Chapter 2
Discrete-Time Signals and Systems
Content
The Discrete-Time Signal: Sequence
The Discrete-Time System
The Discrete-Time Fourier Transform (DTFT)
The Symmetric Properties of the DTFT
System Function and Frequency Response
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The Discrete-Time Signal: Sequences
 Elementary sequences

Unit sample sequence
1, n  0
 ( n)  
0, n  0
1, n  n0
 ( n  n0 )  
0, n  n0
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The Discrete-Time Signal: Sequences

Unit step sequence
1, n  0
u( n)  
0, n  0
1, n  n0
u( n  n0 )  
0, n  n0
 ( n)  u(n)  u( n  1)

u( n)    ( n  m )
m 0
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The Discrete-Time Signal: Sequences

Rectangular sequence
1, 0  n  N  1
RN ( n )  
otherwise
0,
RN (n)  u(n)  u(n  N )
N 1
RN ( n )    ( n  m )
m 0
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The Discrete-Time Signal: Sequences

Sinusoidal sequence
x(n)  A cos( 0 n   ),
A
0

n
amplitude
digital angular frequency
phase
x1 ( n)  1.5  cos(0.05  2 n)
x2 ( n)  1.5  sin( 0.05  2 n)
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The Discrete-Time Signal: Sequences

Real-valued exponential sequence
x( n)  a , n; a  R
n
The
x (n) is convergent when | a | 1
The
x (n) is divergent when | a | 1
x1 ( n)  0.001  1.2
n
x2 ( n)  0.2  0.8
n
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The Discrete-Time Signal: Sequences

Complex-valued exponential sequence
x ( n)  e
(  j 0 ) n
,
n
n
n
x( n)  e cos  0 n  je sin  0 n
 xre ( n)  jx im ( n)

Attenuation factor
x( n)  2e
1 
(   j )n
5 8
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The Discrete-Time Signal: Sequences
 Classification of sequences

Finite-length sequence
x (n) is defined only for a finite time interval:
where    N 1 , N 2  
N1  n  N 2
examples
x ( n)  n ,  8  n  8
y( n)  cos 0.4n
2
The length of a finite-length sequence can be
increased by zero-padding
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The Discrete-Time Signal: Sequences

Right-sided sequence
x (n) has zero-valued samples for n  N 1
where N 1  
If N 1  0 , a right-sided sequence is called a
causal sequence
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The Discrete-Time Signal: Sequences

Left-sided sequence
x (n) has zero-valued samples for n  N 2
where N 2  
If N 2  0 , a left-sided sequence is called a
anti-causal sequence
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The Discrete-Time Signal: Sequences

Two-sided sequence
x (n) is defined for any n
a dual-sided sequence can be seen as the
sum of a right-sided sequence and a left-sided
sequence.
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The Discrete-Time Signal: Sequences

Absolutely summable sequence

 x ( n)  
n  
Example:
 0.3 n ,
x ( n)  
0,
n0
n0

1
0.3 
 1.42857  

1  0.3
n0
n
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The Discrete-Time Signal: Sequences

Square-summable sequence

 x ( n)
2

n  
Example:
sin 0.3n
x ( n) 
n
It is square-summable but not
absolutely summable
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The Discrete-Time Signal: Sequences
 Operations on sequence

Time-shifting operation
y ( n)  x ( n  N )
where
N is an integer
N  0 delaying operation
Unit delay
x (n)
z-1
y( n)  x( n  1)
z
y( n)  x( n  1)
N  0 advance operation
Unit advance x (n)
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The Discrete-Time Signal: Sequences

Time-reversal (folding) operation
y ( n)  x (  n)

Addition operation
Sample-by-sample addition
Adder
y ( n)  x ( n)  w ( n)
y ( n)  x ( n)  w ( n)
x (n)
w(n)
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The Discrete-Time Signal: Sequences

Scaling operation
y( n)  Ax( n)
x (n)
Multiplier

A
y( n)  Ax( n)
Product (modulation) operation
Sample-by-sample multiplication y( n)  x( n)  w( n)
modulator
y ( n)  x ( n)  w ( n)
x (n)
w(n)
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The Discrete-Time Signal: Sequences

Sample summation
n2
 x ( n)  x ( n )    x ( n )
1
n  n1

2
Sample production
n2

n  n1
x( n)  x( n1 )   x( n2 )
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The Discrete-Time Signal: Sequences

Sequence energy
Ex 



 x ( n) x ( n)   | x ( n) |
*
n  
2
n  
Sequence power
1
Px  lim
N  N
N 1
 | x ( n) |
2
n 0
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The Discrete-Time Signal: Sequences

Decimation by a factor D
Every D-th samples of the input sequence
are kept and others are removed:
xd (n)  x( Dn)
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The Discrete-Time Signal: Sequences

Interpolation by a factor I
I -1 equidistant zeros-valued samples are
inserted between each two consecutive
samples of the input sequence.
 n
x ( ), n  0,  I ,  2 I 
x p ( n)   I
 0,
otherwise
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The Discrete-Time Signal: Sequences
 The periodicity of sequence
if
x( n)  x( n  kN )
k : any integer
N: positive integer
then the x(n) is called a periodic sequence,
and the value of N is called the fundamental
period.
a periodic sequence is usually expressed as
x~ ( n)
x(( n)) N
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The Discrete-Time Signal: Sequences

