber o f
bits required to a ch ieve a S N R o f 30 dB is b = 5 bits. H ow ever, w e n eed an
additional 1 0 bits for the accum ulator (th e adder) to accom m odate the increase
in the dynam ic range for addition. A lthou gh w e did not ach ieve any reduction
in the dynam ic range for addition, we have m anaged to reduce the p recision in
m ultiplication from 15 bits to 5 bits, which is highly significant.
6.4.2 Quantization Errors in FFT Algorithms
A s w e have sh ow n , the F F T algorithm s require significantly few er m ultiplications
than the direct com p utation o f the D F T . In view o f this w e m ight con clu d e that the
com p utation o f the D F T via an FFT algorithm w ill result in sm aller quantization
errors. U n fortu n ately, that is n ot the case, as w e will dem onstrate.
L et us con sid er the use o f fixed-point arithm etic in th e com putation o f a
radix-2 F F T algorithm . T o be specific, w e select the radix-2, decim ation-in-tim e
algorithm illustrated in Fig. 6.20 for the case N = S. T h e results on quantiza­
tion errors that w e ob tain for this radix-2 FFT algorithm are typical o f th e results
o b ta in ed w ith o th er radix - 2 and higher radix algorithm s.
W e o b serv e that each butterfly com putation in volves o n e com plex-valued
m ultiplication or, eq u ivalen tly, four real m ultiplications. W e ignore the fact that
so m e butterflies con tain a trivial m ultiplication by ± 1 . If w e consider th e but­
terflies that affect the com p utation o f any on e valu e o f the D F T , w e find that,
in gen eral, there are N /2 in the first stage of the FFT , N / 4 in the secon d stage,
N / 8 in the third state, and so on , until the last stage, w here there is on ly on e.
C on sequ en tly, th e num ber o f butterflies per output point is
2 " - '+ 2 " - 2 + --- + 2 + l = 2v“ ‘ [ l + ( ! ) + ■ • + ( j ) ” ']
= 2 ”[ l - ( j n
= W -l
490
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Stage 1
Stage 2
Chap. 6
Stage 3
For exam ple, the butterflies that affect the com p utation o f A"(3) in the eight-point
FFT algorithm o f Fig. 6.20 are illustrated in Fig. 6.21.
T h e quantization errors introduced in each butterfly propagate to the output.
N o te that the quantization errors introduced in the first stage p ropagate through
(v - 1 ) stages, th ose introduced in the second stage propagate through (v - 2 )
stages, and so on. A s these quantization errors propagate through a num ber of
su bsequent stages, th ey are phase shifted (ph ase rotated) by th e phase factors
W^n. T h ese phase rotations do not change the statistical p rop erties o f the quan­
tization errors and, in particular, the variance o f each q uantization error remains
invariant.
If w e assum e that the quantization errors in each butterfly are uncorrelated
with the errors in other butterflies, then there are 4(W - 1 ) errors that affect the
output o f each point o f the FFT. C on sequ en tly, th e variance o f the total quanti­
zation error at the output is
A 2
f f| = 4 (A r- ! ) — « —
A 2
(6.4.13)
Sec. 6.4
Quantization Effects in the Computation of the DFT
Figure 6.21
491
Butterflies that affect the computation o f X (3).
w here A = 2 h. H en ce
a2= j
■2"“
(6.4.14)
T his is exactly the sam e result that w e ob tain ed for the direct com p utation o f the
DFT.
T h e result in (6.4.14) should n ot b e surprising. In fact, the FFT algorithm
d o es not reduce the num ber o f m ultiplications required to com p ute a single point
o f the D F T . It d o es, h ow ever, exp loit the p eriod icities in W^n and thus reduces
the num ber o f m ultiplications in the com p utation o f the entire block o f N points
in the D F T .
A s in the case o f the direct com p utation o f the D F T , w e m ust scale the
input seq u en ce to prevent overflow . R ecall that if |jc(n) | < \ / N , 0 < n < N —
1, then |X (* )| < 1 for 0 < k < N — 1. T hus overflow is avoided. W ith this
scaling, the relation s in (6.4.7), (6.4.8), and (6.4.9), ob tain ed previously for the
direct com p utation o f the D F T , apply to the F F T algorithm as w ell. C onsequently,
the sam e S N R is o b tain ed for the FFT.
Since the F F T algorithm consists o f a seq u en ce o f stages, w h ere each stage
con tains butterflies that in volve pairs o f points, it is p ossib le to d evise a differ­
en t scaling strategy that is n ot as severe as dividing each input p oin t by N . This
492
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
alternative scaling strategy is m otivated by the observation that the in term edi­
ate values [Xn(/r)| in the n = 1, 2,..., u stages o f th e F F T algorithm satisfy the
conditions (see P roblem 6.35)
m ax[|X n+1 ( * ) U X n+1(/)|] > m ax[|X n( * ) U X B( 0 |]
(6.4.15)
m ax [|X n+1 ( * ) |,|X B+1(/)|] <
2
m ax[jX „(Jt)|,|X n(/)|]
In view o f th ese relations, w e can distribute the total scalin g o f 1 / N in to each
o f the stages o f the F F T algorithm . In particular, if |jr(n)| < 1, w e apply a scale
factor o f 5 in the first stage so that |jr(n)| <
T h en the output o f each subsequent
stage in the FFT algorithm is scaled by | , so that after v stages w e have achieved
an overall scale factor o f ( j ) 1' = 1 //V. Thus overflow in the com p utation o f the
D F T is avoided.
This scaling procedure d o es not affect the signal level at the output o f the
FFT algorithm , but it significantly reduces the variance o f the quantization errors
at the output. Specifically, each factor o f ^ reduces the variance o f a quantization
error term by a factor o f
Thus the 4 ( N / 2 ) quantization errors introduced in
the first stage are reduced in variance by (^ V - 1 , the 4 ( N / 4 ) quantization errors
introduced in the second stage are reduced in variance by ( j ) 1’- 2 . and so on. C on­
sequ en tly, the total variance o f the quantization errors at the output o f the FFT
algorithm is
w here the factor (^ )tJ is negligible.
W e now o b serve that (6.4.16) is n o longer proportional t o N . O n th e other
hand, the signal has th e variance a \ = 1 /3 N , as given in (6.4.8). H e n c e the S N R is
f l = _ L .2 ^
2N
(6.4.17)
_ 22b—v—\
Thus, by distributing th e scaling o f l / N uniform ly throughout th e FFT algorithm ,
w e have achieved an S N R that is inversely proportional to N in stead o f N 2.
Example 6.4.2
Determine the number of bits required to compute an FFT of 1024 points with an
SNR of 30 dB when the scaling is distributed as described above.
Sec. 6.5
Summary and References
Solution
493
The size of the FFT is N = 210. Hence the SNR according to (6.4.17) is
101°gio 22h~v~l = 30
3(2b - 11) = 30
b =
bits)
This can be com pared with the 15 bits required if all the scaling is perform ed in the
first stage of the FFT algorithm.
6.5 SUMMARY AND REFERENCES
T h e fo cu s o f this chapter w as on the efficien t com putation o f the D F T . W e d em on ­
strated that by taking advantage o f the sym m etry and p eriodicity p roperties o f the
ex p on en tial factors W#", w e can reduce the num ber o f com p lex m ultiplications
n eed ed to com p ute the D F T from N 2 to N log 2 N w hen Af is a p ow er o f 2. A s w e
indicated, any seq u en ce can be augm ented with zeros, such that N — 2''.
For d ecad es, FFT -type algorithm s were o f interest to m athem aticians w ho
w ere con cern ed w ith com p utin g values o f F ourier series by hand. H ow ever, it
w as not until C o o ley and T u k ey (1965) published their w ell-k now n paper that the
im pact and significance o f the efficient com putation o f the D F T was recognized.
Since then the C o o le y -T u k e y FFT algorithm and its various form s, for exam ple,
the algorithm s o f S in gleton (1967, 1969), have had a trem en dou s influence on the
use o f the D F T in con v olu tion , correlation, and spectrum analysis. For a historical
p erspective on the F FT algorithm , the reader is referred to the paper by C ooley
et al. (1967).
T h e split-radix FFT (SR F F T ) algorithm d escribed in Section 9.3.5 is due
to D u h a m el and H ollm an n (1 9 8 4 ,1 9 8 6 ). T he “m irror” F F T (M FF T ) and “p h ase”
F F T (PF FT ) algorithm s w ere described to the authors by R. Price. T he exp loitation
o f sym m etry p rop erties in the data to reduce the com putation tim e are described
in a paper by Sw arztrauber (1986).
O ver the years, a num ber o f tutorial papers have b een published on FFT
algorithm s. W e cite the early papers by Brigham and M orrow (1967), Cochran et
al. (1967), B ergland (1969), and C ooley et al. (1967, 1969).
T h e reco g n itio n that the D F T can b e arranged and com p uted as a linear
con volu tion is also highly significant. G o ertzel (1968) indicated that the D F T
can b e com p uted via linear filtering, although the com p utational savings o f this
approach is rath er m odest, as w e have observed. M ore significant is the work
o f B lu estein (1 9 7 0 ), w ho d em onstrated that the com p utation o f the D F T can be
form u lated as a chirp linear filtering operation. T his w ork led to the d evelop m en t
o f th e chirp-z transform algorithm by R ab in er et al. (1969).
In addition to the F F T algorithm s describ ed in this chapter, there are other
efficien t algorithm s for com p utin g the D F T , som e o f w hich further reduce the
494
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
num ber o f m ultiplications, but usually require m ore additions. O f particular im ­
portance is an algorithm due to R ader and B renner (1976), the class o f prim e factor
algorithm s, such as the G o o d algorithm (1971), and the W inograd algorithm (1976,
1978). For a d escription o f these and related algorithm s, the reader m ay refer to
the text by Blahut (1985).
PROBLEMS
6.1 Show that each of the numbers
eja*/NH
o < /t < W - 1
corresponds to an Wth root of unity. Plot these numbers as phasors in the complex
plane and illustrate, by m eans of this figure, the orthogonality property
-ja ir/N M n
_
n=<)
N,
I 0,
I
|
if k ~ 1
if k ^ l
1
6.2 (a) Show that the phase factors can be com puted recursively by
W$ =
(b) Perform this com putation once using single-precision floating-point arithmetic
and once using oniy four significant digits. Note the deterioration due to the
accumulation of round-off errors in the later case.
(c) Show how the results in part (b) can be improved by resetting the result to the
correct value - j . each time gl = N/4.
6 3 Let x(n) be a real-valued N -point (N = 2' ) sequence. Develop a method to compute
an N -point D FT X ’(k), which contains only the odd harmonics [i.e., X'(k) = 0 if Jt is
even] by using only a real A,/2-spoint DFT.
6.4 A designer has available a num ber of eight-point FFT chips. Show explicitly how he
should interconnect three such chips in order to com pute a 24-point DFT.
6.5 The ^-transform of the sequence x(n) = u(n) - u(n - 7) is sampled at five points on
the unit circle as follows
x(k) = X(z) 1- = eJ'2jr*/5
Jt* 0,1 ,2 ,3 ,4
Determ ine the inverse D FT x'(n) of X (Jt). Com pare it with *(/t) and explain the
results.
6.6 Consider a finite-duration sequence x(n), 0 < n < 7, with z-transform X(z). We wish
to com pute X (:) at the following set of values:
zk = 0.8ejf|(2)r*/*,+(’' /8)]
0 < Jfc < 7
(a) Sketch the points {z*} in the complex plane.
D eterm ine a sequence s ( n ) such that its D F T provides the desired samples of
( b )
*(z).
Chap. 6
495
Problems
6.7 Derive the radix-2 decimation-in-time FFT algorithm given by (6.1.26) and (6.1.27)
as a special case of the more general algorithmic procedure given by (6.1.16) through
(6.1.18).
6.8 Com pute the eight-point D FT of the sequence
1.
I 0,
X^
0 < n < 7
otherwise
by using the decimation-in-frequency FFT algorithm described in the text.
6.9 Derive the signal flow graph for the N = 16 point, radix-4 decimation-in-time FFT
algorithm in which the input sequence is in norm al order and the computations are
done in place.
6.10 D erive the signal flow graph for the N = 16 point, radix-4 decimation-in-frequency
FFT algorithm in which the input sequence is in digit-reversed order and the output
D FT is in normal order.
6.11 Com pute the eight-point DFT of the sequence
x{n) = U . i
1,0. 0,0.0
12 2 2 2
using the in-placc radix-2 dccimation-in-time and radix-2 decimation-in-frequency al­
gorithms. Follow exactly the corresponding signal flow graphs and keep track of all
the interm ediate quantities by putting them on the diagrams,
6.12 Com pute the 16-point D FT of the sequence
x(n) = cos
0 < n < 15
using the radix-4 decimation-in-time algorithm.
6.13 Consider the eight-point decimation-in-tim e (D IT) flow graph in Fig. 6.6.
(a) What is the gain of the “signal p ath ” that goes from x(7) to X(2)7
(b) How many paths lead from the input to a given output sample? Is this true for
every output sample?
(c ) Com pute X (3) using the operations dictated by this flow graph.
6.14 Draw the flow graph for the decimation-in-frequency (D IF) SRFFT algorithm for
N = 16. W hat is the num ber of nontrivial multiplications?
6.15 D erive the algorithm and draw the N = 8 flow graph for the D IT SRFFT algorithm.
Com pare your flow graph with the D IF radix-2 FFT flow graph shown in Fig. 6.11.
6.16 Show that the product of two complex numbers (a + j b ) and ( C + j d ) can be perform ed
with three real multiplications and five additions using the algorithm
x R
=
(a
- b ) d + (c - d ) a
xi
=
(a
-
b)d
+ (c + d ) b
where
X =
x R + jx ,
= (a + j b ) ( c + j d )
6.17 Explain how the D FT can be used to com pute N equispaced samples of the ztransform , of an iV-point sequence, on a circle of radius r.
496
Efficient Computation of the DFT: Fast Fourier Transform Algorithms
Chap. 6
6.18 A real-valued A/-point sequence Jt(n) is called D FT bandlim ited if its D FT X(k) = 0
for ko < k < N —An. We insert (L — 1)N zeros in the middle of A'(Jt) to obtain the
following L N -point DFT
X(k),
X'(k) = { 0,
X(k + N - LN) .
0 < i < Ao — 1
Jt0 < Jk < L N - kt,
L N - J t „ + 1 < Jt < L N — 1
Show that
Lx'(Ln) = x(n)
0 < n < N —1
where
x\n)
X (k)
LN
Explain the meaning of this type of processing bv working out an example with N = 4,
L = 1. and A ( J t ) = { 1 , 0 . 0 . 1 ) .
6.19 Let X(k) be the A'-point DFT of the sequence .v(n). 0 < n < A’ - 1. What is the
A-point DFT of the sequence s(n) = X(n). 0 < n < N - 1?
