Lecture
DIGITAL PROCESSING
OF
SPEECH AND IMAGE SIGNALS
RWTH Aachen, WS 2006/7
Prof. Dr.-Ing. H. Ney, Dr.rer.nat. R. Schlüter
Lehrstuhl für Informatik 6
RWTH Aachen
1. System Theory and Fourier Transform
2. Discrete Time Systems
3. Spectral Analysis
4. Fourier Transform and Image Processing
5. LPC Analysis
6. Wavelets
7. Coding
8. Image Segmentation and Contour-Finding
Completions: L. Welling, A. Eiden; April 1997
Completions: J. Dahmen, F. Hilger, S. Koepke; Mai 2000
Completions: F. Hilger, D. Keysers; Juli 2001
Translation: M. Popović, R. Schlüter; April 2003
Corrections: D. Stein; October 2006
Literature:
• A. V. Oppenheim, R. W. Schafer: Discrete Time Signal Processing,
Prentice Hall, Englewood Cliffs, NJ, 1989.
• A. Papoulis: Signal Analysis, McGraw-Hill, New York, NY, 1977.
• A. Papoulis: The Fourier Integral and its Applications, McGraw-Hill
Classic Textbook Reissue Series, McGraw-Hill, New York, NY, 1987.
• W. K. Pratt: Digital Image Processing, Wiley & Sons Inc, New York,
NY, 1991.
Further reading:
• T. K. Moon, W. C. Stirling: Mathematical Methods and Algorithms
for Signal Processing. Prentice Hall, Upper Saddle River, NJ, 2000.
• J. R. Deller, J. G. Proakis, J. H. L. Hansen: Discrete-Time Processing
of Speech Signals, Macmillan Publishing Company, New York, NY,
1993.
• W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery: Numerical Recipes in C, Cambridge Univ. Press, Cambridge, 1992.
• L. Rabiner, B. H. Juang: Fundamentals of Speech Recognition, Prentice Hall, Englewood Cliffs, NJ, 1993.
• T. Lehmann, W. Oberschelp, E. Pelikan, R. Repges: Bildverarbeitung
für die Medizin, Springer Verlag, Berlin, 1997.
• L. Berg: Lineare Gleichungssysteme mit Bandstruktur, VEB Deutscher
Verlag der Wissenschaften, Berlin, 1986.
Contents
1 System Theory and Fourier Transform
1.1 Introduction . . . . . . . . . . . . . . .
1.2 Linear time-invariant Systems . . . . .
1.3 Fourier Transform . . . . . . . . . . . .
1.4 Properties of the Fourier Transform . .
1.5 Parseval Theorem . . . . . . . . . . . .
1.6 Autocorrelation Function . . . . . . . .
1.7 Existence of the Fourier Transform . .
1.8 δ-Function . . . . . . . . . . . . . . . .
1.9 Motivation for Fourier Series . . . . . .
1.10 Time Duration and Band Width . . . .
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2 Discrete Time Systems
2.1 Motivation and Goal . . . . . . . . . . . . . . . . . . . . .
2.2 Digital Simulation using Discrete Time Systems . . . . . .
2.3 Examples of Discrete Time Systems . . . . . . . . . . . . .
2.4 Sampling Theorem (Nyquist Theorem) and Reconstruction
2.5 Logarithmic Scale and dB . . . . . . . . . . . . . . . . . .
2.6 Quantization . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Fourier Transform and z–Transform . . . . . . . . . . . . .
2.8 System Representation and Examples . . . . . . . . . . . .
2.9 Discrete Time Signal Fourier Transform Theorem . . . . .
2.10 Discrete Fourier Transform: DFT . . . . . . . . . . . . . .
2.11 DFT as Matrix Operation . . . . . . . . . . . . . . . . . .
2.12 From Continuous Fourier Transform to Matrix Representation of Discrete Fourier Transform . . . . . . . . . . . . . .
2.13 Frequency Resolution and Zero Padding . . . . . . . . . .
2.14 Finite Convolution . . . . . . . . . . . . . . . . . . . . . .
2.15 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . .
2.16 FFT Implementation . . . . . . . . . . . . . . . . . . . . .
i
1
2
11
16
25
33
34
35
36
41
45
51
52
53
56
61
70
72
74
78
88
90
98
102
104
105
108
118
2.17 Cyclic Matrices and Fourier Transform . . . . . . . . . . .
124
3 Spectral analysis
131
3.1 Features for Speech Recognition . . . . . . . . . . . . . . . 132
3.2 Short Time Analysis and Windowing . . . . . . . . . . . . 135
3.3 Autocorrelation Function and Power Spectral Density . . . 159
3.4 Spectrograms . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.5 Filter Bank Analysis . . . . . . . . . . . . . . . . . . . . . 168
3.6 Mel-frequency scale . . . . . . . . . . . . . . . . . . . . . . 171
3.7 Cepstrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
3.8 Statistical Interpretation of the Cepstrum Transformation
183
3.9 Energy in acoustic Vector . . . . . . . . . . . . . . . . . . 185
4 Fourier Transform and Image Processing
4.1 Spatial Frequencies and Fourier Transform for Images
4.2 Discrete Fourier Transform for Images . . . . . . . .
4.3 Fourier Transform in Computer Tomography . . . . .
4.4 Fourier Transform and RST Invariance . . . . . . . .
5 LPC Analysis
5.1 Principle of LPC Analysis . . . . . . . . .
5.2 LPC: Covariance Method . . . . . . . . . .
5.3 LPC: Autocorrelation Method . . . . . . .
5.4 LPC: Interpretation in Frequency Domain
5.5 LPC: Generative Model . . . . . . . . . .
5.6 LPC: Alternative Representations . . . . .
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187
188
196
197
199
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207
208
212
213
216
221
223
6 Outlook: Wavelet Transform
225
6.1 Motivation: from Fourier to Wavelet Transform . . . . . . 226
6.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.3 Discrete Wavelet Transform . . . . . . . . . . . . . . . . . 229
7 Coding (appendix available as separate document)
233
8 Image Segmentation and Contour-Finding
237
ii
List of Figures
1.1
Oscillograms of three time functions composed as sum of 20
partial oscillations. a) φn = 0, b) φn = π2 , c) φn statistical.
3
Amplitude spectrum of a time function composed as sum of
20 partial tones. . . . . . . . . . . . . . . . . . . . . . . . .
3
from left to right: original photo, low-pass and high-pass
filtered version . . . . . . . . . . . . . . . . . . . . . . . .
3
Phase manipulation for portion of a speech signal (vowel ’o’)
sampled at 8kHz, 25ms analysis window (200 samples), 512
point FFT . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Phase manipulation for portion of a speech signal (consonant
’n’) sampled at 8kHz, 25ms analysis window (200 samples),
512 point FFT . . . . . . . . . . . . . . . . . . . . . . . .
5
1.6
Phase manipulation for a Heaviside–function (step–function)
6
1.7
Schematic representation of the physiological mechanism of
speech production . . . . . . . . . . . . . . . . . . . . . . .
8
2.1
Digital photo . . . . . . . . . . . . . . . . . . . . . . . . .
58
2.2
Gradient image . . . . . . . . . . . . . . . . . . . . . . . .
58
2.3
Several real cases of Laplace Operator subtraction from original image. a) Original image b) Original image minus
Laplace Operator (negative values are set to 0 and values
above the grey scale are set to the highest grade of grey) .
60
Ideal reconstruction of a band-limited signal (from Oppenheim, Schafer)
a) original signal b) sampled signal c) reconstructed signal
64
1.2
1.3
1.4
1.5
2.4
iii
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
Sampling of band-limited signal with different sampling rates:
b) sampling rate higher than Nyquist rate - exact reconstruction possible
c) sampling rate equal to Nyquist rate - exact reconstruction
possible
d) sampling rate smaller than Nyquist rate - aliasing - exact
reconstruction not possible . . . . . . . . . . . . . . . . . .
65
Amplitude spectrum of the voiceless phoneme “s” from the
word “ist” . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Logarithmic amplitude spectrum of the phoneme “s” . . .
71
Amplitude spectrum of the voiced phoneme “ae” from the
word “Äh” . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Logarithmic amplitude spectrum of the phoneme “ae” . . .
71
Amplitude spectrum of a speech pause . . . . . . . . . . .
71
Logarithmic amplitude spectrum of a speech pause . . . .
71
Hanning window . . . . . . . . . . . . . . . . . . . . . . . 103
Example of a linear convolution of two finite length signals:
a) two signals;
b) signal x[n-k] for different values of n:
i) n < 0, no overlap with h[k], therefore convolution y[n] =
0
ii) n between 0 and Nh + Nx − 2, convolution 6= 0
iii) n > Nh + Nx − 2, no overlap with h[k], convolution y[n]
=0
c) resulting convolution y[n]. . . . . . . . . . . . . . . . . . 106
Flow diagram for decomposition of one N -DFT to two N/2–
DFTs with N = 8 . . . . . . . . . . . . . . . . . . . . . . . 110
Flow diagram of an 8–point–FFT using Butterfly operations. 111
Flow diagram of an 8–point–FFT using Butterfly operations. 120
Input and output arrays of an FFT. a) The input array contains N (N is power of 2) complex input values in one real
array of the length 2N . with alternating real and imaginary parts. b) The output array contains complex Fourier
spectrum at N frequency values. Again alternating real and
imaginary parts. The array begins with the zero-frequency
and then goes up to the highest frequency followed with
values for the negative frequencies. . . . . . . . . . . . . . 122
iv
3.1
3.2
3.3
3.4
3.5
Example for the application of the Discrete Fourier Transform (DFT). . . . . . . . . . . . . . . . . . . . . . . . . . .
138
a) signal v[n]; b) DFT-spectrum V [k]; c) Fourier spectrum
V (ejω ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
146
a) signal v[n]; b) DFT-spectrum V [k]; c) Fourier spectrum
V (ejω ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
148
a) DFT of length N = 64; b) DFT of length N = 128; c)
Fourier spectrum V (ejω ). . . . . . . . . . . . . . . . . . . .
151
Influence of the window function:
above: speech signal (vowel “a”); central: 512 point FFT
using rectangle window; below: 512 point FFT using Hamming window . . . . . . . . . . . . . . . . . . . . . . . . .
158
3.6
Fourier Transform of a voiced speech segment:
a) signal progression, b) high resolution Fourier Transform,
c) low resolution Fourier Transform with short Hamming
window (50 sampled values), d) low resolution Fourier Transform using autocorrelation function (19 coefficients), e) low
resolution Fourier Transform using autocorrelation function
(13 coefficients) . . . . . . . . . . . . . . . . . . . . . . . . 162
3.7
Signal progression and autocorrelation function of voiced
(left) and unvoiced (right) speech segment . . . . . . . . .
163
Temporal progression of speech signal and four autocorrelation coefficients . . . . . . . . . . . . . . . . . . . . . . . .
164
3.8
3.9
a) wide-band spectrogram: short time window, high time
resolution (vertical lines), no frequency resolution; for voiced
signals provides information on formant structure b) narrowband spectrogram: long time window, no time resolution,
high frequency resolution (horizontal lines); for voiced signals provides information on fundamental frequency (pitch) 166
3.10 Wide-band and narrow-band spectrogram and speech amplitude for the sentence “Every salt breeze comes from the
sea”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
167
3.11 Above: logarithmized power spectrum of a spoken vowel
(schematic).
Below: corresponding cepstrum (inverse Fourier–transform
of the logarithmized power spectrum). . . . . . . . . . . .
177
v
3.12 Cepstral smoothing: speech signal (vowel “a”), windowed
speech signal (Hamming window), spectrum obtained from
the whole cepstrum (blue) and smoothed spectrum obtained
from the first 13 cepstral coefficients (red). . . . . . . . . .
3.13 Homomorph analysis of a speech segment: signal progression, homomorph smoothed spectrum using 13 and 19 cepstral coefficients . . . . . . . . . . . . . . . . . . . . . . . .
179
4.1
4.2
4.3
4.4
4.5
4.6
TV–image (analog) . . . . . .
Digitized TV–image . . . . . .
Amplitude spectrum of Figure
Low-pass filtered . . . . . . .
High-pass filtered . . . . . . .
High-pass enhancement . . . .
193
193
193
193
194
194
5.1
LPC–analysis of one speech segment
a) signal progression, b) prediction error (K=12), c) LPC–
spectrum with K=12 coefficients, d) spectrum of the prediction error (K=12), e) LPC–spectrum with K=18 coefficients 219
LPC–Spectra for different prediction orders K . . . . . . . 220
5.2
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4.2
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178
List of Tables
2.1
2.2
Fourier transform pairs . . . . . . . . . . . . . . . . . . . .
Fourier transform Theorems . . . . . . . . . . . . . . . . .
87
88
Chapter 1
System Theory and Fourier
Transform
• Overview:
1.1 Introduction
1.2 Linear time-invariant Systems
1.3 Fourier Transform
1.4 Properties of Fourier Transform
1.5 Parseval Theorem
1.6 Autocorrelation Function
1.7 Existence of the Fourier Transform
1.8 δ–Function
1.9 Fourier Series
1.10 Duration and Band Width
Digital Processing of Speech and Image Signals
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WS 2006/2007
1.1
Introduction
What distinguishes the Fourier Transform (FT) from
other transformations?
1. Mathematical property of linear time-invariant systems:
• FT decomposes the time signal into “eigenfunctions”
“eigenfunctions” keep their form by passing the
linear time-invariant system
Ax=λx
• Magnitude of FT: shift invariant
2. Physical observation:
• Human ear produces sort of FT, essentially only magnitude of FT
(strictly speaking: short-time FT)
• Example:
Time functions with different evolution can sound equally.
The human ear either senses sense phase differences of partial
tones of the complete sound of stationary processes very weakly,
or does not sense them at all.
Fourier transform in speech processing:
• Calculation of the spectral components of speech
• Basic method for obtaining observations (features) for
speech recognition
Digital Processing of Speech and Image Signals
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φ=0
0
0
φ = π/2
0
0
random φ
0
0
Figure 1.2: Amplitude spectrum of a time
function composed as sum of 20 partial
tones.
0
Figure 1.1: Oscillograms of three time
functions composed as sum of 20 partial
oscillations. a) φn = 0, b) φn = π2 , c) φn
statistical.
Figure 1.3: from left to right: original photo, low-pass and high-pass filtered version
Digital Processing of Speech and Image Signals
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amplitude spectrum
0
original signal
0
0
inverse FT for phase φ(f ) = 0
0
0
Inverse FT for phase φ(f ) =
π
2
0
0
Inverse FT for random phase φ(f )
0
0
Figure 1.4: Phase manipulation for portion of a speech signal (vowel ’o’) sampled at 8kHz,
25ms analysis window (200 samples), 512 point FFT
Digital Processing of Speech and Image Signals
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amplitude spectrum
0
original signal
0
0
inverse FT for phase φ(f ) = 0
0
0
inverse FT for phase φ(f ) =
π
2
0
0
inverse FT for random phase φ(f )
0
0
Figure 1.5: Phase manipulation for portion of a speech signal (consonant ’n’) sampled at
8kHz, 25ms analysis window (200 samples), 512 point FFT
Digital Processing of Speech and Image Signals
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amplitude spectrum
0
original signal
0
inverse FT for phase φ(f ) = 0
0
0
inverse FT for phase φ(f ) =
π
2
0
0
inverse FT for random phase φ(f )
0
0
Figure 1.6: Phase manipulation for a Heaviside–function (step–function)
Digital Processing of Speech and Image Signals
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Why Fourier?
Roughly:
• Production, description and algorithmic operations on signals (functions or measurement curves over the time axis) can be described very
well in Fourier domain (frequency domain).
Deeper reason:
• Production, description and algorithmic operations on signals are largely
based on linear time-invariant (LTI) operations.
• Fourier Transform: simple representation of LTI-operations (later:
convolution theorem)
Why continuous?
• Real world is continuous
• Computer (digital = time discrete = sampled)
model of the real world
Digital Processing of Speech and Image Signals
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a)
A
glottal
pulses
vocal
tract
filter
T
t
b)
speech [a:]
radiation
from lips
and nose
T
t
|E(f)| [dB]
[a:]
|V(f)| [dB]
|A(f)| [dB]
|S(f)| [dB]
[a:]
1/T
0
*
4
=
*
0
4
0
4
0
NOSE
OUTPUT
NASAL
CAVITY
VELUM
PHARYNX
CAVITY
VOCAL
CORDS
LARYNX
TUBE
MOUTH
CAVITY
TONGUE
HUMP
MOUTH
OUTPUT
TRACHEA AND
BRONCHI
LUNG
VOLUME
MUSCLE
FORCE
Figure 1.7: Schematic representation of the physiological mechanism of speech production
Digital Processing of Speech and Image Signals
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4
signal (speech, image)
feature extraction
(signal analysis)
feature vector
(pattern vector)
(pattern)
comparison
reference data
(vectors, features)
decision
Examples:
• Spoken language
• Written numbers (letters)
• Cell recognition (red blood cells)
Digital Processing of Speech and Image Signals
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Examples of applications of Fourier Transform:
• Electrical switchgears
• Recognition and coding
– Speech and general acoustic signals
– Image signals
• Time series analysis:
– Astronomical measurement curves
– Stock-market course
– ...
• Computer tomography
• Solving differential equations
• Description of image production in optical systems
Digital Processing of Speech and Image Signals
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1.2
Linear time-invariant Systems
Example:
– speech production
– electrical systems
h(t)
input signal
x(t)
output signal
y(t)
S
symbolic:
{t → y(t)} = S {t → x(t)}
simplified:
y(t) = S {x(t)}
• Note: the complete time domain of the function is important, not
individual positions in time t.
more exact:
y = S {x}
LTI–System:
(LTI = Linear Time-Invariant)
• Linear:
Additive:
S {x1 + x2 } = S {x1 } + S {x2 }
Homogeneous:
S {α x} = αS {x} ,
α ∈ IR
• Time-invariant:
{t → y(t − t0 )} = S {t → x(t − t0 )} ,
Digital Processing of Speech and Image Signals
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t0 ∈ IR
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Mathematical theorem:
• Linearity and time invariance result in convolution representation
• Output signal y(t) of LTI system S with input signal x(t):
y(t) =
=
Z∞
−∞
Z∞
−∞
x(t − τ ) h(τ ) dτ
x(τ ) h(t − τ ) dτ
= x(t) ∗ h(t)
• h: impulse response of the system S
e
(t)
∆τ
x (t)
1/∆τ
∆τ
t
τi
t
• system response h∆τ (t) to excitation e∆τ (t):
h∆τ (t) = S {e∆τ (t)}
• signal x(t) is represented as sum of amplitude weighted and time
shifted elementary functions e∆τ (t):
"
#
X
x(t) = lim
x(τi ) e∆τ (t − τi ) ∆τ
∆τ →0
i
Digital Processing of Speech and Image Signals
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Hence the following holds for the output signal y(t):
y(t) = S {x(t)}
= S
=
(
lim
∆τ →0
"
lim
∆τ →0
S
X
(i
additivity:
=
lim
∆τ →0
"
X
x(τi ) e∆τ (t − τi ) ∆τ
X
i
)
x(τi ) e∆τ (t − τi ) ∆τ
S { x(τi ) e∆τ (t − τi ) ∆τ }
i
)#
#
homogeneity (for x(τi ) and ∆τ ):
"
#
X
= lim
x(τi ) S { e∆τ (t − τi ) } ∆τ
∆τ →0
i
time invariance:
=
lim
∆τ →0
"
X
i
x(τi ) h∆τ (t − τi ) ∆τ
#
limiting case ∆τ → 0 :
X
∆τ
τi
h∆τ (t)
−→
−→
−→
−→
Z
dτ
τ
h(t)
result:
y(t) =
Z∞
−∞
h(t):
x(τ ) h(t − τ ) dτ = x(t) ∗ h(t)
impulse response of the system
Digital Processing of Speech and Image Signals
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Examples of LTI-operations:
• Oscillatory systems (electrical or mechanical) with
external excitation:
h(τ )
x(t) −→
y(t) =
Z
−→ y(t)
h(t − τ ) x(τ ) dτ
y ′′ (t) + 2αy ′ (t) + β 2 y(t) = x(t)
α, β: parameters depending on the oscillatory system
• More general electrical engineering systems:
high-pass, low-pass, band-pass
• Sliding average value:
x(t) −→
S
−→ y(t) := x(t)
+T
Z /2
1
x(t) =
T
x(t + τ ) dτ
−T /2
• Differentiator:
x(t) −→
−→ y(t) := x′ (t)
S
• Comb filter: ”hypothesized” period T
x(t) −→
S
−→ y(t) := x(t) − x(t − T )
• In general: linear differential equations with coefficients ck and dl
P
P
ck y (k) (t) = dl x(l) (t)
k
l
[ + further constraints ]
Digital Processing of Speech and Image Signals
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Example of a non-linear system:
system: y(t) = x2 (t)
x(t) = A cos(βt)
A2
(1 + cos(2βt))
=⇒ y(t) = A cos (βt) =
2
2
2
frequency doubling
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1.3
Fourier Transform
Sinusoidal oscillation:
x(t) = A sin ( ω t + ϕ )
amplitude A
phase / null phase ϕ
angular frequency ω = 2 π f
j 2 = −1,
j∈C
Im
1
sin α
α
cos α 1
complex representation:
Re
ej α = cos α + j sin α,
ejα + e−jα
cos α =
2
and
α ∈ IR
ejα − e−jα
sin α =
2j
dimension:
DIM(ω) DIM(t) = 1
DIM(ω) =
1
1
=
= [Hz]
DIM(t) [sec]
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LTI-System
y(t) =
Z∞
−∞
x(t − τ )h(τ )dτ = x(t) ∗ h(t)
• Determine the following specific input signal:
x(t) = A ej(ωt+ϕ)
• For this input signal the output signal becomes:
y(t) =
Z∞
A ej(ω(t−τ )+ϕ) h(τ )dτ
−∞
= A ej(ωt+ϕ)
Z∞
h(τ )e−jωτ dτ
−∞
=
|
{z
}
H(ω) = F {h(τ )}
x(t) · H(ω)
• Definition of the Fourier transform:
Z∞
h(τ )e−jωτ dτ = F {h(τ )} = F {τ → h(τ )}
H(ω) =
−∞
(→
decomposition into e−jωτ )
• H(ω) is called transfer function of the system
Remark about x(t) = A ej(ωt+ϕ) :
• The shape of the input signal x(t), i.e. its frequency ω (“eigenfunction”) remains invariant
• Amplitude (intensity) and phase (time shift) are depending on
H(ω) (“eigenvalue”)
(→
analogy to the problem of eigenvalues in linear algebra)
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Remarks
• FT is complex:
H(ω) = Re {H(ω)} + j Im {H(ω)} = |H(ω)| ejΦ(ω)
• Amplitude (spectrum):
q
Re {H(ω)}2 + Im {H(ω)}2
|H(ω)| =
• Phase (spectrum):