The periodicity of sinusoidal sequence
x(n)  A cos( 0 n   )
If
x(n  N )  A cos( 0 n   0 N   )
2 N
 0 N  2k or

N , k : any integer
0 k
x (n) is a periodic sequence and its period is
2k
2
N min 
N
0
0
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x(n)  A cos( 0 n   )
 If 2 is a integer
N min 
2
0
0
2
 If
is a noninteger rational number
0
2
Q

0 P
 If
2
0
2
Q
N
k k
0
P
N min  Q
is a irrational number
x (n) is an aperiodic sequence
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The Discrete-Time Signal: Sequences

The periodicity of Complex-valued exponential
sequence
x ( n)  e
n
(  j 0 ) n
n
 e cos  0 n  je sin  0 n
when   0 , the periodicity of Complex-valued
exponential sequence is the same as the
sinusoidal sequence
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The Discrete-Time Signal: Sequences

The periodicity of sinusoidal sequence which is developed
by uniformly sampling a continuous-time sinusoidal signal
x(t )  A cos(0 t   )
 0 Analog angular frequency
x( n)  x( t ) t  nT  A cos( 0 nT   )
2 0
 A cos(
n )
T
 A cos( 0 n   )
T
Sampling period
T Sampling angular frequency
 0 Digital angular frequency
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The Discrete-Time Signal: Sequences
f0
1
 0   0T   0  2
fs
fs
Units:
Sampling period
Analog frequency
T:
f0 :
seconds/sample
hertz (Hz)
Analog angular frequency  0 : radians/second
Digital angular frequency  0 : radians/sample
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The Discrete-Time Signal: Sequences
The periodicity:
2
T0
1
1
1
 2 
 2 


0
 0T
2 f 0T f 0T T
T0
T
The period of the continuous-time sinusoidal signal
The sampling period
2
2
Q

If
is a rational number, then
0
0 P
QT  PT0
Q, P
are positive integers
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The Discrete-Time Signal: Sequences
 Sequence synthesis

Unit sample synthesis
Any arbitrary sequence can be synthesized in the timedomain as a weighted sum of delayed (advanced) and
scaled unit sample sequence.
x ( n) 

 x(k ) (n  k )
k  
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The Discrete-Time Signal: Sequences

Even and odd synthesis
Even (symmetric): xe (  n)  xe ( n)
Odd (antisymmetric): xo ( n)   xo ( n)
Any arbitrary real-valued sequence can be
decomposed into its even and odd component:
x ( n)  x e ( n )  x o ( n)
1
xe ( n)  [ x( n)  x(  n)]
2
1
xo ( n)  [ x( n)  x(  n)]
2
return
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The Discrete-Time System
 Introduction
A discrete-time system processes a given input
sequence x(n) to generate an output sequence y(n)
with more desirable properties.
Mathematically, an operation T [ • ] is used.
y(n) = T [ x(n) ]
x(n): excitation, input signal
y(n): response, output signal
example
example
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The Discrete-Time System
 Classification

Linear System

Time-Invariant (Shift-Invariant) System

Linear Time-Invariant (LTI) System

Causal System

Stable System
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The Discrete-Time System
 Linear System
A system is called linear if it has two mathematical
properties: homogeneity and additivity.
T [ax( n)]  aT [ x( n)]
T[ x1 (n)  x2 (n)]  T[ x1 (n)]  T[ x2 (n)]
T[a1 x1 (n)  a2 x2 (n)]  a1T[ x1 (n)]  a2T[ x2 (n)]
Accumulator
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The Discrete-Time System
 Time-Invariant (Shift-Invariant) System
if T [ x( n)]  y( n)
then T [ x( n  n0 )]  y( n  n0 )
Accumulator
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The Discrete-Time System
 Linear Time-Invariant (LTI) System
A system satisfying both the linearity and the timeinvariance properties is called an LTI system.
LTI systems are mathematically easy to analyze and
characterize, and consequently, easy to design.
A accumulator is an LTI system !
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The Discrete-Time System
The output of an LTI system is called
linear convolution sum
y( n)  LTI[ x( n)] 


 x(k )h(n  k ) x(n) * h(n)
k  
An LTI system is completely characterized in the time
domain by the impulse response h(n).
example
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The Discrete-Time System
 Causal System
In a causal system, the n0 -th output sample
depends only on input samples x (n) for n  n0 and
does not depend on input samples for n  n0
e.g.
y(n)  a1 x(n)  a2 x(n  1)  a3 x(n  2)
For a causal system, changes in output samples
do not precede changes in the input samples.
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The Discrete-Time System
An LTI system will be a causal system if and only if :
h(n)  0,
n0
An ideal low-pass filter is not a causal system !
A sequence
x (n) is called a causal sequence if :
x(n)  0,
n0
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The Discrete-Time System
 Stable System
A system is said to be bounded-input bounded-output
(BIBO) stable if every bounded input produces a
bounded output, i.e.
if
x(n)  M  , then y(n)  P  
An LTI system will be a stable system if and only if :
S

 h(n)  
n  
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The Discrete-Time System
 The M-point moving average filter is BIBO stable :
1
y( n) 
M
M 1
 x( n  k )
k 0
prove
 A causal LTI discrete-time system:
h(n)   n u(n)
prove
if |  | 1, the system is stable
if |  | 1, the system is not stable
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The Discrete-Time System
 Causal and Stable System
A system is said to be a causal and stable system if
the impulse response h(n) is causal and absolutely
summable , i.e.
h( n)  h( n)u( n)