6.20 Let X(k) be the A-point DFT of the sequence \(n ), (I < » < N — 1. We define a
2 N -point sequence _v(«) as
V{;I)
1-v^V
««--vcn
n odd
Express the 2 Appoint DFT of y(n) in terms of X(k).
6.21 (a) Determ ine the ^-transform W (-) of the Hanning window w («) = (1 - cos
fl.
(b) Determ ine a formula to compute the JV-point DFT X„(k) of ihe signal .r„.(n) =
w(n)x(n)< 0 < n < N — 1. from the JV-point DFT A'(A ) of the signal .r(n).
6.22 Create a DFT coefficient table that uses only N/4 memory locations to store the first
quadrant of the sine sequence (assume N even).
6.23 D eterm ine the com putational burden of the algorithm given by (6.2.12) and compare
it with the com putational burden required in the 2N -point DFT of g(n). Assume that
the FFT algorithm is a radix-2 algorithm.
6.24 Consider an IIR system described by the difference equation
S'
v(n) = - ^
M
ai-v<n ~
+ y , b k x ( n - k)
*= 1
Describe a procedure that com putes the frequency response H
Jt = 0, 1........
A ' - l using the FFT algorithm ( N = 2‘).
6.25 Develop a radix-3 decimation-in-time FFT algorithm for N = 3’ and draw the corre­
sponding flow graph for N = 9. W hat is the num ber of required com plex multiplica­
tions? Can the operations be perform ed in place?
6.26 Repeat Problem 6.25 for the D IF case.
6.27 FFT input and output pruning In many applications we wish to com pute only a few
points M of the Appoint D FT of a finite-duration sequence of length L (i.e., M « N
and I < < N).
Chap. 6
Problems
497
(a) Draw the flow graph of the radix-2 D IF FFT algorithm for N = 16 and eliminate
[i.e., prune] all signal paths that originate from zero inputs assuming that only
jc(0) and jc(1) are nonzero.
(b) R epeat part (a) for the radix-2 D IT algorithm.
(c) Which algorithm is better if we wish to com pute all points of the D FT? What
happens if we want to compute only the points X (0), X (l), X (2), and X (3)?
Establish a rule to choose between D IT and D IF pruning depending on the
values of M and L.
(d) Give an estim ate of saving in computations in term s of M, L, and N.
6.28 Parallel computation o f the D F T Suppose that we wish to com pute an N = 2P2V
point D FT using 2P digital signal processors (DSPs). For simplicity we assume that
p = v = 2. In this case each DSP carries out all the com putations that are necessary
to com pute 2V D FT points.
(a) Using the radix-2 D IF flow graph, show that to avoid data shuffling, the entire
sequence x(n) should be loaded to the memory of each DSP.
(b) Identify and redraw the portion of the flow graph that is executed by the DSP
that com putes the D FT samples X(2), X(10), X(6), and X(14).
(c) Show that, if we use M = 2'' DSPs, the com putation speed-up S is given by
S= M
log-, N
log2 N - log, M + 2(M - 1)
6.29 Develop an inverse radix-2 D IT FFT algorithm starting with the definition. Draw the
flow graph for com pulation and com parc with the corresponding flow graph for the
direct FFT. Can the IFFT flow graph be obtained from the one for the direct FFT?
6.30 R epeat Problem 6.29 for the D IF case.
6.31 Show that an FFT on data with H erm itian symmetry can be derived by reversing the
flow graph of an FFT for real data.
6.32 D eterm ine the syslem function H(z) and the difference equation for the system that
uses the G oertzel algorithm to compute ihe DFT value X ( N - k).
6.33 (a) Suppose that x(n) is a finite-duration sequence of N = 1024 points. It is desired
to evaluate the z-transform X (;) of the sequence at the points
Zk = eiQ * * ™ *
k = 0 . 100,200 ......1000
by using the most efficient m ethod or algorithm possible. Describe an algorithm
for perform ing this com putation efficiently. Explain how you arrived at your
answer by giving the various options or algorithms that can be used.
(b) R epeat part (a) if X (z) is to be evaluated at
zt = 2(0.9) V '1(2,r/5000*+’r/21
k = 0 ,1 ,2 ........999
634 R epeat the analysis for the variance of the quantization error, carried out in Sec­
tion 6.4.2, for the decimation-in-frequency radix-2 FFT algorithm.
635 The basic butterfly in the radix-2 decimation-in-time FFT algorithm is
*„+,<*) = Xn( * )+ W” X„(l)
498
Efficient Computation of ttie DFT: Fast Fourier Transform Algorithms
Chap. 6
(a) If we require that !X„(Jt)| < j and |Jf„(/)| < 5 , show that
|Re[X „+,(*)]| < 1,
|/te[X n+1(/)]| < 1
| I m [ X „ + 1 (*)]l <
| / m [ A B + 1 (/)]| <
1,
1
Thus overflow does not occur.
(b) Prove that
max[|X,,+i(A)|. |X„+i(/)|] > m ax[|Xn(*)l.l*,,(/)|]
max[|X„+1(*)UX,,+1(/)|] < 2 m ax [|X „(* )|.|* n (/)l]
636* Computation o f the D F T Use an FFT subroutine to compute the following DFTs
and plot the m agnitudes |X(Jt)| of the DFTs.
(a) The 64-point D FT of the sequence
n = 0 ,1 ........15
otherwise
, ,
[1 ,
x(n) = I
10,
(Ni ~ 16)
(b) The 64-point D FT of the sequence
1,
10,
n = 0 ,1 ........... 7
otherwise
(N\ = 8)
(c) The 128-point DFT of the sequence in part (a).
(d) The 64-point D FT of the sequence
| 10f >'*/*>",
x (n) — {
J 0.
„ = 0 ,1 ........63
otherwise
(Ni = 64)
Answer the following questions.
(1) What is the frequency interval between successive samples for the plots in
parts (a), (b). (c), and (d)?
(2) What is the value of the spectrum at zero frequency (dc value) obtained
from the plots in parts (a), (b), (c), (d)?
From the formula
X(k) = Y ^ x (n ) e '
compute the theoretical values for the dc value and check these with the
com puter results.
(3) In plots (a), (b), and (c), what is the frequency interval betw een successive
nulls in the spectrum ? W hat is the relationship between N 1 of the sequence
x( n ) and the frequency interval between successive nulls?
(4) Explain the difference between the plots obtained from parts (a) and (c).
637* Identification o f pole positions in a system
difference equation
Consider the system described by the
y(n) = —r2y(n —2) + x(n)
(a) Let r = 0.9 and x(n) = &(n). G enerate the output sequence y(n) for 0 < n < 127.
Compute the N = 128 point D FT [ I'M ) and plot {|y(Jt)t).
Chap. 6
Problems
499
(b) Com pute the N = 128 point DFT of the sequence
ui(n) = (0.92)- " v(n)
where y(/t) is the sequence generated in part (a). Plot the D FT values | W (t)|.
W hat can you conclude from the plots in parts (a) and (b)?
(c) Let r = 0.5 and repeat part (a).
(d) R epeat part (b) for the sequence
ui(n) = (0.55)~nv(n)
where y( n) is the sequence generated in part (c). W hat can you conclude from
the plots in parts (c) and (d)?
(e) Now let the sequence generated in part (c) be corrupted by a sequence of “mea­
surem ent" noise which is Gaussian with zero mean and variance a 2 = 0.1. Repeat
parts (c) and (d) for the noise-corrupted signal.
Implementation of
Discrete-Time Systems
The focus o f this chapter is on the realization o f linear tim e-invariant discrete­
tim e system s in eith er softw are or hardware. A s w e noted in C h ap ter 2, there are
various configurations or structures for the realization o f any F IR and IIR discrete­
tim e system . In C hapter 2 w e described the sim plest o f these structures, nam ely,
the direct-form realizations. H ow ever, there are other m ore practical structures
that offer som e distinct advantages, especially w hen q uantization effects are taken
into consideration.
O f particular im portance are the cascade, parallel, and lattice structures,
which exhibit robustness in finite-w ord-length im plem en tation s. A lso described
in this chapter is the frequency-sam pling realization for an F IR system , which
often has the advantage o f being com putationally efficient w hen com pared with
alternative FIR realizations. O ther im portant filter structures are ob tain ed by
em ploying a state-sp ace form ulation for linear tim e-invariant system s. A n analysis
o f system s characterized by th e state-variable form is p resen ted in both the tim e
and frequency dom ains.
In addition to describing the various structures for the realization o f discrete­
tim e system s, w e a lso treat problem s associated with q uantization effects in the
im plem en tation o f digital filters using finite-precision arithm etic. T his treatm ent
includes the effects on the filter frequency response characteristics resulting from
coefficient quantization and the round-off noise effects inherent in the digital im­
p lem entation o f d iscrete-tim e system s.
7.1 STRUCTURES FOR THE REALIZATION OF DISCRETE-TIME
SYSTEMS
Let us consider the im portant class o f linear tim e-invariant d iscrete-tim e system s
characterized by the general linear constant-coefficient d ifferen ce eq u ation
N
M
(7.1.1)
500
Sec. 7.1
Structures for the Realization of Discrete-Time Systems
501
A s w e have sh ow n by m eans o f the z-transform , such a class o f linear tim e-invariant
d iscrete-tim e system s are also characterized by the rational system function
M
H(z) =
N
(7.1.2)
which is a ratio o f tw o polyn om ials in z - 1 . From the latter characterization, we
obtain the zero s and p o les o f the system function, which d ep en d on the ch oice o f
the system p aram eters {£>*} and {a*} and which determ ine the frequency response
characteristics o f the system .
O ur fo cu s in this chapter is on the various m eth od s o f im plem en ting (7.1.1)
or (7.1.2) in eith er hardware, or in softw are on a program m able digital com puter.
W e shall sh ow that (7.1.1) or (7.1.2) can be im plem en ted in a variety o f ways
d ep en d ing on the form in which th ese tw o characterizations are arranged.
In g en eral, w e can view (7.1.1) as a com putational p rocedure (an algorithm )
for determ in in g the output seq u en ce v(/t) o f the system from the input sequ en ce
x ( n ) . H o w ev er, in various ways, the com putations in (7.1.1) can be arranged into
eq u ivalen t sets o f d ifferen ce equations. Each set o f eq u ation s defines a com pu­
tational procedure or an algorithm for im plem enting the system . From each set
o f eq u ation s w e can construct a block diagram consisting o f an interconnection o f
d elay elem en ts, m ultipliers, and adders. In Section 2.5 w e referred to such a block
diagram as a realization o f the system or, equivalently, as a structure for realizing
the system .
If the system is to be im plem en ted in softw are, the block diagram or, eq u iv­
alently, the set o f eq u ation s that are obtained by rearranging (7.1.1), can be co n ­
verted in to a program that runs on a digital com puter. A ltern atively, the structure
in block diagram form im plies a hardware configuration for im plem enting the
system .
Perhaps, the o n e issue that m ay not be clear to the reader at this point
is w hy w e are con sid erin g any rearrangem ents o f (7.1.1) or (7.1.2). W hy not
just im plem en t (7.1.1) or (7.1.2) directly w ithout any rearrangem ent? If either
(7.1.1) or (7.1.2) is rearranged in som e m anner, what are the benefits gained in the
corresponding im plem en tation ?
T h ese are the im portant q u estion s which are answ ered in this chapter. A t
this p oin t in our d ev elop m en t, w e sim ply state that the m ajor factors that influ­
en ce our ch o ice o f a specific realization are com putational com p lexity, m em ory
requirem ents, and finite-w ord-length effects in the com putations.
C o m p u t a t i o n a l c o m p le x ity refers to the num ber o f arithm etic op eration s (m u l­
tiplications, d ivisions, and additions) required to com p ute an output valu e y (n ) for
the system . In the past, th ese w ere the on ly item s used to m easure com putational
com p lexity. H o w ev er, w ith recent d evelop m en ts in the design and fabrication o f
rather sop h isticated program m able digital signal p rocessin g chips, oth er factors,
502
Implementation of Discrete-Time Systems
Chap. 7
such as the num ber o f tim es a fetch from m em ory is p erform ed or the num ber of
tim es a com parison betw een tw o num bers is perform ed per output sam ple, have
b ecom e im portant in assessing the com putational com p lexity o f a given realization
o f a system .
M e m o r y requ irem ents refers to the num ber o f memory locations required
to store the system param eters, past inputs, past outputs, and any interm ediate
com puted values.
F inite- wor d-length effects or finite-precision effects refer to the quantization
effects that are inherent in any digital im plem en tation o f the system , either in
hardware or in softw are. T h e param eters o f the system m ust necessarily be repre­
sented with finite precision. T h e com p utations that are p erform ed in the process
o f com puting an output from the system must be rounded- o ff or truncated to fit
within the lim ited precision constraints o f the com p uter or the hardware used in
the im plem en tation . W hether the com p utations are perform ed in fixed-point or
floating-point arithm etic is an oth er consideration. A ll these p rob lem s are usually
called finite-w ord-length effects and are extrem ely im portant in influencing our
ch oice o f a system realization. W e shall see that different structures o f a system,
which are equivalent for infinite precision, exhibit different behavior when finiteprecision arithm etic is used in the im plem entation. T h erefore, it is very important
in practice to select a realization that is not very sensitive to finite-w ord-length
effects.
A lthou gh these three factors are the major o n e s in influencing our ch oice o f
the realization o f a system o f the type described by either (7.1.1) or (7.1.2), other
factors, such as w hether the structure or the realization len d s itself to parallel
processing, or w h eth er the com p utations can b e pip elined, m ay play a role in
our selection o f the specific im plem entation. T h ese additional factors are usually
im portant in the realization o f m ore com p lex digital signal processin g algorithm s.
In our discussion o f alternative realizations, w e con cen trate on the three
m ajor factors just outlined. O ccasionally, w e will include som e additional factors
that m ay be im portant in som e im plem entations.
7.2 STRUCTURES FOR FIR SYSTEMS
In general, an F IR system is described by the differen ce equation
Af-1
bkx (n ~ k)
y(n) =
(7.2.1)
*=0
or, equivalently, by the system function
A f-1
H{z)=*YibkZ~k
*=o
{122)
Furthermore, the unit sample response of the FIR system is identical to the coef-
Sec. 7.2
Structures for FIR Systems
503
ficients {£>*}, that is,
,, ,
\ b„,
'K’,) = | o .
0 < n < Af — 1
,7 , , ,
otherwise
<7-2 '3>
T h e length o f the F IR filter is selected as M to conform w ith the established
n otation in the technical literature.
W e shall present several m eth od s for im plem en ting an F IR system , b egin ­
ning w ith the sim p lest structure, called the direct form. A secon d structure is
the cascade-form realization. T h e third structure that w e shall d escribe is the
frequency-sam pling realization. Finally, w e present a lattice realization o f an FIR
system . In this discussion w e follow the con ven tion often used in the technical
literature, which is to use
for the param eters o f an F IR system .
