Table Of ContentDigital and Statistical Signal
Processing
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Digital and Statistical Signal
Processing
Anastasia Veloni
Nikolaos I. Miridakis
Erysso Boukouvala
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Library of Congress Cataloging‑in‑Publication Data
Names: Veloni, Anastasia, author. | Miridakis, Nikolaos, author. |
Boukouvala, Erysso, author.
Title: Digital and statistical signal processing / Anastasia Veloni, Nikolaos
Miridakis, and Erysso Boukouvala.
Description: Boca Raton : Taylor & Francis, a CRC title, part of the Taylor &
Francis imprint, a member of the Taylor & Francis Group, the academic
division of T&F Informa, plc, 2018. | Includes bibliographical references.
Identifiers: LCCN 2018027136| ISBN 9781138580060 (hardback : acid-free paper)
| ISBN 9780429507526 (ebook)
Subjects: LCSH: Signal processing--Digital techniques.
Classification: LCC TK5102.9 .V45 2018 | DDC 621.382/2--dc23
LC record available at https://lccn.loc.gov/2018027136
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Professor Anastasia Veloni dedicates this book to her husband, Giannis.
Dr. Nikolaos I. Miridakis dedicates this book to his daughter, Ioanna,
his wife, Meni, and his parents, Ioannis and Panayota.
Dr. Erysso Boukouvala dedicates this book to George,
Dimitris, and her parents, Dimitris and Stella.
http://taylorandfrancis.com
Contents
Preface ...........................................................................................................................................xiii
Authors ...........................................................................................................................................xv
List of Acronyms ........................................................................................................................xvii
Section I Topics on Digital Signal Processing
1. Introduction.............................................................................................................................3
1.1 Introduction ...................................................................................................................3
1.2 Advantages of Digital Signal Processing ...................................................................4
1.3 Digitization Steps of Analog Signals .........................................................................5
1.3.1 Sampling ............................................................................................................5
1.3.2 Quantization .....................................................................................................7
1.3.3 Coding................................................................................................................9
1.4 Sampling and Reconstruction of Sinusoidal Signals .............................................10
1.4.1 Proof of the Sampling Theorem and a Detailed Discussion ....................12
1.5 Physical Sampling .......................................................................................................17
1.6 Sampling and Holding ...............................................................................................20
1.7 Non-Accurate Reconstruction of Analog Signals...................................................21
1.8 Solved Problems ..........................................................................................................22
2. Discrete-Time Signals and Systems ..................................................................................45
2.1 Discrete-Time Signals .................................................................................................45
2.2 Basic Discrete-Time Signals .......................................................................................45
2.2.1 Impulse Function ...........................................................................................45
2.2.2 Unit Step Function .........................................................................................47
2.2.3 Ramp Function ...............................................................................................47
2.2.4 Unit Rectangular Function (Pulse Function) .............................................48
2.2.5 Exponential Function .....................................................................................48
2.2.6 The Sinusoidal Sequence ...............................................................................50
2.3 Even and Odd Discrete-Time Signals ......................................................................51
2.4 Energy and Power of a Discrete-Time Signal..........................................................53
2.5 Conversion of the Independent and Dependant Variable.....................................54
2.6 Discrete-Time Systems ...............................................................................................54
2.7 Categories of Discrete-Time Systems .......................................................................55
2.7.1 Linear Discrete Systems ................................................................................55
2.7.2 Time-Invariant Discrete Systems .................................................................56
2.7.3 Discrete Systems with Memory ....................................................................57
2.7.4 Invertible Discrete Systems ...........................................................................57
2.7.5 Casual Discrete Systems ................................................................................57
2.7.6 Stable Discrete Systems .................................................................................58
2.8 System Connections ....................................................................................................58
2.9 Convolution ..................................................................................................................59
2.10 Deconvolution ..............................................................................................................62
vii
viii Contents
2.11 Correlation — Autocorrelation ..................................................................................63
2.12 Difference Equations ..................................................................................................64
2.13 Discrete-Time Systems of Finite Impulse Response ...............................................65
2.14 Solved Problems ..........................................................................................................66
3. z-Transform ..........................................................................................................................113
3.1 Introduction ...............................................................................................................113
3.2 From Laplace Transform to z-Transform ................................................................113
3.2.1 Comparison of the s- and z-Planes into the Region of Convergence .....116
3.3 Properties of z-Transform .........................................................................................117
3.3.1 Time Shift ......................................................................................................117
3.3.2 Linearity.........................................................................................................118
3.3.3 Time Reversal ................................................................................................118
3.3.4 Convolution ...................................................................................................119
3.3.5 Differentiation in z-Plane ............................................................................119
3.3.6 Multiplication by an Exponential Sequence .............................................120
3.3.7 Conjugation of a Complex Sequence .........................................................120
3.3.8 Initial and Final Value Theorem ................................................................121
3.3.9 Correlation of Two Sequences ....................................................................121