 h(n)  
n  
return
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The Discrete-time Fourier Transform (DTFT)
 The transform-domain representation of
discrete-time signal
● Discrete-Time Fourier Transorm (DTFT)
● Discrete-Fourier Transform (DFT)
● z-Transform
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The Discrete-time Fourier Transform (DTFT)
 The definition of DTFT
DTFT: X ( e jw )  DTFT[ x ( n)] 

 jwn
x
(
n
)
e

n 
IDTFT:
1
x ( n)  IDTFT[ X ( e )] 
2
jw

  X (e

jw
)e
jwn
dw

Existence condition:
 | x ( n) |  

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The Discrete-time Fourier Transform (DTFT)
The comparison of x (n) vs. X (e j )
x (n)
X (e j )
Time domain
Frequency domain
discrete
continuous
Real valued
Complex-valued
Summation
integral
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The Discrete-time Fourier Transform (DTFT)
About X (e j )
● It is a periodic function of  with a period of 2

  ~ 
The integral range of   ~ 
The range of
● It can be expressed as
j
j
j
X (e )  X (e ) e
X (e )
magnitude function
 ( )
phase function
X (e j ) and  ( ) are all real function of 
j ( )
example
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The Discrete-time Fourier Transform (DTFT)
DTFT vs. z Transform
j
X ( e )  X ( z ) z  e j


 x ( n )e
 jn
n  
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The Discrete-time Fourier Transform (DTFT)
 The general properties of DTFT

Linearity
The DTFT is a linear transformation
ax(n)  by(n)  aX (e j )  bY (e j )

Time shifting
A shift in the time domain corresponds to the
phase shifting
x( n  m )  e
 jm
j
X (e )
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The Discrete-time Fourier Transform (DTFT)

Frequency shifting
Multiplication by a complex exponential
corresponds to a shift in the frequency domain
e

jn 0
x ( n)  X ( e
j (  0 )
)
Convolution
Convolution in time domain corresponds to
multiplication in frequency domain
x1 (n)  x2 (n)  X 1 (e ) X 2 (e )
jw
jw
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The Discrete-time Fourier Transform (DTFT)

Multiplication

1
x1 ( n)  x2 ( n) 
2

  X (e

j
1
) X 2 (e
j ( w  )
)d
Energy (Parseval’s Theorem)

1
x ( n) 

2
n  
2
j
X (e )



j
2
X (e ) d
2
energy density spectrum
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The Discrete-time Fourier Transform (DTFT)

Multiplied by an exponential sequence
1 j
a  x ( n)  X ( e )
a
n

Sequence weighting
d
j
n  x ( n)  j
[ X ( e )]
d
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The Discrete-time Fourier Transform (DTFT)

Conjugation
Conjugation in the time domain corresponds to the
folding and conjugation in the frequency domain
x  (n)  X  (e  j )

Folding
Folding in the time domain corresponds to the folding
in the frequency domain
 j
x (  n)  X ( e

)
Conjugation and Folding
Conjugation and folding in the time domain corresponds
to the conjugation in the frequency domain
x  ( n)  X  (e j )
return
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The symmetric properties of the DTFT
 Conjugate symmetry of x (n)

Conjugate symmetric sequence:

xe (n)  xe (n)
For real-valued sequence, it is even symmetric:
x e ( n)  x e (  n)

Conjugate antisymmetric sequence:

xo (n)   xo (n)
For real-valued sequence, it is odd symmetric:
x o ( n)   x o (  n)
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The symmetric properties of the DTFT
Any arbitrary sequence can be expressed as
the sum of a conjugate symmetric sequence and
a conjugate antisymmetric sequence

x ( n)  x e ( n )  x o ( n)
1
xe ( n)  [ x ( n)  x  (  n)]
2
1

xo ( n)  [ x ( n)  x (  n)]
2
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The symmetric properties of the DTFT
 Conjugate symmetry of X (e j )
The X (e j ) can be expressed as the sum of the
conjugate symmetric component and the conjugate
antisymmetric component

j
j
j
X (e )  X e (e )  X o (e )
1
X e (e )  [ X (e j )  X  (e  j )]
2
1
j
j

 j
X o (e )  [ X (e )  X (e )]
2
j
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The symmetric properties of the DTFT
j

X e (e )  X e (e－j )
conjugate symmetric
For real-valued function, it is even symmetric
j
X e (e )  X e (e
－j
)
conjugate antisymmetric
j

X o (e )   X o (e  j )
For real-valued function, it is odd symmetric
X o (e j )   X o (e  j )
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The symmetric properties of the DTFT
 The symmetric properties of the DTFT
Re[x( n)] 
j Im[ x ( n)] 
xe ( n)
xo ( n)
j
X e (e )
j
X o (e )
 Re[X (e j )]
j
 j Im[ X (e )]
Implication: If the sequence x (n) is real and even,
then X (e j ) is also real and even.
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The symmetric properties of the DTFT
If the sequence x(n) is real, then
j

X (e )  X (e
 j
)
j
Re[X (e )]  Re[X (e
j
 j
Im[ X (e )]   Im[ X (e
j
X (e )  X (e
j
 j
)]
 j
)]
)
arg[ X (e )]   arg[ X (e
 j
)]
example
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The symmetric properties of the DTFT
 The DTFT of periodic sequences