In addition to the four realizations indicated ab ove, an F IR system can be
realized by m eans o f the D F T , as described in S ection 6.2. From on e point o f view,
the D F T can be considered as a com putational procedure rather than a structure
for an F IR system . H ow ever, when the com putational procedure is im plem ented
in hardw are, there is a corresponding structure for the F IR system . In practice,
hardware im plem en tation s o f the D F T are based on the use o f the fast Fourier
transform (FFT ) algorithm s described in C hapter 6 .
7.2.1 Direct-Form Structure
T he direct-form realization follow s im m ed iately from the nonrecursive difference
eq u ation given by (7.2.1) or, equivalently, by the con volu tion sum m ation
M- 1
y ( n ) = ^ h ( k ) x ( n - k)
t=o
(7.2.4)
T h e structure is illustrated in Fig. 7.1.
W e o b serve that this structure requires Af — 1 m em ory locations for stor­
ing th e Af — 1 previous inputs, and has a com p lexity o f Af m ultiplications and
M — 1 additions p er output point. Since the output con sists o f a w eigh ted linear
com b ination o f Af — 1 past values o f the input and the w eigh ted current value o f
th e input, th e structure in Fig. 7.1, resem b les a tapped d elay line or a transversal
Figwre 7.1
Direct-form realization of FIR system.
504
Implementation of Discrete-Time Systems
Chap. 7
Figure 7.2 Direct-form realization of linear-phase FIR system (Af odd).
system . C on sequ en tly, the direct-form realization is often called a transversal or
tapped-delay-line filter.
W hen the F IR system has linear phase, as described in S ection 8.2, the unit
sam ple response o f the system satisfies either the sym m etry o r asym m etry condition
h(n) = ± h ( M - 1 - n)
(7.2.5)
For such a system the num ber o f m ultiplications is reduced from M to M f l for Af
even and to (Af — l ) / 2 for M odd. For exam p le, the structure that takes advantage
o f this sym m etry is illustrated in Fig. 7.2 for the case in which M is odd.
7.2.2 Cascade-Form Structures
T he cascade realization follow s naturally from the system fun ction given by (7.2.2).
It is a sim ple m atter to factor H ( z ) into secon d-ord er F IR system s so that
K
0-2.6)
fl(z) = [] fl* ( z )
*=i
where
H k (z) = bk0 + bk]Z~l + bk2z~ 2
* = 1 ,2 .........K
(7.2.7)
and K is the in teger part o f ( M + l ) /2 . T h e filter param eter bo m ay be equally
distributed am ong the K filter section s, such that bo = biobw ■• ■b Ka or it m ay be
assigned to a sin gle filter section . T h e zeros o f H ( z ) are grou p ed in pairs to pro­
d uce the secon d-ord er F IR system s o f the form (7.2.7). It is alw ays desirable to
form pairs o f com p lex-con ju gate roots so that th e coefficien ts \bki} in (7.2.7) are
real valued. O n th e other hand, real-valued roots can b e paired in any arbitrary
m anner. T h e cascade-form realization alon g with th e basic secon d -ord er section
are sh ow n in Fig, 7.3.
Sec. 7.2
505
Structures for FIR Systems
jr(n)=jr,(n)
H](z)
>'[(") =
Vjr(n) = ,Y(n)
>'2<n) =
H2(z)
*2<” )
jr,(n)
(a)
Figure 7.3 Cascade realization of an FIR system.
In the case o f linear-phase F IR filters, the sym m etry in h (n) im plies that the
zeros o f H ( z ) a lso exhibit a form o f sym m etry. In particular, if Zk and z*k are a pair
o f co m p lex-con ju gate zeros then 1 f z t and 1 / z \ are also a pair o f com plex-conjugate
zero s (see Sec. 8.2). C on sequ en tly, w e gain som e sim plification by form ing fourthorder section s o f the FIR system as follow s
Hk(z) = c« ,( l - z / t z - ’ K l - c ^ ' K l - z ~ l / z k )(\ - z - ' / z V ,
(7.2.8)
=
Q o + Q - l t " 1 + Ck2Z~ " + Q t Z - ’’ + Z ~ A
w here the coefficien ts {ct i } and (c^ f are functions o f zt- Thus, by com bining
the tw o pairs o f p o les to form a fourth-order filter section , w e have reduced the
n um ber o f m ultiplications from six to three (i.e., by a factor o f 50% ). Figure 7.4
illustrates the b asic fourth-order F IR filter structure.
Figure TA Fourth-order section in a
cascade realization o f an FIR system.
506
Implementation of Discrete-Time Systems
Chap. 7
7.2.3 Frequency-Sampling Structures*
T he frequency-sam pling realization is an alternative structure for an F IR filter
in which the param eters that characterize the filter are the valu es o f the desired
frequency resp onse instead o f the im pulse response h(n). T o d erive the frequencysam pling structure, w e specify the desired frequency response at a set o f equally
spaced frequencies, nam ely
2 tt
cot = — (.k + a )
M
M - 1
A: = 0 , 1 , . . . , — - —
2
M od d
M
A: = 0, 1 , . . . , -------1
M even
2
a = 0 or j
and so lv e for the unit sam ple response h( n) from these equally spaced frequency
specifications. Thus we can write the frequency response as
M- 1
h ( n ) e ~ jum
n —0
and the values o f H(a>) at freq u en cies a>t = ( 2 n / M ) { k + a ) are sim ply
H (k + a ) = H
+ cr)^
(7.2.9)
= £
h (n )e -j2*lk+aWM
k _ 0< 1____ A/ - 1
T he set o f values { //(£ -(-a )} are called the frequency sam ples o f H(a)). In the case
w here a = 0, j//(Jt)} corresponds to th e M -point D F T o f (A(n)}.
It is a sim ple m atter to invert (7.2.9) and express h(n) in term s o f the fre­
quency sam ples. T he result is
i M- 1
h(n) = — J ' H ( k + a ) e j2*(i+a)n/M
n = 0, 1 , . . . , M - 1
(7.2.10)
W hen a = 0, (7.2.10) is sim ply the ID F T o f {//(& )}. N ow if w e use (7.2.10) to
substitute for h (n) in the z-transform H ( z ) , w e have
u- \
H( z ) = £ > ( " ) ; ■ "
n=0
M- 1
-L
■i u - 1
— Y \ H ( k + a ) e j2* il'H' )"/"
M *=0
(7.2.11)
^The reader may also refer to Section 8.2.3 for additional discussion o f frequency-sampling FIR
filters.
Sec. 7.2
507
Structures for FIR Systems
B y interchanging the order o f the tw o sum m ations in (7.2.11) and perform ing
the sum m ation over the index n w e obtain
I M-1
^ ^ej2n(i+a)/M „ - \ y
H( z ) = ^ H ( k + a)
(7.2.12)
} - Z- Mej2na y l
2. 1
M
H ( k + a)
_ e j2nik+a)/M z - 1
Thus the system function H ( z ) is characterized by the set o f frequency sam ples
(W(Jt-t-cr)j instead o f {h(«)).
W e v iew this F IR filter realization as a cascade o f tw o filters [i.e., H { z ) =
/ / i ( z ) / / 2 (s)]- O ne is an all-zero filter, or a com b filter, with system function
H x(z) = — (1 - r wf y2l“ )
M
(7.2.13)
Its zeros are located at equally spaced points on the unit circle at
Zt =
jt = 0 , 1 .........A f - 1
T he second filter with system function
Af —1
H l i z ) — 2 ^ -J _
Jt=0
j2n(t+a)/M
(7.2.1 )
*■
consists o f a parallel bank o f sin gle-p ole filters with resonant frequencies
Pl = e}2* ik+ayM
k = 0 , l .........M -
1
N o te that the p o le locations are identical to the zero locations and that both
occur at a)k = 2 k (k + a ) / M , which are the freq u en cies at which the desired fre­
quency resp onse is specified. T he gains o f the parallel bank o f resonant filters
are sim ply the com p lex-valu ed param eters \ H ( k + a )}. T h is cascade realization is
illustrated in Fig. 7.5.
W hen the desired frequency resp onse characteristic o f the F IR filter is nar­
row band, m ost o f the gain param eters \ H ( k + a )) are zero. C on sequ en tly, the
corresponding resonant filters can be elim inated and on ly the filters with nonzero
gains n eed be retained. T h e n et result is a filter that requires few er com p uta­
tion s (m ultiplications and additions) than the corresponding direct-form realiza­
tion. Thus w e ob tain a m ore efficient realization.
T h e frequency-sam pling filter structure can be sim plified further by exploiting
the s y m m e t r y in H ( k + a ) , nam ely, H ( k ) = H * ( M — k) for a = 0 and
H (* + i ) = H* ( M - k - | )
for a = \
T h ese relations are easily deduced from (7.2.9). A s a result o f this sym m etry, a
pair o f sin g le-p o le filters can b e com b ined to form a sin gle tw o -p o le filter with
508
Implementation of Discrete-Time Systems
Chap. 7
real-valued param eters. Thus for a = 0 the system function Hz (z) reduces to
H ( 0) ,
fiiiz) = t _ — l +
H i 0)
^
t=1
2
A ik ) + B(lc)z- 1
— — ----- 7
2 c o s { 2 n k / M ) z ~ ' + z~2
.
, H iM /2)
, lM^ ~ ]
H^ > = r r p r + T T P - +
£
A{k) + B { k ) z ~ x
~
2cos(2;rfc/Af)z -1 + z ~ 2
w
M odd
M even
(7.2.15)
Sec. 7.2
509
Structures for FIR Systems
w h ere, by definition,
A( k ) = H ( k ) + H ( M - k )
(7.2.16)
B( k) = H { k ) e ~ i2nklM + H ( M Sim ilar exp ression s can b e ob tain ed for a =
Example 7.2.1
Sketch the block diagram for the direct-form realization and the frequency-sampling
realization of the M = 32, a = 0, linear-phase (symmetric) F IR filter which has
frequency samples
1,
■(f)-
* = 0 ,1 ,2
5-
4= 3
0,
* = 4 . 5 ........15
Compare the computational complexity of these two structures.
Solution Since the filter is symmetric, we exploit this symmetry and thus reduce the
num ber of multiplications per output point by a factor of 2, from 32 to 16 in the
direct-form realization. The number of additions per output point is 31. The block
diagram of the direct realization is illustrated in Fig. 7.6.
We use the form in (7.2.13) and (7.2.15) for the frequency-sampling realization
and drop all terms that have zero-gain coefficients |H(k)}. T he nonzero coefficients
are H(k) and the corresponding pairs are H( M - k), for k = 0 ,1 , 2, 3. The block
diagram of the resulting realization is shown in Fig. 7.7. Since H( 0) = 1, the single­
pole filter requires no multiplication. The three double-pole filter sections require
three multiplications each for a total of nine multiplications. The total num ber of
additions is 13. Therefore, the frequency-sampling realization of this FIR filter is
computationally more efficient than the direct-form realization.
Figure 7.6
Direct-form realization of Af = 3 2 FIR filter.
510
Implementation of Discrete-Time Systems
Figare 7.7
Frequency-sampling realization for the FIR filter in Exam ple 7.2.1.
Chap. 7
Sec. 7.2
Structures for FIR Systems
511
7.2.4 Lattice Structure
In this sectio n w e introduce an oth er F IR filter structure, called the lattice filter or
lattice realization. Lattice filters are used exten sively in digital sp eech processing
and in the im plem en tation o f adaptive filters.
L et us b egin the d evelop m en t by considering a seq u en ce o f F IR filters with
system functions
Hm(z) = A m(z)
m = 0 . 1 , 2 .........M - 1
w here, by definition, A m(z) is the polyn om ial
m
A m(z) = 1 + £ a m(k)z~ k
*=i
m > 1
(7.2.17)
(7.2.18)
and /lo(^) = !• T he unit sam ple resp onse o f the m th filter is /j„,(0) = 1 and
h m{k) = a m(k), k = 1, 2 , . . . , m. T he subscript m on the p olyn om ial Am(c) d en otes
the d egree o f the polynom ial. For m athem atical con ven ien ce, w e define a,„ (0) = 1.
If (*(n)} is the input seq u en ce to the filter A,„{z) and (y(n )( is the output
seq u en ce, w e have
m
v(«) = -*(«) + ^ o r m(A:)A-(?f — k)
(7.2.19)
*■=l
T w o direct-form structures o f the F IR filter are illustrated in Fig. 7.8.
Figure 1A
Direct-form realization of the FIR prediction filter.
512
Implementation of Discrete-Time Systems
Chap. 7
In C hapter 11, w e show that the F IR structures sh ow n in Fig. 7.8 are inti­
m ately related w ith the topic o f linear prediction, w here
m
x ( n ) = ~ ^ 2 a m( k )x (n - k)
*=i
(7.2.20)
is the o n e-step forward predicted value o f x (n), based on m past inputs, and
y ( n ) = x (n) — x (n ), given by (7.2.19), rep resents the p red iction error sequ en ce.
In this context, the top filter structure in Fig. 7.8 is called a p re d ictio n er ror filter.
N o w suppose that w e have a filter of order m = 1. T h e ou tp u t o f such a filter
is
>-(«) = x ( n ) - f a i( l)j r ( n - 1)
(7.2.21)
This output can also be ob tain ed from a first-order or sin gle-stage lattice filter,
illustrated in Fig. 7.9, by exciting both o f the inputs by x ( n ) and selectin g the output
from the top branch. Thus the output is exactly (7.2.21), if w e select /li = ori (1).
T he param eter K i in the lattice is called a reflection coefficient and it is identical
to the reflection coefficient introduced in the S ch u r-C ohn stability test described
in Section 3.6.7.
N ext, let us consider an FIR filter for w hich m = 2. In this case the output
from a direct-form structure is
y (n ) = x ( n ) + ff;>(l)jr(/i -
1
) + ct2(2)x (n -
2
)
(7.2.22)
By cascading tw o lattice stages as show n in Fig. 7.10, it is p ossib le to obtain the
sam e output as (7.2.22). Indeed, the output from the first stage is
f i (fl) = -*(n) + K]X{n - 1)
(7.2.23)
£ i(n ) = K {x ( n ) + x ( n -
1
)
T h e output from the secon d stage is
flin ) = f\(n ) +
-
1
)
(7.2.24)
g 2(n) = K 2f i ( n ) + g i ( n - 1)
/o(") = &>(") =■*(«)
/i(«) =/o(") + *i£o{n - 1) = *00 + X , x ( n - 1)
j,{n) = tf,/oOO + SoO1 - 1) = fCix(n) + x(n - 1)
Figirc 7.9
Single-stage lattice filter.
Sec. 7.2
Structures for FIR Systems
513
Figure 7.10 Two-stage lattice filter.
