DIGITAL PROCESSING OF SPEECH AND IMAGE SIGNALS · Lecture DIGITAL PROCESSING OF SPEECH AND IMAGE...

Lecture

DIGITAL PROCESSING

OF

SPEECH AND IMAGE SIGNALS

RWTH Aachen, SS 2003

Prof. Dr.-Ing. H. NeyLehrstuhl fur Informatik VI

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 1997Completions: J. Dahmen, F. Hilger, S. Koepke; Mai 2000Completions: F. Hilger, D. Keysers; Juli 2001Translation: M. Popovic, R. Schluter; April 2003

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-HillClassic 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 Algorithmsfor Signal Processing. Prentice Hall, Upper Saddle River, NJ, 2000.

• J. R. Deller, J. G. Proakis, J. H. L. Hansen: Discrete-Time Processingof Speech Signals, Macmillan Publishing Company, New York, NY,1993.

• W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery: Nu-merical Recipes in C, Cambridge Univ. Press, Cambridge, 1992.

• L. Rabiner, B. H. Juang: Fundamentals of Speech Recognition, Pren-tice Hall, Englewood Cliffs, NJ, 1993.

• T. Lehmann, W. Oberschelp, E. Pelikan, R. Repges: Bildverarbeitungfur die Medizin, Springer Verlag, Berlin, 1997.

• L. Berg: Lineare Gleichungssysteme mit Bandstruktur, VEB DeutscherVerlag der Wissenschaften, Berlin, 1986.

Contents

1 System Theory and Fourier Transform 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Linear time-invariant Systems . . . . . . . . . . . . . . . . 11

1.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 16

1.4 Properties of the Fourier Transform . . . . . . . . . . . . . 25

1.5 Parseval Theorem . . . . . . . . . . . . . . . . . . . . . . . 32

1.6 Autocorrelation Function . . . . . . . . . . . . . . . . . . . 33

1.7 Existence of the Fourier Transform . . . . . . . . . . . . . 34

1.8 δ-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.9 Motivation for Fourier Series . . . . . . . . . . . . . . . . . 40

1.10 Time Duration and Band Width . . . . . . . . . . . . . . . 44

2 Discrete Time Systems 51

2.1 Motivation and Goal . . . . . . . . . . . . . . . . . . . . . 52

2.2 Digital Simulation using Discrete Time Systems . . . . . . 53

2.3 Examples of Discrete Time Systems . . . . . . . . . . . . . 56

2.4 Sampling Theorem (Nyquist Theorem) and Reconstruction 61

2.5 Logarithmic Scale and dB . . . . . . . . . . . . . . . . . . 70

2.6 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.7 Fourier Transform and z–Transform . . . . . . . . . . . . . 74

2.8 Examples of Discrete Time Signal Fourier Transform . . . 78

2.9 Discrete Time Signal Fourier Transform Theorem . . . . . 88

2.10 Discrete Fourier Transform: DFT . . . . . . . . . . . . . . 90

2.11 DFT as Matrix Operation . . . . . . . . . . . . . . . . . . 98

2.12 From Continuous Fourier Transform to Matrix Representa-tion of Discrete Fourier Transform . . . . . . . . . . . . . . 102

2.13 Frequency Resolution and Zero Padding . . . . . . . . . . 104

2.14 Finite Convolution . . . . . . . . . . . . . . . . . . . . . . 105

2.15 Fast Fourier Transform (FFT) . . . . . . . . . . . . . . . . 108

2.16 FFT Implementation . . . . . . . . . . . . . . . . . . . . . 118

i

2.17 Cyclic Matrices and Fourier Transform . . . . . . . . . . . 124

3 Spectral analysis 1313.1 Features for Speech Recognition . . . . . . . . . . . . . . . 1323.2 Short Time Analysis and Windowing . . . . . . . . . . . . 1353.3 Autocorrelation Function and Power Spectral Density . . . 1593.4 Spectrograms . . . . . . . . . . . . . . . . . . . . . . . . . 1653.5 Filter Bank Analysis . . . . . . . . . . . . . . . . . . . . . 1683.6 Mel-frequency scale . . . . . . . . . . . . . . . . . . . . . . 1713.7 Cepstrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 1733.8 Statistical Interpretation of the Cepstrum Transformation 1833.9 Energy in acoustic Vector . . . . . . . . . . . . . . . . . . 185

4 Fourier Transform and Image Processing 1874.1 Spatial Frequencies and Fourier Transform for Images . . . 1884.2 Discrete Fourier Transform for Images . . . . . . . . . . . 1964.3 Fourier Transform in Computer Tomography . . . . . . . . 1974.4 Fourier Transform and RST Invariance . . . . . . . . . . . 199

5 LPC Analysis 2075.1 Principle of LPC Analysis . . . . . . . . . . . . . . . . . . 2085.2 LPC: Covariance Method . . . . . . . . . . . . . . . . . . . 2125.3 LPC: Autocorrelation Method . . . . . . . . . . . . . . . . 2135.4 LPC: Interpretation in Frequency Domain . . . . . . . . . 2165.5 LPC: Generative Model . . . . . . . . . . . . . . . . . . . 2215.6 LPC: Alternative Representations . . . . . . . . . . . . . . 223

6 Outlook: Wavelet Transform 2256.1 Motivation: from Fourier to Wavelet Transform . . . . . . 2266.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276.3 Discrete Wavelet Transform . . . . . . . . . . . . . . . . . 229

7 Coding (appendix available as separate document) 233

8 Image Segmentation and Contour-Finding 237The lecture notes for this chapter are available as a separate document.

ii

List of Figures

1.1 Oscillograms of three time functions composed as sum of 20partial oscillations. a) φn = 0, b) φn = π

2 , c) φn statistical. 3

1.2 Amplitude spectrum of a time function composed as sum of20 partial tones. . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 from left to right: original photo, low-pass and high-passfiltered version . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Phase manipulation for portion of a speech signal (vowel ’o’)sampled at 8kHz, 25ms analysis window (200 samples), 512point FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 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 ofspeech production . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Digital photo . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.2 Gradient image . . . . . . . . . . . . . . . . . . . . . . . . 58

2.3 Several real cases of Laplace Operator subtraction from orig-inal image. a) Original image b) Original image minusLaplace Operator (negative values are set to 0 and valuesabove the grey scale are set to the highest grade of grey) . 60

2.4 Ideal reconstruction of a band-limited signal (from Oppen-heim, Schafer)a) original signal b) sampled signal c) reconstructed signal 64

iii

2.5 Sampling of band-limited signal with different sampling rates:a) sampling rate higher than Nyquist rate - exact reconstruc-tion possibleb) sampling rate equal to Nyquist rate - exact reconstruc-tion possibled) sampling rate smaller than Nyquist rate - aliasing - exactreconstruction not possible . . . . . . . . . . . . . . . . . . 65

2.6 Amplitude spectrum of the voiceless phoneme “s” from theword “ist” . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.7 Logarithmic amplitude spectrum of the phoneme “s” . . . 71

2.8 Amplitude spectrum of the voiced phoneme “ae” from theword “Ah” . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.9 Logarithmic amplitude spectrum of the phoneme “ae” . . . 71

2.10 Amplitude spectrum of a speech pause . . . . . . . . . . . 71

2.11 Logarithmic amplitude spectrum of a speech pause . . . . 71

2.12 Hanning window . . . . . . . . . . . . . . . . . . . . . . . 103

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] =0ii) n between 0 and Nh + Nx − 2, convolution 6= 0iii) n > Nh + Nx − 2, no overlap with h[k], convolution y[n]= 0c) resulting convolution y[n]. . . . . . . . . . . . . . . . . . 106

2.14 Flow diagram for decomposition of one N -DFT to two N/2–DFTs with N = 8 . . . . . . . . . . . . . . . . . . . . . . . 110

2.15 Flow diagram of an 8–point–FFT using Butterfly operations. 111

2.16 Flow diagram of an 8–point–FFT using Butterfly operations. 120

2.17 Input and output arrays of an FFT. a) The input array con-tains N (N is power of 2) complex input values in one real

array of the length 2N . with alternating real and imagi-nary parts. b) The output array contains complex Fourierspectrum at N frequency values. Again alternating real andimaginary parts. The array begins with the zero-frequencyand then goes up to the highest frequency followed withvalues for the negative frequencies. . . . . . . . . . . . . . 122

iv

3.1 Example for the application of the Discrete Fourier Trans-form (DFT). . . . . . . . . . . . . . . . . . . . . . . . . . . 138

3.2 a) signal v[n]; b) DFT-spectrum V [k]; c) Fourier spectrumV (ejω). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

3.3 a) signal v[n]; b) DFT-spectrum V [k]; c) Fourier spectrumV (ejω). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

3.4 a) DFT of length N = 64; b) DFT of length N = 128; c)Fourier spectrum V (ejω). . . . . . . . . . . . . . . . . . . . 151

3.5 Influence of the window function:above: speech signal (vowel “a”); central: 512 point FFTusing rectangle window; below: 512 point FFT using Ham-ming 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 Hammingwindow (50 sampled values), d) low resolution Fourier Trans-form using autocorrelation function (19 coefficients), e) lowresolution Fourier Transform using autocorrelation function(13 coefficients) . . . . . . . . . . . . . . . . . . . . . . . . 162

3.7 Signal progression and autocorrelation function of voiced(left) and unvoiced (right) speech segment . . . . . . . . . 163

3.8 Temporal progression of speech signal and four autocorrela-tion coefficients . . . . . . . . . . . . . . . . . . . . . . . . 164

3.9 a) wide-band spectrogram: long time window, high time res-olution (vertical lines), no frequency resolution; for voicedsignals provides information on formant structure b) narrow-band spectrogram: short time window, no time resolution,high frequency resolution (horizontal lines); for voiced sig-nals provides information on fundamental frequency (pitch) 166

3.10 Wide-band and narrow-band spectrogram and speech am-plitude for the sentence “Every salt breeze comes from thesea”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

3.11 Above: logarithmized power spectrum of a spoken vowel(schematic).Below: corresponding cepstrum (inverse Fourier–transformof the logarithmized power spectrum). . . . . . . . . . . . 177

v

3.12 Cepstral smoothing: speech signal (vowel “a”), windowedspeech signal (Hamming window), spectrum obtained fromthe whole cepstrum (blue) and smoothed spectrum obtainedfrom the first 13 cepstral coefficients (red). . . . . . . . . . 178

3.13 Homomorph analysis of a speech segment: signal progres-sion, homomorph smoothed spectrum using 13 and 19 cep-stral coefficients . . . . . . . . . . . . . . . . . . . . . . . . 179

4.1 TV–image (analog) . . . . . . . . . . . . . . . . . . . . . . 1934.2 Digitized TV–image . . . . . . . . . . . . . . . . . . . . . . 1934.3 Amplitude spectrum of Figure 4.2 . . . . . . . . . . . . . . 1934.4 Low-pass filtered . . . . . . . . . . . . . . . . . . . . . . . 1934.5 High-pass filtered . . . . . . . . . . . . . . . . . . . . . . . 1944.6 High-pass enhancement . . . . . . . . . . . . . . . . . . . . 194

5.1 LPC–analysis of one speech segmenta) signal progression, b) prediction error (K=12), c) LPC–spectrum with K=12 coefficients, d) spectrum of the predic-tion error (K=12), e) LPC–spectrum with K=18 coefficients 219

5.2 LPC–Spectra for different prediction orders K . . . . . . . 220

List of Tables

2.1 Fourier transform pairs . . . . . . . . . . . . . . . . . . . . 872.2 Fourier transform Theorems . . . . . . . . . . . . . . . . . 88

Chapter 1

System Theory and FourierTransform

• 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 SS 2003

1

1.1 Introduction

What distinguishes the Fourier Transform (FT) fromother transformations?

1. Mathematical property of linear time-invariant systems:

• FT decomposes the time signal into “eigenfunctions”“eigenfunctions” keep their form by passing thelinear time-invariant system

A x = λ x

• 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 does not or only very weakly sense phase differ-ences of partial tones of the complete sound of stationary pro-cesses.

Fourier transform in speech processing:

• Calculation of the spectral components of speech

• Basic method for obtaining observations (features) forspeech recognition


2

0

0

φ = 0

0

0

φ = π/2

0

0

random φ

Figure 1.1: Oscillograms of three timefunctions composed as sum of 20 partialoscillations. a) φn = 0, b) φn = π

2, c) φn

statistical.

0

Figure 1.2: Amplitude spectrum of a timefunction composed as sum of 20 partialtones.

Figure 1.3: from left to right: original photo, low-pass and high-pass filtered version


3

amplitude spectrum

0

original signal

0

0

inverse FT for phase φ(f) = 0

0

0

Inverse FT for phase φ(f) = π2

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


4

amplitude spectrum

0

original signal

0

0


0

0

inverse FT for phase φ(f) = π2

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 at8kHz, 25ms analysis window (200 samples), 512 point FFT


5

amplitude spectrum

0

original signal

0


0

0

inverse FT for phase φ(f) = π2

0

0

inverse FT for random phase φ(f)

0

0

Figure 1.6: Phase manipulation for a Heaviside–function (step–function)


6

Why Fourier?

Roughly:

• Production, description and algorithmic operations on signals (func-tions or measurement curves over the time axis) can be described verywell in Fourier domain (frequency domain).

Deeper reason:

• Production, description and algorithmic operations on signals are largelybased 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


7

t

AT

glottal pulses

vocaltractfilter

radiationfrom lipsand nose

t

T

speech [a:]a)

b)

0 4 0 4 0 4 0 4

1/T

|E(f)| [dB] |V(f)| [dB][a:] |A(f)| [dB] |S(f)| [dB]

* * =[a:]

MUSCLEFORCE

LUNGVOLUME

PHARYNXCAVITY

NASALCAVITY

MOUTHCAVITY

TONGUEHUMP

VELUM

VOCALCORDS

TRACHEA ANDBRONCHI

LARYNXTUBE

NOSEOUTPUT

MOUTHOUTPUT

Figure 1.7: Schematic representation of the physiological mechanism of speech production


8

signal (speech, image)

feature extraction(signal analysis)

feature vector(pattern vector)

(pattern)comparison

decision

reference data(vectors, features)

Examples:

• Spoken language

• Written numbers (letters)

• Cell recognition (red blood cells)


9

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


10

1.2 Linear time-invariant Systems

Example:

– speech production

– electrical systems

S

h(t)input signal

x(t)output signal

y(t)

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 xLTI–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) , t0 ∈ IR


11

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) =

∞∫

−∞x(t− τ) h(τ) dτ

=

∞∫

−∞x(τ) h(t− τ) dτ

= x(t) ∗ h(t)

• h: impulse response of the system S

∆τ1/∆τ

∆τe (t)

t t

x (t)

τi

• 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 timeshifted elementary functions e∆τ(t):

x(t) = lim∆τ→0

[ ∑i

x(τi) e∆τ(t− τi) ∆τ

]


12

Hence the following holds for the output signal y(t):

y(t) = S x(t)

= S

lim

∆τ→0

∑i


= lim∆τ→0

[S

∑i


]

additivity:

= lim∆τ→0

[ ∑i

S x(τi) e∆τ(t− τi) ∆τ]

homogeneity (for x(τi) and ∆τ):

= lim∆τ→0

[ ∑i

x(τi) S e∆τ(t− τi) ∆τ

]

time invariance:

= lim∆τ→0

[ ∑i

x(τi) h∆τ(t− τi) ∆τ

]

limiting case ∆τ → 0 :

∑−→

∫

∆τ −→ dτ

τi −→ τ

h∆τ(t) −→ h(t)

result:

y(t) =

∞∫

−∞x(τ) h(t− τ) dτ = x(t) ∗ h(t)

h(t): impulse response of the system


13

Examples of LTI-operations:

• Oscillatory systems (electrical or mechanical) withexternal excitation:

x(t) −→ h(τ) −→ y(t)

y(t) =

∫h(t− τ) x(τ) dτ

y′′(t) + 2αy′(t) + β2y(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)

x(t) =1

T

+T/2∫

−T/2

x(t + τ) dτ

• Differentiator:

x(t) −→ S −→ y(t) := x′(t)

• 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

∑k

cky(k)(t) =

∑l

dlx(l)(t)

[ + further constraints ]


14

Example of a non-linear system:

system: y(t) = x2(t)

x(t) = A cos(βt)

=⇒ y(t) = A2 cos2(βt) =A2

2(1 + cos(2βt))

frequency doubling


15

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

j2 = −1, j ∈ C

cos

sin

αα

α

Im

Re

1

1

complex representation: ej α = cos α + j sin α, α ∈ IR

cos α =ejα + e−jα

2and sin α =

ejα − e−jα

2j

dimension:

DIM(ω) DIM(t) = 1

DIM(ω) =1

DIM(t)=

1

[sec]= [Hz]


16

LTI-System

y(t) =

∞∫

−∞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) =

∞∫

−∞A ej(ω(t−τ)+ϕ)h(τ)dτ

= A ej(ωt+ϕ)

∞∫

−∞h(τ)e−jωτdτ

︸︷︷︸H(ω) = F h(τ)

= x(t) · H(ω)

• Definition of the Fourier transform:

H(ω) =

∞∫

−∞h(τ)e−jωτdτ = F h(τ)

(→ 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 (“eigenfunc-tion”) remains invariant

• Amplitude (intensity) and phase (time shift) are depending onH(ω) (“eigenvalue”)

(→ analogy to the problem of eigenvalues in linear algebra)


17

Remarks

• FT is complex:

H(ω) = Re H(ω) + j Im H(ω) = |H(ω)| ejΦ(ω)

• Amplitude (spectrum):

|H(ω)| =

√Re H(ω)2 + Im H(ω)2

• Phase (spectrum):

Φ(ω) =

arctan

(Im H(ω)Re H(ω)

)Re H(ω) > 0

arctan

(Im H(ω)Re H(ω)

)+ π Re H(ω) < 0

π

2Re H(ω) = 0,

Im H(ω) > 0

−π

2Re H(ω) = 0,

Im H(ω) < 0


18

Examples of Fourier transforms:

1. Rectangle function

h(t) = rect(t

T) =

1, |t| ≤ T/20, |t| > T/2

H(ω) =

∞∫

−∞h(t)e−jωtdt =

T2∫

−T2

e−jωtdt =1

−jω

[e−jω T

2 − ejω T2

]

=2

ωsin(

ωT

2) =

T sin(ωT

2)

ωT

2

(here: Im H(ω) = 0)

t

h(t)

H(ω)

ω


19

2. Double-sided exponential

h(t) = e−α|t| with α > 0

H(ω) =

∞∫

−∞h(t)e−jωtdt

=

∞∫

0

e−(α+jω)tdt +

∞∫

0

e−(α−jω)tdt

=

[e−(α+jω)t

−(α + jω)+

e−(α−jω)t

−(α− jω)

]∞

0

= 0 + 0− 1

−(α + jω)− 1

−(α− jω)

=α− jω + α + jω

α2 + ω2

=2α

α2 + ω2

• Imaginary part equals 0

• Infinite spectrum

• No zeros

h(t)

t

H( )ω

ω

• If h(t) is symmetric (i.e. h(t) = h(−t)), imaginary parts drop awayand the real part is sufficient


20

3. Damped oscillations

h(t) = e−α|t| cos(βt) with α > 0

H(ω) =

∞∫


=

∞∫

0

e−(α+jω)t cos(βt)dt +

∞∫

0

e−(α−jω)t cos(βt)dt

=

∞∫

0

e−(α+jω)t ejβt + e−jβt

2dt +

∞∫

0

e−(α−jω)t ejβt + e−jβt

2dt

= . . . (elementary calculation)

=α

α2 + (ω − β)2 +α

α2 + (ω + β)2

• Limiting case:

H(ω)|ω=±β =1

α+

α

α2 + (2β)2

=⇒ tends towards ∞ or −∞ if α tends towards 0

ω

H( )ω

β−β| |

h(t)

t


21

4. Modulated rectangle function (“truncated cosine”)

h(t) =

cos(β t), |t| ≤ T/2

0, |t| > T/2

H(ω) =

∞∫


=

T2∫

−T2

cos(β t)e−jωtdt

= . . . (elementary calculation)

=T

2

sin

((ω − β)

T

2

)

(ω − β)T

2

+

sin

((ω + β)

T

2

)

(ω + β)T

2

| |

h(t)

t

H( )ω

ω

h(t)

t

| |

ω

ωH( )

β−β


22

Fourier Transform pairs (u = ω/2π)

Rectangle function

1

-1/2 1/2

-1/2 1/2

Triangle function

1

Exponential function

e-α|x|

Gaussian function

e-αx2

Unit impulse

δ(x)

1

Sinc function

Squared sinc function

sin(πu)πu

1

α2+(2πu)2

2α

πu2

αe-πα

1


23

Inverse Fourier–transform

H(ω) =

∞∫


assumption: h(t) =1

2π

∞∫

−∞H(ω)ejωtdω

with: H(ω) =

∞∫

−∞h(τ)e−jωτdτ

inserting H(ω) in h(t):

h(t) =1

2πlim

Ω,T→∞

Ω∫

−Ω

T∫

−T

h(τ) ejω(t−τ) dτ

dω

=1

2πlim

Ω→∞lim

T→∞

T∫

−T

Ω∫

−Ω

ejω(t−τ) dω h(τ) dτ

= limΩ→∞

limT→∞

1

π

T∫

−T

sin (Ω(t− τ))

t− τh(τ) dτ

= limΩ→∞

1

π

∞∫

−∞

sin (Ω(t− τ))

t− τh(τ) dτ

= h(t)

due to:

limΩ→∞

1

π

∞∫

−∞

sin(Ωt)

th(t) dt = h(0)

formal expression:

h(t) =

∞∫

−∞

1

2π

∞∫

−∞ejω(t−τ) dω

︸︷︷︸= δ(t− τ)

h(τ) dτ

(→ distribution theory, see there for stronger proof)


24

1.4 Properties of the Fourier Transform

Symmetry

H(ω) =

∞∫

−∞h(t) e−jωt dt = F h(t)

h(t) =1

2π

∞∫

−∞H(ω) ejωt dω = F−1 H(ω)

F 2h(t) = FH(ω) = 2πh(−t)

F−1 Fh(t) = F−1H(ω) = h(t)

• Time domain and frequency domain are correlated symmetrically.

• Properties of FT are valid in both domains, especially the convolutiontheorem (see later).


25

Theorems for the Fourier transform

H(ω) =

∞∫

−∞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:

∞∫

−∞h(αt) e−jωt dt =

1

|α|

∞∫

−∞h(τ) e−j ω

ατ dτ

Fh(αt) =1

|α| H(ω

α), α ∈ IR\0

Note:

Absolute value, because integral boundaries are swapped for α < 0.

3. Shift: h(t− t0)

∞∫

−∞h(t− t0) e−jωt dt = e−jωt0

∞∫

−∞h(t− t0) e−jω(t−t0) dt

= e−jωt0

∞∫

−∞h(τ) e−jωτ dτ


26

=⇒ Fh(t− t0) = e−jωt0H(ω) t0 ∈ IR

with H(ω) = Fh(t)

important:

| Fh(t− t0) | = | Fh(t) | , because

|e−jωt0| = |e−ju| = | cos u − j sin u|=

√cos2 u + sin2u

= 1

4. Symmetry and antisymmetry:

h(t) = h(−t) results in ImH(ω) = 0

h(t) = −h(−t) results in ReH(ω) = 0

5. Complex conjugation: suppose that h(t) is a complex function

∞∫

−∞h(t) e−jωt dt =

∞∫

−∞h(t) ejωt dt

=

∞∫

−∞h(t) ejωt dt = H(−ω)

Fh(t) = H(−ω) = Fh(t)

Special case: h(t) is real, so h(t) = h(t)

=⇒ H(ω) = H(−ω) =⇒ | H(ω) | = | H(−ω) | = | H(−ω) |


27

6. Differentiation:

dh

dt=

∂

∂t

1

2π

∞∫

−∞H(ω) ejωt dω

=1

2π

∞∫

−∞H(ω) jω ejωt dω

Fdh(t)

dt = jω Fh(t)

Interpretation: differentiation = enhancement of high frequencies(due to the multiplication with ω)

7. Integration:

Ft∫

−∞h(τ)dτ =

1

jωFh(t)

Proof: similar to differentiation or inversion

8. Modulation principle:

Fh(t) cos(ω0t) =

∞∫

−∞h(t) cos(ω0t) e−jωt dt

=1

2

∞∫

−∞h(t) ejω0t e−jωt dt +

∞∫

−∞h(t) e−jω0t e−jωt dt

=1

2

∞∫

−∞h(t) e−j(ω−ω0)t dt +

∞∫

−∞h(t) e−j(ω+ω0)t dt

=1

2[ H(ω − ω0) + H(ω + ω0) ]

and similarly

F h(t) sin(ω0t) =1

2j[ H(ω − ω0) − H(ω + ω0) ]


28

h(t), H(ω) x(t)

X(ω)

y(t)

Y(ω)

Convolution theorem

• Time domain: y(t) = x(t) ∗ h(t) =

∞∫

−∞x(t− τ) h(τ) dτ

• Frequency domain:

Y (ω) =

∞∫

−∞e−jωt

∞∫

−∞h(τ) x(t− τ) dτ

dt

=

∞∫

−∞h(τ)

∞∫

−∞x(t− τ) e−jωt dt

dτ

=

∞∫

−∞h(τ) X(ω) e−jωτ dτ (shifting)

= X(ω)

∞∫

−∞h(τ) e−jωτ dτ

= X(ω) H(ω)

• Convolution in time domain corresponds to multiplication in frequencydomain

• Motivation for the Fourier transform:FT gives the “simplest” representation of the system operation, be-cause every LTI-System can be interpreted as convolution of the inputsignal x(t) and the impulse response of the system h(t). Convolutioncan be then efficiently calculated using FT and convolution theorem.

• Mathematical: eigenfunctions


29

Example: Oscillator with excitation

x(t) −→ Oscillator −→ y(t)

y′′(t) + 2α y′(t) + β2 y(t) = x(t)

x(t) =1

2π

+∞∫

−∞X(ω)ejωtdω

y(t) =1

2π

+∞∫

−∞Y (ω)ejωtdω

y′(t) =1

2π

+∞∫

−∞Y (ω)jω ejωtdω

y′′(t) =1

2π

+∞∫

−∞Y (ω)[−ω2] ejωtdω

+∞∫

−∞[−ω2 + 2αjω + β2]Y (ω)ejωtdω =

+∞∫

−∞X(ω)ejωtdω

+∞∫

−∞

[−ω2 + 2αjω + β2] Y (ω)−X(ω)

︸︷︷︸

=0

ejωtdω = 0 ∀t

In this way we obtain the transfer function of an oscillator:

H(ω) =Y (ω)

X(ω)=

1

−ω2 + 2αjω + β2


30

h(t) =1

2π

+∞∫

−∞H(ω)ejωtdω

(can be given explicitly)

y(t) =

+∞∫

−∞x(t) h(t− τ)dτ

Note:y(t) does not contain the component which corresponds to the homoge-neous differential equation of the oscillator.

FourierTransform

Convolution with h(t)

Multiplication with H(ω) = Fh(t)

Inverse FourierTransform

x(t)

X(ω)

y(t)

Y(ω)


31

1.5 Parseval Theorem

Convolution theorem:

F−1 H(ω) X(ω) =

∞∫

−∞h(t) x(τ − t) dt

(?)1

2π

∞∫

−∞H(ω) X(ω) ejωτ dω = (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

∞∫

−∞H(ω)H(ω) dω =

∞∫

−∞h(t)h(t) dt

1

2π

∞∫

−∞|H(ω)|2 dω =

∞∫

−∞|h(t)|2 dt = E

• Energy E in time domain = Energy E in frequency domain

(up to the factor1

2π; aid: use normalization factor

1√2π

for both

directions of Fourier Transform)

• Physical aspect: energy conservation

• Mathematical aspect: unitary (orthogonal) representation in vectorspace

• |H(ω)|2 is called power spectral density.


32

1.6 Autocorrelation Function

Autocorrelation function

• Autocorrelation function of time continuoussignal or function h(t) is defined as:

R(t) =

∞∫

−∞h(τ) h(t + τ)dτ

• The following equation is valid:

R(t) = h(t) ∗ h(−t) → which results in R(t) = R(−t)

• Fourier transform gives: (”Wiener-Khinchin Theorem”)

FR(t) = H(ω) H(ω) = |H(ω)|2

• Thus: Fourier transform connects autocorrelationfunction R(t) and power spectral density |H(ω)|2

|H(ω)|2 =

∞∫

−∞R(t) e−jωt dt =

∞∫

−∞R(t) cos(ωt) dt

• Remark:autocorrelation is a special case of the cross correlation between sig-nals x(τ) and h(t)

Ch,x =

∞∫

−∞h(τ) x(t + τ)dτ


33

1.7 Existence of the Fourier Transform

Conditions for h(t) for the existence of the Fourier transform

H(ω) =

∞∫

−∞e−jωt h(t) dt , h(t) =

1

2π

∞∫

−∞ejωt H(ω) dω

When are those equations valid?

Sufficient conditions:

1. h(t) is absolutely integrable:

∞∫

−∞|h(t)|dt < ∞

2. h(t) has finite number of jumps, minima and maxima in each intervalof IR

3. h(t) has no infinite jumps

More general conditions are possible (but rather complex set of conditions)

The way out:

• Generalized functions, distributions,definition as functional

• Example: δ-function:

∞∫

−∞δ(t) h(t) dt = h(0) for all functions h


34

Impulse response:

y(t) =

∞∫

−∞h(τ − t)δ(t) dt

= h(t) ∗ δ(t)

= h(t)

Consequence:

h(t) ≡ 1 ⇒∞∫

−∞δ(t) dt = 1

• A function like δ(t) does not “exist”. But it is possible to define thefunctional for each function t → h(t):

[t → h(t)] −→ δ(h) := h(0)

1.8 δ-Function

Starting point: definition of the δ-function as a boundary case ofa function δε(t):

limε→0

+∞∫

−∞f(t) δε(t) dt = f(0) (1.1)

• Possible realizations of δε(t)

a) δε(t) =

1

2εt ∈ [−ε, +ε]

0 otherwise

b) δε(t) =1

π

ε

ε2 + t2


35

c) δε(t) =1

π

sin (t/ε)

t

d) δε(t) =1√2πε2

e−

t2

2ε2

• During inversion of the Fourier transform we have “formally” ob-tained:

δ(t) =1

2π

+∞∫

−∞ejωt dω = lim

Ω→∞1

π

sin (Ωt)

t(1.2)

Fourier transform Fδ(t):

Fδ(t) =

+∞∫

−∞e−jωtδ(t) dt

due to (1.1) the following holds:

Fδ(t) = ejωt|t=0 = 1

• Another derivation using (1.2):

δ(t) =1

2π

+∞∫

−∞ejωt Fδ(t) dω general

=1

2π

+∞∫

−∞ejωt dω according to (1.2)

Comparison results in:

Fδ(t) = 1


36

From this we obtain the following equations:

From symmetry property:

F1 = 2 π δ(ω)

From shifting theorem:

Fejω0t 1 = 2 π δ(ω − ω0)

cos (ω0 t) =1

2

[ejω0t + e−jω0t

]

=1

2

+∞∫

−∞δ(ω − ω0) ejωt dω +

+∞∫

−∞δ(ω + ω0) ejωt dω

= π1

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

2πe−α|t| cos (ω0t)

in the limit α → 0 .