The DTFT of complex-valued exponential sequences
x ( n)  e
X (e
j
j 0 n
)
(   n   )

 2 (  
i  
0
 2i )
(    0   )
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The symmetric properties of the DTFT

The DTFT of constant-value sequences
x ( n)  1
j
X (e ) 
(-  n   )

 2 (  2i )
i  
Copyright © 2005. Shi Ping CUC
The symmetric properties of the DTFT

The DTFT of unit sample sequences
x ( n) 

  (n  iN )
i  
2
X (e ) 
N
j
2
 ( 
k)

N
k  

Copyright © 2005. Shi Ping CUC
The symmetric properties of the DTFT
The DTFT of general periodic sequences

~
x ( n) 


i  
i  
 x( n  iN )  x( n)    ( n  iN )
2
X (e ) 
N
j
2

N

 X (e
k  
j
2
k
N
2
) ( 
k)
N
2
~
X ( k ) ( 
k)

N
k  

return
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 The representation of a LTI system

Impulse response h(n)
N

Difference equation
a
k 0

M
k
y( n  k )   bm x( n  m )
m 0
System function H (z )
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 System function (Transfer function)
The z-transform of the impulse response h(n) of the LTI
system is called system function or transfer function
H ( z )  Ζ [h( n)] 

 h(n)z
n
n  
H (z)  Y (z) X (z)
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 The region of convergence (ROC) for H(z)
An LTI system is stable if and only if the unit circle is in
the ROC of H(z)
An LTI system is causal if and only if the ROC of H(z) is
Rx   | z |  
An LTI system is both stable and causal if and only if
the H(z) has all its poles inside the unit circle, i.e. the
ROC of H(z) is
1| z | 
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 System function vs. difference equation
N
difference equation
a
k 0
take z-transform for both sides
M
k
y( n  k )   bm x( n  m )
N
M
k 0
m 0
k
m
a
z
Y
(
z
)

b
z
 k
 m X (z)
M
M
Y (z)
H (z) 

X (z)
m
b
z
 m
m 0
N
k
a
z
 k
k 0
m 0
K
1
(
1

c
z
 m )
m 1
N
1
(
1

d
z
)

k
k 1
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 Frequency response of an LTI system
The DTFT of an impulse response is called the
frequency response of an LTI system, i.e.

H (e j ) 
j n
h
(
n
)
e

n  
j
j
H (e )  H (e ) e
j arg[ H ( e j )]
H (e j ) magnitude response function
j
example
example
arg[ H (e )] phase response function
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response

Group delays

 d arg[ H (e j )]
 g ( ) 
d

In general, the frequency response H (e j ) is a
complex function of 



H (e j ) is a continuous function of

H (e j ) is a periodic function of  , the period is 2
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response

Response to exponential sequence
x ( n)  e
j 0 n
j
H (e )
y ( n)  e
j 0 n
H (e
j 0
)
The output sequence is the input exponential sequence
modified by the response of the system at frequency 
0
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System Function and Frequency Response

Response to sinusoidal sequences
j
x (n)
H (e )
y(n)
x( n)  A cos( 0 n   )
y ( n)  A | H ( e
j 0
) | cos( 0 n    arg[ H (e
j 0
)])
x( n)   Ak cos( k n   k )
k
y( n)   Ak | H (e j k ) | cos( k n   k  arg[ H (e j k )])
k
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response

Response to arbitrary sequences
x (n)
H (e j ), h(n)
y(n)
y( n)  x( n)  h( n)
j
j
j
Y (e )  X (e ) H (e )
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 Geometric interpretation of frequency response
M
M
Y (z)
H (z) 

X (z)
m
b
z
 m
m 0
N
a z
k
k 0
k
K
1
(
1

c
z
 m )
m 1
N
 (1  d
1
k
z )
k 1
M
 Kz ( N  M )
 (z  c
m
)
m 1
N
 (z  d
k
)
k 1
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
M
j
H (e )  Ke
j ( N  M )

 (e
j
 cm )
 (e
j
 dk )
m 1
N
k 1
j
 H (e ) e
K
j arg[ H ( e j )]
is a real number
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
M
j
H (e )  K
 (e
j
 cm )
 (e
j
 dk )
m 1
N
k 1
j
 cm   m e
j m
zero vector
Cm  e
pole vector
Dk  e j  d k  l k e j k
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
M
arg[ H (e )]  arg[ K ]   arg[ e
j
j
m 1
M
  arg[ e
j
m 1
zero vector
pole vector
 cm ]
 d k ]  ( N  M )
C m  e j   c m   m e j m
Dk  e
j
 d k  lk e
j k
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
M
H ( e j )  K

m
m 1
N
l
k
k 1
M
N
m 1
k 1
arg[ H (e j )]  arg[ K ]    m    k  ( N  M )
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
An approximate plot of the magnitude and phase
responses of the system function of an LTI system can
be developed by examining the pole and zero locations

To highly attenuate signal components in a specified
frequency range, we need to place zeros very close to
or on the unit circle in this range

To highly emphasize signal components in a
specified frequency range, we need to place poles very
close to or on the unit circle in this range

Copyright © 2005. Shi Ping CUC
M
System Function and Frequency Response
 m
H ( e j )  K
m 1
N
l
k
k 1
1
2
2
l1

e j
1
1
l2
2

2

3
2
2

Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 Minimum-Phase and Maximum-Phase system
N
 H (e j )  M
arg 
   m    k  ( N  M )

k 1
 K  m 1
mi
mo
pi
po
the number of zeros inside the unit circle
the number of zeros outside the unit circle
the number of poles inside the unit circle
the number of poles outside the unit circle
M  mi  mo
N  pi  po
Copyright © 2005. Shi Ping CUC