If w e focus our atten tion on f 2{n) and substitute for f \ ( n ) and g\ ( n — 1) from
(7.2.23) in to (7.2.24), w e obtain
f 2(n) = jr(n) + K \ x ( n - 1) +
- 1) + x ( n - 2)]
(7.2.25)
= x ( n ) + ^ i ( l + K 2) x( n - 1) + K 2x {n — 2)
N o w (7.2.25) is identical to the output of the direct-form F IR filter as given by
(7.2.22), if w e eq u ate the coefficients, that is,
a 2 (2) = K 2
a 2( l) = ATi(l + K 2)
(7.2.26)
or, eq u ivalen tly,
K 2 = a 2(2)
Kx =
1 + a 2(2)
(7.2.27)
Thus the reflection coefficients K\ and K 2 o f the lattice can be ob tain ed from the
coefficien ts {am(£)} o f th e direct-form realization.
B y con tinu in g this process, o n e can easily d em onstrate by induction, the
eq u iv a len ce b etw een an m th-order direct-form FIR filter and an w -ord er or m stage lattice filter. T h e lattice filter is gen erally d escribed by the follow in g set o f
order-recursive equations:
/o (n ) = gain) = x ( n )
(7.2.28)
U i n ) = / ffl_ j(« ) + K mgm- X{n - 1)
m = 1 ,2 .........M - 1
(7.2.29)
g m(n) = ^ / B- i W + i . - i ( i i - l )
m = 1,2,..., M — 1
(7.2.30)
T h en the ou tp u t o f th e (A f—l)-sta g e filter corresponds to the output o f an (A f—1)order F IR filter, that is,
y (n ) =
Figure 7.11 illustrates an (Af - l)-sta g e lattice filter in b lock diagram form along
w ith a typical stage that show s the com p utations sp ecified by (7.2.29) and (7.2.30).
A s a co n seq u en ce o f the eq u ivalen ce b etw een an F IR filter and a lattice filter,
th e ou tp u t f m(n) o f an m -stage lattice filter can b e exp ressed as
m
f m( n) = £ <xm( k) x( n - k)
*=o
«m(0) = 1
(7.2.31)
Sin ce (7.2.31) is a con v olu tion sum , it follow s that th e z-transform relationship is
Fm(z) = Am(z)X(z)
Implementation of Discrete-Time Systems
514
Chap. 7
(a)
Figure 7.11 (Af —l)-stage lattice filter.
or, equivalently,
7}
F (?)
F
X(z)
Fo(z)
A m(z) =
(
(7.2.32)
T h e other output com p on en t from the lattice, nam ely, g m(n), can also be
exp ressed in the form o f a con volu tion sum as in (7.2.31), by using another set
o f coefficients, say {^m(Jt)}. T hat this in fact is the case, b ecom es apparent from
ob servation o f (7.2.23) and (7.2.24). From (7.2.23) w e n ote that the filter coeffi­
cients for the lattice filter that produces / i ( n ) are {1 , A’l} = { 1 , ofi( 1 )} w hile the
coefficients for the filter with output g \ ( n ) are (AT], 1} = | a i ( l ) , 1}. W e n ote that
these tw o sets o f coefficients are in reverse order. If w e con sid er the tw o-stage
lattice filter, with th e output given by (7.2.24), w e find that g 2 (n) can be expressed
in the form
g 2(n) = K 2f \ ( n ) + £ i(n - 1)
= K 2[x in) + K \ x { n — 1)] + K \ x { n - 1) + x{ n — 2)
= K 2x{ n ) + K \ ( l + K 2)x{n - 1) -I- x i n - 2)
= cr2 ( 2 )jc(n) + £*2 ( 1 ) * (n -
1
) + xin -
2
)
C onsequently, the filter coefficients are {a 2 (2), a 2 ( l ) , 1}, w h ereas the coefficients
for the filter that produces the output f i i n ) are {1, a 2( \ ) , a 2 (2)}. H ere, again, the
tw o sets o f filter coefficients are in reverse order.
F rom this d ev elo p m en t it follow s that the ou tp u t gm{n) from an m -stage
lattice filter can b e exp ressed by the con volu tion sum o f the form
m
gmin) = £
A *(*M « - k)
(7.2.33)
Sec. 7.2
Structures for FIR Systems
515
w h ere the filter coefficients {& ,(*)} are associated with a filter that produces
f m(n) = y ( n ) but op erates in reverse order. C onsequently,
f}m (k) = a m (m — k)
* = 0 , 1 .........m
(7.2.34)
with p m(m) = 1 .
In the co n tex t o f linear prediction, su p p ose that the data x(n), x{n - 1), . . . ,
x ( n —m + 1) is u sed to linearly predict th e signal valu e x ( n —m ) by u se o f a linear
filter w ith coefficients {—fim(k)}. Thus the predicted value is
m-l
(7.2.35)
Since the data are run in reverse order through the predictor, the prediction per­
form ed in (7.2.35) is called b a c k w a r d predictio n. In contrast, the F IR filter with
system function Am(z) is called a f o r w a r d predictor.
In the z-transform dom ain, (7.2.33) b ecom es
G m(z) = B m( z ) X ( z )
(7.2.36)
or, eq u ivalen tly,
(7.2.37)
w here Bm(z) represents the system function o f the F IR filter with coefficients
{& ,(*)}, that is,
(7.2.38)
4=0
Since fim(k) = a m(m — *), (7.2.38) m ay b e exp ressed as
m
ffl
(7.2.39)
m
T h e relationship in (7.2.39) im plies that the zeros o f the F IR filter w ith system
function B m(z) are sim ply the reciprocals o f the zeros o f A „ ( z ) . H en ce B m(z) is
called the reciprocal or reverse p olyn om ial o f A m(z).
N o w that w e have estab lish ed th ese interesting relationships b etw een the
direct-form F IR filter and th e lattice structure, let us return t o th e recursive lattice
eq u a tio n s in (7.2.28) through (7.2.30) and transfer them to th e z-dom ain. Thus
516
Implementation of Discrete-Time Systems
Chap. 7
w e have
(7.2.40)
F0(z) = G 0(z) = X ( z )
Fm(z) = Fm_ i(z ) + J:mz - 1 Gm_ i(z )
m = 1, 2 , . . . , M - 1
(7.2.41)
G m(z) =
m = 1 , 2 , . . . , M —1
(7.2.42)
+
If w e divide each eq u ation by X (z), w e obtain the desired results in the form
(7.2.43)
A0(z) = B0(z) = 1
A m{z) = A m- X(z) + K „ z ~ } Bm- i ( z )
m = 1 , 2 .........M -
1
(7.2.44)
Bm(z) = K n A n - i W + z - ' B ^ i z )
m = l,2 ,...,M -l
(7.2.45)
T hus a lattice stage is described in the z-dom ain by the m atrix equation
(7.2.46)
B efo re concluding this discussion, it is desirable to d ev elo p the relationships
for converting the lattice param eters
that is, the reflection coefficients, to the
direct-form filter coefficients (a m(Jt)), and vice versa.
Conversion of lattice coefficients to direct-form filter coefficients. The
direct-form F IR filter coefficients {am(Jk)} can b e obtained from the lattice coeffi­
cien ts {AT;} by using the follow ing relations:
(7.2.47)
A 0 (z) = B 0(z) = 1
A m(z) = A m- \ ( z ) + K mz ~ xBm~\(z)
Bm(z) = Z- mA m(z~ l )
m = 1,2- - - M -
m = 1 ,2 .........M -
1
1
(7.2.48)
(7.2.49)
T h e solu tion is o b tain ed recursively, beginning with m = 1 . Thus w e obtain a
seq u en ce o f (M — 1) F IR filters, o n e for each valu e o f m. T h e procedure is best
illustrated by m eans o f an exam ple.
Example 12J2
Given a three-stage lattice filter with coefficients AT] = j, K 2 = 5 , Ky = 5 , determine
the FIR filter coefficients for the direct-form structure.
Solution We solve the problem recursively, beginning with (7.2.48) for m = 1. Thus
we have
Ai(z) = A„<z) + tfiZ ^ B oW
= 1 + K n - 1 = 1 + Jz "1
Hence the coefficients of an FIR filter corresponding to the single-stage lattice are
ori(0) = 1 , ari(l) = Ki = j. Since Bm(z) is the reverse polynomial of A„(z), we have
Sec. 7.2
517
Structures for FIR Systems
Next we add the second stage to the lattice. For m = 2 , (7.2.48) yields
■AjU) = ^ i(i) + KiZ~* B\(z)
— 1 4- J - -1 4- 1 ,- 2
—
1 + Rc T 2'-
Hence the FIR filter param eters corresponding to the two-stage lattice are £*2(0) = 1,
“ 2(1) = | , £*2(2) =
Also,
*2(z) = J + i = +z-2
Finally, the addition of the third stage to the lattice results in the polynomial
M( z ) = * z (;)+ * :3 z _ ,ft(z )
= 1+
z _1 + %z~2 + | z -3
Consequently, the desired direct-form FIR filter is characterized by the coefficients
£*3(0) = 1
a 3(l) = g
of3(2) = 2
o3{3) = i
A s this exam ple illustrates, the lattice structure with param eters K \ , K i .........
K m, corresp on d s to a class o f m direct-form F IR filters with system functions A\ (z),
y4i ( z) , . . . , A„,(z). It is interesting to note that a characterization o f this class o f m
FIR filters in direct form requires m( m + l ) / 2 filter coefficients. In contrast, the
lattice-form characterization requires on ly the m reflection coefficients {A',}. T he
reason that the lattice provides a m ore com pact representation for the class o f m
F IR filters is sim ply du e to the fact that the addition o f stages to the lattice d o es
n ot alter the param eters o f the previous stages. O n the other hand, the addition
o f the mth stage to a lattice with ( m - 1 ) stages results in a F IR filter with system
function A m(z) that has coefficients totally d ifferent from the coefficients o f the
low er-order F IR filter with system function Am_ i(z ).
A form ula for determ ining the filter coefficients [am(Jt)} recursively can be
easily d erived from p olyn om ial relationships in (7.2.47) through (7.2.49). From the
relationship in (7.2.48) w e have
Am(z) = Am_ i(z ) + K mz ~ l B m- i ( z )
m
»_i
m-t
^ a m (k ) z ~ k = ^ a m-] (k)z~ k + K m ^ a m- \ ( m - 1 - k ) z (i+1)
k=0
k=0
(7.2.50)
*=0
B y eq u atin g the coefficients o f equal pow ers o f z - 1 and recalling that a m(0) = 1
for m = 1, 2 .........M - 1, w e obtain the desired recursive eq u ation for the F IR filter
coefficients in the form
£*m(0) = 1
(7.2.51)
a m(m) = K m
a m(k) = ctm- \ ( k ) +
(7.2.52)
- *)
= £*m—i(k) + a m(rn)am-i(rn - k)
^
(7-2. 53)
Implementation of Discrete-Time Systems
518
Chap. 7
W e n o te that (7.2.51) through (7.2.53) are sim ply the L ev in so n -D u rb in recursive
equations given in Chapter 11.
Conversion of direct-form FIR filter coefficients to lattice coefficients.
S u p pose that w e are given the F IR coefficien ts for the direct-form realization or,
equivalently, the polyn om ial A m(z), and w e w ish to determ in e the corresponding
lattice filter param eters { £ ,} . For the m -stage lattice w e im m ed iately obtain the
param eter K m = a m (m). T o obtain K m- \ w e n eed the p olyn om ials Am_ i(z ) since,
in general, K m is ob tain ed from the p olyn om ial A m (z) for m = M —1, Af —2 , . . . , 1.
C onsequently, w e need to com p ute the p olyn om ials A m(z) starting from m = Af —1
and “stepping d o w n ” su ccessively t o m = 1 .
T h e d esired recursive relation for the polyn om ials is easily determ ined from
(7.2.44) and (7.2.45). W e have
A m(z) = A m- \ ( z ) + K mz ~ x B m- \ { z )
= A m- \ ( z ) + K m[Bm(z) - ^ mAm_ i( ;) ]
If w e solve for Am_i (z), w e obtain
A m- i ( z ) =
m = M - h M - 2 .........1
-*"*"■<*>
(7.2.54)
which is just the step-dow n recursion used in the S ch u r-C ohn stability test d e­
scribed in S ection 3.6.7. T hus w e com p ute all low er-d egree p olyn om ials A m(z)
beginning with A M- \ ( z ) and obtain the desired lattice coefficien ts from the rela­
tion K m = a m(m). W e ob serve that th e procedure works as lon g as \ Km \
1 for
m = 1, 2 .........A f - 1 .
Example 7.23
Determ ine the lattice coefficients corresponding to the FIR filter with system function
H(z) = A3(z) = 1 -ISolution
First we note that
-I- f z “2 -I- ±z-J
= a 3(3) = | . Furtherm ore,
# 3 (z) = 5 + jU- 1 +
z~2 + z~y
The step-down relationship in (7.2.54) with m = 3 yields
A2(z) =
A3(z) - K i B ^ z )
1- K
= l + f z - ^ z1 , - 2
Hence K2 — a 2(2) = 5 and B i (z) =
recursion in (7.2.51), we obtain
, , ,
5 +
| z _1 + z_1- By repeating the step-down
A i b ) ~ K2B2(z )
Sec. 7.3
519
Structures for HR Systems
From the step -d ow n recursive eq u ation in (7.2.54), it is relatively easy to
obtain a form ula for recursively com puting K m, begin n ing with m = M — 1 and
stepping dow n to m = 1. For m = M — 1, M — 2 , . . . , 1 w e h ave
K m = a m( m )
Qfm—l (0) = 1
(7.2.55)
a m(k) - K mfim(k)
—
J __ g 2
a m(k) -
otm ( r n ) a m ( m
- k)
1 -al(m )
1 5 k < m —1
(7.2.56)
w hich is again the recursion w e introduced in the S ch u r-C oh n stability test.
A s indicated ab ove, the recursive eq u ation in (7.2.56) breaks dow n if any
lattice param eters | Armj = 1. If this occurs, it is ind icative o f the fact that the
polyn om ial A m- \ ( z ) has a root on the unit circle. Such a root can b e factored out
from Am_](z) and the iterative p rocess in (7.2.56) is carried out for th e reducedorder system .
7.3 STRUCTURES FOR IIR SYSTEMS
In this section w e consider different IIR system s structures d escribed by the dif­
ference eq u ation in (7.1.1) or, eq u ivalen tly, by the system function in (7.1.2). Just
as in the case o f F IR system s, there are several types o f structures or realizations,
including direct-form structures, cascade-form structures, lattice structures, and
lattice-ladder structures. In addition, IIR system s lend th em selv es to a parallelform realization. W e begin by describing tw o direct-form realizations.