Im {H(ω)}


arctan


Re {H(ω)}








Im
{H(ω)}


arctan
+ π


Re
{H(ω)}



Φ(ω) =
π



2










π


−



 2
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Re {H(ω)} > 0
Re {H(ω)} < 0
Re {H(ω)} = 0,
Im {H(ω)} > 0
Re {H(ω)} = 0,
Im {H(ω)} < 0
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Examples of Fourier transforms:
1. Rectangle function
t
h(t) = rect( ) =
T
H(ω) =
Z∞
1,
0,
|t| ≤ T /2
|t| > T /2
T
−jωt
h(t)e
dt =
−∞
=
Z2
−jωt
e
− T2
i
1 h −jω T
jω T2
2
e
−e
dt =
−jω
ωT
)
T sin(
2
ωT
2
sin(
) =
ωT
ω
2
2
(here: Im {H(ω)} = 0)
h(t)
t
H(ω)
ω
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2. Double-sided exponential
h(t) = e−α|t|
H(ω) =
=
Z∞
−∞
Z∞
with α > 0
h(t)e−jωt dt
e−(α+jω)t dt +
0
=
=
=
=
Z∞
e−(α−jω)t dt
0
∞
e−(α−jω)t
e
+
−(α + jω) −(α − jω) 0
1
1
0+0−
−
−(α + jω) −(α − jω)
α − jω + α + jω
α2 + ω 2
2α
α2 + ω 2
−(α+jω)t
• Imaginary part equals 0
• Infinite spectrum
• No zeros
H(ω )
h(t)
ω
t
• If h(t) is symmetric (i.e. h(t) = h(−t)), imaginary parts drop away
and the real part is sufficient
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3. Damped oscillations
h(t) = e−α|t| cos(βt) with α > 0
H(ω) =
=
Z∞
−∞
Z∞
h(t)e−jωt dt
e−(α+jω)t cos(βt)dt +
0
=
Z∞
=
e−(α−jω)t cos(βt)dt
0
e−(α+jω)t
ejβt + e−jβt
dt +
2
0
=
Z∞
Z∞
e−(α−jω)t
ejβt + e−jβt
dt
2
0
...
(elementary calculation)
α
α
+
α2 + (ω − β)2
α2 + (ω + β)2
• Limiting case:
H(ω)|ω=±β =
1
α
+
α
α2 + (2β)2
=⇒ tends towards ∞ or −∞ if α tends towards 0
H(ω )
h(t)
t
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|
|
−β
β
ω
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4. Modulated rectangle function (“truncated cosine”)
cos(β t),
|t| ≤ T /2
h(t) =
0,
|t| > T /2
H(ω) =
Z∞
h(t)e−jωt dt
−∞
T
=
Z2
cos(β t)e−jωt dt
− T2
=
...
=

(elementary calculation)
T
sin
(ω
−
β)
T
2


T
2
(ω − β)
2

T
sin (ω + β)
2 

+

T
(ω + β)
2
h(t)
h(t)
|
|
t
t
H(ω)
H(ω)
ω
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|
|
−β
β
ω
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Fourier Transform pairs (u = ω/2π)
Rectangle function
Sinc function
1
1
-1/2
sin(πu)
πu
1/2
Squared sinc function
Triangle function
1
1
-1/2
1/2
Exponential function
2α
α2+(2πu)2
e-α|x|
Gaussian function
e -αx
2
π - πu
e α
α
2
Unit impulse
δ(x)
1
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Inverse Fourier–transform
Z∞
H(ω) =
h(t)e−jωt dt
−∞
assumption:
with:
h̃(t) =
1
2π
H(ω) =
Z∞
=
=
=
H(ω)ejωt dω
−∞
h(τ )e−jωτ dτ
−∞
inserting H(ω) in h̃(t):
h̃(t) =
Z∞
1
2π
lim
Ω,T →∞

ZΩ
−Ω
1
lim lim
2π Ω→∞ T →∞
1
π
lim lim
Ω→∞ T →∞
lim
Ω→∞
1
π
Z∞
−∞
= h(t)

ZT

h(τ ) ejω(t−τ ) dτ  dω
−T
ZT ZΩ
ejω(t−τ ) dω h(τ ) dτ
−T −Ω
ZT
−T
sin (Ω(t − τ ))
h(τ ) dτ
t−τ
sin (Ω(t − τ ))
h(τ ) dτ
t−τ
due to:
1
lim
Ω→∞ π
Z∞
−∞
(→
sin(Ωt)
h(t) dt = h(0)
t
−∞
formal expression:
h(t) =
Z∞

1
2π
|
Z∞
−∞

ejω(t−τ ) dω  h(τ ) dτ
{z
= δ(t − τ )
}
distribution theory, see there for stronger proof)
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1.4
Properties of the Fourier Transform
Symmetry
H(ω) =
Z∞
h(t) e−jωt dt = F {h(t)}
1
2π
Z∞
−∞
h(t) =
−∞
H(ω) ejωt dω = F −1 {H(ω)}
F 2 {h(t)} = F {H(ω)} = 2πh(−t)
F −1 F {h(t)} = F −1 {H(ω)} = h(t)
• Time domain and frequency domain are correlated symmetrically.
• Properties of FT are valid in both domains, especially the convolution
theorem (see later).
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Theorems for the Fourier transform
H(ω) =
Z∞
e−jωt h(t) dt
−∞
consider the equation:
H(ω) = F {h(t)}
more exact:
{ω → H(ω)} = F {t → h(t)}
1. Linearity:
integral operator is linear
2. Inverse scaling, similarity principle:
Z∞
h(αt) e−jωt dt =
−∞
F {h(αt)} =
1
|α|
Z∞
ω
h(τ ) e−j α τ dτ
−∞
1
ω
H( ),
|α|
α
α ∈ IR\{0}
Note:
Absolute value, because integral boundaries are swapped for α < 0.
3. Shift:
h(t − t0 )
Z∞
−∞
h(t − t0 ) e−jωt dt = e−jωt0
= e−jωt0
Z∞
−∞
Z∞
h(t − t0 ) e−jω(t−t0 ) dt
h(τ ) e−jωτ dτ
−∞
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=⇒ F {h(t − t0 )} = e−jωt0 H(ω) t0 ∈ IR
with H(ω) = F {h(t)}
important:
| F {h(t − t0 )} | = | F {h(t)} |
, because
|e−jωt0 | = |e−ju | = | cos u − j sin u|
p
cos2 u + sin2 u
=
= 1
4. Symmetry and antisymmetry:
h(t) = h(−t)
results in
h(t) = −h(−t)
5. Complex conjugation:
Z∞
Im{H(ω)} = 0
results in
Re{H(ω)} = 0
suppose that h(t) is a complex function
h(t) e−jωt dt
Z∞
=
−∞
−∞
Z∞
=
h(t) ejωt dt
h(t) ejωt dt = H(−ω)
−∞
F {h(t)} = H(−ω) = F {h(t)}
Special case:
h(t) is real, so
h(t) = h(t)
=⇒ H(ω) = H(−ω) =⇒ | H(ω) | = | H(−ω) | = | H(−ω) |
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6. Differentiation:
dh
dt

∂  1
∂t
2π
=
1
2π
=
Z∞
Z∞
−∞

H(ω) ejωt dω 
H(ω) jω ejωt dω
−∞
F{
dh(t)
} = jω F {h(t)}
dt
Interpretation: differentiation = enhancement of high frequencies
(due to the multiplication with ω)
7. Integration:
F{
Zt
h(τ )dτ } =
−∞
Proof:
1
F {h(t)}
jω
similar to differentiation or inversion
8. Modulation principle:
F {h(t) cos(ω0 t)} =
Z∞
−∞
h(t) cos(ω0 t) e−jωt dt

Z∞

Z∞
1 
h(t) ejω0 t e−jωt dt +
h(t) e−jω0 t e−jωt dt 
2
−∞
 −∞

∞
Z
Z∞
1 
h(t) e−j(ω−ω0 )t dt +
h(t) e−j(ω+ω0 )t dt 
=
2
=
−∞
=
−∞
1
[ H(ω − ω0 ) + H(ω + ω0 ) ]
2
and similarly
F { h(t) sin(ω0 t) } =
1
[ H(ω − ω0 ) − H(ω + ω0 ) ]
2j
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y(t)
x(t)
h(t), H(ω)
Y(ω)
X(ω)
Convolution theorem
• Convolution in time domain corresponds to multiplication in frequency
domain
Z∞
Time domain:
y(t) = x(t) ∗ h(t) =
x(t − τ ) h(τ ) dτ
−∞
Frequency domain:
Y (ω) =
=
=
Z∞
−∞
Z∞
−∞
Z∞

e−jωt 
Z∞

h(τ ) x(t − τ ) dτ  dt
 −∞

Z∞
h(τ ) 
x(t − τ ) e−jωt dt  dτ
−∞
h(τ ) X(ω) e−jωτ dτ
−∞
= X(ω)
Z∞
(shifting)
h(τ ) e−jωτ dτ
−∞
= X(ω) H(ω)
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• Likewise, multiplication in time domain corresponds to convolution in
1
):
frequency domain (note the factor 2π
Time domain:
y(t) = a(t) · b(t)
Frequency domain:
Y (ω) =
=
Z∞
−∞
Z∞
−∞
=
=
1
2π
1
2π
a(t) · b(t) e−jωt dt
1
a(t)
2π
Z∞
−∞
Z∞
−∞
=
Z∞
B(ω̃)ej ω̃t e−jωt dω̃ dt
−∞
B(ω̃)
Z∞
a(t)e−j(ω−ω̃)t dt dω̃
−∞
A(ω − ω̃) · B(ω̃)dω̃
1
A(ω) ∗ B(ω)
2π
• Motivation for the Fourier transform:
FT gives the “simplest” representation of the system operation, because every LTI-System can be interpreted as convolution of the input
signal x(t) and the impulse response of the system h(t). Convolution
can be then efficiently calculated using FT and convolution theorem.
• Mathematical: eigenfunctions
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Example: Oscillator with excitation
Oscillator
x(t) −→
−→ y(t)
y ′′ (t) + 2α y ′ (t) + β 2 y(t) = x(t)
Z+∞
1
x(t) =
X(ω)ejωt dω
2π
y(t) =
y ′ (t) =
y ′′ (t) =
1
2π
1
2π
1
2π
−∞
Z+∞
Y (ω)ejωt dω
−∞
Z+∞
Y (ω)jω ejωt dω
−∞
Z+∞
Y (ω)[−ω 2 ] ejωt dω
−∞
Z+∞
Z+∞
[−ω 2 + 2αjω + β 2 ]Y (ω)ejωt dω =
X(ω)ejωt dω
Z+∞
−∞
−∞
−∞
[−ω 2 + 2αjω + β 2 ] Y (ω) − X(ω) ejωt dω = 0
|
{z
}
∀t
=0
In this way we obtain the transfer function of an oscillator:
H(ω) =
Y (ω)
1
=
X(ω) −ω 2 + 2αjω + β 2
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1
h(t) =
2π
Z+∞
H(ω)ejωt dω
−∞
(can be given explicitly)
Z+∞
x(t) h(t − τ )dτ
y(t) =
−∞
Note:
y(t) does not contain the component which corresponds to the homogeneous differential equation of the oscillator.
x(t)
Convolution with
h(t)
Inverse Fourier
Transform
Fourier
Transform
X(ω)
y(t)
Multiplication with
H(ω) = F{h(t)}
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1.5
Parseval Theorem
Convolution theorem:
F −1 {H(ω) X(ω)} =
(⋆)
1
2π
Z∞
H(ω) X(ω) ejωτ dω
−∞
Z∞
−∞
h(t) x(τ − t) dt
= (h ∗ x) (τ )
We make two special assumptions:
i) x(−t) := h(t), then: X(ω) = H(ω)
ii) τ = 0
Inserting in (⋆) results in:
1
2π
Z∞
−∞
1
2π
H(ω)H(ω) dω
Z∞
−∞
Z∞
=
|H(ω)|2 dω
=
−∞
Z∞
−∞
h(t)h(t) dt
|h(t)|2 dt = E
• Energy E in time domain = Energy E in frequency domain
1
1
; aid: use normalization factor √
for both
(up to the factor
2π
2π
directions of Fourier Transform)
• Physical aspect: energy conservation
• Mathematical aspect: unitary (orthogonal) representation in vector
space
• |H(ω)|2 is called power spectral density.
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1.6
Autocorrelation Function
Autocorrelation function
• Autocorrelation function of time continuous
signal or function h(t) is defined as:
R(t) =
Z∞
h(τ ) h(t + τ )dτ
−∞
• The following equation is valid:
R(t) = h(t) ∗ h(−t)
→
• Fourier transform gives:
which results in
R(t) = R(−t)
(”Wiener-Khinchin Theorem”)
F {R(t)} = H(ω) H(ω) = |H(ω)|2
• Thus: Fourier transform connects autocorrelation
function R(t) and power spectral density |H(ω)|2
|H(ω)|2 =
Z∞
R(t) e−jωt dt =
Z∞
R(t) cos(ωt) dt
−∞
−∞
• Remark:
autocorrelation is a special case of the cross correlation between signals x(τ ) and h(t)
Ch,x =
Z∞
h(τ ) x(t + τ )dτ
−∞
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1.7
Existence of the Fourier Transform
Conditions for h(t) for the existence of the Fourier transform
H(ω) =
Z∞
1
h(t) =
2π
e−jωt h(t) dt ,
−∞
Z∞
ejωt H(ω) dω
−∞
When are those equations valid?
Sufficient conditions:
1. h(t) is absolutely integrable:
Z∞
−∞
|h(t)|dt < ∞
2. h(t) has finite number of jumps, minima and maxima in each interval
of IR
3. h(t) has no infinite jumps
More general conditions are possible (but rather complex set of conditions):
• Generalized functions, distributions,
definition as functional
• Example: δ-function:
Z∞
δ(t) h(t) dt = h(0) for all functions h
−∞
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Impulse response:
y(t) =
Z∞
−∞
h(t − τ )δ(τ ) dτ
= h(t) ∗ δ(t)
= h(t)
Consequence:
h(t) ≡ 1 ⇒
Z∞
δ(t) dt = 1
−∞
• A function like δ(t) does not “exist”. But it is possible to define the
functional for each function t → h(t):
[t → h(t)] −→ δ̃(h) := h(0)
1.8
δ-Function
Starting point: definition of the δ-function as a boundary case of
a function δǫ (t):
lim
Z+∞
ǫ→0
−∞
f (t) δǫ (t) dt = f (0)
(1.1)
• Possible realizations of δǫ (t)

 1 t ∈ [−ǫ, +ǫ]
2ǫ
a) δǫ (t) =

0 otherwise
b) δǫ (t) =
1 ǫ
π ǫ2 + t2
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c) δǫ (t) =
1 sin (t/ǫ)
π
t
d) δǫ (t) = √
1
2πǫ2
t2
−
e 2ǫ2
• During inversion of the Fourier transform we have “formally” obtained:
δ(t) =
1
2π
Z+∞
ejωt dω = lim
Ω→∞
1 sin (Ωt)
π
t
(1.2)
−∞
Fourier transform F {δ(t)}:
F {δ(t)} =
Z+∞
e−jωt δ(t) dt
−∞
due to (1.1) the following holds:
F {δ(t)} = ejωt |t=0 = 1
• Another derivation using (1.2):
δ(t) =
=
1
2π
1
2π
Z+∞
−∞
Z+∞
ejωt F {δ(t)} dω
ejωt dω
general
according to (1.2)
−∞
Comparison results in:
F {δ(t)} = 1
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From this we obtain the following equations:
From symmetry property:
F {1} = 2 π δ(ω)
From shifting theorem:
F {ejω0 t 1} = 2 π δ(ω − ω0 )
cos (ω0 t) =
=
1 jω0 t
e
+ e−jω0 t
2 