37

Comb function

• define “comb function” (pulse train, sequence of δ-impulses):

x(t) =+∞∑

n=−∞δ(t− nT )

• Fourier transform of comb function:

X(ω) =

+∞∫

−∞x(t) e−jωt dt

= . . . (long calculation, see Papoulis 1962, p. 44)

=2π

T

+∞∑n=−∞

δ(ω − n2π

T)

• in words:

δ-impulse sequence with period T in time domain

produces

δ-impulse sequence with period 1T in frequency domain

(i.e. 2πT in ω-frequency domain)

comb function is transformed to comb function


38

Comb function

cos(ω0t)

sin(ω0t)

-2π-4π-6π 2π 4π 6πT T T T T T

T 3T-T-3T 6T-6T

δ(t-nT)Σn=-

12j(−δ(ω-ω0)+δ(ω+ω0))

(δ(ω-ω0)+δ(ω+ω0))12

Σ δ(ω-n2π/T)n=-

2πT

−ω0 ω0

−ω0

ω0


39

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 ) for each t ∈ IR

then also x(t) = x(t + kT ) for k ∈ Z

Examples:

• Constant function:

x0(t) = A0

• Harmonic oscillator:

x1(t) = A1 cos (2π

Tt + ϕ1) , A1 > 0

• All higher harmonic:

xn(t) = An cos (n2π

Tt + ϕn) , An > 0

therefore

x(t) =∞∑

n=0

An cos (n ω0 t + ϕn) with ω0 =2 π

T, An ≥ 0

is periodical with period T = 2πω0

• Another notation:

x(t) =∞∑

n=−∞Bn e−j n ω0 t where Bn is a complex number


40

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 asinfinitely prolonged).

• By transition from the finite interval to the complete real axis theFourier series becomes Fourier integral.


41

Calculation of Fourier coefficients:

• Consider a periodical function x(t) with period T = 2πω0

• approach:

x(t) =+∞∑

n=−∞an ej n ω0 t a ∈ C

• multiplication with e−j m ω0 t where m ∈ IN and integration over oneperiod result in:

+T/2∫

−T/2

x(t) e−j m ω0 t dt =+∞∑

n=−∞an

+T/2∫

−T/2

ej (n−m) ω0 t dt

• Due to “orthogonality” holds:

+T/2∫

−T/2

ej (n−m) ω0 t dt =

T if n = m

0 if n 6= m

• Then:

T/2∫

−T/2

x(t) e−j m ω0 t dt = am T

• Result:

an =1

T

+T/2∫

−T/2

x(t) e−j n ω0 t dt

=1

T

+T/2∫

−T/2

x(t) cos (n ω0 t) dt − j1

T

+T/2∫

−T/2

x(t) sin (n ω0 t) dt


42

Spectrum of a periodical function

• If x(t) is periodical with the period T = 2πω0

, then

x(t) =+∞∑

n=−∞an ej n ω0 t, an ∈ C

• The Fourier transform X(ω) is:

X(ω) = Fx(t)

=+∞∑

n=−∞an Fej n ω0 t︸︷︷︸

= 2πδ(ω − nω0)

= 2 π

+∞∑n=−∞

an δ(ω − nω0)

• Note:

This derivation is formal, because the Fourier integral does notexist 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 theform of a line spectrum with the distance ω0 = 2π

T between the com-ponents.


43

1.10 Time Duration and Band Width

1. Similarity principle:

Fh(αt) =1

|α| H(ω

α)

t

h( t)α0<α<1:

h(t)

t

1 H( )_ _α α

ω

ω

H( )

ω

ω

time duration T band width B

T · B = const.

• High resolution in the time domain results in low resolution in thefrequency domain and vice versa


44

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:

h(t) =1

2π

∞∫

−∞H(ω) cos(ωt) dω ≤ 1

2π

∞∫

−∞H(ω) dω = h(0)

define:

T =1

h(0)

∞∫

−∞h(t) dt

B =1

H(0)

∞∫

−∞H(ω) dω

from

T =H(0)

h(0)and B = 2 π

h(0)

H(0)

follows

T · B = 2 π


45

3. In general: normalized impulse h(t) ∈ IR with∞∫

−∞h2(t) dt = 1, h(t) ∈ IR

T 2 :=

∞∫

−∞h2(t) t2 dt

B2 :=

∞∫

−∞|H(ω) |2 ω2 dω

= 2 π

∞∫

−∞[h′(t)]2 dt

• Results in “uncertainty relation”:

T · B ≥√

π

2

• Proof: Cauchy-Schwarz inequality

| xT y | ≤ ||x|| · ||y||

∣∣∣∣∣∣

∞∫

−∞[ t h(t) ] h′(t) dt

∣∣∣∣∣∣

2

︸︷︷︸=1

4

≤∞∫

−∞[ t h(t)]2 dt

︸︷︷︸= T 2

∞∫

−∞[ h′(t) ]

2dt

︸︷︷︸B2

2πFrom: partial integration

∫u′(t) v(t) dt = u(t) v(t)−

∫u(t) v′(t) dt

∫[ h(t) h′(t) ]︸︷︷︸

u′(t)

t︸︷︷︸v(t)

dt =1

2h(t)2 t −

∫1

2h2(t) 1 dt

∞∫

−∞[ h(t) h′(t) ] t dt = 0 − 1

2


46

Equality sign is valid for linear dependency:

h′(t) = −λ t h(t)dh

h= −λ t dt

log(h) = −1

2λ t2 + const., λ > 0

=⇒ Optimum T · B =

√π

2for Gauss’ impulse

h(t) =λ√2π

e−1

2λ t2

Variance: σ2 =1

λ

Quantum Physics: similar statement about position and impulseof a particle


47

4. Finite positive signal

g(t)

≥ 0 0 ≤ t ≤ T

= 0 t < 0 or T < t

0 T

g(t)

t

The following is valid for the amplitude spectrum |G(ω)|:

|G(ω)| =

∣∣∣∣∣∣

+∞∫

−∞g(t) e−jωtdt

∣∣∣∣∣∣

≤+∞∫

−∞|g(t)| |e−jωt|dt

=

+∞∫

−∞|g(t)| dt

because g(t) ≥ 0

= G(0)

Define the band width ωB as:

|G(ωB)|2 =G2(0)

2and |G(ωB)|2 ≤ |G(ω)|2 for |ω| < ωB

Then:

T ωB ≥ π

2


48

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 Examples of Discrete Time Signal Fourier Transform

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


51

2.1 Motivation and Goal

If we want to process a continuous time signal x(t) with a computer, wehave 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 re-spect to their realization;

• in principle arbitrary high accuracy;

• also non-linear methods are possible,in principle even every mathematical method.


52

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], so 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.


53

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], n0 whole number


54

Representation of a LTI System as discrete convolution:

Unit impulse:

δ[n] =

1, n = 00, n 6= 0

The signal x[n] is represented with amplitude weighted and time shiftedunit impulses δ[n]. The system reacts on δ[n] with h[n]:

h[n] = S δ[n]Input signal:

x[n] =∞∑

k=−∞x[k] δ[n− k]

Output signal:

y[n] = S

∞∑

k=−∞x[k] δ[n− k]

Additivity

=∞∑

k=−∞S x[k] δ[n− k]

Homogeneity

=∞∑

k=−∞x[k] S δ[n− k]

Time invariance

=∞∑

k=−∞x[k] h[n− k]

Input signal x[n] and output signal y[n] of a discrete time LTI system arelinked through discrete convolution.h[n] is called impulse response like in continuous time case.


55

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”)

y[n] =1

2M + 1

M∑

k=−M

x[n− k]

• weighted averaging: instead of constant weight

h[n] =1

2M + 1

arbitrary weights can be used:

y[n] =M∑

k=−M

h[k] · x[n− k]

Note: the only difference from general case isfinite 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)


56

• Roberts Cross Operator

gray values x[i, j]

i i+1

j

j+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]|


57

Figure 2.1: Digital photo

Figure 2.2: Gradient image


58

• Laplace Operator ≡ discrete approximation of the second derivation

∇2x[i, j] = 42i x[i, j] +42

jx[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

1

1

1

1

1

1-4

-2 i-1 i i+1

j+1

j

j-1

Image enhancement:

y[i, j] = x[i, j]−∇2x[i, j]

= h[i, j] ∗ x[i, j]


59

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 0and values above the grey scale are set to the highest grade of grey)


60

2.4 Sampling Theorem (Nyquist Theorem) and Re-

construction

The following will be analyzed and derived respectively:

How should we choose the sampling period TS, if we want to represent acontinuous 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):

X(ω) = F x(t) =

∞∫

−∞x(t) e−jωt dt

x(t) = F−1 X(ω) =1

2π

∞∫

−∞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) =∞∑

n=−∞an exp(−jnπ

ω

ωB) (2.2)

• The coefficients an are given by:

an =1

2ωB

ωB∫

−ωB

X(ω) exp(jnπω

ωB) dω (2.3)

• Comparison of the equations (2.1) and (2.3) shows that the coefficientsan are given by the values of the inverse Fourier transform of x(t) atpoints

tn =nπ

ωB(2.4)

The band limitation of X(ω) has to be considered for the integrationlimits in (2.1). Result:

an = x(nπ

ωB) · π

ωB(2.5)


61

• Inserting Eq. (2.5) into Eq. (2.2) and then in Eq. (2.1) results in:

x(t) =1

2π

ωB∫

−ωB

π

ωB

∞∑n=−∞

x(nπ

ωB) exp(−jnπ

ω

ωB) exp(jωt) dω

• After swapping summation and integration and subsequent integra-tion:

x(t) =∞∑

n=−∞x(

nπ

ωB)

sin(ωB (t− nπ

ωB))

ωB (t− nπ

ωB)

• Reconstruction of the signal x(t) from sample values is possible if

equidistant sample values x(nπ

ωB) = x(n · Ts) have the distance TS

TS =π

ωB(2.6)

• 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.


62

• In the Fourier series representation of X(ω) in equation (2.2), theperiod 2 · ωB has been supposed.ωB is the highest frequency component of the signal x(t).

• Since X(ω) is equal to zero for |ω| ≥ ωB, the period 2 · ωB can besubstituted with every period 2 · ωB where ωB ≥ ωB. The previousderivation is also valid for this ωB.

• When

ωB =π

TS

then:

x(t) =∞∑

n=−∞x(nTS)

sin(π (t − nTS)/TS)

π (t − nTS)/TS

(reconstruction formula)

• The condition ωB ≥ 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. Thesampling frequency has to be at least twice as high as the upper limitfrequency of the signal ωB where X(ω) = 0 for |ω| ≥ ωB. If and onlyif this condition is satisfied, an exact reconstruction (without approx-imation) of a continuous signal x(t) from its sample values x(nTS) ispossible.

• Note: The sampling frequency ΩS = 2 · ωB is also calledNyquist frequency.


63

t

x(t)

t

xs(t)

xr(t)

T

T

a)

b)

c)

Figure 2.4: Ideal reconstruction of a band-limited signal (from Oppenheim, Schafer)a) original signal b) sampled signal c) reconstructed signal


64

X(ω)

ωωΒ−ωΒ

a)

ωωΒ−ωΒ

b)

-ΩS ΩS

. . . . . .

XS1(ω) ΩS > 2ωΒ,

XS2(ω)

ωωΒ−ωΒ

c)

, ΩS = 2ωΒ

-ΩS ΩS

. . . . . .

(Nyquist rate)

XS3(ω)

ωΒ−ωΒ

, ΩS < 2ωΒ (aliasing)

. . . . . .

ΩS−ΩS

d)

ω

Figure 2.5: Sampling of band-limited signal with different sampling rates:a) sampling rate higher than Nyquist rate - exact reconstruction possibleb) sampling rate equal to Nyquist rate - exact reconstruction possibled) sampling rate smaller than Nyquist rate - aliasing - exact reconstruction not possible


65

Another proof using delta- and comb-function:

Sampling of the continuous signal x(t) with ΩS = 2πTS

• Band limitation: X(ω) = 0 for |ω| ≥ ωB

(always possible: analog to low-pass with T (ω) = 0 for |ω| ≥ ωB)

• Sampling procedure

= Multiplication with comb-function in time domain

= convolution with comb-function in frequency domain

=⇒ 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


66

• In so-called digital simulation, the signal x(t) is represented by itssampled values x(n · TS) measured at equidistant time points withdistance TS. With a proper sampling period TS an exact reconstruc-tion 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 sampledvalues x(n·TS), then it is possible to perform a discrete time processingof the sampled values x(n · TS) on a computer, which is equivalent tothe continuous time processing of the signal x(t) (digital simulation).

• Continuous time processing:

y(t) =

∞∫

−∞x(τ) h(t− τ) dτ

• Discrete time processing:

– Sampling period TS

– x[n] := x(nTS)

y(nTS) =∞∑

k=−∞x(kTS) h(nTS − kTS) TS, h[n] = h(nTS)

y[n] =∞∑

k=−∞x[k] h[n− k]

• As a result of the convolution theorem (convolution in time domaincorresponds to multiplication in frequency domain), the band limitedinput signal gives an also band limited output signal which is exactlydetermined by its sampled values.


67

Important:

• In the domain |ω| < ΩS/2 the Fourier transform of a continuous timesignal x(t) is identical with the Fourier–transform of the correspondingsampled discrete time signal x(nTS):

X(ω) =

∞∫

−∞x(t) exp(−jωt) dt

for |ω| ≤ ΩS/2 is identical to

TS ·XS(ω) = TS ·∞∑

n=−∞x(nTS) exp(−jωTSn)

= TS ·∞∑

n=−∞x(nTS) exp(−j

2πω

ΩSn)

• Inverse Fourier transform of discrete time signal:

x(nTS) =1

ΩS

ΩS/2∫

−ΩS/2

XS(ω) exp(jωTSn) dω

• One period:

−ΩS

2≤ ω ≤ ΩS

2

−π ≤ 2πω

ΩS≤ π

• The Fourier transform of a discrete time signal is periodic in ω withthe period 2 π/TS = ΩS.

• The Fourier transform of a discrete time signal iscontinuous in ω.


68

Frequency normalization

• Define the normalized frequency ωN :

ωN : = 2πω

ΩS

• Definition: (ω now denotes a normalized frequency)

– Fourier transform of discrete time signal x[n]:

X(ejω) =+∞∑

n=−∞x[n] exp(−jωn)

Note the notation X(ejω).