7.3.1 Direct-Form Structures
T h e rational system function as given by (7.1.2) that ch aracterizes an IIR system
can be v iew ed as tw o system s in cascade, that is,
H ( z ) = H i (z ) H 2( z )
(7.3.1)
w h ere Hi (z) con sists o f th e zeros o f H ( z ) , and H z (z ) con sists o f the p o les o f H {z),
M
H l (z ) = Y ! f bkZ~k
*=o
(7-3.2)
H 2 ( z ) = ------- -- ---------
(7.3.3)
and
1+ X > z"*
*=i
In S ection 2.5.1 w e describe tw o different direct-form realizations, character­
ized by w h eth er H \{z) p reced es H j i z ) , or vice versa. Since H \ ( z ) is an F IR system ,
its direct-form realization w as illustrated in Fig. 7.1. B y attaching th e all-p ole
520
Implementation of Discrete-Time Systems
All-zero system
Chap. 7
All-pole system
Figure 7.12 Direct form I realization.
system in cascade with H\ ( z) , w e obtain the direct form I realization d epicted in
Fig. 7.12. T his realization requires M + N + 1 m ultiplications, M + N additions,
and M + N + 1 m em ory locations.
If th e all-p ole filter f y i z ) is placed b efore th e all-zero filter H i(z), a m ore
com pact structure is ob tain ed as illustrated in S ection 2.5.1. R ecall that the differ­
en ce eq u ation for th e all-pole filter is
N
w(n) = - ^
ai(W{n - k) + jr(n)
(7.3.4)
Since w ( n ) is the input to the all-zero system , its ou tp u t is
M
yW =
bk w (n - k)
(7.3.5)
*=o
W e n o te that both (7.3.4) and (7.3.5) in volve d elayed versions o f the sequ en ce
(u>(n)}- C on sequ en tly, on ly a single d elay line or a sin gle set o f m em ory locations
is required for storing the past values o f {u>(n)}. T h e resulting structure that
im plem en ts (7.3.4) and (7.3.5) is called a direct form II realization and is depicted
in Fig. 7.13. T his structure requires M + N + 1 m ultiplications, M + N additions,
Sec. 7.3
Structures for IIR Systems
Figure 7.13
521
D irect form II realization ( N — Af).
and the m axim um o f {M, N } m em ory locations. Since the direct form II realization
m inim izes the num ber o f m em ory locations, it is said to b e canonic. H ow ever, w e
should indicate that other IIR structures also p ossess this property, so that this
term in ology is perhaps unjustified.
T h e structures in Figs. 7.12 and 7.13 are both called “direct form ” realiza­
tions b ecau se th ey are ob tain ed directly from the system function H ( z ) w ithout
any rearrangem ent o f H ( z ) . U n fortu n ately, both structures are extrem ely sen si­
tive to param eter quantization, in gen eral, and are not recom m en d ed in practical
applications. T his to p ic is discussed in detail in Section 7.6, w h ere w e dem onstrate
that w h en N is large, a sm all change in a filter coefficient d u e to param eter quan­
tization, results in a large change in the location o f th e p o les and zeros o f the
system .
7.3.2 Signal Flow Graphs and Transposed Structures
A signal flow graph provides an alternative,Nbut eq u ivalen t, graphical rep resen ­
tation to a b lock diagram structure that w e have b een using to illustrate various
system realization s. T h e basic elem en ts o f a flow graph are branches and n od es.
A signal flow graph is basically a set o f directed branches that con n ect at n od es.
B y d efinition , th e signal ou t o f a branch is equal to th e branch gain (system func­
tio n ) tim es the signal in to the branch. Furtherm ore, the signal at a n od e o f a
flow graph is eq u al to the sum o f the signals from all branches con n ectin g to the
node.
T o illustrate th e se basic n otion s, let us con sid er the tw o -p o le and tw o-zero
IIR system d ep icted in b lock diagram form in Fig. 7.14a. T h e system block
522
Implementation of Discrete-Time Systems
Chap. 7
(a)
Source node
Sink node
5
(b)
Fignre 7.14 Second-order filter structure (a) and its signal flow graph (b).
diagram can be converted to the signal flow graph show n in F ig. 7.14b. W e note
that the flow graph contains five n od es lab eled 1 through 5. T w o o f the nodes
( 1 ,3 ) are sum m ing n od es (i.e., they contain ad d ers), w hile the other three nodes
represent branching points. Branch transm ittances are ind icated for the branches
in the flow graph. N o te that a d elay is indicated by the branch transm ittance
z- 1 . W h en the branch transm ittance is unity, it is left u n lab eled . T h e input to
the system originates at a so ur ce n o d e and the ou tp u t signal is extracted at a sink
node.
W e observe that the signal flow graph con tains th e sam e b asic inform ation
as the block diagram realization o f the system . T h e on ly ap p aren t d ifference is
that b o th branch p oints and adders in the b lock diagram are rep resen ted by nodes
in th e signal flow graph.
T h e subject o f linear signal flow graphs is an im portant o n e in the treatm ent
o f netw ork s and m any interesting results are available. O n e b asic n otion involves
the transform ation o f o n e flow graph in to a n oth er w ithout changing the basic
in p u t-ou tp u t relationship. Specifically, o n e tech n iq u e that is usefu l in deriving
new system structures for F IR and IIR system s stem s from th e transposition or
flo w - g ra p h reversal theorem. T his th eorem sim ply states that if w e reverse the
Sec. 7.3
523
Structures for IIR Systems
directions o f all brancb transm ittances and interchange the input and output in
the flow graph, the system function rem ains unchanged. T h e resulting structure is
called a transposed s tructure or a tran sp osed f o r m .
F or exam p le, the transposition o f the signal flow graph in Fig. 7.14b is illus­
trated in Fig. 7.15a. T h e corresponding b lock diagram realization o f the transposed
form is d ep icted in Fig. 7.15b. It is interesting to n ote that the transposition o f the
original flow graph resulted in branching n od es b ecom in g ad d er n od es, and vice
versa. In Section 7.5 w e p rovid e a p ro o f o f the transposition theorem by using
state-sp ace techniques.
L et us apply the transposition theorem to the direct form II structure. First,
w e reverse all the signal flow d irections in Fig. 7.13. Second, w e change n od es
into adders and adders into n od es, and Anally, w e interchange the input and the
output. T h ese op eration s result in the transposed direct form II structure show n
in Fig. 7.16. T his structure can b e redrawn as in Fig. 7.17, which show s the input
on the left and the output on the right.
~ ai
s^\
—
(b)
h
Figure 7.15 Signal flow graph of
transposed structure (a) and its
realization (b).
524
Implementation of Discrete-Time Systems
Chap. 7
Figure 7.16 Transposed direct form II
structure.
T h e transposed direct form II realization that w e have ob tain ed can be d e­
scribed by the set o f d ifference equations
(7.3.6)
y ( n ) = wr f n - l ) + b o x ( n )
Wt(n) = wt+i(n - 1)
+ b kx(n)
Jt = 1 ,2 ......... N - 1
w N(n ) = bNx(n) - a Ny(n)
(7.3.7)
(7.3.8)
W ithout loss o f generality, w e have assum ed that Af = W in w riting equ ation s. It
is also clear from observation o f Fig. 7.17 that this set o f d ifferen ce eq u ation s is
equ ivalen t to the sin gle differen ce equation
y ( n ) = ~ ^ 2 a ky (n - k ) + ^ b kx ( n - k )
(7.3.9)
Sec. 7.3
525
Structures for IIR Systems
Figure 7.17 Transposed direct form II
structure.
Finally, w e o b serv e that th e transposed direct form II structure requires the sam e
n um ber o f m ultip lication s, additions, and m em ory locations as the original direct
form II structure.
A lth o u g h our discussion o f transposed structures has b een concerned with
the general form o f an IIR system , it is interesting to n o te that an F IR system ,
ob tain ed from (7.3.9) by setting the a* = 0, k = 1, 2 , . . . , N , also has a transposed
direct form as illustrated in Fig. 7.18. T his structure is sim ply ob tain ed from
Fig. 7.17 by settin g a* = 0, k = 1, 2 , . . . , N . T his transposed form realization m ay
Figure 7.18
Transposed FIR structure.
526
Implementation of Discrete-Time Systems
Chap. 7
be described by the set o f differen ce eq u ation s
w M(n) = b t f x ( n )
(7.3.10)
u>*(/i) = u>*+i(n — 1) + bkx(n)
k = M — 1, M ~ 2, . . . , 1
y ( n ) = w i ( n - 1) +/>&*(«)
(7.3.11)
(7.3.12)
In sum m ary, T ab le 7.1 illustrates the direct-form structures and the corresponding
d ifference eq u ation s for a basic tw o-p ole and tw o-zero IIR system with system
function
x
b0 + b i z ~ l + b 2z ~ 2
H ( z ) = ----------- ------------ 3 -
1 + fli z 1 +azz 1
(7.3.13)
This is th e basic building block in the cascade realization o f h igh-order IIR system s,
as described in the follow in g section. O f the three direct-form structures given in
T ab le 7.1, th e direct form II structures are p referable due to the sm aller number
o f m em ory locations required in their im plem entation.
Finally, w e n o te that in the z-dom ain, the set o f d ifferen ce eq u ation s describ­
ing a linear signal flow graph con stitute a linear set o f equations. A n y rearrange­
m ent o f such a set o f eq u ation s is equ ivalen t to a rearrangem ent o f the signal flow
graph to obtain a n ew structure, and vice versa.
7.3.3 Cascade-Form Structures
Let us con sid er a high-order IIR system with system function given by (7.1.2).
W ithout loss o f generality w e assum e that N > M . T h e system can be factored
into a cascade o f secon d-ord er subsystem s, such that H (z) can b e exp ressed as
K
*=i
w here K is the in teger part o f (N + l ) /2 . Hk (z) has th e general form
rr , ,
bto + b u z ' 1 + bk2Z ' 2
Hk (z) = — --------------------- 3 5 1 + fljtiz 1 + a k2z 2
(7.3.15)
A s in the case o f F IR system s b ased on a cascad e-form realization, the param eter
bo can be distributed equally am ong the K filter se ctio n s so that bo = £>1 0 ^ 2 0 • ■■bxoT he coefficients {a*j} and {*>*, }in the secon d -ord er su b system s are real. This
im plies that in form ing th e secon d-ord er su b system s or quadratic factors in (7.3.15),
w e should group to g eth er a pair o f com p lex-con ju gate p o les and w e should group
togeth er a pair o f com p lex-con ju gate zeros. H ow ever, th e pairing o f tw o com plexconjugate p o les with a pair o f com p lex-con ju gate zeros or real-valu ed zeros to form
a subsystem o f the type given by (7.3.15), can b e d o n e arbitrarily. Furtherm ore,
any tw o real-valued zeros can b e paired to g eth er to form a quadratic factor and,
likew ise, any tw o real-valued p o les can b e paired togeth er to form a quadratic
factor. C on sequ en tly, th e quadratic factor in th e num erator o f (7.3.15) m ay consist
TABLE 7.1
SOME SECOND-ORDER MODULES FOR DISCRETE-TIME SYSTEMS
Structure
Implementation Equations
System Function
xin)
v(n) = fc().t(n) +fc]X(n - I)
-I- i>2.t(rt - 2)
- oi v(n - 1) —<t2\(n —2)
H( z) =
bft + bjZ 1 +biz 2
1 + a u -1 +aiz~2
H( Z) =
bp + bjZ 1 +bjz
1 + a u _1 +aiz~2
H( Z)
bp -t-fri? 1 +bjz 2
1 + c u - * +aiz~2
x<n)
u.’( n )
=
~ d i w ( n — I ) — a 2 w ( n — 2)
+ x(n)
,v(n) = bow(n) -+- fcj utr(»» —1)
+ b2U)(n —2)
x<n)
v(n) = bu.x(n) + wt (n - 1)
= ht x( n) —ai_v(n)
527
w 2 (n)
=
-t- w 2 ( n -
1)
b ix(n) -
a 2y ( n )
528
Implementation of Discrete-Time Systems
Chap. 7
o f either a pair o f real roots or a pair o f com p lex-con ju gate roots. T h e sam e
statem ent applies to the den om in ator o f (7.3.15).
If N > M , so m e o f the secon d-ord er subsystem s h ave num erator coefficients
that are zero, that is, eith er bk2 = 0 or bk\ = 0 or b oth bk2 = bk\ = 0 for som e k. Fur­
therm ore, if N is odd, o n e o f the subsystem s, say Hk (z), m ust h ave a k2 = 0, so that
the subsystem is o f first order. T o p reserve the m odularity in th e im plem en tation
o f H ( z ) , it is o ften preferable to use the basic secon d-ord er su b system s in the cas­
cade structure and h ave som e zero-valu ed coefficients in so m e o f the subsystem s.
E ach o f the secon d-ord er subsystem s with system function o f the form (7.3.15)
can be realized in eith er direct form I, or direct form II, or tran sp osed direct form
II. Since there are m any w ays to pair the p oles and zeros o f H (z ) into a cascade
o f secon d-ord er section s, and several w ays to order the resulting subsystem s, it is
p ossib le to obtain a variety o f cascade realizations. A lth ou gh all cascade realiza­
tions are equ ivalen t for infinite precision arithm etic, the various realizations may
differ significantly w hen im plem en ted with finite-precision arithm etic.
T he general form o f th e cascade structure is illustrated in Fig. 7.19. If we
use the direct form II structure for each of the subsystem s, the com putational
algorithm for realizing the IIR system with system function H ( z ) is described by
the follow in g set o f equations.
>>o(n) = x ( n )
(7.3.16)
w k(n) = - a ki w k(n - 1) - a k2w k (n - 2) + y*_i(n)
k = 1 ,2 ...........K
(7.3.17)
y k(n) = bk0w k(n) + bt i w k(n - 1) -I- bk2w t (n - 2)
k = 1 ,2 ..........K
(7.3.18)
y (n ) = ytcin)
(7.3.19)
(a)
**0
-«*l
S ~
\
> '* ( * ) = X t + 1 ( n )
^ -0
0 -
(b)
Ftg*rc 7.19 Cascade structure of second-order systems and a realization of each
second-order section.
Sec. 7.3
529
Structures for IIR Systems
Thus this set o f eq u ation s p rovides a com p lete description o f the cascade structure
based on direct form II section s.
7.3.4 Parallel-Form Structures
A parallet-form realization o f an IIR system can be ob tain ed by p erform ing a
partial-fraction expansion o f H( z ) . W ithout loss o f generality, w e again assum e
that N > M and that the p o le s are distinct. T hen, by perform ing a partial-fraction
expansion o f H( z ) , w e obtain the result
N
Ai
H ( z ) = C + V ------- — r
t t 1 - P*z~'
(7.3.20)
w here {p*} are the p oles, {A t \are the coefficients (residues) in the partial-fraction
exp an sion , and the constant C is defined as C = b s / a s i . T h e structure im plied
by (7.3.20) is sh ow n in Fig. 7.20. It consists o f a parallel bank o f sin gle-p ole
filters.