A causal stable system
po  0,
pi  N
 H ( e j ) 
arg 
 2m i  2pi  2 ( N  M )

 K    2
 2 m i  2 M  2 mo
A causal stable system with all zeros inside the unit circle
is called a minimum-phase delayed system

 H ( e j ) 
arg 
0

 K    2
A causal stable system with all zeros outside the unit circle
is called a maximum-phase delayed system

 H (e j ) 
 arg 
 2M

 K    2
Copyright © 2005. Shi Ping CUC

An anti-causal stable system
pi  0,
po  N
 H (e j ) 
arg 
 2mi  2 ( N  M )

 K    2
An anti-causal stable system with all zeros inside the unit
circle is called a maximum-phase advanced system

 H ( e j ) 
arg 
 2N  2po

 K    2
An anti-causal stable system with all zeros outside the unit
circle is called a minimum-phase advanced system

 H (e j ) 
 arg 
 2 ( N  M )  2 ( po  mo )

 K    2
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response

Important properties of minimum-phase delayed system
minimum-phase delayed system is often called
minimum-phase system for short.
It plays an important role in telecommunications
 Any nonminimum-phase system can be expressed as
the product of a minimum-phase system function and a
stable all-pass system

For all systems with the identical | H (e j ) |
N 1
N 1
n0
n0
2
2
|
h
(
n
)
|

|
h
(
n
)
|

 min
m
m
2
|
h
(
n
)
|

|
h
(
n
)
|

 min , m  N-1
2
n0
n0
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 All-pass system

Definition
A system that has a constant magnitude response
for all frequencies, that is,
| H ap (e j ) | 1， 0    
The simplest example of an all-pass system is a
pure delay system with system function H ( z )  z  k
This system passes all signals without modification
except for a delay of k samples.
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response

1-th order all-pass system
1

z a
H ap ( z ) 
,
1
1  az
 j
j
a  re , 0  r  1
 j
j (  )
e  re
1  re
 j
H ap (e ) 
e

j  j
 j (  )
1  re e
1  re
j
1  r cos(   )  jr sin(   )

1
1  r cos(   )  jr sin(   )
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response

An alternative form of 1-th order all-pass system
1  ( a  ) 1 z 1
H ap ( z ) 
,
1
1  az
j
H ap (e )  r
1
j
a  re , 0  r  1
j Im[ z ]
a
1
a
Re[z ]
Mirror image symmetry with
respect to the unit circle
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response

2-th order all-pass system
1

1
z a z a
j
H ap ( z ) 

,
a

re
, 0 r 1
1
 1
1  az 1  a z
j
H ap (e )  1
1  ( a  )  1 z  1 1  a 1 z 1
j
H ap ( z ) 

,
a

re
, 0 r 1
1
 1
1  az
1 a z
H ap (e j )  r 2
example
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
example
j
let a  re , 0  r  1
z 1  a  z 1  a
z 2  2rz 1 cos   r 2
H ap ( z ) 


1
 1
1  az 1  a z
1  2rz 1 cos  r 2 z  2
1
2 2
1  2rz cos  r z
2
 2 D( z )
z 
z 
1
2 2
D( z )
1  2rz cos  r z
D( z )  1  2rz cos  r z
1
2 2
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response

N-th order all-pass system
1

k
1
z a
H ap ( z )  
k 1 1  a k z
N
1
 ( N 1 )
N
N
1
d N  d N 1 z    d 1 z
z
z D( z )


1
 ( N 1 )
N
1  d 1 z    d N 1 z
 dN z
D( z )
j

D(e )  D (e
 j
)
H (e j )  1
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 An
alternative form for N-th order all-pass system
z 1  ak N C ( z 1  bk )( z 1  bk )
H ap ( z )  
1 
1
 1
1

a
z
(
1

b
z
)(
1

b
k 1
k 1
k
k
kz )
NR
NR
The number of real poles and zeros
NC
The number of complex-conjugate pair of poles
and zeros
For causal and stable system
| ak | 1, | bk | 1
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response

Application

Phase equalizers
When placed in cascade with a system that has
an undesired phase response, a phase equalizer
is designed to compensate for the poor phase
characteristics of the system and therefore to
produce an overall linear-phase response.
H ( z )  H ap ( z )  H d ( z )
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
j
j
j
H (e )  H ap (e )  H d (e )
j
j
 H ap (e )  H d (e )  e
j [ ap ( )  d ( )]
arg[ H (e j )]   ap ( )   d ( )
Group delays


 d arg[ H (e j )]
 ( ) 
  ap ( )   d ( )   0
d
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
Any causal-stable nonminimum-phase system can
be expressed as the product of a minimum-phase
delayed system cascaded with a stable all-pass system

example
H ( z )  H min ( z )  H ap ( z )
1
1

o
H ( z )  H 1 ( z )  ( z  zo )  ( z  z )
H1 ( z ) a minimum-phase system
1 1
,  a pair of conjugate zeros outside the unit circle
zo zo
| zo | 1
Copyright © 2005. Shi Ping CUC
 1
1
1

z
z
1

z
z
1
1

o
o
H ( z )  H 1 ( z )  ( z  zo )  ( z  zo ) 