– Inverse Fourier transform of discrete time signal x[n]:

x[n] =1

2π

π∫

−π

X(ejω) exp(jωn) dω


69

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

dB := “decibel”

A[dB] ≡ 20 · lg A

A0, A0 = reference amplitude , lg ≡ log10

= 10 · lg A2

A20, A2 = quadratic variable = energy, intensity

because of 210 = 1024 ∼= 103:

1 bit more =

factor 2 for amplitude = 6 dB= factor 4 for intensity

3 dB = factor 2 for intensity


70

0

1

2

3

4

5

6

7

0 1000 2000 3000 4000 5000 6000 7000 8000

A

f / Hz

Phonem: s

Figure 2.6: Amplitude spectrum of thevoiceless phoneme “s” from the word“ist”

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1000 2000 3000 4000 5000 6000 7000 8000

log

A

f / Hz

Phonem: s

Figure 2.7: Logarithmic amplitude spec-trum of the phoneme “s”

0

2

4

6

8

10

12

0 1000 2000 3000 4000 5000 6000 7000 8000

A

f / Hz

Phonem: ae

Figure 2.8: Amplitude spectrum of thevoiced phoneme “ae” from the word“Ah”

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1000 2000 3000 4000 5000 6000 7000 8000

log

A

f / Hz

Phonem: ae

Figure 2.9: Logarithmic amplitude spec-trum of the phoneme “ae”

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000 4000 5000 6000 7000 8000

A

f / Hz

Pause

Figure 2.10: Amplitude spectrum of aspeech pause

-3

-2.5

-2

-1.5

-1

-0.5

0

0 1000 2000 3000 4000 5000 6000 7000 8000

log

A

f / Hz

Pause

Figure 2.11: Logarithmic amplitudespectrum of a speech pause


71

2.6 Quantization

• Uniform quantization

-XMAX XMAX

• Quantisation: x = Q(x)

• B bits correspond to 2B quantisation levels

• Boundaries: x0, x1, . . . , xk, . . . , xK where K = 2B

• Width 4 of one quantisation level using uniform quantisation:

4 =2 ·XMAX

2B

• Quantisation error:

σ2e =

+∞∫

−∞(x− x)2 p(x) dx =

K∑

k=1

xk∫

xk−1

(x− xk)2 p(x) dx

– for uniform quantisation:

a) xk − xk−1 = 4 = const(k)

b) xk = 12(xk−1 + xk)

– uniform distribution with p(x) = const(x) results in:

σ2e =

∑

k

42

12· 1

K=42

12=

X2MAX

3 · 22B


72

• signal-to-noise ratio in dB (general definition):

SNR[dB] := 10 lgσ2

x

σ2n

σ2x = power of the signal x

σ2n = power of the noise n

SNR = signal-to-noise ratio

• signal-to-quantisation noise ratio (special case):

SNR[dB] := 10 lgσ2

x

σ2e

σ2e = power of the noise caused by quantisation errors

• uniform quantisation using B bits:

SNR[dB] = 6.02 B + 4.77− 20 lgXMAX

σx

= 6.02 B − 7.2 for XMAX = 4σx


73

2.7 Fourier Transform and z–Transform

Transfer function and Fourier transform

Eigenfunctions of discrete linear time invariant systems (analog to timecontinuous case):

x[n] = ej ω n −∞ < n < ∞(ω is dimensionless here)

Proof:

x[n] = ej ω n

y[n] =∞∑

k=−∞h[k] ej ω (n−k)

= ej ω n∞∑

k=−∞h[k] e−j ω k

Define: H(ej ω) =∞∑

k=−∞h[k] e−j ω k

Remark:The Fourier transform of a discrete time signal is already introduced asFourier series during the derivation of sampling theorem and reconstruc-tion formula (equation (2.2)).

Result: y[n] = ej ω n H(ej ω)


74

z–transform:

• Fourier transform of a discrete time signal: x[n]

X(ejω) =+∞∑

n=−∞x[n] e−jωn

– 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(z) =+∞∑

n=−∞x[n] z−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:

x[n] =1

2πj

∮X(z) zn−1 dz

formally: z = ejω dz = jzdω

x[n] =1

2π

2π∫

0

X(ejω) ejωn dω


75

Example of Fourier transform and z–transform:

• “Truncated geometric series”

x[n] =

an 0 ≤ n ≤ N − 10 otherwise

• z–transform

X(z) =N−1∑n=0

an z−n =N−1∑n=0

(a z−1)n =1− (a z−1)N

1− a z−1

=1

zN−1

zN − aN

z − a

• Fourier transform

z–transform results in Fourier transformation using substitution

z = ejω

X(ejω) =1− aN e−jωN

1− a e−jω

special case for a = 1 (discrete time rectangle):

= exp

(−j

ω(N − 1)

2

) sin

(ωN

2

)

sin(ω

2

)


76

Proof for the z–transform inversion

• Statement:

x[k] =1

2πj

∮X(z) zk−1 dz

• Cauchy integration rule

1

2πj

∮z−kdz =

1 k = 10 k 6= 1

1

2πj

∮X(z) zk−1dz =

1

2πj

∮ ∑n

x[n] z−n+k−1dz

=∑

n

x[n]1

2πj

∮z−n+k−1dz

︸︷︷︸6= 0 only for n = k

= x[k]

• Fourier:

z = ejω =⇒ dz = j ejω dω

Then:

x[n] =1

2πj

+π∫

−π

X(ejω) (ejω)n−1 j ejωdω

Integration path is unit circle because of ejω

=1

2π

+π∫

−π

X(ejω) ejωn dω


77

2.8 Examples of Discrete Time Signal Fourier Trans-

form

Example 1: Difference calculation

• Difference equation

y[n] = x[n] − x[n− n0], n0 integral number

• Fourier transform gives:∞∑

n=−∞y[n] e−jωn =

∞∑n=−∞

x[n] e−jωn −∞∑

n=−∞x[n− n0] e−jωn

Y (ejω) = X(ejω) −∞∑

n=−∞

(x[n] e−jωn e−jωn0

)

= X(ejω) − e−jωn0 X(ejω)

• Then follows:

H(ejω) =Y (ejω)

X(ejω)

= 1 − e−jωn0

|H(ejω)|2 = (1− cos(ωn0))2 + sin2(ωn0)

= 1 − 2cos(ωn0) + cos2(ωn0) + sin2(ωn0)

= 2 (1 − cos(ωn0))

|H(eiω)|2

0

1

2

3

4

5

ω πn0


78

Example 2: First order difference equation

Delay

y[n]

x[n]

+

y[n-1]

α

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] + · · ·=

αn, n ≥ 00, otherwise

• Fourier spectrum/transfer function H(ejω)

H(ejω) =+∞∑

k=−∞h[k] e−jωk

=+∞∑

k=0

αk e−jωk

=+∞∑

k=0

(α e−jω

)k

=1

1 − α e−jωfor |α| < 1


79

Method 2: Estimation of transfer function H(ejω) usingFourier 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) =Y (ejω)

X(ejω)

=1

1 − α e−jω


80

Example 3: Linear difference equations (with constant coefficients)

• Difference equation:

y[n] =I∑

i=0

b[i] x[n− i]−J∑

j=1

a[j] y[n− j]

• z-transform:

Y (z) = X(z)I∑

i=0

b[i]z−i − Y (z)J∑

j=1

a[j]z−j

• Result:

H(z) =Y (z)

X(z)

=

I∑i=0

b[i] z−i

1 +J∑

j=1a[j] z−j

=+∞∑

n=−∞h[n] z−n

Remark:

If we factorise denominator and numerator polynoms into linear fac-tors, we can obtain a zero-pole-representation of a discrete time LTIsystem:

H(i) =ΠI

i=1(z − vi)

ΠJj=1(z − wj)

with zeros vi ∈ C and poles wj ∈ C.


81

• in general:

h[n] has infinite number of non-zero values

=⇒ IIR–filter: Infinite Impulse Response

• but if: a[j] ≡ 0 ∀jh[n] identical to zero outside of a finite interval

h[n] =

b[n] n = 0, . . . , I0 otherwise

=⇒ FIR–filter: Finite Impulse Response


82

Example 4:Impulse response as “truncated geometric series”

h[n] =

an 0 ≤ n ≤ M a ∈ IR0 otherwise

H(z) =M∑

n=0

an z−n =1− aM+1 z−(M+1)

1− a z−1

system operation:

y[n] =∞∑

k=−∞h[k] x[n− k]

=M∑

k=0

akx[n− k]

or as difference equation (“recursively”)

y[n]− a y[n− 1] = x[n]− aM+1x[n−M − 1]


83

For this example we consider the zero-pole-representation:

H(z) =1− (z

a)−(M+1)

1− (za)−1 α > 0

Zeros: zk = a · ej 2πkM+1 k = 0, 1, . . . , M

Pole: z0 = a

(cancelled by zero z0 = a)

M=11

Re

Im

a


84

Example 5:Fibonacci numbers

Difference equation:

n ≥ 0 h[n + 2] = h[n + 1] + h[n]

h[0] = h[1] = 1

n < 0 h[n] = 0

H(z) =∞∑

n=−∞h[n]z−n

= 1 + z−1 +∞∑

n=0

h[n + 2]z−(n+2)

= 1 + z−1 +∞∑

n=0

h[n + 1]z−(n+2) +∞∑

n=0

h[n]z−(n+2)

= 1 + z−1 + z−1∞∑

n=0

h[n + 1]z−(n+1) + z−2∞∑

n=0

h[n]z−n

= 1 + z−1 (1 +∞∑

n=1

h[n]z−n)

︸︷︷︸H(z)

+ z−2∞∑

n=0

h[n]z−n

︸︷︷︸H(z)

= 1 + z−1H(z) + z−2H(z)

H(z)(1− z−1 − z−2) = 1

H(z) =1

1− z−1 − z−2

=1√5

( a

1− az−1 −b

1− bz−1

)

where a =1 +

√5

2and b =

1−√5

2


85

For a and b the following holds:

1

1− (az )

=∞∑

n=0

( a

z

)n

=∞∑

n=0

anz−n

That results in:

H(z) =∞∑

n=0

1√5

(an+1 − bn+1 )

z−n

!=

∞∑n=0

h[n] z−n

h[n] =1√5

(an+1 − bn+1 )


86

Table 2.1: Fourier transform pairs

signal Fourier–transform

1. δ[n] 1

2. δ[n− n0] e−jωn0

3. 1 (−∞ < n < ∞)∞∑

k=−∞2πδ(ω + 2πk)

4. anu[n] (|a| < 1)1

1− ae−jω

5. u[n]1

1− e−jω+

∞∑

k=−∞πδ(ω + 2πk)

6. (n + 1)anu[n] (|a| < 1)1

(1− ae−jω)2

7.rn sin ωp(n + 1)

sin ωp

u[n] (|r| < 1)1

1− 2r cos ωp e−jω + r2e−j2ω

8.sin ωcn

πnX(ejω) =

1, |ω| < ωc,

0, ωc < |ω| ≤ π

9. x[n] =

1, 0 ≤ n ≤ M

0, otherwise

sin[ω(M + 1)/2]

sin(ω/2)e−jωM/2

10. ejω0n

∞∑

k=−∞2πδ(ω − ω0 + 2πk)

11. cos(ω0n + φ) π

∞∑

k=−∞[ejφδ(ω − ω0 + 2πk) + e−jφδ(ω − ω0 + 2πk)]

u[n] =

1, n ≥ 00, n < 0


87

2.9 Discrete Time Signal Fourier Transform Theo-

rem

Basically there is no difference between FT theorem for the continuoustime and the discrete time case because summation has the same proper-ties as integration. Only differentiation and difference calculation are notcompletely analog, because it is not possible to form a derivative in thediscrete time case.

Table 2.2: Fourier transform Theorems

signal Fourier–transformx[n], y[n] X(ejω), Y (ejω)

1. ax[n] + by[n] aX(ejω) + bY (ejω)

2. x[n− nd], e−jωndX(ejω)nd is integral number

3. ejω0nx[n] X(ej(ω−ω0))

4. x[−n] X(e−jω)X(ejω) if x[n] is real

5. nx[n] jdX(ejω)

dω

6. x[n] ∗ y[n] X(ejω)Y (ejω)

7. x[n]y[n]1

2π

∫ π

−π

X(ejΘ)Y (ej(ω−Θ))dΘ

8. x[n]− x[n− 1] (1− e−jω)X(ejω)|1− e−jω|2 = 2(1− cos ω)

Parseval theorem

9.∞∑

n=−∞|x[n]|2 =

1

2π

∫ π

−π

|X(ejω)|2dω

10.∞∑

n=−∞x[n]y[n] =

1

2π

∫ π

−π

X(ejω)Y (ejω)dω


88

Example 1 corresponding to Theorem 5:

X(ejω) =+∞∑

k=−∞x[k] e−jωk

d

dωX(ejω) =

d

dω

(+∞∑

k=−∞x[k] e−jωk

)

=+∞∑

k=−∞

d

dω

(x[k] e−jωk

)

=+∞∑

k=−∞x[k] (−jk) e−jωk

⇐⇒ jd

dωX(ejω) =

+∞∑

k=−∞k x[k] e−jωk

F n · x[n] = jd

dωF x[n]

Example 2 corresponding to Theorem 8:

F x[n]− x[n− 1] =+∞∑

k=−∞x[k] e−jωk −

+∞∑

k=−∞x[k − 1] e−jωk

=+∞∑

k=−∞x[k] e−jωk −

+∞∑

k=−∞x[k] e−jωk e−jω

= X(ejω)(1− e−jω

)

=⇒ |F x[n]− x[n− 1] |2 = |F x[n] |2 |1 − e−jω|2= |F x[n] |2 2 · (1− cos(ω))


89

2.10 Discrete Fourier Transform: DFT

The Fourier transform for discrete time signals and systems has been ex-plained on the previous pages. For discrete time signals with finite lengththere is also another Fourier representation called “Discrete Fourier Trans-form” (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 thancontinuous frequencies ω.


90

Assume a discrete time signal x[n] with finite length (see also chapter 3.2on page 135):

x[n] =

x[n] 0 ≤ n ≤ N − 10 otherwise

Note: For a continuous time signal it is impossible in the strict sense to beband-limited and time-limited (truncation effect = Windowing).

• The discrete time signal Fourier transform for x[n] is:

X(ejω) =N−1∑n=0

x[n] exp(−jωn)

• ω is a continuous variable. The period is 2π. Frequency discretisationis made by sampling along the frequency axis.

• The Fourier transform X(ejω) is evaluated at

ωk =2π

Nk where k = 0, 1, . . . , N − 1

• Define:

X[k] : = X(ejω)|ω = ωk

N=8

Re

ImC


91

• Discrete Fourier Transform (DFT):

X[k] =N−1∑n=0

x[n] exp(−j2π

Nk n), k = 0, 1, . . . , N − 1

• Inverse DFT:

x[n] =1

N

N−1∑

k=0

X[k] exp(j2π

Nk n), n = 0, 1, . . . , N − 1

• Remark:

This equation can be proven by inserting the equation for X[k] in theequation for x[n] and using the orthogonality:

1

N

N−1∑n=0

exp

(j2π

Nkn

)=

1 k = m N, m is integral number0 otherwise

• Note:

Consider the “analogy” between inverse DFT (above) and inverseFourier transform of discrete time signal:

x[n] =1

2π

2π∫

0

X(ejω) ejωn dω

Under the given conditions the integral is equal to the sum(without approximation!).


92

Remarks:

• DFT coefficients X[k] are not an approximation of the discrete timesignal 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 . Afiner sampling of the discrete time signal Fourier transform is possibleby appending zeros to the signal x[n] (Zero–Padding).

x[n]

nN-1


93

Interpretation of Fourier coefficients

• Fourier transform X(ejω) of the time discrete signal x[n]

|X(e )|

ωπ−π

ωj

• Evaluation at N discrete sampling points

ωk =2π

Nk

yields the DFT coefficients X[k].

At first k lies in the domain k = −N

2+ 1, . . . , 0, . . . ,

N

2.

|X(e )|, |X[k]|

ωπ−π

ωj

-N/2+1 N/21 20-1 k


94

• Because of the periodicity of X(ejω) the coefficients X[k] can also beobtained by shifting the sampling points with negative frequency intothe positive frequency domain (by one period).

Then k = 0, . . . , N/2, . . . , N − 1.