Z+∞
Z+∞
1 
δ(ω + ω0 ) ejωt dω 
δ(ω − ω0 ) ejωt dω +
2
= π
−∞
Z+∞
1
2π
−∞
−∞
[ δ(ω − ω0 ) + δ(ω + ω0 ) ] ejωt dω
F { cos (ω0 t) } = π [ δ(ω − ω0 ) + δ(ω + ω0 ) ]
Note: another derivation:
consider “damped oscillations”
1 −α|t|
e
cos (ω0 t)
2π
in the limit α → 0 .
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Comb function
• define “comb function” (pulse train, sequence of δ-impulses):
+∞
X
x(t) =
n=−∞
δ(t − nT )
• Fourier transform of comb function:
X(ω) =
Z+∞
x(t) e−jωt dt
−∞
=
=
=
Z+∞ X
+∞
δ(t − nT ) e−jωt dt
X
δ(t − nT ) e−jωt dt
−∞ n=−∞
+∞
+∞ Z
n=−∞−∞
+∞
X
−jωnT
e
n=−∞
=
=
...
(see Papoulis 1962, p. 44)
+∞
2π X
2π
δ(ω − n )
T n=−∞
T
• in words:
δ-impulse sequence with period T in time domain
produces
δ-impulse sequence with period T1 in frequency domain
(i.e. 2π
T in ω-frequency domain)
comb function is transformed to comb function
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Comb function
Σ
n=-
-6T
2π
T
δ(t-nT)
-3T -T
T
3T
Σ
n=-
-6π -4π -2π
T T T
6T
δ(ω-n2π/T)
2π 4π 6π
T T T
1(δ(ω-ω )+δ(ω+ω ))
0
0
2
cos(ω0t)
−ω0
ω0
1 j(−δ(ω-ω )+δ(ω+ω ))
0
0
2
sin(ω0t)
ω0
−ω0
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1.9
Motivation for Fourier Series
x:
IR −→ IR
t −→ x(t)
Consider a periodical function x with period T :
x(t) = x(t + T )
then also x(t) = x(t + kT )
for each t ∈ IR
for k ∈ Z
Examples:
• Constant function:
x0 (t) = A0
• Harmonic oscillator:
x1 (t) = A1 cos (
2π
t + ϕ1 ) ,
T
A1 > 0
• All higher harmonic:
xn (t) = An cos (n
2π
t + ϕn ) ,
T
An > 0
therefore
x(t) =
∞
X
An cos (n ω0 t + ϕn ) with ω0 =
n=0
is periodical with period T =
2π
,
T
An ≥ 0
2π
ω0
• Another notation:
x(t) =
∞
X
Bn e−j n ω0 t
where Bn
is a complex number
n=−∞
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Line spectrum representation
Real measured signal has always a ”widespread” spectrum.
Reasons:
• Strictly periodical signal (almost) never exists
– Period can fluctuate
– ”Wave form” within one period can fluctuate
– Only a finite section of the signal is analyzed
(”window function”)
• Only a strictly periodical signal has a sharp line spectrum
Remarks:
• Fourier series are actually not strictly related to periodical functions:
a finite interval of IR is sufficient (the signal is then interpreted as
infinitely prolonged).
• By transition from the finite interval to the complete real axis the
Fourier series becomes Fourier integral.
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Calculation of Fourier coefficients:
• Consider a periodical function x(t) with period T =
2π
ω0
• approach:
x(t) =
+∞
X
an ej n ω0 t
a∈C
n=−∞
• multiplication with e−j m ω0 t where m ∈ IN and integration over one
period result in:
+T
Z /2
x(t) e−j m ω0 t dt =
+∞
X
ej (n−m) ω0 t dt
an
n=−∞
−T /2
+T
Z /2
−T /2
• Due to “orthogonality” holds:
+T
Z /2
j (n−m) ω0 t
e
dt =
−T /2
T
0
if n = m
if n =
6 m
• Then:
ZT /2
x(t) e−j m ω0 t dt = am T
−T /2
• Result:
an
=
1
T
+T
Z /2
x(t) e−j n ω0 t dt
−T /2
=
1
T
+T
Z /2
x(t) cos (n ω0 t) dt − j
−T /2
1
T
+T
Z /2
x(t) sin (n ω0 t) dt
−T /2
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Spectrum of a periodical function
• If x(t) is periodical with the period T =
x(t) =
+∞
X
2π
ω0
an ej n ω0 t ,
n=−∞
, then
an ∈ C
• The Fourier transform X(ω) is:
X(ω) = F {x(t)}
+∞
X
an
=
n=−∞
+∞
X
= 2π
F {ej n ω0 t }
| {z }
= 2πδ(ω − nω0 )
n=−∞
an δ(ω − nω0 )
• Note:
This derivation is formal, because the Fourier integral does not
exist in the “usual sense”;
strict derivation within the scope of distribution theory.
• In words:
a periodic function with the period T has a Fourier transform in the
form of a line spectrum with the distance ω0 = 2π
T between the components.
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1.10
Time Duration and Band Width
1. Similarity principle:
F {h(αt)} =
1
ω
H( )
|α|
α
h(t)
H( ω )
ω
t
0<α<1:
_ H( _
1
ω
α
α)
h(α t)
ω
t
time duration T
band width B
T · B
= const.
• High resolution in the time domain results in low resolution in the
frequency domain and vice versa
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2. Special case: h(t) with
Im {H(ω)} = 0 ( h(t) symmetrical )
and
Re {H(ω)} ≥ 0
h(t) has maximum for t = 0:
1
h(t) =
2π
Z∞
−∞
1
H(ω) cos(ωt) dω ≤
2π
Z∞
H(ω) dω = h(0)
−∞
define:
T
B
=
=
1
h(0)
Z∞
−∞
Z∞
1
H(0)
h(t) dt
H(ω) dω
−∞
from
T
=
H(0)
h(0)
and B = 2 π
h(0)
H(0)
follows
T · B
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= 2π
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3. In general:
normalized impulse h(t) ∈ IR with
Z∞
h2 (t) dt = 1,
h(t) ∈ IR
−∞
T2
B2
Z∞
:=
−∞
Z∞
:=
−∞
h2 (t) t2 dt
| H(ω) |2 ω 2 dω
• Results in “uncertainty relation”:

= 2π
T · B ≥
r
Z∞
−∞

2
[h′ (t)] dt
π
2
• Proof: Cauchy-Schwarz inequality
| xT y | ≤ ||x|| · ||y||
∞
2
Z
Z∞
Z∞
2
2
[ t h(t)] dt
[ h′ (t) ] dt
[ t h(t) ] h′ (t) dt ≤
−∞
|
{z
}
{z
}−∞
{z
}
|
|−∞
2
2
1
B
=T
=
4
2π
From:
partial integration
Z
Z
′
u (t) v(t) dt = u(t) v(t) −
Z
u(t) v ′ (t) dt
1
t dt = h(t)2 t −
[ h(t) h′ (t) ] |{z}
{z
}
|
2
v(t)
u′ (t)
Z∞
−∞
Z
[ h(t) h′ (t) ] t dt = 0 −
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1 2
h (t) 1 dt
2
1
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Equality sign is valid for linear dependency:
h′ (t) = −λ t h(t)
dh
= −λ t dt
h
1
log(h) = − λ t2 + const.,
2
=⇒
Optimum T · B =
r
π
2
λ > 0
for Gauss’ impulse
1 2
λ − λt
h(t) = √ e 2
2π
Variance: σ 2 =
1
λ
Quantum Physics: similar statement about position and impulse
of a particle
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4. Finite positive signal
≥0
0≤t≤T
g(t)
=0
t < 0 or T < t
g(t)
0
T
t
The following is valid for the amplitude spectrum |G(ω)|:
+∞
Z
−jωt
g(t) e
dt |G(ω)| = ≤
=
−∞
+∞
Z
|g(t)| |e−jωt |dt
−∞
Z+∞
−∞
|g(t)| dt
because g(t) ≥ 0
= G(0)
Define the band width ωB as:
|G(ωB )|2 =
G2 (0)
2
and
|G(ωB )|2 ≤ |G(ω)|2 for |ω| < ωB
Then:
T ωB ≥
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Proof:
The following inequalities are valid:
(a − b)2
a +b ≥
2
| sin α| + | cos α| ≥ 1
2
2
∀ a, b ∈ IR
∀ α ∈ IR
For the Fourier-Transform of g(t) holds:
Re{G(ω)} =
ZT
g(t) cos ωt dt
0
Im{G(ω)} = −
ZT
g(t) sin ωt dt
0
π
holds:
cos ωt ≥ 0, sin ωt ≥ 0
2
and therefore:
cos ωt + sin ωt = | cos ωt| + | sin ωt| ≥ 1
For 0 ≤ ω t ≤
Re{G(ω)} − Im{G(ω)} =
ZT
g(t) [cos ωt + sin ωt] dt
≥
ZT
g(t) 1 dt
0
0
= G(0)
|G(ω)|2 = Re2 {G(ω)} + Im2 {G(ω)}
[Re{G(ω)} − Im{G(ω)}]2
≥
2
1 2
G (0) ≡ |G(ωB )|2
≥
2
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Chapter 2
Discrete Time Systems
• Overview:
2.1 Motivation and Goal
2.2 Digital Simulation using Discrete Time Systems
2.3 Examples of Discrete Time Systems
2.4 Sampling Theorem and Reconstruction
2.5 Logarithmic Scale and dB
2.6 Quantization
2.7 Fourier Transform and z–Transform
2.8 System Representation and Examples
2.9 Discrete Time Signal Fourier Transform Theorems
2.10 Discrete Fourier Transform (DFT)
2.11 DFT as Matrix Operation
2.12 From continuous FT to Matrix Representation of DFT
2.13 Frequency Resolution and Zero Padding
2.14 Finite Convolution
2.15 Fast Fourier Transform (FFT)
2.16 FFT Implementation
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2.1
Motivation and Goal
If we want to process a continuous time signal x(t) with a computer, we
have to sample it at discrete equidistant time points
tn = n · TS
where TS is called sampling period.
Terminology:
“time discrete” is often called “digital”, where this adjective often
(but not always) denotes the amplitude quantization,
i.e. the quantization of the value x(n · TS ).
Advantages of digital processing in comparison to analog components:
• independent of analog components and technical difficulties with respect to their realization;
• in principle arbitrary high accuracy;
• also non-linear methods are possible,
in principle even every mathematical method.
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2.2
Digital Simulation using Discrete Time Systems
Task definition:
• Given:
Analog system with input signal x(t) and output signal y(t);
Sampling with sampling period TS
• Wanted:
Discrete System with input signal x[n] and output signal y[n], such
that
x[n] = x(nTS )
results in
y[n] = y(nTS )
• For which signals is such a digital simulation possible?
• The sampling theorem gives (most of) the answer.
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LTI System (analog to continuous time case):
• Linearity:
Homogeneity:
S {α x[n]} = α S {x[n]}
Additivity:
S {x1 [n] + x2 [n]} = S {x1 [n]} + S {x2 [n]}
• Shift invariance:
S {x[n − n0 ]} = y[n − n0 ],
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n0
whole number
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Representation of an LTI System as discrete convolution:
Unit impulse:
δ[n] =
1,
0,
n = 0
n 6= 0
The signal x[n] is represented with amplitude weighted and time shifted
unit impulses δ[n]. The system reacts on δ[n] with h[n]:
h[n] = S {δ[n]}
Input signal:
∞
X
x[n] =
k=−∞
x[k] δ[n − k]
Output signal:
y[n] = S
(
∞
X
k=−∞
x[k] δ[n − k]
)
Additivity
=
∞
X
S { x[k] δ[n − k] }
∞
X
x[k] S { δ[n − k] }
k=−∞
Homogeneity
=
k=−∞
Time invariance
=
∞
X
k=−∞
x[k] h[n − k]
Input signal x[n] and output signal y[n] of a discrete time LTI system are
linked through discrete convolution.
h[n] is called impulse response like in continuous time case.
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2.3
Examples of Discrete Time Systems
• Difference calculation:
y[n] = x[n] − x[n − n0 ]
• “1-2-1”-averaging:
y[n] = 0.5 · x[n − 1] + x[n] + 0.5 · x[n + 1]
• sliding window averaging (“smoothing”)
M
X
1
y[n] =
x[n − k]
2M + 1
k=−M
• weighted averaging: instead of constant weight
h[n] =
1
2M + 1
arbitrary weights can be used:
y[n] =
M
X
k=−M
h[k] · x[n − k]
Note: the only difference from general case is
finite length of the convolution kernel h[n].
• First order difference equation:
(recursive averaging, averaging with memory)
y[n] − α y[n − 1] = x[n]
• (Digital) resonator (second order difference equation)
y[n] − α y[n − 1] − β y[n − 2] = x[n]
• Image processing:
Gradient calculation and image enhancement
(Roberts Operator, Laplace Operator)
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• Roberts Cross Operator
gray values x[i, j]
j+1
j
i
i+1
2
|∇x[i, j]|2 = (x[i, j] − x[i + 1, j + 1])2 + (x[i, j + 1] − x[i + 1, j])2
Note: non-linear operation
simplified:
|∇x[i, j]| = |x[i, j] − x[i + 1, j + 1]| + |x[i, j + 1] − x[i + 1, j]|
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Figure 2.1: Digital photo
Figure 2.2: Gradient image
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• Laplace Operator ≡ discrete approximation of the second derivation
∇2 x[i, j] = △2i x[i, j] + △2j x[i, j]
= x[i + 1, j] − 2x[i, j] + x[i − 1, j] +
x[i, j + 1] − 2x[i, j] + x[i, j − 1]
= x[i + 1, j] + x[i − 1, j] + x[i, j + 1] + x[i, j − 1] − 4x[i, j]
1
-2
1
j+1
1
1
-4
1
-2
1
j-1
1
i-1
i
j
i+1
1
Image enhancement:
y[i, j] = x[i, j] − ∇2 x[i, j]
= h[i, j] ∗ x[i, j]
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Figure 2.3: Several real cases of Laplace Operator subtraction from original image. a)
Original image b) Original image minus Laplace Operator (negative values are set to 0
and values above the grey scale are set to the highest grade of grey)
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2.4
Sampling Theorem (Nyquist Theorem) and Reconstruction
The following will be analyzed and derived respectively:
How should we choose the sampling period TS , if we want to represent a
continuous signal x(t) with its sample values x(nTS ) so that the signal x(t)
can be exactly reconstructed from its sample values?
• Fourier transform of the continuous time signal x(t):
Z∞
X(ω) = F { x(t) } =
x(t) e−jωt dt
−∞
1
x(t) = F −1 { X(ω) } =
2π
Z∞
X(ω) ejωt dω
(2.1)
−∞
• Signal x(t) has limited bandwidth with upper limit ωB , which means:
X(ω) = 0
for all |ω| ≥ ωB
Note: X(ωB ) = 0
• X(ω) in domain −ωB < ω < ωB can be represented as Fourier Series:
∞
X
ω
an exp(−jnπ )
X(ω) =
(2.2)
ω
B
n=−∞
• The coefficients an are given by:
ZωB
ω
1
X(ω) exp(jnπ ) dω
an =
2ωB
ωB
(2.3)
−ωB
• Comparison of the equations (2.1) and (2.3) shows that the coefficients
an are given by the values of the inverse Fourier transform of x(t) at
points
nπ
tn =
(2.4)
ωB
The band limitation of X(ω) has to be considered for the integration
limits in (2.1). Result:
π
nπ
(2.5)
an = x( ) ·
ωB ωB
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• Inserting Eq. (2.5) into Eq. (2.2) and then in Eq. (2.1) results in:
1
x(t) =
2π
ZωB
−ωB
∞
nπ
ω
π X
x( ) exp(−jnπ ) exp(jωt) dω
ωB n=−∞ ωB
ωB
• After swapping summation and integration and subsequent integration:
x(t) =
∞
X
x(
n=−∞
nπ
)
ωB
nπ
))
ωB
nπ
)
ωB (t −
ωB
sin(ωB (t −
• Reconstruction of the signal x(t) from sample values is possible if
nπ
equidistant sample values x( ) = x(n · Ts ) have the distance TS
ωB
π
(2.6)
TS =
ωB
• The sampling period TS corresponds to the sampling frequency ΩS :
ΩS =
2π
TS
• Equation (2.6) shows that if the sampling frequency is
ΩS := 2 ωB
the original signal x(t) can be reconstructed exactly.
• In the Fourier series representation of X(ω) in equation (2.2), the
period 2 · ωB has been supposed.
ωB is the highest frequency component of the signal x(t).
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• Since X(ω) is equal to zero for |ω| ≥ ωB , the period 2 · ωB can be
substituted with every period 2 · ω
eB where ω
eB ≥ ωB . The previous
derivation is also valid for this ω
eB .
• When
ω
eB =
then:
x(t) =
∞
X
x(n TS )
n=−∞
π
TS
sin(π (t − n TS )/TS )
π (t − n TS )/TS
(reconstruction formula)
= 1 (l’Hôpital’s rule)
Note: limt→0 sin(t)
t
• The condition ω
eB ≥ ωB results in:
TS ≤
π
ωB
(2.7)
for the sampling period TS and in:
ΩS ≥ 2 · ωB
(2.8)
for the sampling frequency ΩS .
• The equations (2.7) and (2.8) are denoted as sampling theorem. The
sampling frequency has to be at least twice as high as the upper limit
frequency of the signal ωB where X(ω) = 0 for |ω| ≥ ωB . If and only
if this condition is satisfied, an exact reconstruction (without approximation) of a continuous signal x(t) from its sample values x(nTS ) is
possible.
• Note: The sampling frequency ΩS = 2 · ωB is also called
Nyquist frequency.
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a)
x(t)
t
b)
xs(t)
t
T
c)
xr(t)
T
Figure 2.4: Ideal reconstruction of a band-limited signal (from Oppenheim, Schafer)
a) original signal b) sampled signal c) reconstructed signal
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X(ω)
a)
−ωΒ
ωΒ
ω
XS1(ω) , ΩS > 2ωΒ
b)
...
...
−ωΒ
-ΩS
ΩS
ωΒ
ω
XS2(ω) , ΩS = 2ωΒ (Nyquist rate)
c)
...
...
-ΩS
−ωΒ
ωΒ
ΩS
ω
XS3(ω) , ΩS < 2ωΒ (aliasing)
d)
...
...
ωΒ ΩS
Β
−Ω−ω
S
ω
Figure 2.5: Sampling of band-limited signal with different sampling rates:
b) sampling rate higher than Nyquist rate - exact reconstruction possible
c) sampling rate equal to Nyquist rate - exact reconstruction possible
d) sampling rate smaller than Nyquist rate - aliasing - exact reconstruction not possible
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Another proof using delta- and comb-function:
Sampling of the continuous signal x(t) with ΩS =
2π
TS
• Band limitation: X(ω) = 0 for |ω| ≥ ωB
(always possible: analog to low-pass with T (ω) = 0 for |ω| ≥ ωB )
• Sampling procedure
Multiplication of a function with a comb-function in time domain
xs (t) = Ts x(t) ·
+∞
X
n=−∞
δ(t − nTs )
results in a convolution with a comb-function in frequency domain:
+∞
1
2πn
2π X
Xs (ω) = Ts · X(ω) ∗
δ ω − ω̃
2π
Ts n=−∞
Ts
Z+∞
+∞
X
2πn
dω̃
δ ω − ω̃
X(ω̃)
=
T
s
n=−∞
−∞
+∞
X
2π
X ω−n
=
Ts
n=−∞
=⇒ sampled signal has periodical Fourier spectrum
(Analogy to Fourier series: periodical signal has line spectrum, i.e.
discrete spectrum)
No overlap if:
ωB ≤ ΩS − ωB
2ωB ≤ ΩS
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• In so-called digital simulation, the signal x(t) is represented by its
sampled values x(n · TS ) measured at equidistant time points with
distance TS . With a proper sampling period TS an exact reconstruction of the signal x(t) from the sampled values x(n · TS ) is possible.
• If it is possible to exactly reconstruct the signal x(t) from the sampled
values x(n·TS ), then it is possible to perform a discrete time processing
of the sampled values x(n · TS ) on a computer, which is equivalent to
the continuous time processing of the signal x(t) (digital simulation).
• Continuous time processing:
y(t) =
Z∞
−∞
x(τ ) h(t − τ ) dτ
• Discrete time processing:
– Sampling period TS
– x[n] := x(nTS )
y(nTS ) =
y[n] =
∞
X
k=−∞
∞
X
k=−∞
x(kTS ) h(nTS − kTS ) TS ,
h̃[n] = h(nTS )
x[k] h̃[n − k]
• As a result of the convolution theorem (convolution in time domain
corresponds to multiplication in frequency domain), the band limited
input signal gives an also band limited output signal which is exactly
determined by its sampled values.
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Important:
• In the domain |ω| < ΩS /2 the Fourier transform of a continuous time
signal x(t) is identical with the Fourier–transform of the corresponding
sampled discrete time signal x(nTS ):
Z∞
X(ω) =
x(t) exp(−jωt) dt
−∞
for |ω| ≤ ΩS /2 is identical to
∞
X
x(nTS ) exp(−jωTS n)
TS · XS (ω) = TS ·
= TS ·
n=−∞
∞
X
x(nTS ) exp(−j
n=−∞
2πω
n)
ΩS
• Inverse Fourier transform of discrete time signal:
x(nTS ) =
1
ΩS
Ω
ZS /2
XS (ω) exp(jωTS n) dω
−ΩS /2
• One period:
ΩS
ΩS
≤ ω ≤
2
2
2πω
≤ π
−π ≤
ΩS
−
• The Fourier transform of a discrete time signal is periodic in ω with
the period 2 π/TS = ΩS .
• The Fourier transform of a discrete time signal is
continuous in ω.
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Frequency normalization
• Define the normalized frequency ωN :
ωN : = 2π
• Definition:
ω
ΩS
(ω now denotes a normalized frequency)
– Fourier transform of discrete time signal x[n]:
+∞
X
jω
X(e ) =
x[n] exp(−jωn)
n=−∞
Note the notation X(ejω ).
– Inverse Fourier transform of discrete time signal x[n]:
1
x[n] =
2π
Zπ
X(ejω ) exp(jωn) dω
−π
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2.5
Logarithmic Scale and dB
Why?
– large dynamic range for the amplitude values of a signal
x(t) = A cos βt
A :=
amplitude
(pressure, velocity, inclination, current, voltage, ... )
“linear” variable
A0 :=
reference amplitude
predefined value for calibration
dB := “decibel”
A[dB] ≡ 20 · lg
A
,
A0
A2
= 10 · lg 2 ,
A0
lg ≡ log10
A2 = quadratic variable = energy, intensity
because of 210 = 1024 ∼
= 103 :
1 bit more =
ˆ
factor 2 for amplitude =
ˆ 6 dB
= factor 4 for intensity
3 dB =
ˆ factor 2 for intensity
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Phonem: s
Phonem: s
7
2
6
1.5
1
5
0.5
A
log A
4
3
0
-0.5
2
-1
1
0
-1.5
0
1000
2000
3000
4000
f / Hz
5000
6000
7000
-2
8000
Figure 2.6: Amplitude spectrum of the
voiceless phoneme “s” from the word
“ist”
0
1000
2000
3000
4000
f / Hz
5000
6000
7000
8000
Figure 2.7: Logarithmic amplitude spectrum of the phoneme “s”
Phonem: ae
Phonem: ae
12
2.5
2
10
1.5
1
log A
A
8
6
0.5
0
4
-0.5
2
0
-1
0
1000
2000
3000
4000
f / Hz
5000
6000
7000
-1.5
8000
Figure 2.8: Amplitude spectrum of the
voiced phoneme “ae” from the word
“Äh”
0
1000
2000
3000
4000
f / Hz
5000
6000
7000
8000
Figure 2.9: Logarithmic amplitude spectrum of the phoneme “ae”
Pause
Pause
1
0
0.9
-0.5
0.8
0.7
-1
log A
A
0.6
0.5
-1.5
0.4
-2
0.3
0.2
-2.5
0.1
0
0
1000
2000
3000
4000
f / Hz
5000
6000
7000
-3
8000
Figure 2.10: Amplitude spectrum of a
speech pause
0
1000
2000
3000
4000
f / Hz
5000
6000
7000
8000
Figure 2.11: Logarithmic amplitude
spectrum of a speech pause
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2.6
Quantization
• Uniform quantization
-X MAX
XMAX
• Quantisation: x̂ = Q(x)
• B bits correspond to 2B quantisation levels
• Boundaries:
x0 , x1 , . . . , xk , . . . , xK
where
K = 2B
• Width △ of one quantisation level using uniform quantisation:
△ =
2 · XM AX
2B
• Quantisation error:
σe2
Z+∞
Zxk
K
X
=
(x − x̂)2 p(x) dx =
(x − x̂k )2 p(x) dx
k=1 x
k−1
−∞
– for uniform quantisation:
a)
b)
xk − xk−1 = △ = const(k)
x̂k = 12 (xk−1 + xk )
– uniform distribution with p(x) = const(x) results in:
σe2
=
X △2
k
2
1
△2
XM
AX
·
=
=
12 K
12
3 · 22B
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• signal-to-noise ratio in dB (general definition):
σx2
SN R[dB] := 10 lg 2
σn
σx2 = power of the signal x
σn2 = power of the noise n
SN R = signal-to-noise ratio
• signal-to-quantisation noise ratio (special case):
σx2
SN R[dB] := 10 lg 2
σe
σe2 = power of the noise caused by quantisation errors
• uniform quantisation using B bits:
SN R[dB] = 6.02 B + 4.77 − 20 lg
XM AX
σx
• if signal amplitude has Gaussian distribution, only 0.064% of samples
have amplitude greater than 4σx :
SN R[dB] = 6.02 B − 7.2
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for XM AX = 4σx
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2.7
Fourier Transform and z–Transform
Transfer function and Fourier transform
Eigenfunctions of discrete linear time invariant systems (analog to time
continuous case):
x[n] = ej ω n
−∞ < n < ∞
(ω is dimensionless here)
Proof:
x[n] = ej ω n
∞
X
y[n] =
h[k] ej ω (n−k)
k=−∞
∞
X
jωn
= e
h[k] e−j ω k
k=−∞
Define:
H(ej ω ) =
∞
P
h[k] e−j ω k
k=−∞
Remark:
The Fourier transform of a discrete time signal is already introduced as
Fourier series during the derivation of sampling theorem and reconstruction formula (equation (2.2)).
Result:
y[n] = ej ω n H(ej ω )
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z–transform:
• Fourier transform of a discrete time signal: x[n]
+∞
X
jω
X(e ) =
x[n] e−jωn
n=−∞
–
periodic in ω
–
ω is normalized frequency, thence:
−π < ω ≤ π
–
X is evaluated on the unit circle (ejω )
• Generalization: X is evaluated for any complex values z.
• That results in z–transform:
+∞
X
X(z) =
x[n] z −n
n=−∞
• Reasons for z–transform
1. analytically simpler, function theory methods are applicable
2. better handling of convergence problem:
– convergence of finite signal, i.e. x[n] = 0 for each n > N0
– convergence of infinite signal depends on z
• Inverse z–transform:
I
1
x[n] =
2πj
formally: z = ejω
X(z) z n−1 dz
dz = jzdω
x[n] =
1
2π
Z2π
X(ejω ) ejωn dω
0
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Example of Fourier transform and z–transform:
• “Truncated geometric series”
n
a
x[n] =
0
• z–transform
N
−1
X
X(z) =
n
a z
−n
0≤n≤N −1
otherwise
=
n=0
1
=
z N −1
N
−1
X
−1 n
(a z )
n=0
N
N
z −a
z−a
1 − (a z −1 )N
=
1 − a z −1
• Fourier transform
z–transform results in Fourier transformation using substitution
z = ejω
jω
X(e ) =
1 − aN e−jωN
1 − a e−jω
special case for a = 1 (discrete time rectangle):
ωN
sin
ω(N − 1)
2
ω = exp −j
2
sin
2
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Proof for the z–transform inversion
• Statement:
1
x[k] =
2πj
I
X(z) z k−1 dz
• Cauchy integration rule
I
1
1
k=1
z −k dz =
0
k 6= 1
2πj
I
I X
1
1
x[n] z −n+k−1 dz
X(z) z k−1 dz =
2πj
2πj
n
I
X
1
x[n]
=
z −n+k−1 dz
2πj
n
|
{z
}
6= 0 only for n = k
= x[k]
• Fourier:
z = ejω
=⇒
dz = j ejω dω
Then:
x[n] =
1
2πj
Z+π
X(ejω ) (ejω )n−1 j ejω dω
−π
Integration path is unit circle because of ejω
=
1
2π
Z+π
X(ejω ) ejωn dω
−π
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2.8
System Representation and Examples
Example 1: Difference calculation
• Difference equation
y[n] = x[n] − x[n − n0 ],
• Fourier transform gives:
∞
X
−jωn
y[n] e
n=−∞
=
∞
X
n0 integral number
−jωn
x[n] e
n=−∞
Y (ejω ) = X(ejω ) −
jω
∞
X
n=−∞
−jωn0
= X(e ) − e
• Then follows:
−
∞
X
n=−∞
x[n − n0 ] e−jωn
x[n] e−jωn e−jωn0
X(ejω )
Y (ejω )
H(e ) =
X(ejω )
= 1 − e−jωn0
jω
|H(ejω )|2
= (1 − cos(ωn0 ))2 + sin2 (ωn0 )
= 1 − 2cos(ωn0 ) + cos2 (ωn0 ) + sin2 (ωn0 )
= 2 (1 − cos(ωn0 ))
|H(eiω )|2
5
4
3
2
1
0
ω
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Example 2: First order difference equation
x[n]
y[n]
+
α
Delay
y[n-1]
x[n] + α y[n − 1] = y[n]
y[n] − α y[n − 1] = x[n]
⇐⇒
Method 1: Estimation of transfer function H(ejω )
from impulse response h[n]:
• From the Eq. above with y[n] = h[n] and x[n] = δ[n] follows:
h[n] = δ[n] + α h[n − 1]
= δ[n] + α δ[n − 1] + α2 δ[n − 2] + · · ·
n
α ,
n≥0
=
0,
otherwise
• Fourier spectrum/transfer function H(ejω )
jω
H(e ) =
=
=
+∞
X
h[k] e−jωk
k=−∞
+∞
X
αk e−jωk
k=0
+∞
X
α e−jω
k=0
=
1
1 − α e−jω
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for |α| < 1
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Method 2: Estimation of transfer function H(ejω ) using
Fourier transform of difference equation:
• Difference equation:
y[n] − α y[n − 1] = x[n]
• Fourier–transform:
Y (ejω ) − α e−jω Y (ejω ) = X(ejω )
• Result:
H(ejω ) =
=
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Y (ejω )
X(ejω )
1
1 − α e−jω
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Example 3: Linear difference equations (with constant coefficients)
• Difference equation:
y[n] =
I
X
i=0
b[i] x[n − i] −
• z-transform:
Y (z) = X(z)
I
X
b[i]z
−i
i=0
• Result:
H(z) =
=
Y (z)
X(z)
I
P
J
X
a[j]z −j
j=1
b[i] z −i
1+
j=1
=
j=1
a[j] y[n − j]
− Y (z)
i=0
J
P
+∞
X
J
X
a[j] z −j
h[n] z −n
n=−∞
Using the definition of H(z) we can optain the impulse response as a
function of the coefficients of the difference equation in the above term.
Remark:
If we factorise denominator and numerator polynoms into linear factors, we can obtain a zero-pole-representation of a discrete time LTI
system:
ΠI (z − vi )
H(i) = Ji=1
Πj=1 (z − wj )
with zeros vi ∈ C and poles wj ∈ C.
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• in general:
h[n] has infinite number of non-zero values
=⇒ IIR–filter: Infinite Impulse Response
• but if: a[j] ≡ 0
∀j
h[n] identical to zero outside of a finite interval
h[n] =
b[n]
0
n = 0, . . . , I
otherwise
=⇒ FIR–filter: Finite Impulse Response
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Example 4:
Impulse response as “truncated geometric series”
h[n] =
H(z) =
an
0
M
X
0≤n≤M
otherwise
n
a z
−n
n=0
a ∈ IR
1 − aM +1 z −(M +1)
=
1 − a z −1
system operation:
y[n] =
∞
X
k=−∞
=
M
X
k=0
h[k] x[n − k]
ak x[n − k]
or as difference equation (“recursively”)
y[n] − a y[n − 1] = x[n] − aM +1 x[n − M − 1]
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For this example we consider the zero-pole-representation:
1 − ( az )−(M +1)
H(z) =
1 − ( az )−1
Zeros:
Pole:
zk
z0
α>0
2πk
k = 0, 1, . . . , M
= a · ej M +1
= a
(cancelled by zero z0 = a)
Im
M=11
a
Re
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Example 5:
Fibonacci numbers
Difference equation:
n≥0
h[n + 2] = h[n + 1] + h[n]
h[0]
= h[1] = 1
h[n]
= 0
n<0
H(z) =
∞
X
h[n]z −n
n=−∞
= 1 + z
= 1 + z
−1
−1
+
+
∞
X
n=0
∞
X
h[n + 2]z −(n+2)
h[n + 1]z
−(n+2)
∞
X
+
n=0
= 1 + z
−1
+ z
−1
= 1 + z −1 (1 +
|
h[n]z −(n+2)
n=0
∞
X
n=0
∞
X
h[n + 1]z
h[n]z −n ) + z −2
{z
H(z)
= 1 + z −1 H(z) + z −2 H(z)
=
H(z)
=
=
+ z
−2
∞
X
h[n]z −n
n=0
n=1
H(z)(1 − z −1 − z −2 )
−(n+1)
}
∞
X
h[n]z −n
|n=0 {z
H(z)
}
1
1
1 − z −1 − z −2
1 a
b
√
−
1 − bz −1
5 1 − az −1
√
√
1− 5
1+ 5
and b =
where a =
2
2
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For a and b the following holds:
∞
∞ X
X
a n
1
=
=
an z −n
a
1 − (z )
z
n=0
n=0
That results in:
∞
X
1
√
an+1 − bn+1 z −n
H(z) =
5
n=0
∞
X
!
=
h[n] z −n
n=0
h[n] =
1
√
5
an+1 − bn+1
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Table 2.1: Fourier transform pairs
signal
Fourier–transform
1.
δ[n]
1
2.
δ[n − n0 ]
e−jωn0
3.
1
4.
∞
X
(−∞ < n < ∞)
an u[n]
2πδ(ω + 2πk)
k=−∞
1
1 − ae−jω
(|a| < 1)
∞
X
1
+
πδ(ω + 2πk)
1 − e−jω k=−∞
5.
u[n]
6.
(n + 1)an u[n]
7.
rn sin ωp (n + 1)
u[n]
sin ωp
8.
sin ωc n
πn
(
1, |ω| < ωc ,
X(e ) =
0, ωc < |ω| ≤ π
9.
(
1, 0 ≤ n ≤ M
x[n] =
0, otherwise
sin[ω(M + 1)/2] −jωM/2
e
sin(ω/2)
10. ejω0 n
(|a| < 1)
(|r| < 1)
1
(1 − ae−jω )2
1
1 − 2r cos ωp e−jω + r2 e−j2ω
jω
∞
X
k=−∞
11. cos(ω0 n + φ)
π
2πδ(ω − ω0 + 2πk)
∞
X
[ejφ δ(ω − ω0 + 2πk) + e−jφ δ(ω − ω0 + 2πk)]
k=−∞
u[n] =
1, n ≥ 0
0, n < 0
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2.9
Discrete Time Signal Fourier Transform Theorem
Basically there is no difference between FT theorem for the continuous
time and the discrete time case because summation has the same properties as integration. Only differentiation and difference calculation are not
completely analog, because it is not possible to form a derivative in the
discrete time case.
Table 2.2: Fourier transform Theorems
signal
x[n], y[n]
Fourier–transform
X(ejω ), Y (ejω )
1.
ax[n] + by[n]
aX(ejω ) + bY (ejω )
2.
x[n − nd ],
nd is integral number
e−jωnd X(ejω )
3.
ejω0 n x[n]
X(ej(ω−ω0 ) )
4.
x[−n]
X(e−jω )
X(ejω ) if x[n] is real
5.
nx[n]
j
6.
x[n] ∗ y[n]
X(ejω )Y (ejω )
x[n]y[n]
1
2π
7.
8.
Parseval theorem
9.
1
|x[n]| =
2π
n=−∞
∞
X
2
Z
π
X(ejΘ )Y (ej(ω−Θ) )dΘ
−π
(1 − e−jω )X(ejω )
|1 − e−jω |2 = 2(1 − cos ω)
x[n] − x[n − 1]
∞
X
dX(ejω )
dω
Z
1
10.
x[n]y[n] =
2π
n=−∞
π
−π
Z
|X(ejω )|2 dω
π
X(ejω )Y (ejω )dω
−π
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Example 1 corresponding to Theorem 5:
+∞
X
jω
X(e ) =
x[k] e−jωk
k=−∞
k=−∞
+∞
X
=
j
k=−∞
+∞
X
d
X(ejω ) =
dω
x[k] e−jωk
k=−∞
+∞
X
=
⇐⇒
+∞
X
d
dω
d
X(ejω ) =
dω
d
dω
!
x[k] e−jωk
x[k] (−jk) e−jωk
k x[k] e−jωk
k=−∞
F {n · x[n]} = j
d
F {x[n]}
dω
Example 2 corresponding to Theorem 8:
+∞
X
F {x[n] − x[n − 1]} =
k=−∞
+∞
X
=
k=−∞
jω
=⇒
|F {x[n] − x[n − 1]} |2
−jωk
x[k] e
−
x[k] e−jωk −
= X(e ) 1 − e−jω
+∞
X
k=−∞
+∞
X
x[k − 1] e−jωk
x[k] e−jωk e−jω
k=−∞
= |F {x[n]} |2 |1 − e−jω |2
= |F {x[n]} |2 2 · (1 − cos(ω))
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2.10
Discrete Fourier Transform: DFT
The Fourier transform for discrete time signals and systems has been explained on the previous pages. For discrete time signals with finite length
there is also another Fourier representation called “Discrete Fourier Transform” (DFT).
The DFT plays a central role in digital signal processing.
Decisive reasons:
• fast algorithms exist for DFT calculation
(Fast Fourier Transform, FFT).
• discrete frequencies ωk can be better represented in the computer than
continuous frequencies ω.
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Assume a discrete time signal x[n] with finite length (see also chapter 3.2
on page 135):
x[n] =
x[n]
0
0≤n≤N −1
otherwise
Note: For a continuous time signal it is impossible in the strict sense to be
band-limited and time-limited (truncation effect =
ˆ Windowing).
• The discrete time signal Fourier transform for x[n] is:
jω
X(e ) =
N
−1
X
x[n] exp(−jωn)
n=0
• ω is a continuous variable. The period is 2π. Frequency discretisation
is made by sampling along the frequency axis.
• The Fourier transform X(ejω ) is evaluated at
ωk =
2π
k
N
where k = 0, 1, . . . , N − 1
• Define:
X[k] : = X(ejω )|ω = ωk
Im
C
N=8
Re
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• Discrete Fourier Transform (DFT):
X[k] =
N
−1
X
x[n] exp(−j
n=0
2π
k n),
N
k = 0, 1, . . . , N − 1
• Inverse DFT:
N −1
1 X
2π
x[n] =
k n),
X[k] exp(j
N
N
k=0
n = 0, 1, . . . , N − 1
• Remark:
This equation can be proven by inserting the equation for X[k] in the
equation for x[n] and using the orthogonality:
N −1
1 X
2π
1
exp j kn =
0
N n=0
N
k = m N,
otherwise
m is integral number
• Note:
Consider the “analogy” between inverse DFT (above) and inverse
Fourier transform of discrete time signal:
x[n] =
1
2π
Z2π
X(ejω ) ejωn dω
0
Under the given conditions the integral is equal to the sum
(without approximation!).
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Remarks:
• DFT coefficients X[k] are not an approximation of the discrete time
signal Fourier transform X(ejω ). On the contrary:
X[k] = X(ejω )|ω = ωk
• Number of the coefficients X[k] depends on the signal length N . A
finer sampling of the discrete time signal Fourier transform is possible
by appending zeros to the signal x[n] (Zero–Padding).
x[n]
N-1
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Interpretation of Fourier coefficients
• Fourier transform X(ejω ) of the time discrete signal x[n]
jω
|X(e
)|
−π
π
ω
• Evaluation at N discrete sampling points
2π
k
N
ωk =
yields the DFT coefficients X[k].
At first k lies in the domain k = −
N
N
+ 1, . . . , 0, . . . , .
2
2
|X(e
−π
-N/2+1
-1
0
1
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jω
)|, |X[k]|
π
ω
N/2
k
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• Because of the periodicity of X(ejω ) the coefficients X[k] can also be
obtained by shifting the sampling points with negative frequency into
the positive frequency domain (by one period).
Then k = 0, . . . , N/2, . . . , N − 1.
X[k] =
N
−1
X
x[n] exp(−j
n=0
|X(e
jω
2π
k n)
N
)|, |X[k]|
−π
π
0
1
ω
k
N-1
2
• Interpretation of coefficients for general signal x[n]:
k=0
N
−1
1≤k≤
2
N
k=
2
N
+1≤k ≤N −1
2
←→
f =0
←→
0<f <
←→
±
←→
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fS
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Symmetric relations by real signals:
• For DFT coefficients X[k] of a real signal x[n] the following holds:
X[k] = X[N − k]
Re(X[k]) =
Re(X[N − k])
Im(X[k]) = −Im(X[N − k])
• For the amplitude spectrum | X[k] | the following holds:
| X[k] |2
= Re2 {X[k]} + Im2 {X[k]}
= | X[N − k] |2
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Realization of DFT:
/*
/*
/*
/*
PI = 3.14159265358979
x:
input signal
N: length of input signal
Xre, Xim: real and imaginary part of DFT coefficients
void
dft (int N, float x[], float Xre[], float Xim[]) {
int
n, k;
float
SumRe, SumIm;
for (k=0; k<=N-1; k++)
{
SumRe = 0.0;
SumIm = 0.0;
for (n=0; n<=N-1; n++)
{
SumRe += x[n]*cos(2*PI*k*n/N);
SumIm -= x[n]*sin(2*PI*k*n/N);
}
Xre[k] = SumRe;
Xim[k] = SumIm;
}
}
Remark:
• discrete realization
2πj
2πj
• Reduction of “Fourier powers” e− N ·kn to e− N ·l (l = 0, 1, . . . , N − 1)
is possible, because they are periodical (on the unit circle).
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*/
*/
*/
*/
2.11
DFT as Matrix Operation
• Notation with unit roots
X[k]
N
−1
X
=
x[n] exp (−
n=0
N
−1
X
=
2πj
k n)
N
x[n] WNkn
n=0
where
WN := exp (−
2πj
)
N
N=12
W 0N =1
W 1N
W
3
N
W 2N
• Periodicity of WN
unit root:
2π
k) = (WN )k
N
2π
:= exp (−j )
N
exp (−j ωk ) = exp (−j
WN
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Note:
1.
WNr
= WNr mod N
2.
WNkN
= (WNN )k = 1k = 1
3.
WN2
= [exp (−
k∈Z
2πj
2πj 2
)] = exp (−
2)
N
N
= exp (−
2πj
) = WN/2
N/2
N/2
= exp (−
2πj N
) = exp (−πj) = −1
N 2
r+N/2
= WN
4.
WN
5.
WN
N/2
N even
WNr = −WNr
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DFT as matrix multiplication
X[k] =
=
=
N
−1
X
n=0
N
−1
X
n=0
N
−1
X
n=0
x[n] exp (−
2πj
k n)
N
WNkn x[n]
{WN }kn x[n]
with the matrix {WN } and the matrix elements:
{WN }kn := WNkn
Inversion:
N −1
2πj
1 X
k n)
X[k] exp (
x[n] =
N
N
=
=
1
N
1
N
k=0
N
−1
X
(WN−1 )kn X[k]
k=0
N
−1
X
k=0
{WN−1 }kn X[k]
Therefore for the matrix {WN }−1 holds:
{WN }−1 :=
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{WN−1 }
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DFT matrix operation: properties
• DFT: invertible linear mapping
N complex signal values ↔ N complex Fourier components
N real signal values ↔
N
complex Fourier components
2
(due to symmetry)
in words:
DFT causes no “information loss” in the signal.
• Parseval theorem for DFT
general Fourier:
N
−1
X
n=0
1
=
2π
|x[n]|2
Z+π
|X(ejω )|2 dω
−π
special DFT: (recalculate for yourself!)
N
−1
X
n=0
2
|x[n]|
N −1
1 X
=
|X[k]|2
N
k=0
in words:
1
, the DFT is a norm conserving (=
ˆ energy
N
conserving) transformation (mathematical terminology: “unitary”).
Disregarding the factor
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2.12
From Continuous Fourier Transform to Matrix
Representation of Discrete Fourier Transform
Assumption: band-limited signal x(t)
Fourier transform of the continuous time signal x(t):
X(ω) = F {x(t)} =
Z
∞
x(t) e−jωt dt
(2.9)
−∞
For the exact reconstruction (without approximation) of the continuous
time signal from sampled values, the samples x[n] = x(n · Ts ) must have
the distance of at most
Ts =
π
ωB
(sampling theorem).
This results in the Fourier transform of the discrete time signal x[n]:
jω
X(e ) =
∞
X
x[n] e−jωn
(2.10)
−∞
where ω is frequency “normalised on Ts′′
Functions (2.9) and (2.10) agree in interval [−ΩS /2, +ΩS /2] = [−ωB , +ωB ].
| X (ω)|
−Ω S
ωB
−ω B
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ΩS
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1
0.8
0.6
0.4
0.2
0
0
N-1
Figure 2.12: Hanning window
The signal x[n] is further decomposed by applying a window function w[n]
(windowing):
(
...
n = 0, . . . , N − 1
w[n] =
0
otherwise
Windowed signal y[n]:
y[n] = w[n] · x[n]
can be analyzed using Fourier transform or DFT.
jωk
Y (e
) =
N
−1
X
y[n] e−jωk
n=0
DFT:
ωk
=
Y [k] =
2πk
N
N
−1
X
where k = 0, . . . , N − 1
2π
y[n] e− N kn
n=0
Matrix representation:
 

Y [0]
..
 