 1
1
1  zo z 1  zo z
1
1

z

z
z

z
o
o
 H 1 ( z )  (1  zo z 1 )  (1  zo z 1 ) 

 1
1
1  zo z 1  zo z
 H min ( z )  H ap ( z )
H 1 ( z )  (1  zo z 1 )  (1  zo z 1 )
z 1  zo z 1  zo

 1
1  z o z 1  z o z 1
is a minimum-phase system
is a 2-th all-pass system
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
By cascading an all-pass system an unstable
system can be made stable without changing its
magnitude response

1
example

1
z a z a
H ap ( z ) 

1
1  az 1  a  z 1
H ( z )  H ( z )  H ap ( z )
H (z ) unstable system
H (z ) stable system
Copyright © 2005. Shi Ping CUC
System Function and Frequency Response
 Relationships between system representations
H (z )
Express H(z) in
z-1 cross multiply
and take inverse
Difference
Equation
ZT
take ZT solve
for Y/X
h(n)
substitute
j
ze
Take DTFT
solve for Y/X
Inverse ZT
Inverse DTFT
DTFT
H (e j )
return
Copyright © 2005. Shi Ping CUC
 (n)
unit sample sequence
Amplitude
1
0.8
0.6
0.4
0.2
0
-10
-5
0
5
n
10
15
20
-5
0
5
n
10
15
20
 ( n  5)
Amplitude
1
0.8
0.6
0.4
0.2
0
-10
return
Copyright © 2005. Shi Ping CUC
unit step sequence
u(n)
Amplitude
1
0.8
0.6
0.4
0.2
0
-5
0
5
u( n  5)
10
15
20
10
15
20
n
Amplitude
1
0.8
0.6
0.4
0.2
0
-5
0
5
n
return
Copyright © 2005. Shi Ping CUC
R10 ( n)
Rectangular
sequence
unit step sequence
Amplitude
1
0.8
0.6
0.4
0.2
0
-5
0
5
10
15
20
10
15
20
n
R10 ( n  5)
Amplitude
1
0.8
0.6
0.4
0.2
0
-5
0
5
n
return
Copyright © 2005. Shi Ping CUC
1.5  cos(
0
.
05

2

n
)
2
Sinusoidal sequence
Amplitude
1
0
-1
-2
0
10
20
1.5  sin( 0.05  2 n)
30
n
40
50
60
30
n
40
50
60
2
Amplitude
1
0
-1
-2
0
10
20
return
Copyright © 2005. Shi Ping CUC
Real-valued
exponential
sequence
Sinusoidal
sequence
0.25
0.001  1.2
Amplitude
0.2
n
0.15
0.1
0.05
0
0
5
10
15
n
20
0.2
30
0.2  0.8
0.15
Amplitude
25
n
0.1
0.05
0
0
5
10
15
n
20
25
30
return
Copyright © 2005. Shi Ping CUC
Complex-valued exponential sequence
real part
2
2e
Amplitude
1.5
1 
(   j )n
5 8
1
0.5
0
-0.5
0
5
10
n
20
25
30
35
25
30
35
imaginary part
1.5
Amplitude
15
1
0.5
0
-0.5
0
5
10
15
20
n
return
Copyright © 2005. Shi Ping CUC
n2 R1780( n  8)
finite-length sequence
Amplitude
60
40
20
0
-20
-15
-10
cos(0.14n)
-5
0
5
10
15
20 n
10
15
20 n
infinite-length sequence
Amplitude
0.5
0
-0.5
-1
-20
-15
-10
-5
0
5
return
Copyright © 2005. Shi Ping CUC
right-sided sequence
0.8
0.2  0.8 u(n  5)
n
Amplitude
0.6
0.4
0.2
0
-10
-5
0
5
10
n
15
20
25
30
causal sequence
0.2
0.2  0.8n u( n)
Amplitude
0.15
0.1
0.05
0
-10
-5
0
5
10
n
15
20
25
30
return
Copyright © 2005. Shi Ping CUC
left-sided sequence
0.8
0.2  0.8 n u(n  5)
Amplitude
0.6
0.4
0.2
0
-30
-25
-20
-15
-10
n
-5
0
5
10
0.2  0.8 n u(n)
anti-causal sequence
0.2
Amplitude
0.15
0.1
0.05
0
-30
-25
-20
-15
-10
n
-5
0
5
10
return
Copyright © 2005. Shi Ping CUC
0.2  0.8
|n|
two-sided sequence
0.2
Amplitude
0.15
0.1
0.05
0
-20
-15
-10
-5
0
n
5
10
15
20
0.2  cos(0.3n)
two-sided sequence
0.2
Amplitude
0.1
0
-0.1
-0.2
-20
-15
-10
-5
0
n
5
10
15
20
return
Copyright © 2005. Shi Ping CUC
absolutely summable sequence
1
n
0.3 u( n)
Amplitude
0.8
0.6
0.4
0.2
0
-5
0
5
10
15
n
20
25
30
35
absolutely summable sequence
1
0.85n u(n)
Amplitude
0.8
0.6
0.4
0.2
0
-5
0
5
10
15
n
20
25
30
35
return
Copyright © 2005. Shi Ping CUC
square-summable sequence
sin 0.3n
n
0.15
Amplitude
0.1
0.05
0
-0.05
-20
-15
-10
-5
0
n
5
10
15
square-summable sequence
sin 0.6n
n
0.2
0.15
Amplitude
20
0.1
0.05
0
-0.05
-20
-15
-10
-5
0
n
5
10
15
20
return
Copyright © 2005. Shi Ping CUC
Time-shifting operation
0.2  0.8 u( n)
n
original sequence
Amplitude
0.2
0.1
0
-10
-5
0
5
Amplitude
0.2
20
25
0.2  0.8
n5
30
n
u( n  5)
0.1
0
-10
-5
0
0.2
Amplitude
10
15
delayed sequence
5
10
15
advanced sequence
20
0.2  0.8
25
n 5
30 n
u( n  5)
0.1
0
-10
-5
0
5
10
15
20
25
30
n
return
Copyright © 2005. Shi Ping CUC
folding operation
original sequence
1
n
0.8 u( n)
Amplitude
0.8
0.6
0.4
0.2
0
-20
-15
-10
-5
0
n
5
10
15
20
folding sequence
1
n
0.8 u( n)
Amplitude
0.8
0.6
0.4
0.2
0
-20
-15
-10
-5
0
n
5
10
15
20
return
Copyright © 2005. Shi Ping CUC
addition operation
x1(n)
Amplitude
1
n
0.8 u( n)
0.5
0
0
5
10
15
Amplitude
1
25
20
x1(n)+x2(n)
25
30
35
40
n
cos(0.2n)u( n)
0
-1
0
5
10
15
30
35
40 n
0.8 u(n)  cos(0.2n)u(n)
2
Amplitude
20
x2(n)
n
1
0
-1
0
5
10
15
20
25
30
35
40 n
return
Copyright © 2005. Shi Ping CUC
modulation operation
0.1sin 0.0125n
x1(n)
Amplitude
0.1
0
-0.1
0
20
40
60
80
x2(n)
100
120
Amplitude
160
sin 0.125n
1
0
-1
0
20
40
60
80
x1(n)*x2(n)
100
120
0
20
40
60
80
100
120
0.1
Amplitude
140
140
160
140
160
x1 (n)  x2 (n)
0
-0.1
return
Copyright © 2005. Shi Ping CUC
periodic sequence
periodic sequence
sin(
1