In gen eral, som e o f the p oles o f H ( z ) may be com plex valued. In such a case,
the corresp on d ing coefficients A t are also com plex valued. T o avoid m ultiplica­
tions by com p lex num bers, w e can com b ine pairs o f com p lex-con ju gate p oles to
form tw o -p o le subsystem s. In addition, w e can com b ine, in an arbitrary m anner,
c
Figure 7.20
Parallel structure of IIR system.
530
Implementation of Discrete-Time Systems
Figure 7*21
Chap. 7
Structure of second-order section in a parallel IIR system realization.
pairs o f real-valued p o les to form tw o-p ole subsystem s. Each o f th ese subsystem s
has the form
Hk{z) =
1
(7.3.21)
+ a kiz ] + a k2r
w here the coefficients {bk,} and (at;] are real-valued system param eters. T he over­
all function can n ow be expressed as
(7.3.22)
H( z) = C + J 2 h*<*>
w here K is the in teger part o f ( N + 1)/2. W hen N is odd, on e o f the H k (z) is really
a sin gle-p ole system (i.e., bk\ = a k 2 = 0 ).
T h e individual second-order section s which are th e basic b uilding blocks for
H ( z ) can be im plem en ted in eith er o f the direct form s or in a transposed direct
form. T h e direct form II structure is illustrated in Fig. 7.21. W ith this structure as
a basic building block, the parallel-form realization o f the F IR system is described
by the follow in g set o f equations
wk(n) = —aki w k(n - 1) - ak2wk(n - 2) + x(n)
* = 1,2,...,
(7.3.23)
yt(n) = bk0wk(n ) + bktwt (n - 1)
k = 1 ,2 .........K
(7.3.24)
K
y(n) = Cx ( n ) + ^ y t ( n )
(7.3.25)
Example 7.3.1
Determine the cascade and parallel realizations for the system described by the system
function
10(1 -
H(z) =
1)(1 — l z - 1 ) ( l + 2 z-1)
(1 - |z _1)(l - 5Z-1)[1 - (j + ; j ) z -1][l - (5 -
Sec. 7.3
Structures for IIR Systems
531
Solution The cascade realization is easily obtained from this form. One possible
pairing of poles and zeros is
1 - lz~*
= i _ 2 .- i + 2, z- 2
1
8'
T
32 ‘
l + l z - '- z " 2
H2<c) = i1 - z ~ l—
+ \rz~~2;
and hence
H(z) = 10//,(z)Jfe(z)
The cascade realization is depicted in Fig. 7.22a.
To obtain the parallel-form realization, H(z) must be expanded in partial frac­
tions. Thus we have
Ai
Ai
" (2 ) =
■+
1-fz-1
1-iz-1
i - ( i + yi)z-i
i - ( l - y i );-i
where j4t , A 2, A 3, and A j are to be determ ined. A fter some arithmetic we find that
A { = 2.93,
A, = -17.68,
A3 = 12.25 - yl4.57,
A\ = 12.25 + >14.57
upon recom bining pairs of poles, we obtain
„ % —14.75 —12.90z_l
24.50 + 26.82;"'
— ;----- 5— i— I- 1 _ -- I -L J.--2
H (z) — -------1
1 _ Z--1 _1_ 2 . 7 - 1
x*- + 324
*■
2^
The parallel-form realization is illustrated in Fig. 7.22b.
7.3.5 Lattice and Lattice-Ladder Structures for IIR
Systems
In Section 7.2.4 w e d ev elop ed a lattice filter structure that is eq u ivalen t to an F IR
system . In this sectio n w e exten d the d evelop m en t to IIR system s.
L et us b egin with an all-p ole system with system function
H (z) = ------- ^ ----------- =■ —
A
t
a n (z )
1+ ^cis(k)z
k=l
(7. 3. 26)
T h e direct form realization o f this system is illustrated in Fig. 7.23. T h e difference
eq u ation for this IIR system is
N
y( n ) = ~ ^ a N (k) y( n - k) + x { n)
*=i
(7.3.27)
It is in terestin g to n ote that if w e interchange the roles o f input and output
[i.e., interchange x ( n ) w ith y(n ) in (7.3.27)], w e obtain
x(n) = - ^ T a N(k)x(n - k) 4- y(ri)
Implementation of Discrete-Time Systems
532
Chap. 7
(a)
(b)
Figure 7.22 Cascade and parallel realizations for the system in Example 7.3.1.
or, equivalently,
N
y ( n ) = x ( n ) 4- ^ a w(fc)*(n - k)
*=i
(7.3.28)
W e n o te that the eq u ation in (7.3.28) describes an F IR system having the
system function H ( z ) = A N (z), w h ile the system describ ed by th e differen ce equa­
tion in (7.3.27) rep resen ts an IIR system with system function H ( z ) = lM w(z)*
Sec. 7.3
533
Structures for IIR Systems
yi.n)
*(«)
Figure 7.23 Direct-form realization of an all-pole system.
O n e system can b e o b tain ed from the other sim ply by interchanging th e roles o f
th e input and output.
B ased on this o b servation , w e shall use the all-zero (F IR ) lattice describ ed in
S ection 7.2.4 to ob tain a lattice structure for an all-p ole IIR system by interchanging
the ro les o f the input and output. First, w e take the all-zero lattice filter illustrated
in Fig. 7.11 and th en redefine the input as
x ( n ) = f N(n)
(7.3.29)
v(n) = fo(n)
(7.3.30)
and the ou tp u t as
T h ese are exactly the o p p osite o f the definitions for the all-zero lattice filter. T h ese
d efinitions d ictate that the qu an tities { / m(n)l be com p uted in d escen din g ord er [i.e.,
/ v ( n ) , / v _ i ( n ) , . . . ) . T h is com p utation can be accom plished by rearranging the
recursive eq u ation in (7.2.29) and thus solvin g for / m_\ (n) in term s o f f m(n ), that is,
K mgm- i ( n - 1 )
/ m - i ( n ) = f m( n) -
m =
N, N - 1 ,..., 1
T h e eq u a tio n (7.2.30) for gm(n) rem ains unchanged.
T h e result o f th ese ch an ges is th e set o f equations
f s ( n ) = x (n)
(7.3.31)
f m - \ ( n ) = f m(n) - K mgm. ] ( n - 1)
gm(n) = K mf m- i ( n ) + g m- \ ( n - 1)
m = N , N - 1,.
(7.3.32)
m = N , N - 1,
(7.3.33)
y ( n ) = /o (n ) = go(n)
w hich corresp on d to the structure sh ow n in Fig. 7.24.
Input
Figure 7.24
Lattice structure for an all-pole IIR system.
(7.3.34)
Implementation of Discrete-Time Systems
534
Chap. 7
T o d em onstrate that the set o f eq u ation s (7.3.31) through (7.3.34) represent
an all-pole IIR system , let us con sid er th e case w h ere N = 1. T h e eq u ation s
reduce to
* (n ) = /i( n )
fain) = h i n ) - K\goin - 1)
g i(n ) = ATi/o(n) + go(n - 1)
(7.3.35)
?(«) = foin)
= xin ) - K \yin - 1 )
Furtherm ore, the eq u ation for g i(/i) can be exp ressed as
£ i(n ) = ATi^n) + y in - 1)
(7.3.36)
W e observe that (7.3.35) represents a first-order all-p ole IIR system w hile (7.3.36)
represents a first-order F IR system . T h e p o le is a result o f the feed b ack introduced
by the solution o f th e ( / m(n)} in descen din g order. T his feed b ack is dep icted in
Fig. 7.25a.
Forward
(a)
Forward
Reverse
(b)
Figure 125
Single-pole and two-pole lattice system.
Sec. 7.3
535
Structures for IIR Systems
N ex t, let us con sid er the case N = 2, which corresponds to the structure in
Fig. 7.25b. T h e eq u ation s corresponding to this structure are
f 2(n) = x (n)
f i ( n ) = f 2(n) - K 2g i(n - 1)
g2(n) = K 2f i ( n ) + g i(n -
1
)
(7.3.37)
/o (n ) = / i ( n ) - K igo in - 1 )
gi (n) = K ifo (n ) + g o ( n - l )
y (n ) = /o (n ) = go(«)
A fter so m e sim ple substitutions and m anipulations w e obtain
y (n ) =
+ K 2)y (n — 1) - K 2y (n - 2) + x ( n )
g2(n) = K 2v(n ) + K XQ + K 2)y (n - 1) + v(n - 2)
(7.3.38)
(7.3.39)
C learly, the differen ce equation in (7.3.38) represents a tw o-p ole IIR system , and
the relation in (7.3.39) is the in p u t-ou tp u t equation for a tw o-zero F IR system .
N o te that the coefficients for the F IR system are identical to th ose in the IIR
system excep t that they occu r in reverse order.
In general, th ese con clu sion s hold for any N . Indeed, with the definition of
Am( i) given in (7.2.32). the system function for the all-p ole IIR system is
„ , ,
Y (z)
Fo(z)
1
Ha (z) = -------= ----------= ---------X (z )
Fm(z)
Am( z)
7 , ,,
(7.3.40)
Sim ilarly, the system function o f the all-zero (F IR ) system is
H b(z) =
Y (z )
G 0(Z)
= B m(z) = z~ mA m( z ~ l )
(7.3.41)
w h ere w e used the p reviously established relationships in (7.2.36) through (7.2.42).
Thus the coefficients in the F IR system H b(z) are identical to the coefficients in
Am(z), excep t that they occur in reverse order.
It is interesting to n o te that the all-p ole lattice structure has an all-zero path
with input go(n) and output g x ( n ), w hich is identical to its counterpart all-zero
path in th e all-zero lattice structure. T h e polynom ial Bm (z), w hich represents the
system function o f the all-zero path com m on to b oth lattice structures, is usually
called the b a c k w a rd sy stem fu n c tio n , b ecau se it provid es a backward path in the
a ll-p ole lattice structure.
From this discussion th e reader should ob serve that the all-zero and all-pole
lattice structures are characterized by th e sam e set o f lattice param eters, nam ely,
K \, K 2, . . K n . T h e tw o lattice structures differ on ly in the in tercon n ection s o f
their signal flow graphs. C on sequ en tly, the algorithm s for con vertin g b etw een the
system param eters (a m(fc)} in th e direct form realization o f an F IR system , and the
p aram eters o f its lattice counterpart apply as w ell to the all-p ole structure.
536
implementation of Discrete-Time Systems
Chap. 7
W e recall that the roots o f the polynom ial A N (z) lie in sid e th e unit circle if
and o n ly if the lattice param eters | ^ m| < 1 for all m = 1, 2 .........N . T h erefore, the
all-p o le lattice structure is a stable system if and on ly if its param eters \K m\ < 1
for all m.
In practical applications the all-pole lattice structure has b een used to m odel
the hum an vocal tract and a stratified earth. In such cases the lattice param eters,
{K m) have the physical significance o f b ein g identical to reflection coefficients in
the physical m edium . T his is the reason that the lattice param eters are o ften called
reflection coefficients. In such applications, a stable m od el o f the m edium requires
that the reflection coefficients, obtained by perform ing m easu rem en ts o n output
signals from the m edium , b e less than unity.
T h e all-p ole lattice provides the basic b uilding block for lattice-type structures
that im plem en t IIR system s that contain both p oles and zeros. T o d evelop the
appropriate structure, let us consider an IIR system with system function
M
H (z ) = -------------------------=
A
,
1 + J 2 a N(k)z ~ L
k= 1
* n (z )
(1 3 A 2 )
w here the notation for the num erator polynom ial has b een changed to avoid con­
fusion with our previous develop m en t. W ithout loss o f generality, w e assum e that
N > M.
In the direct form II structure, the system in (7.3.42) is described by the
difference equations
N
w (n ) = — ^ a tf(fc )u > (n — k) + x (n )
*=i
M
y (n ) = £ c M(fc)w(n - k)
*=o
(7.3.43)
(7.3.44)
N o te that (7.3.43) is the in p ut-ou tp u t o f an all-p ole IIR system and that (7.3.44) is
the in p u t-o u tp u t o f an all-zero system . Furtherm ore, w e ob serve that the output of
the all-zero system is sim ply a linear com bination o f delayed ou tp u ts from the all­
p o le system . This is easily seen by observing th e direct form II structure redrawn
as in Fig. 7.26.
Since zeros result from form ing a linear com bination o f previous outputs we
can carry o ver this observation to construct a p o le -z e r o IIR system using the all­
p ole lattice structure as the basic building block. W e have already observed that
gm(n ) is a linear com bination o f present and past outputs. In fact, the system
Hh(z) =
= Bm(z)
Y (z)
Sec. 7.3
537
Structures for IIR Systems
0>
^ - 0 - ------0 -------0
w (n )
w(n - 1 )
w(n - 2)
w(n - Af + 1) ------
w ( n - M)
t - M ------------- ------
e« 0 )
c*<2 )
cu ( M -
1)
c^M)
yirt)
0 —
- 0 - -------0 ------- 0 ^
Figure 7.26 D irect form II realization of IIR system.
is an all-zero system . T h erefore, any linear com bination o f {gm(n)} is also an
all-zero system .
Thus w e begin with an all-p ole lattice structure with param eters K m, 1 <
m < N , and w e add a la dder part by taking as the output a w eigh ted linear
com bination o f (£ * (n )}. T h e result is a p o le-zero IIR system w hich has the latticela d d e r structure show n in Fig. 7.27 for M = N . Its output is
M
(7.3.45)
msrO
w here (um) are the param eters that determ ine the zeros o f the system . T h e system
Figure 7.27
Lattice-ladder structure for the realization o f a p ole-zero system.
Implementation of Discrete-Time Systems
538
Chap. 7
function corresponding to (7.3.45) is
H(z) =
Y (z )
X (z)
(7.3.46)
G m(z)
-Z> X { z )
Since X ( z ) = F N(z) and fb (z) = G q( z ), (7.3.46) can be written as
rr/ ^
H(z) =
G m{z) Fq( z )
> u m— — — 7 ^
G 0 (z) FN (z)
Bm{z)
a n (z )
(7.3.47)
M
y , vmBm{z)
A \(z)
If w e com pare (7.3.41) with (7.3.47), w e con clu d e that
M
C M(z) = ' £ / vmBm(z)
m=0
(7.3.48)
T his is the desired relationship that can be used to determ in e the w eighting coef­
ficients {um). Thus, w e have dem onstrated that the coefficients o f the num erator
polyn om ial C*f(z) determ in e the ladder param eters {um}, w h ereas the coefficients
in the d en om in ator polyn om ial A ^(z) determ ine the lattice p aram eters {Km\.
G iven the polyn om ials C « (z ) and A n ( z ), w here N > M , th e param eters of
the all-pole lattice are d eterm ined first, as described p reviou sly, by the conver­
sion algorithm given in Section 7.2.4, which con verts the direct form coefficients
in to lattice param eters. B y m ean s o f the step -d ow n recursive relations given by
(7.2.54), w e obtain the lattice param eters {Km} and th e p olyn om ials Bm(z), m = 1,
2 ,..., N.