X[k] =N−1∑n=0

x[n] exp(−j2π

Nk n)

|X(e )|, |X[k]|

ωπ−π

ωj

1 20 N-1 k

• Interpretation of coefficients for general signal x[n]:

k = 0 ←→ f = 0

1 ≤ k ≤ N

2− 1 ←→ 0 < f <

fS

2

k =N

2←→ ±fS

2N

2+ 1 ≤ k ≤ N − 1 ←→ −fS

2< f < 0


95

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 = Re2X[k] + Im2X[k]= |X[N − k] |2


96

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

• Reduction of “Fourier powers” e−2πjN ·kn to e−

2πjN ·l (l = 0, 1, . . . , N − 1)

is possible, because they are periodical (on the unit circle).


97

2.11 DFT as Matrix Operation

• Notation with unit roots

X[k] =N−1∑n=0

x[n] exp (−2πj

Nk n)

=N−1∑n=0

x[n] W knN

where WN := exp (−2πj

N)

N=12

W =10N

W 1N

W 2NW 3N

• Periodicity of WN

unit root:

exp (−j ωk) = exp (−j2π

Nk) = (WN)k

WN := exp (−j2π

N)


98

Note:

1. W rN = W r mod N

N

2. W kNN = (WN

N )k = 1k = 1 k ∈ Z

3. W 2N = [exp (−2πj

N)]2 = exp (−2πj

N2)

= exp (− 2πj

N/2) = WN/2 N even

4. WN/2N = exp (−2πj

N

N

2) = exp (−πj) = −1

5. Wr+N/2N = W

N/2N W r

N = −W rN


99

DFT as matrix multiplication

X[k] =N−1∑n=0

x[n] exp (−2πj

Nk n)

=N−1∑n=0

W knN x[n]

=N−1∑n=0

WNkn x[n]

with the matrix WN and the matrix elements:

WNkn := W knN

Inversion:

x[n] =1

N

N−1∑

k=0

X[k] exp (2πj

Nk n)

=1

N

N−1∑

k=0

(W−1N )kn X[k]

=1

N

N−1∑

k=0

W−1N kn X[k]

Therefore for the matrix WN−1 holds:

WN−1 :=1

NW−1

N


100

DFT matrix operation: properties

• DFT: invertible representation

N complex signal values ↔ N complex Fourier components

N real signal values ↔ N

2complex Fourier components

(due to symmetry)

in words:DFT causes no “information loss” in the signal.

• Parseval theorem for DFTgeneral Fourier:

N−1∑n=0

|x[n]|2 =1

2π

+π∫

−π

|X(ejω)|2dω

special DFT: (recalculate for yourself!)

N−1∑n=0

|x[n]|2 =1

N

N−1∑

k=0

|X[k]|2

in words:

Disregarding the factor1

N, the DFT is a norm conserving (= energy

conserving) transformation (mathematical terminology: “unitary”).


101

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) = Fx(t) =

∫ ∞

−∞x(t) e−jωtdt (2.9)

For the exact reconstruction (without approximation) of the continuoustime signal from sampled values, the samples x[n] = x(n · Ts) must havethe distance

Ts =π

ωB

(sampling theorem).

This results in the Fourier transform of the discrete time signal x[n]:

X(ejω) =∞∑−∞

x[n] e−jωn (2.10)

where ω is frequency “normalised on T ′′s

Functions (2.9) and (2.10) agree in interval [−ΩS/2, +ΩS/2] = [−ωB, +ωB].

X

ω

(ω)||

−ω ωBB

ΩS−ΩS


102

0

0.2

0.4

0.6

0.8

1

0 N-1

Figure 2.12: Hanning window

The signal x[n] is further decomposed by applying a window function w[n](windowing):

w[n] =

. . . n = 0, . . . , N − 1

0 otherwise

Windowed signal y[n]:y[n] = w[n] · x[n]

can be analyzed using Fourier transform or DFT.

Y (eiωk) =N−1∑n=0

y[n] e−jωk

DFT:

ωk =2πk

Nwhere k = 0, . . . , N − 1

Y [k] =N−1∑n=0

y[n] e−2πN kn

Matrix representation:

K = N

Y [0]...

Y [k]...

Y [K − 1]

=

...

e−2πiN ·n·k

...

y[0]...

y[n]...

y[N − 1]


103

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

Kk, where k = 0, 1, . . . , K − 1 and K > N

Inserting the definitions:

X(ejωk) =N−1∑n=0

x[n] exp (−2πj

Kk n)

=K−1∑n=0

x[n] exp (−2πj

Kk n)

where x[n] =

x[n] n = 0, . . . , N − 10 n = N, . . . ,K

i.e. Zero Padding (appending zeros).

Matrix representation:

X[0]...

...X[K − 1]

=

W nkK

x[0]...

...x[N − 1]

0...0

n = 0

n = N − 1n = N

n = K

Note:“Zero Padding” does not introduce any additional information into the sig-nal. This is only a trick so that DFT and particularly FFT (Fast FourierTransform) can be performed with a

higher frequency resolution .


104

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] =∞∑

k=−∞h[k] x[n− k]

=

Nh−1∑

k=0

h[k] x[n− k]

k

k

h[k]

x[-k]

N-1

-(N-1)x

h

0

n=0

• Altogether: Nx + Nh − 1 positions with “overlap”

• Therefore only Nx + Nh − 1 values of output signal can be differentfrom zero:

y[n] =

0 n > Nx + Nh − 2. . . n = 0, 1, . . . , Nx + Nh − 20 n < 0


105

k0 Nh-1=12

1

k0 Nx-1=4

x[k]

1 0.80.6

0.40.2

k0-Nx

x[n-k] n=-1,

k0

x[n-k] n=m (m>0 & m<Nh+Nx-2),

k0

x[n-k] n=Nh+Nx-1,

n0

y[n]

-1 Nh-1

Nh-1

Nh+Nx-1

mm-NX+1

Nh+Nx-2

Nh-1

a)

b)

c)

i)

ii)

iii)

Nh-1

1

1.8

2.4

2.8

2

1.2

0.60.2

3 3

Nx-1

h[k]

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] = 0ii) n between 0 and Nh + Nx − 2, convolution 6= 0iii) n > Nh + Nx − 2, no overlap with h[k], convolution y[n] = 0c) resulting convolution y[n].


106

Finite convolution using DFT

Convolution theorem:

y[n] =∞∑

k=−∞h[k] x[n− k]

Fourier:

Y (ejω) = H(ejω) X(ejω), 0 ≤ ω ≤ 2π

Also valid for sample frequencies:

ωk :=2π

Nk, 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 − 1frequency 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 ofcosts for realization of the finite convolution by DFT and FFT can befound at the end of paragraph 2.15.


107

2.15 Fast Fourier Transform (FFT)

Principle of FFT:Calculation of the DFT can be done by successive decomposition intosmaller DFT calculations. In this way, number of elementary operations(multiplications and additions) is dramatically reduced:

FFT:

N 2 → N

2ld N operations

N = 1024 :2 ·Nld N

=2 · 1024

10= 200 factor of velocity gain

The matrix is decomposed into a product of sparse matrices, therefore Nwith 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 Tukey1942 Danielson and Lanczos1905 Runge1805 Gauss


108

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∑n=0

x[n] exp(−j2π

Nk n) where k = 0, 1, . . . , N − 1

=N−1∑n=0

x[n] W nkN where W nk

N = exp(−j2π

Nk n)

• Decomposition of the sum over n into the sums over even and odd n:

X[k] =

N/2−1∑r=0

x[2r] W 2rkN +

N/2−1∑r=0

x[2r + 1] W(2r+1)kN

=

N/2−1∑r=0

x[2r] (W 2N)rk + W k

N

N/2−1∑r=0

x[2r + 1] (W 2N)rk

• Because of

W 2N = exp(−2j

2π

Nk n) = exp(−j

2π

N/2k n) = WN/2

for k = 0, . . . , N − 1 holds:

X[k] =

N/2−1∑r=0

x[2r] W rkN/2 + W k

N

N/2−1∑r=0

x[2r + 1] (WN/2)rk

= G[k] + W kN 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 W k

N .


109

Complexity:The complexity O(N 2) of one-dimensional FT can be reduced by adequateresorting values from two FTs with length N

2 and complexity O(2 · (N2 )2) =

N2

2 . By successive application of this resorting the complexity can be re-duced 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 equa-tion.

• Because of the DFT–length N2 = 4:

H[4] = H[0] and G[4] = G[0]

And then:

X[4] = G[0] + W 4N 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]

N/2-point

DFT

N/2-point

DFT

x[0]

x[2]

x[4]

x[6]

x[1]

x[3]

x[5]

x[7]

G[0]

G[1]

G[2]

G[3]

H[0]

H[1]

H[2]

H[3]

X[0]

X[1]

X[2]

X[3]

X[4]

X[5]

X[6]

X[7]

W0N

W1N

W2N

W3N

W4N

W5N

W6N

W7N

Figure 2.14: Flow diagram for decomposition of one N -DFT to two N/2–DFTs withN = 8


110

• Further analogous decomposition, until only DFT’s with the lengthN = 2 remain (so called Butterfly Operation)

• Resulting flow diagram of the FFT:

x[n] X[k]x[0]

x[4]

x[2]

x[6]

x[1]

x[5]

x[3]

x[7]

W0N

W0N

W0N

W0N

-1

-1

-1

-1

W0N

W2N

-1

-1

-1

-1

W0N

W1N

W2N

W3N

X[0]

X[1]

X[2]

X[3]

X[4]

X[5]

X[6]

X[7]

W0N

W2N

-1

-1

-1

-1

Figure 2.15: Flow diagram of an 8–point–FFT using Butterfly operations.


111

Complexity reduction

• Number of complex multiplications in FFTis N/2 · ld N .

• Comparison:Direct application of the DFT definition needsN 2 complex multiplications.

• Example: N = 1024 = 210

N 2

N/2 · ld N≈ 200

Complexity reduction of factor 200

FFT with the base 2 is not minimal according to number of addi-tions, FFT with the base 4 can be better.


112

Matrix representation of the FFT principle

• The complex Fourier matrix can be decomposed into the product ofr = ld N matrices, each of them having only two non-zero elements ineach column.

• The following graph shows the decomposition of the Fourier matrix inthe case of inverse transformation.

• w corresponds to W−1N

x = |wnk|X = T3 · T2 · T1 · TS ·XThis is how the decomposition into r + 1 = 4 matrices looks like(w4 = −1, w8 = 1):

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 w w2 w3 w4 w5 w6 w7 1 w w2 w3 w4 w5 w6 w7

1 w2 w4 w6 w8 w10 w12 w14 1 w2 w4 w6 1 w2 w4 w6

1 w3 w6 w9 w12 w15 w18 w21 1 w3 w6 w w4 w7 w2 w5

|wnk| = 1 w4 w8 w12 w16 w20 w24 w28 = 1 w4 1 w4 1 w4 1 w4

1 w5 w10 w15 w20 w25 w30 w35 1 w5 w2 w7 w4 w w6 w3

1 w6 w12 w18 w24 w30 w36 w42 1 w6 w4 w2 1 w6 w4 w2

1 w7 w14 w21 w28 w35 w42 w49 1 w7 w6 w5 w4 w3 w2 w


113

Signal flow diagram

The calculation operations which correspond to the matrix representationof FFT can be showed in a signal flow diagram.

T3 T2 T1 TS

1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 00 1 0 0 0 w 0 0 0 1 0 w2 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 00 0 1 0 0 0 w2 0 1 0 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 00 0 0 1 0 0 0 w3 0 1 0 −w2 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0 1 0

1 0 0 0 −1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 00 1 0 0 0 −w 0 0 0 0 0 0 0 1 0 w2 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 0 00 0 1 0 0 0 −w2 0 0 0 0 0 1 0−1 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 00 0 0 1 0 0 0 −w3 0 0 0 0 0 1 0 −w2 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1

TS T1 T2 T3

-1

-1

-1

-1

-1

-1

-1

ω2

ω2

-1

ω

ω2

ω3

−1

−1

−1

−1

X[0]

X[1]

X[2]

X[3]

X[4]

X[5]

X[6]

X[7]

x[0]

x[1]

x[2]

x[3]

x[4]

x[5]

x[6]

x[7]


114

• Matrices T1, T2 and T3 contain exactly two non-zero elements in eachrow.

• Non-zero elements are realizing the Butterfly Operation.

• Matrix T1: step width of the Butterfly Operation is 1Matrix T2: step width of the Butterfly Operation is 2Matrix 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


115

Butterfly Operation

• Signal flow diagram and matrix representation of the FFT are basedon the following basic operation:

Xm-1[p]

Xm-1[q]

Xm[p]

Xm[q]-1

WrN

• For two input values Xm−1[p] and Xm−1[q] this operation producestwo output values Xm[p] and Xm[q]. The output values are thereby alinear combination of the input values.

• Because of the flow graph, the operation is called“Butterfly Operation”.

Xm[p] = Xm−1[p] + W rN Xm−1[q]

Xm[q] = Xm−1[p]−W rN Xm−1[q]

[Xm[p]Xm[q]

]=

[1 W r

N

1 −W rN

] [Xm−1[p]Xm−1[q]

]


116

Bit Reversal

• The matrix representation of the FFT uses a sorting matrix, i.e. thesignal which is to be transformed is at first resorted.

• Example for N = 8:

n Binary representation Reversed n’

0 000 000 01 001 100 42 010 010 23 011 110 64 100 001 15 101 101 56 110 011 37 111 111 7

• Bit Reversal is a necessary part of the FFT–Algorithm.

Bit Reversal for N = 23


117

2.16 FFT Implementation

• Fortran version

C adapted from: Oppenheim, Schafer p. 608C SUBROUTINE FFT DecimationInTime (X, ld n) **********************************C ****************************************************************************

PARAMETER PI = 3.14159265358979PARAMETER N max = 2048

COMPLEX X(N max) ! array for input AND outputCOMPLEX Temp ! temporary storageCOMPLEX W uni ! root of unityCOMPLEX W pow ! powers of W uniINTEGER N, ld N, ip, iq, iqbeq, j, k, i exp, istp

N = 2**ld nIF (N.GT.N max) STOP

C BIT Reversed Sorting *********************************************************j = 1DO 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

ENDIFk = N/2DO WHILE (k.LT.j)

j = j - kk = k / 2

ENDDOj = j + k

ENDDOC End of Bit Reversed Sorting **************************************************

C FFT Butterfly Operations *****************************************************DO i=1, ld N

i exp = 2**i ! exponentistp = i exp/2 ! stepsizeW pow = (1.0,0.0)W uni = CMPLX (COS (PI/FLOAT(istp)), -SIN(PI/FLOAT(istp)))DO iqbeg = 1, istp

DO iq = iqbeg, N, i expip = iq + istpTemp = X(ip) * W powX(ip) = X(iq) - TempX(iq) = X(iq) + Temp

ENDDOW pow = W pow * W uni

ENDDOENDDO

C End of FFT Butterfly Operations **********************************************RETURNEND


118

Explanations about Fortran Program

Two program parts:

1. Bit Reversal

2. Butterfly Operations

• 3 loops with variables i, iqbeg, iq are controlling the execution ofthe 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 withinone level.

• middle loop, iqbeg:iqbeg: goes over the “nested” Butterfly operationsi=1: iqbeg=1i=2: iqbeg=1,2i=3: iqbeg=1,2,3,4iqbeg: specifies the sequence of starting points for inner loop

• inner loop, iq:iq: specifies the first element of the Butterfly operationistp: step width of the Butterfly operationip=iq+istp: specifies the second element for Butterfly operationinner loop is “started” once per “nesting”


119

x[0]

x[4]

x[2]

x[6]

x[1]

x[5]

x[3]

x[7]

W0N

W0N

W0N

W0N

-1

-1

-1

-1

W0N

W2N

-1

-1

-1

-1

W0N

W1N

W2N

W3N

X[0]

X[1]

X[2]

X[3]

X[4]

X[5]

X[6]

X[7]

W0N

W2N

-1

-1

-1

-1

Figure 2.16: Flow diagram of an 8–point–FFT using Butterfly operations.