n)
8
Amplitude
0.5
0
-0.5
-1
0
10
20
30
40
50
60
70
80
90
n
periodic sequence
sin(
1
Amplitude
0.5

16
n)
0
-0.5
-1
0
10
20
30
40
50
60
70
80
90
n
return
Copyright © 2005. Shi Ping CUC
Periodicity of sequence
x1(n)
sin(
Amplitude
1
0
10
20
30
40
x2(n)
50
60
70
80
3
n)
10
90
sin(
1
Amplitude
8
n)
0
-1
0
-1
0
10
20
30
40
x3(n)
50
60
70
80
90
sin( 0.4n)
1
Amplitude

0
-1
0
10
20
30
40
50
60
70
80
90
return
Copyright © 2005. Shi Ping CUC
T0 32

T
3
Periodicity of sequence 2
1
Amplitude
0.5
0
-0.5
-1
0
10
20
30
40
50
60
T0 10

T
3
1
Amplitude
0.5
0
-0.5
-1
0
10
20
30
40
50
60
return
Copyright © 2005. Shi Ping CUC
 ( n)  4 ( n  1)
 3 ( n  2)  6 ( n  3)
 8 ( n  4)  10 ( n  5)
 11 ( n  6)  9 ( n  7)
12
11
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
return
Copyright © 2005. Shi Ping CUC
x(n)
0 .9
Amplitude
10
5
0
-20
-15
-10
-5
0
xe(n)
5
10
15
20
-15
-10
-5
0
xo(n)
5
10
15
20
-15
-10
-5
0
5
10
15
20
Amplitude
6
4
2
0
-20
5
Amplitude
n
0
-5
-20
return
Copyright © 2005. Shi Ping CUC
Accumulator
n
n 1
l  
l  
y ( n) 
 x(l )   x(l )  x(n)  y(n  1)  x(n)
The output y (n) at time instant n is the sum of the input
sample x (n) at time instant n and the previous output y( n  1)
at time instant n-1, which is the sum of all previous input
sample values from   to n-1
The input-output relation can also be written in the form:
y ( n) 
1
n
n
l  
l 0
l 0
 x(l )   x(l )  y(1)   x(l ), n  0
This form is used for a causal input sequence, in which
case y(-1) is called the initial condition
return
Copyright © 2005. Shi Ping CUC
M-point moving-average system
1
y ( n) 
M
M 1
 x( n  k )
k 0
An application: consider x( n)  s( n)  d ( n)
Where s(n) is the signal, and d (n) is a random noise
s( n)  2n0.9n
M 8
1 7
y ( n)   x ( n  k )
8 k 0
return
Copyright © 2005. Shi Ping CUC
s( n)  2n0.9n
s(n),d(n)
Amplitude
10
5
0
0
5
10
15
20
25
x(n)
30
35
40
45
50
0
5
10
15
20
25
y(n)
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
Amplitude
10
5
0
Amplitude
10
5
0
return
Copyright © 2005. Shi Ping CUC
n
y( n) 
Accumulator
 x( l )
l  
T [ x1 ( n)] 
n
 x (l ),
l  
1
T [ax1 ( n)  bx 2 ( n)] 
n
n
l  
l  
T [ x2 ( n)] 
n
 x (l )
l  
2
n
 [ax ( l )  bx
l  
1
2
( l )]
 a  x1 ( l )  b  x 2 ( l )  aT [ x1 ( n)]  bT [ x 2 ( n)]
Hence, the above system is linear
return
Copyright © 2005. Shi Ping CUC
y( n) 
Accumulator
n
 x( l )
l  
n k
T [ x( n  k )] 
 x( l )
l  
y( n  k ) 
n k
 x(l )  T [ x(n  k )]
l  
Hence, the above system is time-invariant
return
Copyright © 2005. Shi Ping CUC
x(n)  R10 (n)
x(n)
Amplitude
1.5
1
0.5
0
0
5
10
15
20
25
h(n)
30
35
Amplitude
1.5
40
45
50
h( n)  0.9 u( n)
n
1
0.5
0
0
5
10
15
20
25
y(n)
30
35
Amplitude
10
40
45
50
y( n)  x( n)  h( n)
5
0
0
5
10
15
20
25
30
35
40
45
50
return
Copyright © 2005. Shi Ping CUC
The M-point moving average filter
1
y ( n) 
M
M 1
 x( n  k )
k 0
For a bounded input
1
y( n) 
M
M 1
x(n)  Bx , we have
1
x( n  k ) 