3.4 Inverse z-Transform ..................................................................................................121
3.4.1 Method of Power Series Expansion (Division Method) .........................122
3.4.2 Method of Partial Fraction Expansion ......................................................122
3.4.3 Method of Complex Integration .................................................................123
3.5 z-Transform in System Analysis..............................................................................124
3.5.1 Transfer Function of Discrete-Time Signal ..............................................124
3.5.2 Causality of Discrete-Time Systems ..........................................................124
3.5.3 Stability of Discrete-Time Systems ............................................................125
3.5.4 Transfer Function of Connected Systems .................................................126
3.5.5 Transfer Function of Discrete-Time Systems ...........................................127
3.6 Formula Tables ...........................................................................................................129
3.7 Solved Problems ........................................................................................................130
4. Structures for the Realization of Discrete-Time Systems ..........................................179
4.1 Introduction ...............................................................................................................179
4.2 Block Diagrams .........................................................................................................179
4.3 Realization Structures ..............................................................................................181
4.3.1 Implementation Structures of IIR Discrete Systems ...............................183
4.3.2 Implementation Structures of FIR Discrete Systems ..............................186
4.4 Signal Flow Graphs ...................................................................................................188
4.4.1 Mason’s Gain Formula .................................................................................189
4.5 Solved Problems ........................................................................................................190
5. Frequency Domain Analysis ............................................................................................211
5.1 Introduction ...............................................................................................................211
5.2 Discrete-Time Fourier Transform (DTFT) ..............................................................212
5.3 Discrete Fourier Series (DFS) ...................................................................................214
5.3.1 Periodic Convolution ...................................................................................215
5.3.2 The Relation of the DFS Components and the DTFT over a Period .....216
5.4 Discrete Fourier Transform ......................................................................................216
Contents ix
5.4.1 Properties of the DFT ...................................................................................218
5.4.1.1 Linearity .........................................................................................218
5.4.1.2 Circular Shift.................................................................................218
5.4.1.3 Circular Convolution ...................................................................219
5.4.1.4 Multiplication of Sequences ........................................................220
5.4.1.5 Parseval’s Theorem .......................................................................220
5.5 Fast Fourier Transform .............................................................................................221
5.5.1 FFT Equations ...............................................................................................221
5.5.2 Computation of the IDFT Using FFT .........................................................228
5.5.3 Fast Convolution ..........................................................................................228
5.5.3.1 Overlap and Add Method ...........................................................228
5.5.3.2 Overlap and Save Method ...........................................................229
5.6 Estimation of Fourier Transform through FFT .....................................................229
5.7 Discrete Cosine Transform ......................................................................................229
5.8 Wavelet Transform ....................................................................................................231
5.8.1 Wavelet Transform Theory .........................................................................233
5.9 Solved Problems ........................................................................................................236
6. Design of Digital Filters ....................................................................................................287
6.1 Introduction ...............................................................................................................287
6.2 Types of Digital Filters ..............................................................................................288
6.3 Digital Filter Design Specifications .........................................................................288
6.4 Design of Digital IIR Filters .....................................................................................290
6.5 Indirect Methods of IIR Filter Design ....................................................................292
6.5.1 The Impulse Invariant Method ..................................................................292
6.5.2 Step Invariant Method (or z-Transform Method with Sample
and Hold) .......................................................................................................293
6.5.3 Backward Difference Method ....................................................................295
6.5.4 Forward Difference Method .......................................................................296
6.5.5 Bilinear or Tustin Method ...........................................................................297
6.5.6 Matched Pole-Zero Method ........................................................................299
6.6 Direct Methods of IIR Filter Design .......................................................................300
6.6.1 Design of H(ejω)2 Method ........................................................................300
6.6.2 The Method of Calculating h[n] .................................................................301
6.7 IIR Filter Frequency Transformations ....................................................................301
6.8 FIR Filters ....................................................................................................................303
6.9 FIR Linear Phase Filters ............................................................................................304
6.10 Stability of FIR Filters ...............................................................................................307
6.11 Design of FIR Filters ..................................................................................................307
6.12 The Moving Average Filters ....................................................................................307
6.13 FIR Filter Design Using the Frequency Sampling Method .................................309
6.14 FIR Filter Design Using the Window Method .......................................................311
6.15 Optimal Equiripple FIR Filter Design ....................................................................317
6.16 Comparison of the FIR Filter Design Methods .....................................................319
6.17 Solved Problems ........................................................................................................320