120

• 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 replacesdata[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 MUSTbe 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 recurrencewi=wi*wpr+wtemp*wpi+wi;

mmax=istep


121

• Input and output arrays (C version)

1 real

2 imag

3 real

4 imag

2N-3

2N-2

2N-1

2N

real

imag

real

imag

a)

t = 0

t = ∆

t = (N-2)∆

t = (N-1)∆

1 real

imag

3 real

4 imag

b)

f = 0

f =

2

1N∆

2N-1

2N

real

imagf = 1

N∆

N+3

N+4

real

imag

N+1

N+2

real

imag

N-1

N

real

imagf = N/2 - 1

N∆

f = N/2 - 1N∆

f = 12∆

Figure 2.17: Input and output arrays of an FFT. a) The input array contains N (N ispower of 2) complex input values in one real array of the length 2N . with alternatingreal and imaginary parts. b) The output array contains complex Fourier spectrum at Nfrequency values. Again alternating real and imaginary parts. The array begins with thezero-frequency and then goes up to the highest frequency followed with values for thenegative frequencies.


122

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

2 log2(Nx + Nh)

Nx ·Nh multiplication in frequency domain

Nx + Nh Nx + Nh

inverse transformation

(Nx + Nh)2 Nx+Nh

2 log2(Nx + Nh)


123

2.17 Cyclic Matrices and Fourier Transform

The Fourier transform plays a significant role for so-called cyclic matrices:

0 n N − 1

H =

h0 h1 h2 . . . hN−1

hN−1. . . . . . . . . hN−2

hN−2. . . . . . . . . . . . ...

. . . . . . . . . . . .... . . . . . . . . . h2

. . . . . . h1

h1 hN−1 h0

0

m

N − 1

so: Hnm = 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. periodicconvolutions and to build a cyclic variant of the system theory.

The eigenvectors of a cyclic matrix can be obtained from the columns ofDFT matrix:

0 n N − 1

0

m

N − 1

w0·0 · · · wn·0 · · · w(N−1)·0... wn·1 ......

...

· · · ... · · · ......

...

· · · ... · · · ...... wn·(N−1) ...

where w = e−2πjN


124

The eigenvalues λk of a cyclic matrix with the kernel vector (h0, h1 . . . , hN−1)T

are:

λk =N−1∑n=0

hn e−2πjN ·k·n where k = 0, 1 . . . , N − 1

The representation is oriented on L. Berg: Linear equation systems withband structure (page 52 ff).

The special case of a cyclic matrix when the kernel vector h is symmetricand real is especially interesting for many applications:

hn = hN−n and hn ∈ IR

That means that the matrix is real, symmetric and cyclic:

0 N − 1

0

N − 1

h0 h1 h2 . . . h2 h1

h1. . . . . . . . . h2. . . . . . . . . . . . h3

. . . . . . . . . . . . ...... . . . . . . . . .

h2 h3. . . . . . h1

h1 h2 h1 h0


125

For such a matrix is valid:

a) the eigenvalues λk are:

λk =N−1∑n=0

hn cos( 2πkn

N

)

b) The eigenvectors can be obtained from the columns of the “discretecos–matrix”:

n

m

...

...

...cos

( 2πnmN )

...

...

...

Application: diagonalisation of covariance matrices, e.g. for codingof image and speech signals.


126

Proof:

We will prove that

vk =

cos 2πk·0N

cos 2πk·1N

...

...

...

cos 2πk·(N−1)N

are eigenvectors of the given matrix with eigenvalues λk.

For a symmetric cyclic matrix is valid:

Hnm := h(n−m)modN where hn = hN−n

One row results in (for odd N):

∑n

Hmn · vkn = h0 · cos2πkm

N+

N−12∑

l=1

hl·(

cos2πk(m− l)

N+ cos

2πk(m + l)

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:

∑n

Hmn · vkn = h0 · cos2πkm

N+ 2

N−12∑

l=1

hl·(

cos2πkl

N· cos

2πkm

N

)

=[

h0 + 2

N−12∑

l=1

hl cos2πkl

N

]

︸︷︷︸λk

· cos2πkm

N︸︷︷︸vkm

= λk · vkm


127

Excursion: Toeplitz matrices

We consider only quadratic real matrices:

H ∈ IRN×N with Hnm ∈ IR

a) H is a (general) Toeplitz matrix , if:

Hnm = hn−m

i.e.

H =

h0 h1 h2 . . . hN−2 hN−1

h−1. . . . . . . . . hN−2

h−2. . . . . . . . . . . . ...

... . . . . . . . . . . . . . . .

. . . . . . . . . . . . h2

h2−N. . . . . . . . . h1

h1−N h2−N h−2 h−1 h0

b) For a symmetric Toeplitz matrix then holds:

Hnm = h|n−m|

i.e.

H =

h0 h1 h2 . . . hN−2 hN−1

h1. . . . . . . . . hN−2

h2. . . . . . . . . . . . ...

... . . . . . . . . . . . . . . .

. . . . . . . . . . . . h2

hN−2. . . . . . . . . h1

hN−1 hN−2 h2 h1 h0


128

c) A cyclic matrix can be obtained by special choice:

Hnm = h(n−m)modN

H =

h0 h1 h2 . . . hN−2 hN−1

hN−1. . . . . . . . . hN−2

hN−2. . . . . . . . . . . . ...

. . . . . . . . . . . .

... . . . . . . . . . h2

h2. . . . . . h1

h1 h2 hN−1 h0

Also valid: each cyclic matrix is a Toeplitz matrix.

d) A symmetric cyclic matrix can be obtained by the following choice ofthe 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


129

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


131

3.1 Features for Speech Recognition

Architecture of an automatic speech recognition system

speech signal

short-timeanalysis

each 10 ms(using FFT)

sequence ofacoustic vectors

patterncomparison

decision

reference modelfor each word

in the vocabulary


132

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)


133

Goal:

• Ideally: Real features for the recognition

• In practice: Data reduction, i.e. compact descriptionof the speech signal (amplitude spectrum)

Side effect:

• Method also enables coding of speech signals using lowest possiblenumber of bits

Key words:

• Fourier transform: wide band/narrow band, autocorrelation function

• Filter bank

• Cepstrum

• Linear Predictive Coding (LPC) analysis

• Fundamental frequency analysis


134

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, |n| ≤ N/20, otherwise

• The windowed signal x[n] is analyzed with a Fourier Transform orDFT.

• The multiplication of the original signal s[n] with the window functionw[n] in the time domain corresponds to the convolution of the spectraof two signals S(ejω) and window function W (ejω) in the frequencydomain:

X(ejω) =1

2π

π∫

−π

S(ejθ) W (ej(ω−θ)) dθ

• This convolution performs a (spectral) smearing in the frequency do-main (leakage).


135

Window function: Impulse response:

0

0.2

0.4

0.6

0.8

1

0 N-1

Rectangle

-60

-50

-40

-30

-20

-10

0

-0.5 fs 0 0.5 fs

dB

0

0.2

0.4

0.6

0.8

1

0 N-1

Triangle

-60

-50

-40

-30

-20

-10

0

-0.5 fs 0 0.5 fs

dB

0

0.2

0.4

0.6

0.8

1

0 N-1

Hanning

-60

-50

-40

-30

-20

-10

0

-0.5 fs 0 0.5 fs

dB

0

0.2

0.4

0.6

0.8

1

0 N-1

Hamming

-60

-50

-40

-30

-20

-10

0

-0.5 fs 0 0.5 fs

dB


136

Window function: Impulse response:

0

0.2

0.4

0.6

0.8

1

0 N-1

Nuttall

-60

-50

-40

-30

-20

-10

0

-0.5 fs 0 0.5 fs

dB

0

0.2

0.4

0.6

0.8

1

0 N-1

Gauss

-60

-50

-40

-30

-20

-10

0

-0.5 fs 0 0.5 fs

dB

0

0.2

0.4

0.6

0.8

1

0 N-1

Chebyshev

-60

-50

-40

-30

-20

-10

0

-0.5 fs 0 0.5 fs

dB


137

πT

- πT

-Ω0 Ω0

XC(Ω)

0

0

1 H(Ω)

πT

πT

-

0

SC(Ω)

-Ω0 Ω0 Ω

Ω

Ω

2π0 ω0=ΩT−ω0−π π−2π

ω

X(ejω)

0 π-π 2πω

−2π

W(ejω)

ω2π−2π −π π0

V(ejω), V[k]

2πn

Fourier Transformof a continuous

time signal

Frequency graphof anti-aliasinglow-pass filter

Fourier Transformof filtered signal

Fourier Transform of sampled signal

Fourier Transform of window function

Fourier Transform of windowed signaland sampled values

of continuous spectrumobtained using DFT

Figure 3.1: Example for the application of the Discrete Fourier Transform (DFT).


138

Properties of short-time DFT–analysis

Important effects:

• Picket FenceIf not enough sampled values of continuous spectrum are available,spectral sampling can yield delusive results. This problem can be re-duced using Zero Padding (inter-space between the coefficients S[k]becomes smaller, i.e. frequency resolution becomes larger)

• Leakage: Spreading of the line spectrumBecause the window function is limited in time, a spreaded spectrumis measured instead of the spectrum of the original signal unlimited intime. That means, the line spectrum even becomes spreaded for puresinusoidal signals.


139

3 examples of DFT analysis

• we observe a continuous time signal x(t) composed of twosinusoids:

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 = Ω0TS and ω1 = Ω1TS

• with the window function w[n]:

v[n] = A0 w[n] cos(ω0 n) + A1 w[n] cos(ω1 n)

Intermediate calculations:

v[n] =A0

2w[n] exp(j ω0 n) +

A0

2w[n] exp(−j ω0 n)

+A1

2w[n] exp(j ω1 n) +

A1

2w[n] exp(−j ω1 n)

also modulation principle

• Fourier Transform of the windowed signal:

V (ejω) =A0

2W (ej(ω−ω0)) +

A0

2W (ej(ω+ω0))

+A1

2W (ej(ω−ω1)) +

A1

2W (ej(ω+ω1))


140

• Assume:

Ω0 =2π

14· 10kHz, Ω1 =

4π

15· 10kHz

1/TS = 10kHz, rectangle window with N = 64, A0 = 1, A1 = 0.75

• The windowed signal v[n] for the discrete time signal x(n) is therefore:

v[n] =

cos(2π

14n) + 0.75 cos(

4π

15n) : 0 ≤ n ≤ 63

:0 : otherwise

-1

0

1

2

63

n

v[n]


141

• Fourier Transform W (ejω) of the rectangle window function

0

64

π−π

Example 1: Leakage Effect

Variation of ω0 and ω1 resp. Ω0 and Ω1

Difference between frequencies ω0 and ω1 is reduced gradually

Case 1a:

Ω0 =2π

6104 Hz, Ω1 =

2π

3104 Hz

ω0 = Ω0 TS =2π

6104 Hz 10−4 s =

2π

6

ω1 = Ω1 TS =2π

3104 Hz 10−4 s =

2π

3


142

Case 1a (continued): ω0 =2π

6ω1 =

2π

3

0

32

π−π 2π2π3 6

2π 2π6 3

V(ω)

ω

Case 1b: ω0 =2π

14ω1 =

4π

15

0 π−π ω4π15

2π14

2π14

4π15

32

V(ω)


143

Case 1c: ω0 =2π

14ω1 =

2π

12

0

30

-π π ω2π 2π12 14

V(ω)

Case 1d: ω0 =2π

14ω1 =

4π

25

0

40

V(ω)

−π π


144

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]:

v[n] =

cos(

2π

14n) + 0.75 cos(

4π

15n) : 0 ≤ n ≤ 63

0 : otherwise

• DFT of the length N = 64 without Zero Padding


145

a)

-1

0

1

2

63

n

v[n]

b)

0

30

V(k)

k63

c)

0

32

ωπ 2π

V(ω)

Figure 3.2: a) signal v[n]; b) DFT-spectrum V [k]; c) Fourier spectrum V (ejω).


146

Case 2b:

• In contrast to case 2a, the frequencies of sinusoids are changed onlyslightly.

• Windowed signal v[n]:

v[n] =

cos(

2π

16n) + 0.75 cos(

2π

8n) : 0 ≤ n ≤ 63

0 : otherwise

• DFT of the length N = 64 without Zero Padding


147

a)

-1

0

1

n

v(n)

63

b) 30

0 k

V(k)

63

c)

0

32

V(ω)

π 2π ω

Figure 3.3: a) signal v[n]; b) DFT-spectrum V [k]; c) Fourier spectrum V (ejω).


148

Analysis of Example 2:

• The manifestation of the DFT can be put down to the spectral sam-pling. Although in Case 2b the windowed signal v[n] contains a sig-nificant number of frequencies beyond ω0 and ω1, they do not show inthe DFT spectrum of length N = 64.

• Using a rectangle window, the DFT of the sinusoidal signal gives sharpspectral lines, if the period N of the transformation is a whole multipleof the signal period and no Zero Padding is applied.

• Explanation for the case of a complex exponential function:

– Assume the signal x[n]:

x[n] =1

Nexp(j

2π

n0n)

– 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:

V [k] = X[k] ∗ W [k] =

sin

(π(k − N

n0)

)

sin

(π(k − N

n0)/N

)

– In case of

N

n0∈ IN

only the DFT coefficient k =N

n0is non-zero.


149

Example 2 (continued)

• Assume signal v[n] of Case 2b:

v[n] =

cos(

2π

16n) + 0.75 cos(

2π

8n) : 0 ≤ n ≤ 63

0 : otherwise

• In contrast to Case 2b, a DFT with length N = 128 is applied (ZeroPadding).

• Result:

Using finer sampling, existing additional frequency components emerge.


150

a) 30

0 k

V(k)

63

b)

0 k127

V(k)

32

c)

0

32

V(ω)

π 2π ω

Figure 3.4: a) DFT of length N = 64; b) DFT of length N = 128; c) Fourier spectrumV (ejω).


151

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 graduallyincreased up to N = 1024 (Zero Padding).

• Finally: DFT spectrum for the case N = 1024 and L = 64.

The Kaiser window is defined as:

wK [n] =

I0

[β

(1− [(n− α) /α]2

)1/2]

I0(β): 0 ≤ n ≤ L− 1

0 : otherwise

In this example:

β = 0.8 and α =L− 1

2

The windowed signal v[n]:

v[n] = wK [n] cos(2π

14n) + 0.75 wK [n] cos(

4π

15n)


152

Example 3: (continued)DFT length N = 64, window length L = 64

• Windowed signal

-1

0

1

n

v(n)

63

• DFT spectrum

30

0 k

V(k)

63


153

Example 3: (continued)DFT length N = 32, window length L = 32

(N and L halved)

• Windowed signal

1

n310

v(n)

• DFT spectrum

8

V(k)

0 k31


154

Example 3: (continued)

Effect of changing DFT length N at constant window length L = 32 (ZeroPadding)

• DFT length N = 32, window length L = 32

8

V(k)

0 k31


8

V(k)

0 k63


155



8

V(k)

0 k127


8

V(k)

0 k1024


156


Increasing the window length (L)


16

V(k)

0 k1024


157

Example 4: Window function influence on the spectrum

0

0

speech signalphoneme "a"

0

amplitude spectrum- rectangle window -

0

amplitude spectrum- Hamming window -

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


158

3.3 Autocorrelation Function and Power Spectral Den-

sity

Definition of Autocorrelation Function (ACF) analog to the continuoustime case:

R[k] : =∞∑

n=−∞x[n] x[n + k]

For a signal x[n] assume (e.g. after some suitable windowing):

x[n] =

x[n] 0 ≤ n ≤ N − 1

0 otherwise

In this case the ACF gives:

R[k] =N−1−k∑

n=0

x[n] x[n + k] because x[n] = 0 for n < 0 and n ≥ N

“triangular effect”

N k-N

number of terms in R[k]

Cross correlation:

Rxy[k] =∞∑

n=−∞x[n] · y[n− k]

In contrast to convolution:

Oxy[k] =∞∑

n=−∞x[n] · y[k − n]


159

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)

=∞∑

k=−∞x[k] exp(−jωk) ·

∞∑

l=−∞x[l] exp(jωl)

=∞∑

k=−∞

∞∑

l=−∞x[k] x[l] exp(−jωk) exp(jωl)

=∞∑

k=−∞

∞∑

l=−∞x[k + l] x[l] exp(−jωk) exp(−jωl) exp(jωl)

=∞∑

k=−∞

( ∞∑

l=−∞x[k + l] x[l]

)exp(−jωk)

=∞∑

k=−∞R[k] exp(−jωk)

Note: The phase spectrum is removed.