.
 

 

 Y [k]  = 
 

..
 

.
Y [K − 1]
K=N
..
.
2πj
e− N
·n·k
..
.
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






y[0]
..
.
y[n]
..
.
y[N − 1]







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2.13
Frequency Resolution and Zero Padding
Task: signal x[n] with finite length N is given.
Wanted: Fourier transform X(ejωk ) at
ωk =
2π
k,
K
where k = 0, 1, . . . , K − 1 and K > N
Inserting the definitions:
jωk
X(e
)
=
=
N
−1
X
n=0
K−1
X
x[n] exp (−
2πj
k n)
K
x̃[n] exp (−
2πj
k n)
K
n=0
where
x̃[n] =
x[n]
0
n = 0, . . . , N − 1
n = N, . . . , K − 1
i.e. Zero Padding (appending zeros).
Matrix representation:
 

X[0]
..
 

.
 

 

 

 

 

=

 

 

 

 

..
 

.
X[K − 1]

x[0]

..

.




..

.

  x[N − 1]


0

..

.
0
WKnk














n=0
n=N −1
n=N
n=K −1
Note:
“Zero Padding” does not introduce any additional information into the signal. This is only a trick so that DFT and particularly FFT (Fast Fourier
Transform) can be performed with a
higher frequency resolution
.
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2.14
Finite Convolution
Input signal and convolution kernel have finite duration
Consider “finite” convolution:
• Impulse response:
h[n] ≡ 0
for
n 6∈ {0, 1, 2, . . . , Nh − 1}
• Input signal:
x[n] ≡ 0
for
n 6∈ {0, 1, 2, . . . , Nx − 1}
• Output signal:
y[n] =
=
∞
X
k=−∞
N
h −1
X
k=0
h[k] x[n − k]
h[k] x[n − k]
h[k]
N-1
h
k
x[-k]
n=0
-(N-1)
x
• Altogether:
k
0
Nx + Nh − 1 positions with “overlap”
• Therefore only Nx + Nh − 1 values of output signal can be different
from zero:

n > N x + Nh − 2
0
y[n] =
...
n = 0, 1, . . . , Nx + Nh − 2

0
n<0
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a)
h[k]
1
0
Nh-1=12
k
x[k]
1
0.8
0.6
0.4
0.2
0
b)
Nx-1=4
k
x[n-k] , n=-1
i)
-Nx
-1 0
Nh-1
k
x[n-k] ,n=m (m>0 & m<Nh+Nx-2)
ii)
m-NX+1
0
m
Nh-1
k
x[n-k] , n=Nh+Nx-1
iii)
0
c)
Nh+Nx-1
Nh-1
k
y[n]
2.8 3
3
2.4
2
1.8
1.2
1
0.6
0.2
0
Nx-1
Nh-1
Nh+Nx-2
n
Figure 2.13: Example of a linear convolution of two finite length signals: a) two signals;
b) signal x[n-k] for different values of n:
i) n < 0, no overlap with h[k], therefore convolution y[n] = 0
ii) n between 0 and Nh + Nx − 2, convolution 6= 0
iii) n > Nh + Nx − 2, no overlap with h[k], convolution y[n] = 0
c) resulting convolution y[n].
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Finite convolution using DFT
Convolution theorem:
y[n] =
∞
X
k=−∞
h[k] x[n − k]
Fourier:
Y (ejω ) = H(ejω ) X(ejω ),
0 ≤ ω ≤ 2π
Also valid for sample frequencies:
ωk :=
2π
k,
N
k = 0, . . . , N − 1 for any N
Notation: Y [k] = H[k] X[k]
• Question:
How to choose the length N of the DFT ?
– Reminder: different “lengths”
– x[n]: Nx non-zero values
– h[n]: Nh non-zero values
– y[n]: Ny = Nx + Nh − 1 non-zero values
• Answer:
– The convolution theorem is certainly correct for any N > 0.
– If we want to calculate the output signal completely from Y [k],
we have to know Y [k] for
at least N = Nx + Nh − 1
frequency values k = 0, 1, . . . , N − 1.
– In words: for the DFT length N must be valid:
N ≥ Nx + Nh − 1
Method:
Zero Padding, i.e. appending zeros.
Note: The FFT will be introduced on the next pages. A comparison of
costs for realization of the finite convolution by DFT and FFT can be
found at the end of paragraph 2.15.
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2.15
Fast Fourier Transform (FFT)
Principle of FFT:
Calculation of the DFT can be done by successive decomposition into
smaller DFT calculations. In this way, the number of elementary operations (multiplications and additions) is dramatically reduced:
FFT:
N2 →
N = 1024 :
N
ld N
2
2·N
2 · 1024
=
= 200
ld N
10
operations
factor of velocity gain
The matrix is decomposed into a product of sparse matrices, therefore N
with lot of prime factors is convenient (not necessarily only powers of two).
Terminology for different variants of FFT:
• in time ↔ in frequency
• in place:
yes/no
• radix 2 ↔ radix 4
• decomposition to prime factors instead of N = 2n
History
1965 Cooley and Tukey
1942 Danielson and Lanczos
1905 Runge
1805 Gauss
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Algorithms which are based on a decomposition of the signal x[n] are called
“decimation–in–time algorithms”.
The case N = 2ν is considered in the following.
X[k] =
=
N
−1
X
n=0
N
−1
X
x[n] exp(−j
2π
k n)
N
x[n] WNnk
where k = 0, 1, . . . , N − 1
where WNnk = exp(−j
n=0
2π
k n)
N
• Decomposition of the sum over n into the sums over even and odd n:
N/2−1
N/2−1
X[k] =
X
x[2r]
WN2rk
X
+
r=0
r=0
N/2−1
=
X
(2r+1)k
x[2r + 1] WN
x[2r]
(WN2 )rk
+
N/2−1
WNk
r=0
X
x[2r + 1] (WN2 )rk
r=0
• Because of
WN2 = exp(−2j
2π
2π
) = exp(−j
) = WN/2
N
N/2
for k = 0, . . . , N − 1 holds:
N/2−1
X[k] =
X
N/2−1
x[2r]
rk
WN/2
+
r=0
= G[k] +
WNk
X
x[2r + 1] (WN/2 )rk
r=0
WNk
H[k]
• Each of the two sums corresponds to the DFT with the length N/2.
The first sum is a N/2–DFT of the even indexed signal values x[n],
the second sum is a N/2–DFT of the odd indexed values.
• The DFT of the length N can be obtained by getting the two N/2–
DFT’s together, with the factor WNk .
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Complexity:
The complexity O(N 2 ) of one-dimensional FT can be reduced by adequate
resorting values from two FTs with length N2 and complexity O(2 · ( N2 )2 ) =
N2
2 . By successive application of this resorting the complexity can be reduced to O(N log N ).
The case N = 23 = 8 is considered in the following.
• X[4] can be obtained from H[4] and G[4] according to previous equation.
• Because of the DFT–length
N
2
= 4:
H[4] = H[0] and G[4] = G[0]
And then:
X[4] = G[0] + WN4 H[0]
The values X[5], X[6] and X[7] can be obtained analogously.
Flow diagram for decomposition of one N -DFT into two N/2–DFTs:
x[n]
X[k]
G[0]
x[0]
X[0]
G[1]
x[2]
x[4]
0
N
W
X[1]
N/2-point
G[2]
DFT
1
N
W
X[2]
G[3]
2
N
W
x[6]
X[3]
W3N
x[1]
X[4]
H[0]
x[3]
x[5]
4
N
W
X[5]
N/2-point
H[1]
5
N
W
DFT
X[6]
H[2]
6
N
W
x[7]
X[7]
H[3]
7
N
W
Figure 2.14: Flow diagram for decomposition of one N -DFT to two N/2–DFTs with
N =8
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• Further analogous decomposition, until only DFT’s with the length
N = 2 remain (so called Butterfly Operation)
• Resulting flow diagram of the FFT:
x[n]
X[k]
X[0]
x[0]
W0N
x[4]
-1
X[1]
0
N
W
x[2]
W
W
x[6]
X[2]
-1
2
N
0
N
-1
-1
X[3]
W0N
x[1]
1
N
0
N
W
W
x[5]
-1
2
N
0
N
W
W
x[3]
W2N
W0N
x[7]
-1
-1
-1
W3N
X[4]
-1
X[5]
-1
X[6]
-1
X[7]
-1
Figure 2.15: Flow diagram of an 8–point–FFT using Butterfly operations.
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Complexity reduction
• Number of complex multiplications in FFT
is N/2 · ld N .
• Comparison:
Direct application of the DFT definition needs
N 2 complex multiplications.
• Example:
N = 1024 = 210
N2
≈ 200
N/2 · ld N
Complexity reduction by factor 200
FFT with the base 2 is not minimal according to number of additions, FFT with the base 4 can be better.
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Matrix representation of the FFT principle
• The complex Fourier matrix can be decomposed into the product of
r = ld N matrices, each of them having only two non-zero elements in
each column.
• The following graph shows the decomposition of the Fourier matrix in
the case of inverse transformation.
• w corresponds to WN−1
X = |wnk |X = T3 · T2 · T1 · TS · x
This is how the decomposition into r + 1 = 4 matrices looks like
(w4 = −1, w8 = 1):
1
1
1
1
nk
|w | = 1
1
1
1
1
w
w2
w3
w4
w5
w6
w7
1
w2
w4
w6
w8
w10
w12
w14
1
w3
w6
w9
w12
w15
w18
w21
1
w4
w8
w12
w16
w20
w24
w28
1
w5
w10
w15
w20
w25
w30
w35
1
w6
w12
w18
w24
w30
w36
w42
1
1 1 1
w7
1 w w2
w14
1 w2 w4
w21
1 w3 w6
w28 = 1 w4 1
w35
1 w5 w2
w42
1 w6 w4
w49
1 w7 w6
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1
w3
w6
w
w4
w7
w2
w5
1
w4
1
w4
1
w4
1
w4
1
w5
w2
w7
w4
w
w6
w3
1
w6
w4
w2
1
w6
w4
w2
1
w7
w6
w5
w4
w3
w2
w
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Signal flow diagram
The calculation operations which correspond to the matrix representation
of FFT can be showed in a signal flow diagram.
T3
1000 1 0 0
0
0100 0 w 0
0
0 0 1 0 0 0 w2 0
0 0 0 1 0 0 0 w3
1 0 0 0 −1 0 0
0
0 1 0 0 0 −w 0
0
0 0 1 0 0 0 −w2 0
0 0 0 1 0 0 0 −w3
T2
10 1 0
0 1 0 w2
1 0 -1 0
0 1 0 −w2
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
00 0 0
10 1 0
0 1 0 w2
1 0 −1 0
0 1 0 −w2
TS
T1
T1
1100
1 -1 0 0
0011
0 0 1 -1
0000
0000
0000
0000
TS
0 0 0 0 1000
0 0 0 0 0000
0 0 0 0 0010
0 0 0 0 0000
1 1 0 0 0100
1 -1 0 0 0 0 0 0
0 0 1 1 0001
0 0 1 -1 0 0 0 0
T2
0000
1000
0000
0010
0000
0100
0000
0001
T3
x[0]
x[1]
X[0]
X[1]
-1
x[2]
ω2
x[3]
-1
X[2]
-1
X[3]
-1
x[4]
x[5]
ω
-1
x[6]
ω2
x[7]
-1
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ω2
-1
ω3
-1
−1
X[4]
−1
X[5]
−1
X[6]
−1
X[7]
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• Matrices T1 , T2 and T3 contain exactly two non-zero elements in each
row.
• Non-zero elements are realizing the Butterfly Operation.
• Matrix T1 : step width of the Butterfly Operation is 1
Matrix T2 : step width of the Butterfly Operation is 2
Matrix T3 : step width of the Butterfly Operation is 4
• step widths can be found:
– in signal flow diagram
– distance between the non-zero elements in T1 , T2 and T3
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Butterfly Operation
• Signal flow diagram and matrix representation of the FFT are based
on the following basic operation:
Xm[p]
Xm-1[p]
WrN
Xm-1[q]
Xm[q]
-1
• For two input values Xm−1 [p] and Xm−1 [q] this operation produces
two output values Xm [p] and Xm [q]. The output values are thereby a
linear combination of the input values.
• Because of the flow graph, the operation is called
“Butterfly Operation”.
Xm [p] = Xm−1 [p] + WNr Xm−1 [q]
Xm [q] = Xm−1 [p] − WNr Xm−1 [q]
Xm [p]
Xm [q]
=
1
WNr
1 − WNr
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Xm−1 [p]
Xm−1 [q]
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Bit Reversal
• The matrix representation of the FFT uses a sorting matrix, i.e. the
signal which is to be transformed is at first resorted.
• Example for N = 8:
n Binary representation Reversed n’
0
000
000
0
1
001
100
4
2
010
010
2
3
011
110
6
4
100
001
1
5
101
101
5
6
110
011
3
7
111
111
7
• Bit Reversal is a necessary part of the FFT–Algorithm.
Bit Reversal for N = 23
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2.16
FFT Implementation
• Fortran version
C
adapted from: Oppenheim, Schafer p. 608
C SUBROUTINE FFT DecimationInTime (X, ld n) **********************************
C ****************************************************************************
PARAMETER PI = 3.14159265358979
PARAMETER N max = 2048
COMPLEX
COMPLEX
COMPLEX
COMPLEX
INTEGER
X(N max)
! array for input AND output
Temp
! temporary storage
W uni
! root of unity
W pow
! powers of W uni
N, ld N, ip, iq, iqbeq, j, k, i exp, istp
N = 2**ld n
IF (N.GT.N max) STOP
C BIT Reversed Sorting *********************************************************
j = 1
DO i = 1, N-1
IF (i.LT.j) THEN
! swap X(j) and X(i)
Temp = X(j)
X(j) = X(i)
X(i) = Temp
ENDIF
k = N/2
DO WHILE (k.LT.j)
j = j - k
k = k / 2
ENDDO
j = j + k
ENDDO
C End of Bit Reversed Sorting **************************************************
C FFT Butterfly Operations *****************************************************
DO i=1, ld N
i exp = 2**i
! exponent
istp = i exp/2
! stepsize
W pow = (1.0,0.0)
W uni = CMPLX (COS (PI/FLOAT(istp)), -SIN(PI/FLOAT(istp)))
DO ipbeg = 1, istp
DO ip = ipbeg, N, i exp
iq = ip + istp
Temp = X(iq) * W pow
X(iq) = X(iq) - Temp
X(ip) = X(iq) + Temp
ENDDO
W pow = W pow * W uni
ENDDO
ENDDO
C End of FFT Butterfly Operations **********************************************
RETURN
END
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Explanations about Fortran Program
Two program parts:
1. Bit Reversal
2. Butterfly Operations
• 3 loops with variables i, ipbeg, ip are controlling the execution of
the Butterfly operations
• outer loop i:
i specifies the level of the FFT
• With exception of the first level, Butterfly operations are “nested”.
Therefore two loops are used for the Butterfly operations within
one level.
• middle loop, ipbeg:
ipbeg: goes over the “nested” Butterfly operations
i=1: ipbeg=1
i=2: ipbeg=1,2
i=3: ipbeg=1,2,3,4
iqbeg: specifies the sequence of starting points for inner loop
• inner loop, ip:
ip: specifies the first element of the Butterfly operation
istp: step width of the Butterfly operation
iq=ip+istp: specifies the second element for Butterfly operation
inner loop is “started” once per “nesting”
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x[0]
X[0]
0
N
W
x[4]
-1
X[1]
0
N
W
x[2]
0
N
W
x[6]
X[2]
-1
2
N
W
-1
-1
X[3]
0
N
W
x[1]
0
N
1
N
W
x[5]
W
-1
W0N
W2N
x[3]
W0N
x[7]
-1
W2N
-1
-1
W3N
-1
-1
-1
-1
X[4]
X[5]
X[6]
X[7]
Figure 2.16: Flow diagram of an 8–point–FFT using Butterfly operations.
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• C version (from Numerical Recipes in C)
#include <math.h>
#define SWAP(a,b) tempr=(a);(a)=(b);(b)=tempr
void four(float data[], unsigned long nn, int isign)
Replaces data[1..2*nn] by its discrete Fourier transform, if isign is input as 1; or replaces
data[1..2*nn] by nn times its inverse discrete Fourier transform if insign is input as -1.
data is a complex array of lenght nn or, equivalently, a real array of lenght 2*nn. nn MUST
be an whole number power of 2 (this is not checked for!).
{
unsigned long n, mmax, m, j, istep, i;
double wtemp, wr, wpr, wpi, wi, theta;
Double prec. for the trigonometric recurrences.
float tempr, tempi;
n=nn << 1;
j=1;
for (i=1; i<n; i+=2) {
This is the bit-reversal section of the routine.
if (j > i) {
SWAP (data[j], data[i]);
Exchange the two complex numbers.
SWAP (data[j+1], data[i+1]);
}
m=n >> 1;
while (m >= 2 && j > m) {
j -= m;
m >>= 1;
}
j += m;
}
Here begins the Danielson-Lanczos section of the routine.
mmax=2;
while (n > mmax) {
Outer loop executed log2 nn times
istep=mmax << 1;
theta=isign∗(6.28318530717959/mmax); Initialise the trigonometric recurrence.
wtemp=sin(0.5*theta):
wpr = -2.0*wtemp*wtemp;
wpi=sin(theta);
wr=1.0;
wi=0.0;
for (m=1;m <mmax;m+=2) {
Here are the two nested loops.
for (i=m;i<=n;i+=istep) {
j=i+mmax;
This is the Danielson-Lanczos formula:
tempr=wr*data[j]-wi*data[j+1];
tempi=wr*data[j+1]+wi*data[j];
data[j]=data[i]-tempr;
data[j+1]=data[i+1]-tempi;
data[i] += tempr;
data[i+1] += tempi;
}
wr=(wtemp=wr)*wpr-wi*wpi+wr;
Trigonometric recurrence
wi=wi*wpr+wtemp*wpi+wi;
}
mmax=istep
}
}
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• Input and output arrays (C version)
a)
1
real
2
imag
3
real
4
imag
2N-3
real
}
}
b)
1
real
2
imag
3
real
4
imag
N-1
real
N
imag
N+1
real
t=0
t=∆
}
}
N+2
imag
N+3
real
N+4
imag
2N-1
real
}
}
}
}
}
f=0
f= 1
N∆
f = N/2 - 1
N∆
f=
1
2∆
f=
N/2 - 1
N∆
f=
1
N∆
t = (N-2)∆
2N-2
imag
2N-1
real
t = (N-1)∆
2N
imag
2N
imag
}
Figure 2.17: Input and output arrays of an FFT. a) The input array contains N (N is
power of 2) complex input values in one real array of the length 2N . with alternating
real and imaginary parts. b) The output array contains complex Fourier spectrum at N
frequency values. Again alternating real and imaginary parts. The array begins with the
zero-frequency and then goes up to the highest frequency followed with values for the
negative frequencies.
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Finite convolution: complexity by the application of FFT
Estimation of number of necessary multiplications for a convolution of x[n]
and h[n]
x[n]: Nx non-zero values
h[n]: Nh non-zero values
Realisation
direct implementation
DFT
FFT
transformation
(Nx + Nh )2
Nx · Nh
Nx +Nh
2
log2 (Nx + Nh )
multiplication in frequency domain
Nx + Nh
Nx + Nh
inverse transformation
(Nx + Nh )2
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Nx +Nh
2
log2 (Nx + Nh )
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2.17
Cyclic Matrices and Fourier Transform
The Fourier transform plays a significant role for so-called cyclic matrices
(cf. chapter 3.8):
0






H = 





n
h0
h1 h2
hN −1 . . . . . .
hN −2 . . . . . .
...
..
.
N −1
...
...
... ...
... ... ...
... ... ...
... ...
h1
so:
hN −1
hN −1
hN −2
..
.
h2
h1
h0

0




m





N −1
Hmn = h(n−m)modN
with so-called kernel vector (h0 , h1 . . . , hN −1 )T and hn ∈ C, mostly hn ∈ IR.
Remark: using cyclic matrices it is possible to define cyclic i.e. periodic
convolutions and to build a cyclic variant of the system theory.
The eigenvectors of a cyclic matrix can be obtained from the columns of
DFT matrix:
0
0
m
N −1












n
N −1
w0·0 · · ·
wn·0
· · · w(N −1)·0
..
..
.
.
wn·1
..
..
.
.
..
..
.
.
···
···
..
..
.
.
..
..
···
.
···
.
..
..
.
wn·(N −1)
.
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





 where w = e− 2πj
N





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The eigenvalues λk of a cyclic matrix with the kernel vector (h0 , h1 . . . , hN −1 )T
are:
λk =
N
−1
X
2πj
hn e− N
·k·n
n=0
where k = 0, 1 . . . , N − 1
The representation is oriented on L. Berg: Linear equation systems with
band structure (page 52 ff).
The special case of a cyclic matrix when the kernel vector h is symmetric
and real is especially interesting for many applications:
hn = hN −n and hn ∈ IR
That means that the matrix is real, symmetric and cyclic:
0
0
N −1












h0 h1 h2
h1 . . . . . . . . .
... ... ...
... ...
..
...
.
h2 h3
h1 h2
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...
...
...
...
...
N −1

h2 h1
h2 


h3 
. . . ... 