T h e ladder p aram eters are d eterm in ed from (7.3.48), w hich can be expressed
as
TO—1
C m( z) = £ vk B k(z) + vmB m(z)
(7.3.49)
or, equivalently, as
C m(z) = Cm_ ,(z ) + vmB m( z)
(7.3.50)
Thus Cm(z) can b e com p uted recursively from the reverse p olyn om ials Bm(z), m =
1 , 2 , . . . , M . Since
— 1 for all m, th e param eters vm, m = 0, 1 , . . . , M can be
determ ined by first noting that
vm = cm(m)
m = 0,1,..., M
(7.3.51)
Sec. 7.4
State-Space System Analysis and Structures
539
T h en , by rew riting (7.3.50) as
Cm- \ (z) = Cm (z ) - iv, Bm(z)
(7.3.52)
and running this recursive relation backward in m (i.e., m = M , M — 1 , . , , , 2 ), w e
o b tain cm(m ) and therefore the ladder param eters according to (7.3.51).
T h e lattice-ladder filter structures that w e h ave p resen ted require the m in­
im um am ount o f m em ory but not the m inim um num ber o f m ultiplications. A l­
thou gh lattice structures w ith only on e m ultiplier per lattice stage exist, the tw o
m ultiplier-per-stage lattice that w e have described, is by far th e m ost w idely used in
practical applications. In con clu sion , th e m odularity, the built-in stability charac­
teristics em b o d ied in the coefficients {ATm}, and its robustness to finite-w ord-length
effe cts m ake th e lattice structure very attractive in m any practical applications,
including sp eech processing system s, adaptive filtering, and geophysical signal pro­
cessing.
7.4 STATE-SPACE SYSTEM ANALYSIS AND STRUCTURES
U p to this point our treatm ent o f linear tim e-invariant system s has b een lim ited
to an in p u t-o u tp u t or external description of the characteristics o f the system . In
other w ords, the system was characterized by m athem atical eq u ation s that relate
the input signal to the output signal. In this section w e introduce the basic concepts
in the state-sp ace description o f linear tim e-invariant causal system s. A lth ou gh the
state-space or in tern a l description of the system still in volves a relationship betw een
the input and output signals, it also in volves an additional set o f variables, called
state variables. Furtherm ore, the m athem atical eq u ation s describing the system ,
its input, and its output are usually divided into tw o parts:
1. A set o f m athem atical eq u ation s relating th e state variables to the input
signal.
2. A seco n d set o f m athem atical eq u ation s relating the state variables and the
current input to the output signal.
T h e state variables provide inform ation ab ou t all th e internal signals in the
system . A s a result, the state-sp ace description provides a m ore d etailed descrip­
tion o f the system than the in p u t-ou tp u t description. A lth o u g h our treatm ent o f
state-sp ace analysis is confined prim arily to sin gle in p u t-sin g le output linear tim einvariant causal system s, the state-sp ace tech n iqu es can also b e applied to non­
linear system s, tim e-variant system s, and m ultip le in p u t-m u ltip le ou tp u t system s.
In fact, it is in the characterization and analysis o f m ultip le in p u t-m u ltip le output
system s that th e p ow er and im portance o f state-sp ace m eth od s are clearly evident.
B o th in p u t-o u tp u t and state-variable descriptions o f a system are useful in
practice. T h e description w e use d ep en d s on the p rob lem , the available inform a­
tion, and the q u estion s to b e answ ered. In our p resen tation , the em phasis is on
540
Implementation of Discrete-Time Systems
Chap. 7
the use o f state-sp ace tech n iqu es in system analysis, and in the d evelop m en t o f
state-sp ace structures for th e realization o f discrete-tim e system s.
7.4.1 S ta te -S p a ce D e sc rip tio n s o f S y s te m s C haracterized
by D ifference E qu ations
A s w e have already o b served , the determ ination o f the output o f a system requires
that w e know the input signal and the set o f initial con d ition s at th e tim e the input
is applied. If a system is n ot relaxed initially, say at tim e no, th en k n ow led ge o f
the input signal x ( n ) for n > n Q is not sufficient to un iqu ely d eterm in e the output
y(n) for n > no- T h e initial con d ition s o f the system at n = no m ust also b e know n
and taken in to account. T his set o f initial con d ition s is called th e state o f the
system at n = no- H en ce we defin e the state o f a sy ste m a t tim e no a s the a m o u n t o f
in fo rm a tio n that m u s t be p r o v id e d at tim e n0, w hich, together w ith th e in p u t signal
x ( n ) f o r n > no, u n iq u e ly d eterm in e the o u tp u t o f th e sy stem f o r a ll n > noFrom this definition w e infer that the con cep t o f state lead s to a d ecom p o­
sition o f a system in to tw o parts, a part that contains m em ory, and a m em oryless
com p onent. T he inform ation stored in the m em ory co m p on en t con stitu tes the set
o f initial conditions and is called the state o f the system . T h e current ou tp u t o f the
system then b eco m es a function o f the current value o f the input and the current
state. T hus, to d eterm in e the output o f the system at a given tim e, w e n eed the
current value o f the state and the current input. Since the current value o f the
input is available, w e only n eed to provide a m echanism for upd atin g the state o f
the system recursively. C on sequ en tly, the state o f the system at tim e no + 1 should
d epend on the state o f the system at tim e n 0 and the value o f th e input signal x (n)
at n = noT h e follow in g exam p le illustrates the approach in form ulating a state-space
description o f a system . L et us con sid er a linear tim e-invariant causal system
described by the d ifferen ce equation
3
3
(7.4.1)
T h e direct form II realization for the system is show n in Fig. 7.28.
A s state variables, w e u se the con ten ts o f the system m em ory registers, cou n t­
ing them from the b o ttom , as show n in Fig. 7.28. W e recall that th e output o f a
delay elem en t rep resen ts the present value stored in th e register and the input
represents the next v alu e to b e stored in th e m em ory. C on seq u en tly, with the aid
o f Fig. 7.28, w e can w rite
t>i(n + 1 ) = V2 (n)
v i (n + 1 ) = U3 («)
v$(n + 1 ) = - a 3U i(n ) - 0 2 « 2 (n ) - o m 3( n ) + x(n)
(7.4.2)
Sec. 7.4
541
State-Space System Analysis and Structures
<*>
o
-
©
-« l
v2(«)
0
■6
-
U,(«)
"« 3
3
Figure 7.28 Direct form II realization of system described by the difference equa­
tion in (7.5.1).
It is in terestin g to n o te that the state-variable form ulation for the third-order
system o f (7.4.1) in v o lv es three first-order difference eq u ation s given by (7.4.2). In
general, an n th-order system can be described by n first-order differen ce equations.
T h e ou tp u t eq u ation , w hich exp resses y ( n ) in term s o f the state variables and
the presen t input v alu e x (n ), can also be ob tain ed by referring to Fig. 7.28. W e have
y (n ) = i>oVi(n + 1 ) + />3 Ui(n) + b2v2 (n) + 6 it>3 (/i)
W e can elim in a te v$(n + 1 ) by using the last equation in (7.4.2). Thus w e obtain
the desired ou tp u t eq u ation
y (n ) = (f >3 - boa2) v i (n ) + (£ 2 — boa2) v2(n) + (b\ - i>o<ai)u3 (n) + b0x ( n )
(7.4.3)
If w e put (7.4.2) and (7.4.3) in to m atrix form w e have
’ V[(n + 1 )-
V2 (n + 1) =
_ v 3 (n + 1 ) .
■ ‘ t>] ( n ) -
0
1
0
0
0
.-0 3
-a 2
1
-a \.
-
V2 (n)
-i> 3 (n )-
and
y ( n ) = [(b3 - b0a 3) ( ^ - M 2 ) (i>i ~ M l ) ]
-0 0
+
x(n)
(7.4.4)
.1 .
~ v i(n ) ’
v2(n) ■+■box(n)
(7.4.5)
-V 3 ( * ) -
T h e eq u a tio n s (7.4.4) and (7.4.5) provide a com p lete description o f the sys­
tem . F urtherm ore, th e variables v i(n ), V2 (n), and 113( 0 ), w hich sum m arize all the
n ecessary past inform ation, are the state variables o f the system . W e also observe
that as in d icated previou sly, eq u ation s (7.4.4) and (7.4.5) split the system in to tw o
co m p o n en t parts, a dynam ic (m em ory) subsystem and a static (m em oryless) sub­
system . W e say that this se t o f eq u ation s provides a state-space description o f the
system .
Implementation of Discrete-Time Systems
542
Chap. 7
B y generalizing the previous exam p le, it can easily be se en that the A'th-order
system described by
N
Af
,( « ) = - £ a ty(n ~
*=i
bkX^n ~ k)
+
(7.4.6)
Jt=0
can be expressed as a linear tim e-invariant state-space realization by the relations
State equation
v(n -j- 1 ) = Fv(n) + q x(n )
(7.4.7)
>■(«) = gf v ( n ) + d x ( n )
(7.4.8)
Output equation
w here the elem en ts o f F, q, g, and d are con stan ts (i.e., they d o n ot change as a
function o f the tim e index n ), given by
•
-
“
0
1
0
•
0
0
1
0
0
0
0
1
.-a s
~a„ -1
-a2
~a\ .
-
0
"0 "
0
0
q=
F =
(7.4.9)
0
b N — boaN
b fj - 1 — bo a n-\
g=
b\ — boa\
A n y discrete-tim e system w h ose input x (n ), ou tp u t y( n ) , and state v{n), for
all n > no, are related by the state-sp ace eq u ation s ab ove, w h ere F, q, g, and d are
arbitrary but fixed quantities, will be called linear and tim e invariant. If at least
o n e o f the quantities in F, q, g, or d d ep en d s on tim e, the system b ecom es time
variant.
W e will refer to (7.4.7) through (7.4.8) as the linear tim e-in va rian t state-space
m o d el, which can b e represented by the sim ple vector-m atrix block diagram in
Fig. 7.29. In this figure the d ou b le lin es represent vector qu an tities and the blocks
represent the v ector or m atrix coefficients.
Example 7.4.1
Determine the state-space equations for the transposed direct form II structure shown
in Fig. 7.30.
Solution
The validity of this structure can be seen if we rewrite (7.4.1) as
y(n) =
- k) - aky(n - * )] + box(n)
Sec. 7.4
State-Space System Analysis and Structures
543
Figure 7.29 General state-space description of a linear time-invariant system.
*(«)
Figure 7JO State-space realization for the system described by (7.4.1).
Due to the linearity and time invariance of the system, instead of first delaying the
signals x(n) and y(n) and then computing the terms bkx(n - k) — aky(n — k) as in
Fig. 7.28, we first compute the terms bkx(n) — at y(n) and then delay them.
If we use the state variables indicated in Fig. 7.30, we obtain
'v i(n + 1 )"
Vi (n + 1 ) =
. v3(n + 1 ) .
‘0
0
1
0
.0
1
y(n) = [0
0
’ 'M n )‘
‘ bi —boaj "
-a 2
i>2 (n) = * 2 - M 2 x(n)
-b\ — bod\ .
- a i . .t>B(n).
— 03
(7.4.10)
Di(n)
1 ]
l> 2 (fl)
-f box(n)
(7.4.11)
V3(n) J
T h e state-sp ace description specified by (7.4.4) and (7.4.5) is know n as a ty p e
1 state-sp ace realization, w h ereas the o n e described by (7.4.10) and (7.4.11) is
ca lled a typ e 2 state-sp ace realization.
7.4 .2 S o lu tio n o f th e S ta te -S p a ce E q u ation s
T h ere are several m eth od s for solvin g the state-sp ace eq u ation s. H ere w e discuss
a recursive solu tion w hich m akes u se o f th e fact that the state-sp ace eq u ation s are
a se t o f linear first-order differen ce equations.
544
Implementation of Discrete-Time Systems
Chap. 7
For the jV-dim ensional state-sp ace m odel
v(« + 1) = F v(/i) + qjr(n)
(7.4.12)
y(n) = g'v(n) + d x(n )
(7.4.13)
and given the initial co n d ition v (« 0), we have for n > n0,
v(n 0 + 1) = Fv(n0) + q*(n)
v(no + 2 ) = F v(n 0 + 1 ) + qjr(n0 + 1 )
= F 2 v(n0) + F q j(« o ) + q-^(«o + 1)
w here F 2 represents the matrix product FF and Fq is the product o f the m atrix F
and the vector q. If w e continue as in the on e-dim en sion al case, w e obtain, for
n > n0,
(7.4.14)
The matrix F° is defined as the N x N identity matrix, having unity on the
main diagonal and zeros elsew h ere. T he m atrix F'- -' is often d en o te d as 4>(/ - j ) ,
that is,
* (/ - j) = F ~ J
(7.4.15)
for any p ositive in tegers i > j . T his m atrix is called the state transition matrix of
the system .
T h e output o f the system is obtained by substituting (7.4.14) in to (7.4.13).
T he result o f this substitution is
y(n) = g'F" nov(n0) + £
g'F" 1 kq x ( k ) + d x { n )
*=«0
(7.4.16)
n- 1
=
- floM no) + ^
2
-
1 - &)q*(*) + d x ( n )
From this general result, w e can determ ine the output for tw o special cases.
First, the zero-input resp onse o f the system is
yzM ) = g'F" n“v(n0) = g '$ ( n - floM no)
(7.4.17)
O n the other hand, the zero-state response is
n—1
yzs(n) =
g' $ { n - 1 - fc)q*(/:) + d x ( n )
(7.4.18)
Clearly, the A'-dim ensional state-sp ace system is zero-in p u t linear, zero-state
linear, and since y ( n ) = y Zi(n) + y zs(n), it is linear. F urtherm ore, sin ce any system
described by a linear con stan t-coefficien t d ifferen ce eq u ation can be put in the
state-space form, it is linear, in agreem en t with the results ob tain ed in S ection 2.4.
Sec. 7.4
State-Space System Analysis and Structures
545
7.4.3 Relationships Between Input-Output and
State-Space Descriptions
From our p reviou s discussion w e h ave seen that there is n o unique ch oice for the
state variables o f a causal system . Furtherm ore, d ifferent ch o ices for th e state
vecto r lead to differen t structures for the realization o f the sam e system . H en ce,
in g en eral, the in p u t-o u tp u t relationship d oes not un iqu ely d escrib e the internal
structure o f th e system .