M
k 0
M 1
 x( n  k )
k 0
1

( MB x )  B x
M
Hence, the M-point moving average filter is BIBO stable
return
Copyright © 2005. Shi Ping CUC
A causal LTI discrete-time system
h(n)   u(n)
n


1
S   | a | u( n)   | a | 
1 a
n  
n0
n
n
if | a | 1
if |  | 1, the system is stable
if |  | 1, the system is not stable
return
Copyright © 2005. Shi Ping CUC
real part
DTFT[0.9n e jn8 / 3 R11 (n)]
Amplitude part
8
6
Amplitude
Amplitude
6
4
2
0
-2
4
2
-1
0
pi
1
0
-2
2
-1
phase part
2
1
2
5
Amplitude
phase(pi)
1
imaginary part
0.5
0
-0.5
-2
0
pi
-1
0
pi
1
2
0
-5
-2
-1
0
pi
return
Copyright © 2005. Shi Ping CUC
Amplitude part
8
DTFT[0.9n 8R11 (n)]
6
Amplitude
Amplitude
6
4
2
0
-1
4
2
-0.5
0
pi
0.5
0
-1
1
phase part
0
pi
0.5
1
5
Amplitude
phase(pi)
-0.5
imaginary part
0.5
0
-0.5
-1
real part
-0.5
0
pi
0.5
1
0
-5
-1
-0.5
0
pi
0.5
1
return
Copyright © 2005. Shi Ping CUC
z 1
H (z)  2
z  0.9 z  0.81
1
0.8
0.6
Imaginary Part
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1
-0.5
0
Real Part
0.5
1
Copyright © 2005. Shi Ping CUC
Amplitude response
Amplitude
15
z 1
H (z)  2
z  0.9 z  0.81
10
5
0
0
0.1
0.2
0.3
0.4
0.5
pi
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
phase response
1
phase(pi)
0.5
0
-0.5
-1
0
0.1
0.2
0.3
0.4
0.5
pi
0.6
return
Copyright © 2005. Shi Ping CUC
7 19 1
 z
H ( z )  12 24
5
1  z 1  z  2
2
1
Imaginary Part
0.5
0
-0.5
-1
-1
-0.5
0
0.5
Real Part
1
1.5
2
Copyright © 2005. Shi Ping CUC
Amplitude response
0.55
Amplitude
0.5
0.45
7 19 1
 z
H ( z )  12 24
5
1  z 1  z  2
2
0.4
0.35
0.3
0
0.1
0.2
0.3
0.4
0.5
pi
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
phase response
0
phase(pi)
-0.1
-0.2
-0.3
-0.4
0
0.1
0.2
0.3
0.4
0.5
pi
0.6
return
Copyright © 2005. Shi Ping CUC
1

1
Imaginary Part
z a
z a
H ap ( z ) 

1
 1
1  az 1  a z
1.5
1

(r  ,   )
21
3
1
a
0.5
a
0
a
-0.5

-1
Mirror image symmetry
1
a
-1.5
-2
-1.5
-1
-0.5
0
Real Part
0.5
1
1.5
2
return
Copyright © 2005. Shi Ping CUC
1  2.61z 1  3.08 z 2  1.85 z 3  0.83 z 4
H (z) 
1
2
1  1.12 z  0.48 z
1
0.8
1
z o
0.6
Imaginary Part
0.4
0.2
2
0
-0.2
-0.4
1
zo
-0.6
-0.8
-1
-1
-0.5
0
Real Part
0.5
1
return
Copyright © 2005. Shi Ping CUC
1
0.8
1
z o
0.6
Imaginary Part
0.4
zo
0.2
2
0
-0.2
zo
-0.4
1
zo
-0.6
-0.8
-1
-1
-0.5
0
Real Part
0.5
1
return
Copyright © 2005. Shi Ping CUC
1  0.83 z 1  0.35 z 2
H (z) 
1
2
 2.26 z
11  2.41z
1

a
1
0.8
0.6
z o
Imaginary Part
0.4
a
0.2
0
a
-0.2

-0.4
-0.6
1
a
-0.8
-1
-1
-0.5
0
Real Part
0.5
1
return
Copyright © 2005. Shi Ping CUC