...

... h 
1 
h1 h0
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For such a matrix is valid:
a) the eigenvalues λk are:
λk =
N
−1
X
n=0
2πkn hn cos
N
b) The eigenvectors can be obtained from the columns of the “discrete
cos–matrix”:
n




..
.




..
.








.
..









 cos 2πnm ) 
m
N




..




.








.
..




..
.
Application: diagonalisation of covariance matrices, e.g. for coding
of image and speech signals.
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Proof:
We will prove that





vk = 




cos 2πk·0
N
2πk·1
cos N
..
.
..
.
..
.
2πk·(N −1)
cos
N










are eigenvectors of the given matrix with eigenvalues λk .
For a symmetric cyclic matrix is valid:
Hmn := h(n−m)modN
where hn = hN −n
One row results in (for odd N ):
X
n
N −1
Hmn · vkn
2πkm
= h0 · cos
+
N
2
X
l=1
hl ·
2πk(m − l)
2πk(m + l) cos
+ cos
N
N
For even N only one term for l = (N − 1)/2 can go into the sum.
According to addition theorem:
cos(x + y) + cos(x − y) = 2 · cos x · cos y
With x = 2πkm/N and y = 2πkl/N follows:
X
n
N −1
Hmn · vkn
2πkm
+ 2
= h0 · cos
N
N −1
2
2
X
l=1
hl ·
2πkm 2πkl
· cos
cos
N
N
2πkm
2πkl i
· cos
= h0 + 2
hl cos
N
| {zN }
l=1
|
{z
}
vkm
h
X
λk
= λk · vkm
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Excursion: Toeplitz matrices
We consider only quadratic real matrices:
H ∈ IRN ×N
with Hmn ∈ IR
a) H is a (general) Toeplitz matrix , if:
Hmn = hn−m
i.e.









H = 








h0
h−1
h−2
..
.
h1
...
h2
...
... ...
... ... ...
... ... ...
...
...
... ... ...
... ...
h2−N
h1−N h2−N
h−2

hN −2 hN −1 

hN −2 


..

.


...



...

h2



...
h1


h−1 h0
b) For a symmetric Toeplitz matrix then holds:
i.e.









H = 








Hmn = h|n−m|
h0
h1
h2
..
.
h1
...
...
...
h2
. . . hN −2
... ...
... ... ...
... ... ... ...
... ... ... ...
... ... ...
hN −2
hN −1 hN −2
h2 h1
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
hN −1 

hN −2 


..

.






h2



h1


h0
WS 2006/2007
c) A cyclic matrix can be obtained by special choice:
Hmn = h(n−m)modN









H = 








h0
hN −1
hN −2
..
.
h1 h2
. . . hN −2
... ... ...
... ... ... ...
... ... ... ...
... ... ...
... ...
h2
h1
h2
hN −1

hN −1 

hN −2 


..

.






h2



h1


h0
Also valid: each cyclic matrix is a Toeplitz matrix.
d) A symmetric cyclic matrix can be obtained by the following choice of
the kernel vectors h:
hn = hN −n
for n = 0, . . . , N
For example, we obtain for N = 8:









H = 







h0 h1 h2 h3 h4 h3 h2 h1


h1 h0 h1 h2 h3 h4 h3 h2 


h2 h1 h0 h1 h2 h3 h4 h3 

h3 h2 h1 h0 h1 h2 h3 h4 


h4 h3 h2 h1 h0 h1 h2 h3 


h3 h4 h3 h2 h1 h0 h1 h2 

h2 h3 h4 h3 h2 h1 h0 h1 

h1 h2 h3 h4 h3 h2 h1 h0
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Chapter 3
Spectral analysis
• Overview:
3.1 Features for Speech Recognition
3.2 Short Time Analysis and Windowing
3.3 Autocorrelation function and Power Spectral Density
3.4 Spectrograms
3.5 Filter Bank Analysis
3.6 Mel–Scale
3.7 Cepstrum
- Cepstrum Calculation from Filter Bank Output
- Mel–Cepstrum according to Davis and Mermelstein
3.8 Statistical Interpretation of Cepstrum Transformation
3.9 Energy in acoustic Vector
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3.1
Features for Speech Recognition
Architecture of an automatic speech recognition system
speech signal
short-time
analysis
each 10 ms
(using FFT)
sequence of
acoustic vectors
reference model
for each word
in the vocabulary
pattern
comparison
decision
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Short time analysis:
• window length 10–40ms
• sampling period 10–20ms
• in case of sampling rate of 10kHz:
– Window: 100–400 samples
– sampling period (frame shift): 100–200 samples
Recommended windows:
• Hamming
• Kaiser
• Blackman
Model parameters:
• Energy, intensity (“loudness”)
• Fundamental frequency (“height”)
• Spectral parameters (“colour”, “smoother” amplitude spectrum)
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Goal:
• Ideally: Real features for the recognition
• In practice: Data reduction, i.e. compact description
of the speech signal (amplitude spectrum)
Side effect:
• Method also enables coding of speech signals using lowest possible
number of bits
Key words:
• Fourier transform: wide band/narrow band, autocorrelation function
• Filter bank
• Cepstrum
• Linear Predictive Coding (LPC) analysis
• Fundamental frequency analysis
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3.2
Short Time Analysis and Windowing
• The DFT is defined for signals with finite duration.
• Speech signal s[n]:
quasi stationary, i.e. properties do not change within 20-50 ms.
• Window function w[n]:
Decomposition of the original signal s[n] into (overlapping)
segments using a window function w[n]:
x[n] = s[n] · w[n]
where for example
w[n] =
1,
0,
|n| ≤ N/2
otherwise
• The windowed signal x[n] is analyzed with a Fourier Transform or
DFT.
• The multiplication of the original signal s[n] with the window function
w[n] in the time domain corresponds to the convolution of the spectra
of two signals S(ejω ) and window function W (ejω ) in the frequency
domain:
1
X(ejω ) =
2π
Zπ
S(ejθ ) W (ej(ω−θ) ) dθ
−π
• This convolution performs a (spectral) smearing in the frequency domain (leakage).
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Window function:
Impulse response:
0
1
-10
0.8
-20
dB
0.6
0.4
-30
-40
0.2
-50
0
0
-60
-0.5 fs
N-1
0
0.5 fs
0
0.5 fs
0
0.5 fs
0
0.5 fs
Rectangle
0
1
-10
0.8
-20
dB
0.6
0.4
-30
-40
0.2
-50
0
0
-60
-0.5 fs
N-1
Triangle
0
1
-10
0.8
-20
dB
0.6
0.4
-30
-40
0.2
-50
0
0
-60
-0.5 fs
N-1
Hanning
0
1
-10
0.8
-20
dB
0.6
0.4
-30
-40
0.2
-50
0
0
-60
-0.5 fs
N-1
Hamming
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Window function:
Impulse response:
0
1
-10
0.8
-20
dB
0.6
0.4
-30
-40
0.2
-50
0
0
-60
-0.5 fs
N-1
0
0.5 fs
0
0.5 fs
0
0.5 fs
Nuttall
0
1
-10
0.8
-20
dB
0.6
0.4
-30
-40
0.2
-50
0
0
-60
-0.5 fs
N-1
Gauss
0
1
-10
0.8
-20
dB
0.6
0.4
-30
-40
0.2
-50
0
0
-60
-0.5 fs
N-1
Chebyshev
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Fourier Transform
of a continuous
time signal
SC(Ω)
-Ω0
Ω0
0
Frequency graph
of anti-aliasing
low-pass filter
Ω
H(Ω)
1
-π
T
0
Ω
π
T
Ω
XC(Ω)
Fourier Transform
of filtered signal
-π
T
π
T
-Ω0
0
Ω0
Fourier Transform of
sampled signal
X(ejω)
ω
−2π
−π
−ω0
0
Fourier Transform
of window function
π
ω0=ΩT
2π
W(ejω)
ω
−2π
-π
−π
2π
2π
n
Fourier Transform
of windowed signal
and sampled values
of continuous spectrum
obtained using DFT
−2π
π
0
V(ejω), V[k]
0
π
ω
2π
Figure 3.1: Example for the application of the Discrete Fourier Transform (DFT).
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Properties of short-time DFT–analysis
Important effects:
• Picket Fence
If not enough sampled values of continuous spectrum are available,
spectral sampling can yield delusive results. This problem can be reduced using Zero Padding (inter-space between the coefficients S[k]
becomes smaller, i.e. frequency resolution becomes better)
• Leakage: Spreading of the line spectrum
Because the window function is limited in time, a spreaded spectrum
is measured instead of the spectrum of the original signal unlimited in
time. That means, the line spectrum even becomes spreaded for pure
sinusoidal signals.
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3 examples of DFT analysis
• we observe a continuous time signal x(t) composed of two
sinusoids:
x(t) = A0 cos(Ω0 t) + A1 cos(Ω1 t)
−∞<t<∞
• sampling according to sampling theorem
(with negligible quantization errors)
• discrete time signal x[n]:
x[n] = A0 cos(ω0 n) + A1 cos(ω1 n)
−∞<n<∞
where ω0 = Ω0 TS and ω1 = Ω1 TS
• with the window function w[n]:
v[n] = A0 w[n] cos(ω0 n) + A1 w[n] cos(ω1 n)
Intermediate calculations:
v[n] =
+
also modulation principle
A0
A0
w[n] exp(j ω0 n) +
w[n] exp(−j ω0 n)
2
2
A1
A1
w[n] exp(j ω1 n) +
w[n] exp(−j ω1 n)
2
2
• Fourier Transform of the windowed signal:
V (ejω ) =
+
A0
A0
W (ej(ω−ω0 ) ) +
W (ej(ω+ω0 ) )
2
2
A1
A
1
W (ej(ω−ω1 ) ) +
W (ej(ω+ω1 ) )
2
2
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• Assume:
4π
2π
· 10kHz, Ω1 =
· 10kHz
14
15
1/TS = 10kHz, rectangle window with N = 64, A0 = 1, A1 = 0.75
Ω0 =
• The windowed signal v[n] for the discrete time signal x(n) is therefore:

4π
2π

 cos( n) + 0.75 cos( n) : 0 ≤ n ≤ 63
14
15
v[n] =
:


0 :
otherwise
v[n]
2
1
63
0
n
-1
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• Fourier Transform W (ejω ) of the rectangle window function
64
−π
Example 1:
π
0
Leakage Effect
Variation of ω0 and ω1 resp. Ω0 and Ω1
Difference between frequencies ω0 and ω1 is reduced gradually
Case 1a:
Ω0 =
2π 4
10 Hz,
6
Ω1 =
2π 4
10 Hz
3
ω0 = Ω0 TS =
2π 4
2π
10 Hz 10−4 s =
6
6
ω1 = Ω1 TS =
2π
2π 4
10 Hz 10−4 s =
3
3
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Case 1a (continued):
ω0 =
2π
6
ω1 =
2π
3
V(ω)
32
−π
Case 1b:
2π
3
ω0 =
2π
6
2π
14
2π
6
0
ω1 =
2π
3
π
ω
4π
15
V(ω)
32
−π
4π
15
2π
14
0
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2π
14
4π
15
π
ω
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Case 1c:
ω0 =
2π
14
ω1 =
2π
12
V(ω)
30
-π
Case 1d:
2π 2π
12 14
0
ω0 =
2π
14
ω1 =
π
ω
4π
25
V(ω)
40
−π
0
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π
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Example 2:
Picket Fence Effect
DFT gives sampled values of the spectrum of the windowed signal.
Spectral sampling can yield delusive results.
Case 2a:
• Windowed signal v[n]:
(
2π
4π
cos( n) + 0.75 cos( n) : 0 ≤ n ≤ 63
v[n] =
14
15
0 :
otherwise
• DFT of the length N = 64 without Zero Padding
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a)
v[n]
2
1
63
0
n
-1
b)
V(k)
30
0
c)
63
k
2π
ω
V(ω)
32
0
π
Figure 3.2: a) signal v[n]; b) DFT-spectrum V [k]; c) Fourier spectrum V (ejω ).
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Case 2b:
• In contrast to case 2a, the frequencies of sinusoids are changed only
slightly.
• Windowed signal v[n]:
(
2π
2π
cos( n) + 0.75 cos( n) : 0 ≤ n ≤ 63
v[n] =
16
8
0 :
otherwise
• DFT of the length N = 64 without Zero Padding
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a)
v(n)
1
0
63
n
-1
b)
V(k)
30
0
c)
k
63
V(ω)
32
0
π
2π
ω
Figure 3.3: a) signal v[n]; b) DFT-spectrum V [k]; c) Fourier spectrum V (ejω ).
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Analysis of Example 2:
• The manifestation of the DFT can be put down to the spectral sampling. Although in Case 2b the windowed signal v[n] contains a significant number of frequencies beyond ω0 and ω1 , they do not show in
the DFT spectrum of length N = 64.
• Using a rectangle window, the DFT of the sinusoidal signal gives sharp
spectral lines, if the period N of the transformation is a whole multiple
of the signal period and no Zero Padding is applied.
• Explanation for the case of a complex exponential function:
– Assume the signal x[n]:
1
2π
n)
exp(j
N
n0
x[n] =
– Then:
X[k] = δ(k −
N
)
n0
– For the DFT of rectangle window holds:
W [k] =
sin(πk)
sin(πk/N )
– Convolution theorem for windowed signal v[n] gives:
N
sin π(k − )
n0
V [k] = X[k] ∗ W [k] =
N
sin π(k − )/N
n0
– In case of
N
∈ IN
n0
only the DFT coefficient k =
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N
is non-zero.
n0
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Example 2 (continued)
• Assume signal v[n] of Case 2b:
(
2π
2π
cos( n) + 0.75 cos( n) : 0 ≤ n ≤ 63
v[n] =
16
8
0 :
otherwise
• In contrast to Case 2b, a DFT with length N = 128 is applied (Zero
Padding).
• Result:
Using finer sampling, existing additional frequency components emerge.
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a)
V(k)
30
0
b)
k
63
V(k)
32
0
c)
127
k
V(ω)
32
0
π
2π
ω
Figure 3.4: a) DFT of length N = 64; b) DFT of length N = 128; c) Fourier spectrum
V (ejω ).
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Example 3:
Explanation of following illustrations:
• Assume: signal of Example 2, Case 2a.
• Window: Kaiser window is applied instead of rectangle window.
• First: window length L = 64 and DFT length N = 64.
• Then: window length L and DFT length N are halved.
• Afterwards: for the case L = 32, the DFT length N is gradually
increased up to N = 1024 (Zero Padding).
• Finally: DFT spectrum for the case N = 1024 and L = 64.
The Kaiser window is defined as:

1/2 2


 I0 β 1 − [(n − α) /α]
wK [n] =
: 0≤n≤L−1

I0 (β)