T o illu strate these assertions, let us consider an N -dim en sion al system with
the sta te-sp a ce rep resen tation
v(n + 1) = Fv(rt) 4- qjf(n)
(7.4.19)
y(n ) = g'v(n) -1- d x ( n )
(7.4.20)
L et P b e any N x N matrix w h ose inverse m atrix P-1 exists. W e d efine a new
state v ecto r v(n) as
(7.4.21)
(7.4.22)
(7.4.23)
(7.4.24)
N o w , w e d efine a n ew system param eter matrix F and the vectors q and g as
F = PFP-1
(7.4.25)
W ith th ese d efinition s, th e state eq u ation s can b e exp ressed in term s o f th e new
system qu an tities as
v(rt 4 1) — Fv(n) 4 - q *(n )
(7.4.26)
y ( n ) = §fv(n) -j- d x ( n )
(7.4.27)
If w e com p are (7.4.19) and (7.4.20) with (7.4.26) and (7.4.27), w e ob serve
that by a sim p le linear transform ation o f the state variables, w e have gen erated
a n ew set o f sta te eq u ation s and an output eq u ation , in w hich th e input x ( n ) and
the ou tp u t y ( n ) are unchanged. Since there is an infinite n um ber o f ch oices o f the
transform ation m atrix P, there is also an infinite num ber o f state-sp ace eq u ation s
Implementation of Discrete-Time Systems
546
Chap. 7
and structures for a system . S om e o f these structures are different, w hile som e
others are very sim ilar, differing o n ly by scale factors.
A ssociated with any state-sp ace realization o f a system is the con cep t o f a
m i n i m a l realization. A state-sp ace realization is said to b e m i n i m a l if the d im ension
o f the state sp a ce (th e num ber o f state variab les) is th e sm allest o f all p ossible
realizations. Since each state variable rep resen ts a quantity that m ust be stored
and updated at every tim e instant n , it fo llo w s that a m inim al realization is o n e
that requires the sm allest num ber o f d elays (storage registers). W e recall that the
direct form II realization requires the sm allest num ber o f storages registers, and
con sequ en tly, a state-sp ace realization based on the con ten ts o f the d elay elem en ts
results in a m inim al realization. Sim ilarly, an F IR system realized as a direct form
structure leads to a m inim al state-sp ace realization if th e valu es o f the storage
registers are defined as the state variables. O n the other hand, the direct form I
realization o f an IIR system d oes n ot lead to a m inim al realization.
N o w , let us determ in e the im pulse resp on se o f the system from the statespace realization. T he im pulse resp onse provid es o n e o f the links betw een the
in p u t-o u tp u t and state-space description o f system s.
B y definition the im pulse resp onse h ( n ) o f a system is the zero-state re­
sponse o f the system to the excitation x ( n ) = 6 (n). H e n c e it can be obtained from
equation (7.4.16) if w e set no = 0 (th e tim e w e apply th e input), v(0) = 0, and
x ( n ) — S(n). T hus the im pulse resp onse o f the system describ ed by (7.4.19) and
(7.4.20) is given by
h (n) — g, Fn~ 1 q«(n — 1) + d 8(n)
(7.4.28)
= g '$ ( n — l)qw (n — 1 ) + d 8 ( n )
G iven a state-sp ace description, it is straightforward to d eterm in e the im pulse re­
sp on se from (7.4.28). H ow ever, the inverse is n ot easy since there is an infinite
number o f state-sp ace realizations for the sam e in p u t-o u tp u t description.
The transpose system. T h e transpose o f a m atrix F is ob tain ed by inter­
changing its colum ns and rows, and it is d en oted by F . F or exam p le,
■ f\N '
r /n
/12
•
/21
/2 2
• ■
-/v 1
fs2
f2N
• fsN -
F* =
r /11
/21
/12
/2 2
• •
• ■
/V 2
-flN
flN
• •
f NN -
fm '
N o w define the tra nspose s y s t e m (7 .4 .1 9 )-(7 .4.20) as
v'tn + 1) = F v * (rt) + g x ( n )
(7.4.29)
y '( n ) = q V (n ) - \-d x ( n )
(7.4.30)
A ccording to (7.4.28), the im pulse resp on se o f this sy stem is given as
t i ( n ) = qr(F')"-1 g«(n - 1) + d S ( n )
(7.4.31)
Sec. 7.4
State-Space System Analysis and Structures
547
From m atrix algebra w e know that (F )" 1 = ( F -1 )'- H en ce
h '( n ) ~ q ' ( F _ 1 ), gw(n - 1 ) + d5(n)
W e claim that h'(n) = h(n). In d eed , the term q ' ( F -1 )'g is a scalar. H en ce it
is eq u al to its transpose. C onsequently,
[ q ' ( F - 1 )'g]' = g'(F y - ' q
S ince this is true, it fo llow s that (7.4.31) is identical to (7.4.28) and, therefore,
h '(n) = h (n). T hus a single in p ut-sin g le o u t p u t sy stem a n d its trans po se have i d e n ­
tical im p u lse respons es a n d h en ce the s a m e i n p u t - o u t p u t relationship. T o support
this claim further, w e n ote that the type 1 and type 2 state-sp ace realizations,
d escribed by (7.4.3), (7.4.4), (7.4.10), and (7.4.11) are transpose structures, which
stem from the sam e in p ut-ou tp u t relationship (7.4.1).
W e have introduced the transpose structure b ecau se it provides an easy
m eth o d for generating a new structure. H ow ever, som etim es this new structure
m ay either differ trivially or be identical to the original on e.
The diagonal system. A closed -form solu tion o f the state-space equations
is easily ob tain ed w hen the system m atrix F is diagonal. H en ce, by finding a m atrix
P so that F = P F P - 1 is diagonal, the solu tion of the state eq u ation s is sim plified
considerably. T h e d iagonalization o f the m atrix F can be accom plished by first
d eterm ining the eigen valu es and eigen vectors o f the matrix.
A num ber A is an eigenvalue o f F and a n on zero vector u is the associated
eigen vecto r if
Fu = Au
(7.4.32)
T o determ ine the eigen valu es o f F, w e n ote that
(F - Xl)u = 0
(7.4.33)
T his eq u ation has a (nontrivial) non zero solu tion u if the m atrix F - XI is singular
[i.e., if (F — Al) is n oninvertible], which is the case if the determ inant o f (F — /.I)
is zero, that is, if
d et (F - XI) = 0
(7.4.34)
This determ inant in (7.4.34) yield s the characteristic p o l y n o m i a l o f the m atrix
F. For a n N x N m atrix F, the characteristic polyn om ial o f F is degree N and h en ce
it has jV roots, say X,, i = 1, 2 .........N . T h e roots m ay b e distinct or som e roots
m ay b e repeated. In any case, for each root A,, w e can d eterm ine a vector u (,
called the eigen vector corresponding to the eigen valu e X,, from the equation
FU; = X(U;
T h ese eigen vectors are orthogonal, that is, uju, = 0, for i ^ j .
If w e form a m atrix U w h ose colum ns consist o f the eigen vectors {u, }, that is,
U =
f t
t
Ul
U2
- 1 1
t ■
•••
UjV
i J
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Implementation of Discrete-Time Systems
Chap. 7
then the m atrix F = U -1 F U is diagonal. Thus w e have solved for the m atrix that
diagonalizes F.
T h e follow in g exam ple illustrates the procedure o f d iagon alizin g F.
Example 7.4.2
The Fibonacci sequence, which is the sequence {1,1,2, 3, 5 ,8 .1 3 ....} , can be gener­
ated as the impulse response of the system that satisfies the state-space equations
v ( « + 1) =
j j v < n ) + j ^ J .x ( n )
y(rt) = [ 1
1 ] v ( n ) + x(n)
Determine the impulse response {/»(«)} of the system.
Solution
Now we wish to determine an equivalent system
v(n + 1) = Fv(n) + qx(n)
y(n) = g'v(n) + dx(n)
such that the matrix F is diagonal. From (7.4.25) we recall that the two systems are
equivalent if
F = PFP“'
q = P?
g '= g 'P " 1
Given F. the problem is to determine a matrix P such that F = PFP - 1 is a diagonal
matrix.
First, we compute the determinant in (7.4.34). We have
d e t(F -A l) = d e t [ “ j
—X — 1 = 0
or
1 + V5
2—
x, = -
1 - Vs
X2 = - 1 -
To find the eigenvector U] corresponding to A.], we have
[?
°r »'=[i]
Similarly, we obtain
We observe that ur,u 2 = 1 +
= 0 (i.e., the eigenvectors are orthogonal). Now
matrix U, whose columns are the eigenvectors of F, is
Then the matrix U -1FU is diagonal. Indeed, it easily follows that
Xj
0 1
Sec. 7.4
State-Space System Analysis and Structures
549
and since the transformation matrix is P = U -1, we have
p =
__ !__ r ^
-M
A.2 —A.i L—A-i
1 J
Thus the diagonal matrix F has the form
M o':]
where the diagonal elements are the eigenvalues of the characteristic polynomial.
Furthermore, we obtain
_1_
vl
q = Pq =
L 'V 5 J
and
r =
'3 + V s 3 - V 5
2
2
The impulse response of this equivalent diagonal system is
fi(n) = g'Fqu(n - 1) +d&(n)
(^K^y
m
u(n - 1 ) + 5(n)
m
which is the general formula for the Fibonacci sequence.
An alternative expression can be found by noting that the Fibonacci sequence
can be considered as the zero-input response of the system described by the difference
equation
y(n) = y(n - 1 ) + y(n - 2 ) + x (n)
with initial conditions j><—1) = 1, y (—2) = —1. From the type 1 state-space realization,
we note that U](0) = y { - 2 ) = - 1 and t^(0) = ;y (-l) = 1. Hence
r * (° n
Lt>2 (0 ) J
p r ^ n
Lv2 <0 )J
zi
5
r - 3 + V5 n
2
3 + VS
and the zero-input response is
y n(") = r ^ ^ ( 0 )
(»)
This is the more familiar form for the Fibonacci sequence, where the first term of the
sequence is zero, that is, the sequences is {0 , 1 , 1 , 2 , 3 , 5 , 8 , . . .)•
550
Implementation of Discrete-Time Systems
Chap. 7
This exam ple illustrates the m eth od for diagonalizing the matrix F. The
diagonal system y ield s a set o f N d ecou p led , first-order linear d ifferen ce equations
that are easily solved to yield the state and the ou tp u t o f the system .
It is im portant to n ote that the eigen valu es o f the matrix F are identical to the
roots o f the characteristic polynom ial, which are ob tain ed from th e h om ogen eou s
d ifferen ce eq u ation that characterizes the system . F or exam p le, the system that
g en erates the F ibonacci seq u en ce is characterized by the h o m o g e n e o u s difference
equation
y(n) - y(n - I) - y{n - 2) = 0
(7.4.35)
R ecall that the solu tion is ob tain ed by assum ing that the h o m o g e n e o u s solution
has the form
yh(n) = kn
Substitution o f this solu tion into (7.4.35) yield s the characteristic polynom ial
A2 - / - 1 = 0
B ut this is exactly the sam e characteristic polynom ial ob tain ed from the determ i­
nant o f (F - XI).
Since the state-variable realization o f the system is not u n iq u e, the matrix
F is also not unique. H ow ever, the eigen valu es o f the system are unique, that is,
they are invariant to any nonsingular linear transform ation o f F. C onsequently,
the characteristic polynom ial o f F can be determ in ed eith er from evaluating the
determ inant o f ( F - A l ) or from the d ifferen ce eq u ation characterizing the system .
In conclusion, th e state-sp ace description provides an alternative character­
ization o f the system that is eq u ivalen t to the in p u t-ou tp u t description. O n e ad­
vantage o f the state-variable form ulation is that it provid es us with the additional
inform ation concerning the internal (state) variables o f the system , inform ation
that is not easily ob tain ed from the in p u t-ou tp u t description. Furtherm ore, the
state-variable form ulation o f a linear tim e-invariant system allow s us to represent
the system by a set o f (usually cou p led ) first-order differen ce eq u ation s. T he d e­
coupling o f the eq u ation s can be ach ieved by m eans o f a linear transform ation that
can b e ob tain ed by solvin g for the eigen valu es and eigen vectors o f the system . The
d ecou p led eq u ation s are then relatively sim ple to solve. M ore im portant, how ever,
the state-space form ulation provides a pow erful, yet straightforward m eth od for
d ealing with system s that have m ultiple inputs and m ultiple ou tp u ts (M IM O ). A l­
though w e have n ot con sid ered such system s in our study, it is in the treatm ent of
M IM O system s w h ere the true p ow er and the b eau ty o f the sp ace-sp ace form ula­
tion can be fully appreciated.
7.4.4 State-Space Analysis in the z-Domain
T h e state-sp ace analysis in th e previous section s has b een perform ed in the time
d om ain. H o w ev er, as w e have ob served previously, the analysis o f linear timeinvariant discrete-tim e system s can also b e carried ou t in the z-transform
Sec. 7.4
State-Space System Analysis and Structures
551
d om ain , often w ith greater ease. In this section w e treat the state-sp ace rep­
resen tation o f linear tim e-invariant d iscrete-tim e system s in the z-transform d o ­
main.
L et us con sid er the state-sp ace eq u ation
v(rt + 1) = Fv(n) + <pr(n)
(7.4.36)
If w e define the v ecto r V (z) as
* \(z ) 1
V2(z)
V (z) =
(7.4.37)
LVV(z).
then (7.4.36) can b e ex p ressed in m atrix form as
zV (z) = F V (z) + q * ( z )
(7.4.38)
T h e tw o term s in volving V(z) can be collected togeth er and the resulting equation
can b e used to so lv e for V(z). Thus
(zi - F)V(z) = q*(z)
(7.4.39)
V(z) = ( z I - F r V ( z )
T h e inverse z-transform o f (7.4.39) yield s the solu tion for the state equations.
N ex t, w e turn our atten tion to the output eq u ation , which is given as
j (n ) = g*v (n ) + d x ( n )
(7.4.40)
y ( z ) = g ' V ( z ) + d X (z)
(7.4.41)
T h e z-transform o f (7.4.40) is
B y using the solu tion in (7.4.39) w e can elim in ate th e state vector V (z) in
(7.4.41). Thus w e obtain
y (z ) = [g, ( z I - F ) ~ 1q + d ]X (z)
w hich is the z-transform o f the zero-state resp onse o f th e system .
fun ction is easily ob tain ed from (7.4.42) as
H (z ) = — y = g ' ( z I - F ) - 1q + <f
(7.4.42)
T h e system
(7.4.43)
T h e state eq u ation given by (7.4.39), th e ou tp u t eq u ation given by (7.4.42) and the
system function given by (7.4.43) all h ave in com m on the factor (z i — F )- 1 . This
is a fun d am en tal quantity that is related to the z-transform o f the state transition
m atrix o f th e system . T h e relationship is easily estab lish ed by com puting the
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Implementation of Discrete-Time Systems
Chap. 7
z-transform o f the im pulse resp on se h (n), which is g iv en by (7.4.28). T hus w e
have
00
H (z) = X > (« )z ~ "
n=0
oc
- 1) + d i ( n ) \ z ~ n
=
(7.4.44)
T h e term in p a ren theses in (7.4.44) can b e w ritten as
00
= z - H l + F z ” 1 + F 2 z - 2 + ---)
(7.4.45)
= z -H l-F z " 1)" 1 = ( z I - F ) ' 1
If w e substitute the result in (7