160

5. Because of the symmetry R[k] = R[−k] the DFT becomes the cosinetransform:

| X(ejω) |2 =∞∑

k=−∞R[k] exp(−jωk)

=N−1∑

k=−(N−1)

R[k] exp(−jωk)

= R[0] +N−1∑

k=1

R[k] (exp(−jωk) + exp(jωk))

= R[0] + 2 ·N−1∑

k=1

R[k] cos(ωk) because R[k] = R[−k]

6. The intensity spectrum | X(ejω) |2 is a polynom of cos(ω) with gradeN − 1.

Reason: Moivre formula:

cos(ωk) = cosk(ω) −(

k

2

)cosk−2(ω) sin2(ω) +

(k

4

)cosk−4(ω) sin4(ω) − . . .


161

Example 1: Spectral analysis using ACF

0

0


0


0

amplitude spectrum- short hamming window -

0

amplitude spectrum- 19 ACF-coefficients -

0

amplitude spectrum- 13 ACF-coefficients -

Figure 3.6: Fourier Transform of a voiced speech segment:a) signal progression, b) high resolution Fourier Transform, c) low resolution FourierTransform with short Hamming window (50 sampled values), d) low resolution FourierTransform using autocorrelation function (19 coefficients), e) low resolution Fourier Trans-form using autocorrelation function (13 coefficients)


162

Example 2: ACF of voiced and unvoiced speech segments

0

0


0

0

speech signalphoneme "s"

0

0

autocorrelation- rectangle window -

0

0

autocorrelation- rectangle window -

0

0

autocorrelation- Hamming window -

0

0

autocorrelation- Hamming window -

Figure 3.7: Signal progression and autocorrelation function of voiced (left) and unvoiced(right) speech segment


163

Example 3: Temporal progression of autocorrelation coefficients

0

0

speech signal - digit sequence 0861909

0

ACF - coefficient for index 0 (energy)

0

0

ACF - coefficient for index 3

0

0


0

0


Figure 3.8: Temporal progression of speech signal and four autocorrelation coefficients


164

3.4 Spectrograms

Using DFT

• Wide-band: in frequency domain:

– short time window

– “interaction” in the “synchronization” betweentime 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


165

Example 1: speech spectrograms

Figure 3.9: a) wide-band spectrogram: long time window, high time resolution (verti-cal lines), no frequency resolution; for voiced signals provides information on formantstructure b) narrow-band spectrogram: short time window, no time resolution, highfrequency resolution (horizontal lines); for voiced signals provides information on funda-mental frequency (pitch)


166

Example 2: speech spectrograms

Figure 3.10: Wide-band and narrow-band spectrogram and speech amplitude for thesentence “Every salt breeze comes from the sea”.


167

3.5 Filter Bank Analysis

• History:Decomposition of the signal using a “bank” of band-pass filters andenergy calculation in each frequency band

transferfunction

f

• Today digitally:

– Digital filters:

yk[n] =∞∑

m=−∞hk[n−m] x[m] , k = 1, . . . , K

– FIR: Finite Impulse ResponseIIR: 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 respec-

tively


168

DFT/FFT filter bank:

transferfunction

f

transferfunction

f


169

• Averaging:summation should be as smooth as possible over all channelsForm: 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

fk= const. for all k

– frequency groups of the ear (total number 24):

f < 500Hz : ∆f = 100

f ≥ 500Hz :∆f

f= 20%

– adjusted to vowels or sounds


170

3.6 Mel-frequency scale

The frequency resolution of the human ear is decreasing on the higherfrequencies. This empirical dependency results in the definition of the Melscale, which is approximately calculated as (from: Hidden Markov Toolkit,Cambridge University Engineering Departement, S.J.Young):

fMEL = 2595 log10 (1 +f

700Hz)

7000 f / Hz

2700

fMEL

Compression of the high frequencies

f

fMEL

A filter bank with constant band-widths can be used on the Mel scale:

fMEL


171

Table: MEL Scale:

f/Hz fMEL

65 100136 200213 300298 400391 500492 600603 700724 800856 9001000 10001158 11001330 12001519 13001724 14001949 15002195 16002464 17002757 18003078 19003429 20003812 21004230 22004688 23005187 24005734 25006331 26006984 2700


172

3.7 Cepstrum

The Cepstrum is the Fourier series expansion of the logarithm of the spec-trum.Comparison: autocorrelation function is a Fourier series of the normalspectrum.

We consider:

y[n] =∞∑

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)


173

• Another notation:

y[n] = x[n] + h[n]

using the definition of the cepstrum for x[n](analogous for y[n] and h[n])

x[n] = F−1 log X(ejω)

=1

2π

π∫

−π

exp(jωn) log X(ejω) dω

=1

2π

π∫

−π

exp(jωn) log

[∑m

x[m] exp(−jωm)

]dω

= C x[n]

• Note:

– Cepstrum = artificial word derived from “spectrum”

– Cepstrum is located in time domain


174

Through the cepstrum transformation

x[n] −→ x[n] = C x[n]the convolution comes down to a simple addition.

In the cepstrum domain, a linear operation L (time invariance is not nec-essary) on y[n] is performed separately on h[n] and x[n]:

y[n] =∞∑

k=−∞h[n− k] x[k]

y[n] = h[n] + x[n]

L y[n] = L

h[n]

+ L x[n]

With the definition GL for the concatenation of the cepstrum, the operationL, 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)


175

Complex cepstrum:

x[n] =1

2π

π∫

−π

exp(jωn) logX(ejω) dω

Note: complex logarithm

Simple cepstrum (real cepstrum):

x[n] =1

2π

2π∫

0

exp(jωn) log|X(ejω)| dω

• Cepstrum: Fourier coefficients of the logarithmized power spectraldensity

• ACF: Fourier coefficients of Fourier series of the power spectral density

Setting cepstral coefficients x[n] to zero for high n results in smoothing ofthe power spectral density.

Implementation:

Fourier Transform via N–FFT (N = 512, 1024, 2048)(But: discretisation error):

x[n] :=1

N

N−1∑

k=0

ej2π

Nkn

log |X(ej2π

Nk)|


176

Example 1: Real cepstrum

Fine structure of power spectral density with the period 1/T results in asingle peak in the cepstrum at time T .

frequency

log|F(ω)|2

0 1T

F-1(log|F(w)|2)

time0 T

Figure 3.11: Above: logarithmized power spectrum of a spoken vowel (schematic).Below: corresponding cepstrum (inverse Fourier–transform of the logarithmized powerspectrum).


177

Example 2: Smoothing

0

0


0

0

windowed phoneme "a"- Hamming window -

0

spectrum from cepstrum whole cepstrum

first 13 coefficients

Figure 3.12: Cepstral smoothing: speech signal (vowel “a”), windowed speech signal(Hamming window), spectrum obtained from the whole cepstrum (blue) and smoothedspectrum obtained from the first 13 cepstral coefficients (red).


178

Example 3: Smoothing with different numbers of cepstral coefficients

0

0


0



0



0


first 19 coefficients first 13 coefficients

Figure 3.13: Homomorph analysis of a speech segment: signal progression, homomorphsmoothed spectrum using 13 and 19 cepstral coefficients


179

Cepstrum calculation using Filter Bank Output

• Filter bank outputs A[k] for k = 1, . . . , KNote: k = 0 is missing.

• We complete the outputs symmetrically:

A A AA AA-K+1 -1 0 1 2 K

• Symmetry A−k+1 = Ak for all k = 1, . . . , K.


180

Inverse DFT a[n] of the symmetric sequence A−K+1, . . . , AK :

a[n] =1

2K

K∑

k=−K+1

Ak exp

(2πj

2Knk

)

=1

2K

K∑

k=1

Ak

[exp

(2πj

2Knk

)+ exp

(2πj

2Kn(−k + 1)

)]

= exp

(2πj

2K0.5

)1

2K

K∑

k=1

Ak

[exp

(2πj

2Kn(k − 0.5)

)+ exp

(−2πj

2Kn(k − 0.5)

)]

= exp

(2πj

2K0.5

)1

K

K∑

k=1

Ak cos(πn

K(k − 0.5)

)

The phase term exp(2πj

2K 0.5)

depends on the position of the symmetry axisaround k = 0.5.

Cepstrum is defined as:

a[n] =1

K

K∑

k=1

Ak cos(πn

K(k − 0.5)

)


181

Mel Cepstrum according to Davis and Mermelstein

ff = 100 f = 300

k = 1 k = 3 k = K

MEL MEL MEL

Filter bank:

• overlapping band-pass filters triangular shape,

• all channels have equal band width, and filter positioning is equidis-tant on a Mel scale.

Calculation of the filter bank outputs:

• magnitude of DFT coefficients,

• for each channel summation of the magnitudes according to triangularweight function,

• for each channel logarithm of the sum.

Thus the filter outputs A[k] with k = 1, . . . , K are obtained. Using thefilter bank outputs, the cepstrum is calculated using a cosine transform.(see previous description)


182

3.8 Statistical Interpretation of the Cepstrum Trans-

formation

We consider the filter bank outputs log|Xk|.

s p kN/20

log |X |k

Assumption: The correlation between the outputs s and p, i.e. the elementCsp of the covariance matrix does not depend directly on s or p, but onlyon their difference. Because the spectrum is periodical there is no distancegreater 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.


183

Example: the covariance matrix for N = 8:

C =

c0 c1 c2 c3 c4 c3 c2 c1

c1 c0 c1 c2 c3 c4 c3 c2

c2 c1 c0 c1 c2 c3 c4 c3

c3 c2 c1 c0 c1 c2 c3 c4

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 thesymmetry) (see excursion in chapter 2.17).


184

3.9 Energy in acoustic Vector

The energy is usually added as zeroth (or first) component to the acousticvector.

For the logarithmic energy we have:

log E =1

2π

π∫

−π

log|X(ejω)|2 dω

For the (short time) spectrum or cepstrum is approximately holds:

log E ≈ 1

K

K∑

k=1

log|Xk|2

Spectra are usually normalized with log E:

logY 2k = log|Xk|2 − log E

such that:

K∑

k=1

logY 2k ≡ 0

The cepstral coefficient x[0] is the logarithmized energy.


185

Chapter 4

Fourier Transform and ImageProcessing

• 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


187

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. Discretiza-tion 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)

=

∞∫

−∞

∞∫

−∞g(x, y) e−2πi(fxx+fyy) dxdy

The arguments fx and fy are called spatial frequencies.


188

The two-dimensional Fourier–transform can be obtained by using two one-dimensional Fourier–transforms.We consider one “image row” with a constant value of y:

x → g(x, y)

Corresponding Fourier–transform Gy(fx) :

Gy(fx) = Fx → g(x, y)

=

∞∫

−∞g(x, y) e−2πifxx dx

Then we compute the Fourier–transform of the function:

y → Gy(fx)

and obtain:

Fy → Gy(fx) =

+∞∫

−∞Gy(fx) e−2πifyy dy

=

+∞∫

−∞

+∞∫

−∞g(x, y) e−2πi(fxx+fyy) dxdy

In this way we get the result:

G(fx, fy) = Fy → Fx → g(x, y)For the inverse FT we have (as can be expected):

g(x, y) = F−1(fx, fy) → G(fx, fy)

=

+∞∫

−∞

+∞∫

−∞G(fx, fy) e2πi(fxx+fyy) dfxdfy


189

We would like to interpret the two-dimensional FT visually. For this pur-pose, we consider the exponential factor in the FT and require the followingcondition:

e2πi(fxx+fyy) != 1

⇒ 2πi(fxx + fyy) = 2πn for n ∈ IN

⇒ y = −fx

fyx +

n

fy

y

x

∆Lθ

1/fy

1/fx

spatial period ∆L =1√

f 2x + f 2

y


190

Special case:|G(fx, fy)| has a large value only at “one” point (u, v) = (fx, fy) in thespatial frequency plane

fx

fy

u-u

v

-v

|G(fx,fy)|

x

y


191

Since G(fx, fy) = G(−fx,−fy) for a real image g(x, y), we have twodominant 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 greyvalues

really: straight line through (u, v) and (−u,−v) represents abrupt changesof the grey values


192

Figure 4.1: TV–image (analog) Figure 4.2: Digitized TV–image

Figure 4.3: Amplitude spectrum of Fig-ure 4.2

Figure 4.4: Low-pass filtered


193

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 = high amplitudenote:1. strong components along the axes= vertical and horizontal image edges2. concentration around (fx, fy) = (0, 0)= regions with constant grey values

Figure 4.4: Low-pass filter:H(fx, fy) = [cos(πfx) · cos(πfy)]

16

0 5 H 5 1Figure 4.5: High pass filter:

H(fx, fy) = 1.5− [cos(πfx) · cos(πfy)]4

0.5 5 H 5 1.5Figure 4.6: High pass enhancement:

H(fx, fy) = 2.0− [cos(πfx) · cos(πfy)]4

1.0 5 H 5 2.0


194

Following general rules for G(fx, fy) ensue:

Edges in the image g(x, y):

• An image edge produces “strong” spatial frequency components alongone straight line in the spatial frequency plane which is orthogonal tothe edge.

• The “sharper” the edge is, the “longer” is the corresponding line inthe 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 “DCcomponent” (average grey value, DC=direct current).


195

4.2 Discrete Fourier Transform for Images

The analog image g(x, y) is discretized (“sampled”) along both axes. Weobtain 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(e2πiN ·u, e

2πiN ·v) =

N−1∑

j=0

N−1∑

k=0

g[j, k]e−2πiN (uj + vk) where u, v = 0, 1, . . . , N − 1

Discretization is written as:

G[u, v] =N−1∑j=0

N−1∑

k=0

g[j, k] e−2πiN (uj + vk)

=N−1∑j=0

e−2πiN uj·

(N−1∑

k=0

g[j, k] e−2πiN vk

)

Interpretation:Fourier–transform of the image is first performed row by row, then columnby column.Using usual definition of the “Fourier matrix” W (i.e. Wvk = (e−

2πiN )vk),

we obtain the matrix representation of Fourier–transform.

Using the notation:g ∈ IRNxN

W ∈ CNxN

G ∈ CNxN

we obtainG = [W g W ]

g =1

N 2 · [W−1 G W−1]

Note: In the corresponding definition of the Fourier–transform, instead ofthe factors 1 and 1/N 2 we have the factors 1/N and 1/N or 1/N 2 and 1.