0 :
otherwise
In this example:
β = 0.8
and
α=
L−1
2
The windowed signal v[n]:
v[n] = wK [n] cos(
4π
2π
n) + 0.75 wK [n] cos( n)
14
15
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Example 3: (continued)
DFT length N = 64, window length L = 64
• Windowed signal
v(n)
1
0
63
n
-1
• DFT spectrum
V(k)
30
0
63
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Example 3: (continued)
DFT length N = 32, window length L = 32
(N and L halved)
• Windowed signal
v(n)
1
0
31
n
• DFT spectrum
V(k)
8
0
31
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Example 3: (continued)
Effect of changing DFT length N at constant window length L = 32 (Zero
Padding)
• DFT length N = 32, window length L = 32
V(k)
8
0
31
k
63
k
• DFT length N = 64, window length L = 32
V(k)
8
0
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Example 3: (continued)
• DFT length N = 128, window length L = 32
V(k)
8
0
127
k
1024
k
• DFT length N = 1024, window length L = 32
V(k)
8
0
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Example 3: (continued)
Increasing the window length (L)
• DFT length N = 1024, window length L = 64
V(k)
16
0
1024
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Example 4: Window function influence on the spectrum
speech signal
phoneme "a"
0
0
amplitude spectrum
- rectangle window -
0
amplitude spectrum
- Hamming window -
0
Figure 3.5: Influence of the window function:
above: speech signal (vowel “a”); central: 512 point FFT using rectangle window; below:
512 point FFT using Hamming window
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3.3
Autocorrelation Function and Power Spectral Density
Definition of Autocorrelation Function (ACF) analog to the continuous
time case:
R[k] : =
∞
X
x[n] x[n + k]
n=−∞
For a signal x[n] assume (e.g. after some suitable windowing):
x[n]
0≤n≤N −1
x[n] =
0
otherwise
In this case the ACF gives:
R[k] =
NX
−1−k
x[n] x[n + k]
because x[n] = 0 for n < 0 and n ≥ N
n=0
“triangular effect”
number of terms in R[k]
-N
N
k
Cross correlation:
Rxy [k] =
∞
X
x[n] · y[n − k]
∞
X
x[n] · y[k − n]
n=−∞
In contrast to convolution:
Oxy [k] =
n=−∞
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Properties of ACF:
1. R[k] = R[−k]
2. R[k] ≤ R[0]
for each k ∈ IN (R[0]: energy, intensity)
3. If x[n] −→ R[k], then α x[n] −→ α2 R[k]
4. Intensity spectrum is the Fourier Transform of the ACF:
| X(ejω ) |2
= X(ejω ) · X(ejω )
∞
∞
X
X
=
x[k] exp(−jωk) ·
x[l] exp(jωl)
=
=
=
=
k=−∞
∞
X
l=−∞
∞
X
k=−∞ l=−∞
∞
∞
X
X
x[k] x[l] exp(−jωk) exp(jωl)
x[k + l] x[l] exp(−jωk) exp(−jωl) exp(jωl)
k=−∞ l=−∞
∞
∞
X
X
k=−∞
∞
X
x[k + l] x[l]
l=−∞
!
exp(−jωk)
R[k] exp(−jωk)
k=−∞
Note:
The phase spectrum is removed.
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5. Because of the symmetry R[k] = R[−k] the DFT becomes the cosine
transform:
jω
2
| X(e ) |
=
=
∞
X
R[k] exp(−jωk)
k=−∞
N
−1
X
k=−(N −1)
= R[0] +
R[k] exp(−jωk)
N
−1
X
R[k] (exp(−jωk) + exp(jωk))
k=1
N
−1
X
= R[0] + 2 ·
R[k] cos(ωk)
because
R[k] = R[−k]
k=1
6. The intensity spectrum | X(ejω ) |2 is a polynom of cos(ω) with grade
N − 1.
Reason:
Moivre formula:
k
k
cosk−4 (ω) sin4 (ω) − . . .
cosk−2 (ω) sin2 (ω) +
cos(ωk) = cosk (ω) −
4
2
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Example 1: Spectral analysis using ACF
speech signal
phoneme "a"
0
0
amplitude spectrum
- Hamming window -
amplitude spectrum
- short hamming window -
0
0
amplitude spectrum
- 19 ACF-coefficients -
amplitude spectrum
- 13 ACF-coefficients -
0
0
Figure 3.6: Fourier Transform of a voiced speech segment:
a) signal progression, b) high resolution Fourier Transform, c) low resolution Fourier
Transform with short Hamming window (50 sampled values), d) low resolution Fourier
Transform using autocorrelation function (19 coefficients), e) low resolution Fourier Transform using autocorrelation function (13 coefficients)
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Example 2: ACF of voiced and unvoiced speech segments
speech signal
phoneme "s"
speech signal
phoneme "a"
0
0
0
0
autocorrelation
- rectangle window -
autocorrelation
- rectangle window -
0
0
0
0
autocorrelation
- Hamming window -
autocorrelation
- Hamming window -
0
0
0
0
Figure 3.7: Signal progression and autocorrelation function of voiced (left) and unvoiced
(right) speech segment
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Example 3: Temporal progression of autocorrelation coefficients
speech signal - digit sequence 0861909
0
0
ACF - coefficient for index 3
ACF - coefficient for index 0 (energy)
0
0
0
0
0
ACF - coefficient for index 6
ACF - coefficient for index 9
0
0
Figure 3.8: Temporal progression of speech signal and four autocorrelation coefficients
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3.4
Spectrograms
Using DFT
• Wide-band:
in frequency domain:
– short time window
– “interaction” in the “synchronization” between
time window and “pitch impulses”
– vertical lines
– no resolution of spectral fine structure
• Narrow-band:
in frequency domain:
– long time window
– good resolution of the spectral fine structure
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Example 1: speech spectrograms
Figure 3.9: a) wide-band spectrogram: short time window, high time resolution (vertical lines), no frequency resolution; for voiced signals provides information on formant
structure b) narrow-band spectrogram: long time window, no time resolution, high
frequency resolution (horizontal lines); for voiced signals provides information on fundamental frequency (pitch)
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Example 2: speech spectrograms
Figure 3.10: Wide-band and narrow-band spectrogram and speech amplitude for the
sentence “Every salt breeze comes from the sea”.
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3.5
Filter Bank Analysis
• History:
Decomposition of the signal using a “bank” of band-pass filters and
energy calculation in each frequency band
transfer
function
f
• Today digitally:
– Digital filters:
yk [n] =
∞
X
m=−∞
hk [n − m] x[m] ,
k = 1, . . . , K
– FIR: Finite Impulse Response
IIR: Infinite Impulse Response (recursive filters)
– DFT (FFT) + further processing
• DFT/FFT Method:
– Window function
– Appending zeros for desired “resolution” (zero padding)
– FFT
– “Energy” calculation:
|X(ejω )|, |X(ejω )|2 , log |X(ejω )|
– Weighted averaging for each channel and frequency band respectively
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DFT/FFT filter bank:
transfer
function
f
transfer
function
f
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• Averaging:
summation should be as smooth as possible over all channels
Form: rectangle, triangle, trapeze, etc.
• Choosing the central frequencies fk :
– constant:
∆fk = const. for all k
e.g. 20 channels with ∆f = 200Hz for 0 − 4 kHz
– constant relative band width:
∆fk
= const. for all k
fk
– frequency groups of the ear (total number 24):
f
< 500Hz :
f
≥ 500Hz :
∆f = 100
∆f
= 20%
f
– adjusted to vowels or sounds
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3.6
Mel-frequency scale
The frequency resolution of the human ear is decreasing on the higher
frequencies. This empirical dependency results in the definition of the Mel
scale, which is approximately calculated as (from: Hidden Markov Toolkit,
Cambridge University Engineering Departement, S.J.Young):
f
)
fMEL = 2595 log10 (1 +
700Hz
f
MEL
2700
7000
f / Hz
Compression of the high frequencies
f
fMEL
A filter bank with constant band-widths can be used on the Mel scale:
f
MEL
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Table: MEL Scale:
f /Hz fMEL
65
100
136
200
213
300
298
400
391
500
492
600
603
700
724
800
856
900
1000 1000
1158 1100
1330 1200
1519 1300
1724 1400
1949 1500
2195 1600
2464 1700
2757 1800
3078 1900
3429 2000
3812 2100
4230 2200
4688 2300
5187 2400
5734 2500
6331 2600
6984 2700
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3.7
Cepstrum
The Cepstrum is the Fourier series expansion of the logarithm of the spectrum.
Comparison: autocorrelation function is a Fourier series of the normal
spectrum.
We consider:
y[n] =
∞
X
k=−∞
h[n − k] x[k]
Goal:
Separating the kernel h[n] from the input signal x[n].
This problem is also called inversion or deconvolution.
• Convolution theorem:
Y (ejω ) = H(ejω ) X(ejω )
• Logarithm (complex):
log Y (ejω ) = log H(ejω ) + log X(ejω )
• Inverse Fourier Transform:
F −1 log Y (ejω ) = F −1 log H(ejω ) + F −1 log X(ejω )
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• Another notation:
ŷ[n] = x̂[n] + ĥ[n]
using the definition of the cepstrum for x[n]
(analogous for ŷ[n] and ĥ[n])
x̂[n] = F −1 log X(ejω )
Zπ
1
exp(jωn) log X(ejω ) dω
=
2π
−π
#
"
Zπ
X
1
x[m] exp(−jωm) dω
=
exp(jωn) log
2π
m
−π
= C {x[n]}
• Note:
– Cepstrum = artificial word derived from “spectrum”
– Cepstrum is located in time domain
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Through the cepstrum transformation
x[n]
−→
x̂[n] = C {x[n]}
the convolution comes down to a simple addition.
In the cepstrum domain, a linear operation L (time invariance is not necessary) on ŷ[n] is performed separately on ĥ[n] and x̂[n]:
y[n] =
∞
X
k=−∞
h[n − k] x[k]
ŷ[n] = ĥ[n] + x̂[n]
o
n
L {ŷ[n]} = L ĥ[n] + L {x̂[n]}
With the definition GL for the concatenation of the cepstrum, the operation
L, and the inverse cepstrum
GL := C −1 ◦ L ◦ C
we obtain
GL {h[n] ∗ x[n]} = GL {h[n]} ∗ GL {x[n]} .
Such a transformation GL acts on h[n] and x[n] separately, and is called:
homomorph (structure preserving)
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Complex cepstrum:
1
x̂[n] =
2π
Zπ
exp(jωn) logX(ejω ) dω
−π
Note: complex logarithm
Simple cepstrum (real cepstrum):
1
x̃[n] =
2π
Z2π
exp(jωn) log|X(ejω )| dω
0
• Cepstrum: Fourier coefficients of the logarithmized power spectral
density
• ACF: Fourier coefficients of Fourier series of the power spectral density
Setting cepstral coefficients x̂[n] to zero for high n results in smoothing of
the power spectral density.
Implementation:
Fourier Transform via N –FFT (N = 512, 1024, 2048)
(But: discretisation error):
2π
2π
N −1
j k
1 X j kn
x̂[n] :=
log |X(e N )|
e N
N
k=0
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Example 1: Real cepstrum
Fine structure of power spectral density with the period 1/T results in a
single peak in the cepstrum at time T .
log|F(ω)|2
0
1
T
frequency
F-1(log|F(w)|2)
T
0
time
Figure 3.11: Above: logarithmized power spectrum of a spoken vowel (schematic).
Below: corresponding cepstrum (inverse Fourier–transform of the logarithmized power
spectrum).
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Example 2: Smoothing
speech signal
phoneme "a"
0
0
windowed phoneme "a"
- Hamming window -
0
0
spectrum from cepstrum
whole cepstrum
first 13 coefficients
0
Figure 3.12: Cepstral smoothing: speech signal (vowel “a”), windowed speech signal
(Hamming window), spectrum obtained from the whole cepstrum (blue) and smoothed
spectrum obtained from the first 13 cepstral coefficients (red).
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Example 3: Smoothing with different numbers of cepstral coefficients
speech signal
phoneme "a"
0
0
spectrum from cepstrum
whole cepstrum
first 13 coefficients
0
spectrum from cepstrum
whole cepstrum
first 19 coefficients
0
spectrum from cepstrum
whole cepstrum
first 19 coefficients
first 13 coefficients
0
Figure 3.13: Homomorph analysis of a speech segment: signal progression, homomorph
smoothed spectrum using 13 and 19 cepstral coefficients
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Cepstrum calculation using Filter Bank Output
• Filter bank outputs A[k] for k = 1, . . . , K
Note: k = 0 is missing.
• We complete the outputs symmetrically:
A
-K+1
A
-1
A
A
A A
0
1
2
K
• Symmetry A−k+1 = Ak for all k = 1, . . . , K.
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Inverse DFT a[n] of the symmetric sequence A−K+1 , . . . , AK :
1
a[n] =
2K
K
X
k=−K+1
2πj
nk
Ak exp
2K
K
2πj
2πj
1 X
nk + exp
n(−k + 1)
Ak exp
=
2K
2K
2K
k=1
K
2πj
2πj
1 X
2πj
= exp
0.5
n(k − 0.5) + exp −
n(k − 0.5)
Ak exp
2K
2K
2K
2K
k=1
K
πn
1 X
2πj
0.5
(k − 0.5)
Ak cos
= exp
2K
K
K
k=1
The phase term exp
around k = 0.5.
2πj
2K
0.5 depends on the position of the symmetry axis
Cepstrum is defined as:
a[n] =
K
πn
1 X
(k − 0.5)
Ak cos
K
K
k=1
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Mel Cepstrum according to Davis and Mermelstein
f
= 100
MEL
k=1
f
f
= 300
MEL
MEL
k=K
k=3
Filter bank:
• overlapping band-pass filters triangular shape,
• all channels have equal band width, and filter positioning is equidistant on a Mel scale.
Calculation of the filter bank outputs:
• magnitude of DFT coefficients,
• for each channel summation of the magnitudes according to triangular
weight function,
• for each channel logarithm of the sum.
Thus the filter outputs A[k] with k = 1, . . . , K are obtained. Using the
filter bank outputs, the cepstrum is calculated using a cosine transform.
(see previous description)
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3.8
Statistical Interpretation of the Cepstrum Transformation
We consider the filter bank outputs log|Xk |.
log |X k|
0
s
p
N/2
k
Assumption: The correlation between the outputs s and p, i.e. the element
Csp of the covariance matrix does not depend directly on s or p, but only
on their difference. Because the spectrum is periodical there is no distance
greater than N :
Csp = c(s−p)modN
It is further assumed that the correlation is locally symmetric:
Cs,s+n = Cs,s−n
Then:
c(s−s−n)modN = c(s−s+n)modN
c(−n)modN = c(+n)modN
With 0 ≤ n ≤ N follows:
cn = cN −n
i.e. we have a symmetric cyclic matrix with the kernel vector c.
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Example: the covariance matrix for N

c0 c1 c2 c3
c c c c
 1 0 1 2
c c c c
 2 1 0 1

c c c c
C =  3 2 1 0
 c4 c3 c2 c1

 c3 c4 c3 c2

 c2 c3 c4 c3
c1 c2 c3 c4
= 8:
c4
c3
c2
c1
c0
c1
c2
c3
c3
c4
c3
c2
c1
c0
c1
c2
c2
c3
c4
c3
c2
c1
c0
c1
c1
c2
c3
c4
c3
c2
c1
c0













Such a covariance matrix will be diagonalised using the cosine transform
(or Fourier Transform, which results in the cosine transform due to the
symmetry) (see excursion in chapter 2.17).
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3.9
Energy in acoustic Vector
The energy is usually added as zeroth (or first) component to the acoustic
vector.
For the logarithmic energy we have:
log E =
1
2π
Zπ
log|X(ejω )|2 dω
−π
For the (short time) spectrum or cepstrum it approximately holds:
K
1 X
log E ≈
log|Xk |2
K
k=1
Spectra are usually normalized with log E:
logYk2 = log|Xk |2 − log E
such that:
K
X
k=1
logYk2 ≡ 0
The cepstral coefficient x̂[0] is the logarithmized energy.
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186
Chapter 4
Fourier Transform and Image
Processing
• Overview:
4.1 Spatial Frequencies and Fourier Transform for Images
4.2 Discrete Fourier Transform for Images
4.3 Fourier Transform in Computer Tomography
4.4 Fourier Transform and RST Invariance
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4.1
Spatial Frequencies and Fourier Transform for
Images
A grey-valued image g(x, y) can be interpreted as:
g : IR2 → [0, ∞[
(x, y) → g(x, y)
Space coordinates (x, y) are at first considered as continuous. Discretization and DFT will be analyzed later.
Convention:
g(x, y) ≡ 0
outside of the image.
The Fourier–transform G(fx , fy ) of the image g(x, y) is defined as:
G(fx , fy ) = F {(x, y) → g(x, y)}
Z∞ Z∞
g(x, y) e−2πj(fx x+fy y) dxdy
=
−∞ −∞
The arguments fx and fy are called spatial frequencies.
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The two-dimensional Fourier–transform can be obtained by using two onedimensional Fourier–transforms.
We consider one “image row” with a constant value of y:
x → g(x, y)
Corresponding Fourier–transform Gy (fx ) :
Gy (fx ) = F {x → g(x, y)}
Z∞
g(x, y) e−2πjfx x dx
=
−∞
Then we compute the Fourier–transform of the function:
y → Gy (fx )
and obtain:
F {y → Gy (fx )} =
=
Z+∞
Gy (fx ) e−2πjfy y dy
−∞
Z+∞ Z+∞
g(x, y) e−2πj(fx x+fy y) dxdy
−∞ −∞
In this way we get the result:
G(fx , fy ) = F {y → F {x → g(x, y)}}
For the inverse FT we have (as can be expected):
g(x, y) = F −1 {(fx , fy ) → G(fx , fy )}
Z+∞ Z+∞
G(fx , fy ) e2πj(fx x+fy y) dfx dfy
=
−∞ −∞
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We would like to interpret the two-dimensional FT visually. For this purpose, we consider the exponential factor in the FT and require the following
condition:
!
e2πj(fx x+fy y) = 1
⇒ 2πj(fx x + fy y) = 2πn
⇒
y = −
for n ∈ IN
fx
n
x +
fy
fy
y
1/fy
∆L
θ
1/fx
x
1
∆L = q
fx2 + fy2
spatial period
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Special case:
|G(fx , fy )| has a large value only at “one” point (u, v) = (fx , fy ) in the
spatial frequency plane
fy
|G(fx,fy)|
v
-u
u
fx
-v
y
x
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Since G(fx , fy ) = G(−fx , −fy ) for a real image g(x, y), we have two
dominant frequency pairs in the Fourier–transform integral:
|G(u, v)| · [e2πj(ux+vy) + e−2πj(ux+vy) ] = 2|G(u, v)| · cos 2π(ux + vy)
This function describes a black-white cosine wave pattern with
(fx , fy ) = (u, v)
Where is the value of G(fx , fy ) large ?
ideally: points (u, v) and (−u, −v) represent cosine–variant of the grey
values
really: straight line through (u, v) and (−u, −v) represents abrupt changes
of the grey values
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Figure 4.1: TV–image (analog)
Figure 4.2: Digitized TV–image
Figure 4.3: Amplitude spectrum of Figure 4.2
Figure 4.4: Low-pass filtered
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Figure 4.5: High-pass filtered
Figure 4.6: High-pass enhancement
Explanation for figures 4.1–4.6 (from Duda & Hart 1973, pp. 310–312):
Figure 4.1: TV–image (analog)
Figure 4.2: digitized TV–image
- 120×120 pixels
- grey values from 0 (black) to 15 (white)
Figure 4.3: Fourier–transform of the image from Figure 4.2 (amplitude spectrum)
log|G(fx , fy )|: black =
ˆ high amplitude
note:
1. strong components along the axes
=
ˆ vertical and horizontal image edges
2. concentration around (fx , fy ) = (0, 0)
=
ˆ regions with constant grey values
Figure 4.4: Low-pass filter:
H(fx , fy ) = [cos(πfx ) · cos(πfy )]16
0≦H≦1
Figure 4.5: High pass filter:
H(fx , fy ) = 1.5 − [cos(πfx ) · cos(πfy )]4
0.5 ≦ H ≦ 1.5
Figure 4.6: High pass enhancement:
H(fx , fy ) = 2.0 − [cos(πfx ) · cos(πfy )]4
1.0 ≦ H ≦ 2.0
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Following general rules for G(fx , fy ) ensue:
Edges in the image g(x, y):
• An image edge produces “strong” spatial frequency components along
one straight line in the spatial frequency plane which is orthogonal to
the edge.
• The “sharper” the edge is, the “longer” is the corresponding line in
the spatial frequency domain.
Regions with constant grey values:
• Regions with constant grey values increase the values of |G(fx , fy )|
around the origin (fx , fy ) = (0, 0). (fx , fy ) = (0, 0) is called “DC
component” (average grey value, DC=direct current).
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4.2
Discrete Fourier Transform for Images
The analog image g(x, y) is discretized (“sampled”) along both axes. We
obtain the discrete image:
g[j, k] := g(j · ∆x, k · ∆y) where j, k = 0, 1, . . . , N − 1
√
Change in notation: i = −1 instead of j
G(e
2πi
N ·u
,e
2πi
N ·v
) =
−1
N
−1 N
X
X
2πi
g[j, k]e− N (uj + vk)
where u, v = 0, 1, . . . , N − 1
j=0 k=0
Discretization is written as:
−1
N
−1 N
X
X
G[u, v] =
2πi
g[j, k] e− N (uj + vk)
j=0 k=0
N
−1
X
=
j=0
− 2πi
N uj
e
·
N
−1
X
− 2πi
N vk
g[j, k] e
k=0
!
Interpretation:
Fourier–transform of the image is first performed row by row, then column
by column.
2πi
Using usual definition of the “Fourier matrix” W (i.e. Wvk = (e− N )vk ),
we obtain the matrix representation of Fourier–transform.
Using the notation:
g ∈ IRN xN
W ∈ CN xN
G ∈ CN xN
we obtain
G =
g
=
[W g W ]
1
· [W −1 G W −1 ]
2
N
Note: In the corresponding definition of the Fourier–transform, instead of
the factors 1 and 1/N 2 we have the factors 1/N and 1/N or 1/N 2 and 1.
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4.3
Fourier Transform in Computer Tomography
b
y
x
y=ax+b,
a const.
We consider a projection of the image g(x, y) along the straight line:
y = ax + b
We produce a set of straight lines by keeping a constant and varying b.
Projection:
ga (b) =
Fourier–transform:
Z
g(x, ax + b) dx
Z
ga (b) e−2πjfb b db
Z Z
=
g(x, ax + b) e−2πjfb b db dx
Ga (fb ) =
We substitute b = y − ax and obtain:
Z Z
Ga (fb ) =
g(x, y) e−2πj(yfb −xafb ) dydx
= G(−afb , fb )
= Fourier–transform G(fx , fy ) of g(x, y) along
1
the spatial frequency straight line (fx , fy ) with fy = − fx
a
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Remarks:
a. Straight line in spatial frequency domain: (fx , fy ) = (−afb , fb ) is
orthogonal to y = ax + b:
1
y = ax + b => in Fourier–transform fy = − fx
a
The angle
between these straight lines is a right angle because
1
a· − a = −1.
In general:
y1 (x) = m1 x + b1
y2 (x) = m2 x + b2
y1 (x) ⊥ y2 (x) ⇔ m1 · m2 = −1
b. The value Ga (fb ) is independent of the “offset” b and depends only
on the orientation a of the straight line. Therefore, if we calculate the projection for many different inclinations a and apply the
one-dimensional FT, we obtain the two-dimensional FT of the image
g(x, y).
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4.4
Fourier Transform and RST Invariance
We will investigate invariance of the Fourier–transform to
R : Rotation
S : Scaling
T : Translation
We will use vector notation for the two-dimensional Fourier–transform:
x
coordinates:
z
=
∈ IR2
y
image grey values: g(z) = g(x, y) ∈ IR+
spatial frequency:
f
fx
=
fy
∈ IR2
We ignore the discretization.
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Translation
z → z + z0
x0
with translation vector z0 =
∈ IR2
y0
Image : g(z) → g̃(z) := g(z + z0 )
FT : G̃(f ) = exp (i[fx x0 + fy y0 ]) · G(f )
Rotation
z
→ Dϑ z
with rotation matrix Dϑ =
cos ϑ
sin ϑ
− sin ϑ cos ϑ
y
ϑ
x
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Scaling
z →
Image : g(z) →
FT : G̃(f ) =
α·z
with scaling factor α > 0
g̃(z) = g(α · z)
f
1
·
G
α2
α
basically: similarity principle (S.??) for one-dimensional FT
transferred to two dimensions
Invertible linear mapping
z
Image : g(z)
FT :
→
A·z
where A ∈ IR2×2 invertable
→
G̃(f ) =
=
Proof:
g̃(z) = g(Az)
...
1
· G̃((A−1 )T f )
det(A)
transformation of two-dimensional integration variables
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We apply two basic rules to obtain the RST-invariance:
1. Invariance to translation (=T) can be obtained by using the square of
the absolute value
g(z) → G(f ) → |G(f )|2
2. To obtain RS-invariance we transfer to polar coordinates in the spatial
frequency domain. We write in complex notation:
fz ≡ fx + i fy = r eiφ = exp (ln r + iφ)
∈ C2
Complex logarithm:
fz′ := ln fz = ln r + iφ
We already know:
a) rotation by angle φ0 in spatial domain
=
ˆ rotation by angle φ0 in spatial frequency domain
b) scaling with factor α in spatial domain
1 1 2
=
ˆ scaling with factor
respectively in spatial
and
α
α
frequency domain
fz′
scaling and
−→
rotation
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f˜z′
= ln z̃
= ln r̃ + iφ̃
r = ln
+ i(φ − φ0 )
α
= ln r + i φ − ln α − i φ0
= fz′ − ln α − i φ0
= translation with the shift vector (− ln α − i φ) ∈ C2
in logarithmic polar coordinates of the spatial frequency plane
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RST-invariant features can therefor be obtained as follows:
g(x, y)
image: (x, y) ∈ IR2
first Fourier–transform
G(fx , fy )
= F {g(x, y)}
mit (fx , fy ) ∈ IR2
squared absolute value
|G(fx , fy )|2
logarithmic polar coordinates: (ln r, φ)
|G(ln r, φ)|2
second Fourier–transform
F (|G(ln r, φ)|2 )
squared absolute value
|F (|G(ln r, φ)|2 )|2
Next page:
Analysis of the RST-invariant features.
Original grey valued images are identical up to a 90◦ –rotation.
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y
6
- x
original image
fy
6
- f
x
|FFT|
φ
6
-
log–polar
ln r
f˜y
6
- f˜
x
|FFT|
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Warning:
a) Invariant observations are not necessarily good for classification.
b) Observations that are calculated using the two-dimensional Fourier–
transform are not complete, i.e. the original image cannot be reconstructed completely.
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Chapter 5
LPC Analysis
• Overview:
5.1 Principle of LPC Analysis
5.2 LPC: Covariance Method
5.3 LPC: Autocorrelation Method
5.4 LPC: Interpretation in Frequency Domain
5.5 LPC: Generative Model
5.6 LPC: Alternative Representations
The acronym LPC stands for
Linear Predictive Coefficients / Coding
and is utilized in signal processing and frequency analysis, as well as in
signal coding.
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5.1
Principle of LPC Analysis
n-2 n
time
We consider a discrete time signal x[n], possibly multiplied with a window
function. The goal of an LPC analysis is to predict each signal value x[n]
by its preceding values x[n − 1], x[n − 2], ..., x[n − K]. We distinguish:
x[n] :
x̂[n] :
signal value
predicted value
We assume the predicted value x̂[n] to be a linear combination of the
preceding values of x[n]:
x̂[n] :=
K
X
k=1
αk x[n − k]
with at first unknown coefficients αk , k = 1, ..., K, which are called
LPC–coefficients or prediction coefficients.
The value K is called prediction order, e.g. K = 8, . . . , 10 at a sampling
frequency of 4 kHz (about 2 coefficients per kHz).
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Outlook
Starting point: “coding” in time domain (goal: bit reduction)
↓
Parseval Theorem
parametric model for power spectrum of Fourier–transform
(more exact: rough structure of power spectrum for speech signal)
LPC analysis applications:
• speech coding
(ADPCM = adaptive differential pulse code modulation)
• signal processing:
parametric modelling with autoregressive or all-pole models (order K)
• time curves:
resonance and oscillator curves, sun spots, stock-market course, ...
• image coding
• also: interpretation as Maximum Entropy Approach
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The coefficients αk are unknown at first. To estimate these, we define the
prediction error for each point n in time:
e[n] := x[n] − x̂[n]
K
X
= x[n] −
αk x[n − k]
k=1
For a reliable set of LPC–coefficients we calculate the squared error criterion E as sum of the squared prediction errors e[n]:
X
e2 (n)
E =
n
=
X
n
!
=
"
x[n] −
K
X
k=1
αk x[n − k]
#2
minimum with respect to α1 , . . . , αk , . . . , αK
∂
Taking the derivative ∂α
for l = 1, . . . , K results in:
l
P
P
!
x[n] − αk x[n − k] x[n − l] = 0
n
P
k
k
P
P
αk x[n − k]x[n − l] = x[n − l]x[n]
n
n
Here, the summation limits are not specified on purpose.
If the squared error criterion E is considered as a function of LPC–coefficients,
the following properties ensue:
• E is quadratic in α1 , . . . , αk , . . . , αK ; it is guaranteed to be nonnegative and it has a single well-defined minimum.
• The optimal LPC–coefficients are invariant
to linear scaling of the signal values x[n].
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Minimization of the squared error criterion with respect to the LPC–
coefficients results either from taking the derivative or from the “quadratic
complement” (recalculate for yourself!). The linear equation system for
the LPC–coefficients αk ensues:
l = 1, . . . , K :
K
P
k=1
αk ·
P
n
x[n − k] x[n − l] =
X
n
x[n − l] x[n]
with still unspecified summation limits over n. We consider two methods
for the choice of summation limits:
1. covariance method
2. autocorrelation method
Warning: terminology is not consistent.
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5.2
LPC: Covariance Method
known values
predicted value
0
n
N-1
Covariance Method
• No window function is applied, such that we obtain the following
summation limits:
X
2
e (n) =
n
N
−1
X
e2 (n)
n=0
i.e. we also use signal values x[n] with n < 0 for prediction.
• The resulting equation system for LPC–coefficients:
l = 1, . . . , K :
K
X
αk Φ(l, k) = Φ(l, 0)
k=1
with the definition:
Φ(l, k) :=
N
−1
X
n=0
x[n − l] x[n − k]
For the above terms hold:
– they describe a kind of cross correlation between two “signals”
– they are similar to a covariance matrix
Computational complexity for solving the equation system:
O(K 3 ) + O(N K)
– autocorrelation method has more favorable complexity: O(K 2 )
– but: calculation of auto/cross-correlation function dominates
In contrast to covariance method, autocorrelation method offers an interpretation in the frequency domain and therefore is often preferred.
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5.3
LPC: Autocorrelation Method
window
function
0
n
N-1
We consider the signal after multiplication with a convenient window function, usually Hamming window:
In principle, the summation limits now are
X
2
e [n] =
n=+∞
X
e2 [n] .
n=−∞
n
Since, due to windowing the signal x[n] is identical to zero outside the
window function, i.e.
x[n] ≡ 0
for n < 0 or N − 1 < n
we obtain the following for the prediction error e[n]:
e[n] ≡ 0
for n < 0 or N − 1 + K < n
Therefore, the total error E becomes:
E =
NX
+K−1
e2 [n]
n=0
The prediction error e[n] can become “large” on the window function
boundaries:
- Beginning: prediction from ”zeros”
- End:
prediction of ”zeros”
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Inserting the summation limits:
X
x[n − k] x[n − l] = R(|l − k|),
n
where R(|l − k|) =
R(|l|) =
NX
−1−l
n=0
X
n
X
n
x[n] x[n − l] = R(|l|)
x[n − k] x[n − l]
x[n] x[n − l] =
NX
−1−l
n=0
x[n] x[n − l]
In this way we obtain the following equation system for the LPC–coefficients
αk :
l = 1, ..., K :
K
X
k=1
αk R(|l − k|) = R(l)
or in matrix form:














R(0)
R(1)
R(1)
..
.
R(0)
..
.
R(K − 1) R(K − 2)

 

α1
R(1)



α2 
...
R(K − 2) 
  R(2) 

 



 

..

.
 



 


=



..
.
.
...
  ..   .. 

.

 


 


 

R(1)  





. . . R(1)
R(0)
αK
R(K)
...
R(K − 1)
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Note that this equation system is completely determined by the autocorrelation coefficients
R(0), ..., R(k), ..., R(K).
Hence, the autocorrelation coefficients will “only” be converted to obtain
the LPC–coefficients
α1 , ..., αk , ..., αK .
The matrix of this equation system has the following properties:
- Toeplitz structure (follows from time invariance)
- solution: Durbin–algorithm with complexity O(K 2 )
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5.4
LPC: Interpretation in Frequency Domain
The LPC autocorrelation method allows prediction error conversion from
time domain into frequency domain using Parseval theorem so that LPC
analysis can be interpreted as adaptation of parametric model spectrum to
the observed signal spectrum.
We start with the prediction error e[n]:
e[n] = x[n] −
K
X
k=1
αk x[n − k]
and apply the z–transform to this equation. The z–transform is restricted
to the unit circle.
z = ejω
∈C
For the z-transforms E(z) and X(z) we obtain:
"
#
K
X
E(z) = X(z) · 1 −
αk z −k
k=1
The total error Etot for the squared error criterion becomes:
Etot =
=
=
=
NX
+K−1
e2 [n]
n=0
Z+π
1
2π
1
2π
1
2π
|E(ejω )|2 dω
−π
Z+π
(Parseval Theorem)
2
K
X
−jωk αk e
· |X(ejω )|2 dω
1 −
−π
Z+π
−π
k=1
P (ejω )2 · |X(ejω )|2 dω
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with the so-called predictor polynom:
jω
P (e ) := 1 −
K
X
αk e−jωk
k=1
Squared absolute value of the predictor polynom
2
K
X
−jωk P (ejω )2 = 1 −
αk e
k=1
= ...
=
K
X
k=1
Bk · cos(ωk)
(with suitable coefficients Bk resulting from the predictor coefficients) is a
polynom with respect to cos(ω), which can be obtained via application of
trigonometric transformations.
The predictor polynom tries to “compensate” for |X(ejω )|2 – especially at
maxima – and to generate a “white” spectrum for the prediction error e[n].
The complex predictor polynom P (z) with z ∈ C has exactly K zeros in
the complex plane and therefore can be factorised into linear factors:
K
Y
P (z) =
(z − zk )
k=1
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Observations:
• These zeros are complex conjugated pairs because αk ∈ IR.
jω 2
• The zeros can cause ”minima” of P (e ) . The minima of |P (ejω )|2
approximately correspond to the maxima of the smoothed spectrum
|X(ejω )|2 , because for minimization of the error integral it is first of
all necessary to “compensate” for the maxima of the signal spectrum.
The LPC analysis could therefore be used to describe of the speech
signal formant structure.
|P(e iω )|2
|X(e iω )|2
ω
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windowed phoneme "a"
- Hamming window -
prediction error
- 12 LPC-coefficients -
0
0
0
0
LPC-spectrum
- 12 coefficients -
spectrum of
prediction error
(12 LPC-coefficients)
0
0
LPC-spectrum
- 18 coefficients -
0
Figure 5.1: LPC–analysis of one speech segment
a) signal progression, b) prediction error (K=12), c) LPC–spectrum with K=12 coefficients, d) spectrum of the prediction error (K=12), e) LPC–spectrum with K=18 coefficients
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windowed phoneme "a"
- Hamming window -
amplitude spectrum
- Hamming window -
0
0
0
LPC-spectrum
- 4 coefficients -
LPC-spectrum
- 8 coefficients -
0
0
LPC-spectrum
- 12 coefficients -
LPC-spectrum
- 16 coefficients -
0
0
LPC-spectrum
- 18 coefficients -
LPC-spectrum
- 20 coefficients -
0
0
Figure 5.2: LPC–Spectra for different prediction orders K
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5.5
LPC: Generative Model
e(n)
x(n)
recursive
filter
αk
For the prediction error e[n] and its z–transform holds:
e[n] = x[n] −
K
X
αk x[n − k]
k=1
K
X
E(z) = X(z) −
αk X(z) z −k
k=1
= X(z) · [1 −
K
X
αk z −k ]
k=1
If we consider prediction error as input signal, we can also interpret the
LPC–theorem as generative model which generates an output signal x[n]
from an adequate “input signal” e[n]:
x[n] = e[n] +
K
X
k=1
αk x[n − k] .
For the signal spectrum X(z) holds:
X(z) =
E(z)
K
P
αk z −k
1−
k=1
This model is called autoregressive model. The excitation has to be chosen
such that E(z) is “white”, i.e. it does not have fine structure due to the
fundamental frequency (”pitch–frequency”).
In other words:
E(z) = G = const. (”gain”)
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Special case:
E[n] = G · δ[n]
Then for LPC model spectrum X(z) holds:
X(z) =
G
K
P
αk z −k
1−
k=1
This spectrum is often interpreted as LPC model spectrum X(z) of observed signal. It is reasonable to set (without explanation):
#
"
K
K
X
X
R(k)
G2 = R(0) −
αk R(k) = R(0) 1 −
αk
R(0)
k=1
k=1
This LPC model spectrum does not have any zeros, it has only poles, and
therefore is also called all–pole model.
Remarks:
• stability problems by solving the equation system
(←− truncation error in autocorrelation)
• way out: preemphasis through difference calculation
• absolute rule for choice of order K:
1 formant needs 2 LPC–coefficients
1 formant per kHz
+ excitation pulse shape + radiation: 2 LPC–coefficients
=⇒ rule of thumb:
bandwidth
4 kHz
5 kHz
6 kHz
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5.6
LPC: Alternative Representations
• so far:
G
gain
αk
LPC–coefficients
• impulse response of generative model
• impulse response of squared absolute value of ”predictor polynom”
• cepstrum
• poles / zeros of synthesis model / ”predictor polynom”
=⇒ formants / bandwidths
problem: noise susceptible
• PARCOR–coefficients: partial correlation
• Area–coefficients: cross-section surfaces Ak
• reflexion coefficients ∼ PARCOR; tube model
A1
A2
A3
Glottis
A4
A5
Lips
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Chapter 6
Outlook: Wavelet Transform
• Overview:
6.1 Motivation: from Fourier to Wavelet Transform
6.2 Definition
6.3 Discrete Wavelet Transform
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6.1
Motivation: from Fourier to Wavelet Transform
Fourier transform uses infinitely extended basis functions
e−jωt
and therefore does not have any time resolution, i.e. there is no information about the localization along the time axis.
Therefore, a window function often is used
t → w(t),
complex in general
This function has finite support such that it is possible to investigate a
segment of a function of interest.
We define a short–time Fourier transform Fb (w) of time signal t → f (t)
at position b ∈ IR in time:
Fb (w) :=
Z+∞
−∞
f (t)w(t − b)e−jωt dt
where w(t) denotes the complex conjugated value of w(t).
The wavelet–transform can be derived from this equation in two steps:
a) we ignore the basis function e−jωt and we define the window function
as the new basis function.
b) in addition to the localization parameter b, we also introduce a scaling
parameter a > 0.
We consider the family of window functions Wab (t):
t−b wab (t) := w
a
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6.2
Definition
The following notation is usually used for the Wavelet–transform:
t−b
1
ψab (t) := √ · ψ
a
a
which is the so-called Mother-Wavelet t → ψ(t).
Like the window function for the short–time Fourier transform the MotherWavelet should be “localized” as much as possible.
Example:
Mexican-Hat Function:
1 2
ψ(t) = (1 − t2 )e− 2 t
ψ (t)
t
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The wavelet–transform of f (t) with respect to ψ(t) is defined as:
1
Fψ (a, b) =
a
Z+∞
−∞
t−b f (t) · ψ
dt
a
with scaling parameter a > 0 and localization parameter b ∈ IR.
For the inverse transformation holds:
Z+∞ Z+∞
t−b 1 1
f (t) =
·
· da db Fψ (a, b) · ψ
Cψ a2
a
−∞
0
with
Cψ :=
Z∞
0
|ψ(ω)|2
dω < ∞
ω
Proof (principle only):
The proof uses the (generalized) Parseval Theorem:
Fψ (a, b)
1
= √
a
1
= √
a
with
Z+∞
−∞
Z+∞
−∞
1
ψab (t) = √ · ψ
a
f (t) · ψ ab (t) dt
F (ω) · ψab (ω) dω
t−b
a
using further conversions.
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6.3
Discrete Wavelet Transform
For the scaling parameters a > 0 we choose:
a = α0m
where α0 > 1 and m ∈ Z.
The values of m determine the width of wavelet ψab (t).
In order to adjust the localization parameter properly, we define:
b = n b0 am
0
where b0 > 0 and n ∈ Z.
Thus we constrain the Wavelet transform to discrete values:
Fψ (a, b)
→
Fψ (n, m)
The choice of the function ψ(t) is still open.
It is useful to choose ψ(t) such that the function system {ψmn |m, n ∈
Z, m > 0}
with
−m
2
ψmn (t) := a0
· ψ(a−m
0 t − nb0 )
represents an orthonormal basis for functions t → f (t)
∈ L2 (IR).
Note: The scalar product < f (t), g(t) > of two functions f (t) and g(t) is
defined as:
Z
< f (t), g(t) > =
f (t) · g(t) dt
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In this way we obtain the following representation for the discrete Wavelet–
transform:
Fψ (m, n) =
Z+∞
−∞
−m
f (t)a0 2 ψ(a−m
0 t − nb0 ) dt
= < f (t), ψmn (t) >
Due to the orthogonality it is possible to convert the integral of the inverse
Wavelet–transform into an infinite series:
f (t) =
=
1 XX
Fψ (m, n)ψmn (t)
Cψ m n
1 XX
−m
Fψ (m, n) a0 2 ψ(a−m
0 t − nb0 )
Cψ m n
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Example:
Haar –function and Haar –basis
special choice: a0 = 2
b0 = 1
The Haar –function is defined as:

0 ≦ t ≦ 12

1
ψ(t) = −1 12 ≦ t < 1


0
otherwise
This defines the Haar –basis
ψ(t) | m, n ∈ Z, m > 0 :
1
ψmn (t) = √ · ψ
2m
2m t − n
It is easy to see that for increasing m a increasingly finer resolution is obtained and that n determines localization in time.
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Chapter 7
Coding
The following types of coding are distinguished:
• source coding (data compression)
goal: transmission (storage) using as few bits as possible without or
with few errors
• channel coding
goal: preferably faultless data transmission (storage)
e.g. error-recognizing and error-correcting codes
• simultaneous source and channel coding
goal: simultaneous optimization
The following data types are distinguished:
• discrete alphabet
• continuous signal (audio, video, . . . )
Source coding
• lossless coding (compression)
usually discrete sources, e.g. text compression
• lossy coding
usually continuous signals
notation:
rate - distortion theory
distortion, error
bit rate
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Three effects can be utilized for signal coding:
a) statistical redundancy and correlation:
samples are not independent.
b) perceptive properties of the receiver (ear and eye):
some fine structures in the signal are irrelevant to the receiver
c) signal distortion:
coded signal differs from the original signal without significant quality
deterioration.
signal
Q
T
C
transmission
reconstructed
signal
T -1
Q -1
C -1
T:
transformation, e.g. DCT
Q:
quantization, e.g. vector quantization
C:
mapping of bit representation
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References:
• Ze-Nian Li: CMPT 365 Multimedia Systems. Simon Fraser
University, British Columbia, Canada, fall 1999, Version Jan.2000;
http://www.cs.sfu.ca/CourseCentral/365/li/index_prev.html.
• Peter Noll: MPEG Digital Audio Coding. IEEE Signal Processing
Magazine, pp.59-81, Sep. 1997.
• Thomas Sikora: MPEG Digital Video-Coding Standards. IEEE Signal
Processing Magazine, pp.82-100, Sep. 1997.
• A. Ortega, K. Ramchandran: Rate-Distortion Methods for Image
and Video Compression. IEEE Signal Processing Magazine, pp.2350, Nov. 1998.
• G. J. Sullivan, Th. Wiegand: Rate-Distortion Optimization for Video
Compression. IEEE Signal Processing Magazine, pp.74-90, Nov. 1998.
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Chapter 8
Image Segmentation and
Contour-Finding
The lecture notes for this chapter are available as a separate document.
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