196

4.3 Fourier Transform in Computer Tomography

y=ax+b,a const.

x

yb

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) =

∫g(x, ax + b) dx

Fourier–transform:

Ga(fb) =

∫ga(b) e−2πjfbb db

=

∫ ∫g(x, ax + b) e−2πjfbb db dx

We substitute b = y − ax and obtain:

Ga(fb) =

∫ ∫g(x, y) e−2πj(yfb−xafb) dydx

= G(−afb, fb)

= Fourier–transform G(fx, fy) of g(x, y) along

the spatial frequency straight line (fx, fy) with fy = −1

afx


197

Remarks:

a. Straight line in spatial frequency domain: (fx, fy) = (−afb, fb) isorthogonal to y = ax + b:

y = ax + b => in Fourier–transform fy = −1

afx

The angle between these straight lines is a right angle becausea· ( −1

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 onlyon the orientation a of the straight line. Therefore, if we calcu-late the projection for many different inclinations a and apply theone-dimensional FT, we obtain the two-dimensional FT of the imageg(x, y).


198

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:

coordinates: z =

(x

y

)∈ IR2

image grey values: g(z) = g(x, y) ∈ IR+

spatial frequency: f =

(fx

fy

)∈ IR2

We ignore the discretization.


199

Translation

z → z + z0

with translation vector z0 =

(x0

y0

)∈ IR2

Image : g(z) → g(z) := g(z + z0)

FT : G(f) = exp (i[fxx0 + fyy0]) ·G(f)

Rotation

z → Dϑz

with rotation matrix Dϑ =

[cos ϑ sin ϑ

− sin ϑ cos ϑ

]

y

x

ϑ


200

Scaling

z → α · zwith scaling factor α > 0

Image : g(z) → g(z) = g(α · z)

FT : G(f) =1

α2 · G

(f

α

)

basically: similarity principle (S.??) for one-dimensional FT

transferred to two dimensions

Invertible linear mapping

z → A · zwhere A ∈ IR2×2 invertable

Image : g(z) → g(z) = g(Az)

FT : G(f) = . . .

=1

det(A)· G((A−1)Tf)

Proof: transformation of two-dimensional integration variables


201

We apply two basic rules to obtain the RST-invariance:

1. Invariance to translation (=T) can be obtained by using the square ofthe absolute value

g(z) → G(f) → |G(f)|2

2. To obtain RS-invariance we transfer to polar coordinates in the spatialfrequency domain. We write in complex notation:

fz ≡ fx + i fy = r eiφ = exp (ln r + iφ) ∈ C2

Complex logarithm:f ′z := ln fz = ln r + iφ

We already know:

a) rotation by angle φ0 in spatial domain= rotation by angle φ0 in spatial frequency domain

b) scaling with factor α in spatial domain

= scaling with factor( 1

α

)and

( 1

α

)2respectively in spatial

frequency domain

f ′zscaling and−→

rotationf ′z


202

f ′z = ln z

= ln r + iφ

= ln( r

α

)+ i(φ − φ0)

= ln r + i φ − ln α − i φ0

= f ′z − ln α − i φ0

= translation with the shift vector (− ln α − i φ) ∈ C2

in logarithmic polar coordinates of the spatial frequency plane


203

original image-

6

x

y

|FFT|-

6

fx

fy

log–polar-

6

ln r

φ

|FFT|-

6

fx

fy


205

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 recon-structed completely.


206

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 insignal coding.


207

5.1 Principle of LPC Analysis

timenn-2

We consider a discrete time signal x[n], possibly multiplied with a windowfunction. 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] : signal value

x[n] : predicted value

We assume the predicted value x[n] to be a linear combination of thepreceding values of x[n]:

x[n] :=K∑

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 samplingfrequency of 4 kHz (about 2 coefficients per kHz).


208

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


209

The coefficients αk are unknown at first. To estimate these, we define theprediction error for each point n in time:

e[n] := x[n]− x[n]

= x[n]−K∑

k=1

αk x[n− k]

For a reliable set of LPC–coefficients we calculate the squared error crite-rion E as sum of the squared prediction errors e[n]:

E =∑

n

e2(n)

=∑

n

[x[n]−

K∑

k=1

αk x[n− k]

]2

!= minimum with respect to α1, . . . , αk, . . . , αK

Taking the derivative ∂∂αl

for l = 1, . . . , K results in:

∑n

(x[n]−∑

k

αk x[n− k]

)x[k − l]

!= 0

∑k

αk

∑n

x[n− k]x[k − l] =∑n

x[k − l]x[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 non-negative and it has a single well-defined minimum.

• The optimal LPC–coefficients are invariantto linear scaling of the signal values x[n].


210

Minimization of the squared error criterion with respect to the LPC–coefficients results either from taking the derivative or from the “quadraticcomplement” (recalculate for yourself!). The linear equation system forthe LPC–coefficients αk ensues:

l = 1, . . . , K :K∑

k=1αk ·

∑n

x[n− k] x[n− l] =∑

n

x[n− l] x[n]

with still unspecified summation limits over n. We consider two methodsfor the choice of summation limits:

1. covariance method

2. autocorrelation method

Warning: terminology is not consistent.


211

5.2 LPC: Covariance Method

n0 N-1

known values predicted value

Covariance Method

• No window function is applied, such that we obtain the followingsummation limits:

∑n

e2(n) =N−1∑n=0

e2(n)

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∑

k=1

αk Φ(l, k) = Φ(l, 0)

with the definition:

Φ(l, k) :=N−1∑n=0

x[n− l] x[n− k]

For the above terms holds:

– 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(K3) + O(NK)

– autocorrelation method has more favorable complexity: O(K2)

– but: calculation of auto/cross-correlation function dominates

In contrast to covariance method, autocorrelation method offers an inter-pretation in the frequency domain and therefore is often preferred.


212

5.3 LPC: Autocorrelation Method

n0 N-1

windowfunction

We consider the signal after multiplication with a convenient window func-tion, usually Hamming window:

In principle, the summation limits now are

∑n

e2[n] =n=+∞∑n=−∞

e2[n] .

Since, due to windowing the signal x[n] is identical to zero outside thewindow 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 =N+K−1∑

n=0

e2[n]

The prediction error e[n] can become “large” on the window functionboundaries:

- Beginning: prediction from ”zeros”- End: prediction of ”zeros”


213

Inserting the summation limits:∑

n

x[n− k] x[n− l] = R(|l − k|),∑

n

x[n] x[n− l] = R(|l|)

where R(|l − k|) =N−1−l∑

n=0

x[n− k] x[n− l]

R(|l|) =∑

n

x[n] x[n− l] =N−1−l∑

n=0

x[n] x[n− l]

In this way we obtain the following equation system for the LPC–coefficientsαk:

l = 1, ..., K :K∑

k=1

αkR(|l − k|) = R(l)

or in matrix form:

R(0) R(1) . . . R(K − 1)

R(1) R(0) . . . R(K − 2)

......

. . . ...

... R(1)

R(K − 1) R(K − 2) . . . R(1) R(0)

α1

α2

...

αK

=

R(1)R(2)

...

R(K)


214

Note that this equation system is completely determined by the autocor-relation coefficients

R(0), ..., R(k), ..., R(K).

Hence, the autocorrelation coefficients will “only” be converted to obtainthe 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(K2)


215

5.4 LPC: Interpretation in Frequency Domain

The LPC autocorrelation method allows prediction error conversion fromtime domain into frequency domain using Parseval theorem so that LPCanalysis can be interpreted as adaptation of parametric model spectrum tothe observed signal spectrum.We start with the prediction error e[n]:

e[n] = x[n]−K∑

k=1

αk x[n− k]

and apply the z–transform to this equation. The z–transform is restrictedto the unit circle.

z = ejω ∈ CFor the z-transforms E(z) and X(z) we obtain:

E(z) = X(z) ·[1−

K∑

k=1

αkz−k

]

The total error Etot for the squared error criterion becomes:

Etot =N+K−1∑

n=0

e2[n]

=1

2π

+π∫

−π

|E(ejω)|2 dω (Parseval Theorem)

=1

2π

+π∫

−π

∣∣∣∣∣1−K∑

k=1

αk e−jωk

∣∣∣∣∣

2

· |X(ejω)|2 dω

=1

2π

+π∫

−π

∣∣P (ejω)∣∣2 · |X(ejω)|2 dω


216

with the so-called predictor polynom:

P (ejω) := 1−K∑

k=1

αk e−jωk

Squared absolute value of the predictor polynom

∣∣P (ejω)∣∣2 =

∣∣∣∣∣1−K∑

k=1

αk e−jωk

∣∣∣∣∣

2

= ...

=K∑

k=1

Bk · cos(ωk)

(with suitable coefficients Bk resulting from the predictor coefficients) is apolynom with respect to cos(ω), which can be obtained via application oftrigonometric transformations.

The predictor polynom tries to “compensate” for |X(ejω)|2 – especially atmaxima – 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 inthe complex plane and therefore can be factorised into linear factors:

P (z) =K∏

k=1

(z − zk)


217

Observations:

• These zeros are complex conjugated pairs because αk ∈ IR.

• The zeros can cause ”minima” of∣∣P (ejω)

∣∣2. The minima of |P (ejω)|2approximately correspond to the maxima of the smoothed spectrum|X(ejω)|2, because for minimization of the error integral it is first ofall necessary to “compensate” for the maxima of the signal spectrum.The LPC analysis could therefore be used to describe of the speechsignal formant structure.

|P(e )|2iω

ω

|X(e )|2iω


218

0

0


0

0

prediction error- 12 LPC-coefficients -

0

LPC-spectrum- 12 coefficients -

0

spectrum ofprediction error

(12 LPC-coefficients)

0


Figure 5.1: LPC–analysis of one speech segmenta) signal progression, b) prediction error (K=12), c) LPC–spectrum with K=12 coeffi-cients, d) spectrum of the prediction error (K=12), e) LPC–spectrum with K=18 coeffi-cients


219

0

0


0


0


0


0


0


0


0


Figure 5.2: LPC–Spectra for different prediction orders K


220

5.5 LPC: Generative Model

e(n) x(n)

recursivefilter

αk

For the prediction error e[n] and its z–transform holds:

e[n] = x[n]−K∑

k=1

αk x[n− k]

E(z) = X(z)−K∑

k=1

αk X(z) z−k

= X(z) · [1−K∑

k=1

αk z−k]

If we consider prediction error as input signal, we can also interpret theLPC–theorem as generative model which generates an output signal x[n]from an adequate “input signal” e[n]:

x[n] = e[n] +K∑

k=1

αk x[n− k] .

For the signal spectrum X(z) holds:

X(z) =E(z)

1−K∑

k=1αk z−k

This model is called autoregressive model. The excitation has to be chosensuch that E(z) is “white”, i.e. it does not have fine structure due to thefundamental frequency (”pitch–frequency”).In other words:

E(z) = G = const. (”gain”)


221

Special case:

E[n] = G · δ[n]

Then for LPC model spectrum X(z) holds:

X(z) =G

1−K∑

k=1αk z−k

This spectrum is often interpreted as LPC model spectrum X(z) of ob-served signal. It is reasonable to set (without explanation):

G2 = R(0)−K∑

k=1

αk R(k) = R(0)

[1−

K∑

k=1

αkR(k)

R(0)

]

This LPC model spectrum does not have any zeros, it has only poles, andtherefore 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 K = 105 kHz K = 126 kHz K = 14


222

5.6 LPC: Alternative Representations

• so far:

G gainαk 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 A4 A5

Glottis Lips


223

Chapter 6

Outlook: Wavelet Transform

• Overview:

6.1 Motivation: from Fourier to Wavelet Transform

6.2 Definition

6.3 Discrete Wavelet Transform


225

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 informa-tion 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 asegment 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) :=

+∞∫

−∞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 functionas the new basis function.

b) in addition to the localization parameter b, we also introduce a scalingparameter a > 0.

We consider the family of window functions Wab(t):

wab(t) := w( t− b

a

)


226

6.2 Definition

The following notation is usually used for the Wavelet–transform:

ψab(t) :=1√a· ψ

(t− b

a

)

which is the so-called Mother-Wavelet t → ψ(t).

Like the window function for the short–time Fourier transform the Mother-Wavelet should be “localized” as much as possible.

Example:

Mexican-Hat Function: ψ(t) = (1− t2)e−12 t2

t

ψ (t)


227

The wavelet–transform of f(t) with respect to ψ(t) is defined as:

Fψ(a, b) =1

a

+∞∫

−∞f(t) · ψ

( t− b

a

)dt

with scaling parameter a > 0 and localization parameter b ∈ IR.

For the inverse transformation holds:

f(t) =1

Cψ· 1

a2 ·+∞∫

0

da

+∞∫

−∞db Fψ(a, b) · ψ

( t− b

a

)

with

Cψ :=

∞∫

0

|ψ(ω)|2ω

dω < ∞

Proof (principle only):

The proof uses the (generalized) Parseval Theorem:

Fψ(a, b) =1√a

+∞∫

−∞f(t) · ψab(t) dt

=1√a

+∞∫

−∞F (ω) · ψab(ω) dω

with ψab(t) =1√a· ψ

(t− b

a

)

using further conversions.


228

6.3 Discrete Wavelet Transform

For the scaling parameters a > 0 we choose:

a = αm0 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 am0 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 ψmn(t) := a−m

20 · 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) isdefined as:

< f(t), g(t) > =

∫f(t) · g(t) dt


229

In this way we obtain the following representation for the discrete Wavelet–transform:

Fψ(m,n) =

+∞∫

−∞f(t)a

−m2

0 ψ(a−m0 t − nb0) dt

= < f(t), ψmn(t) >

Due to the orthogonality it is possible to convert the integral of the inverseWavelet–transform into an infinite series:

f(t) =1

Cψ

∑m

∑n

Fψ(m,n)ψmn(t)

=1

Cψ

∑m

∑n

Fψ(m,n) a−m

2

0 ψ(a−m0 t − nb0)


230

Example: Haar–function and Haar–basis

special choice: a0 = 2b0 = 1

The Haar–function is defined as:

ψ(t) =

1 0 5 t 5 12

−1 12 5 t < 1

0 otherwise

This defines the Haar–basis

ψ(t) |m,n ∈ Z,m > 0:

ψmn(t) =1√2m

· ψ(

2mt − n)

It is easy to see that for increasing m a increasingly finer resolution is ob-tained and that n determines localization in time.


231


232

Chapter 7

Coding

The following types of coding are distinguished:

• source coding (data compression)goal: transmission (storage) using as few bits as possible without orwith few errors

• channel codinggoal: preferably faultless data transmission (storage)e.g. error-recognizing and error-correcting codes

• simultaneous source and channel codinggoal: 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 codingusually continuous signalsnotation: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 qualitydeterioration.

signal

reconstructedsignal

T Q C

T Q C-1 -1 -1

trans-mission

T: transformation, e.g. DCT

Q: quantization, e.g. vector quantization

C: mapping of bit representation


234

References:

• Ze-Nian Li: CMPT 365 Multimedia Systems. Simon FraserUniversity, 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 ProcessingMagazine, pp.59-81, Sep. 1997.

• Thomas Sikora: MPEG Digital Video-Coding Standards. IEEE SignalProcessing Magazine, pp.82-100, Sep. 1997.

• A. Ortega, K. Ramchandran: Rate-Distortion Methods for Imageand Video Compression. IEEE Signal Processing Magazine, pp.23-50, Nov. 1998.

• G. J. Sullivan, Th. Wiegand: Rate-Distortion Optimization for VideoCompression. IEEE Signal Processing Magazine, pp.74-90, Nov. 1998.


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Chapter 8

Image Segmentation andContour-Finding

The lecture notes for this chapter are available as a separate document.

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