Digital Signal Processing

Digital Signal Processing Theory, Analysis and Digital-filter Design B. Somanathan Nair

DIGITAL SIGNAL PROCESSING Theory, Analysis and Digital-filter Design B. SOMANATHAN NAIR Principal SHM Engineering College Kadakkal Kerala Formerly Director Centre for Continuing Education Kerala Delhi-110092 2013

DIGITAL SIGNAL PROCESSING: Theory, Analysis and Digital-filter Design B. Somanathan Nair 2004 by PHI Learning Private Limited, Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-2595-1 The export rights of this book are vested solely with the publisher. Eighth Printing º º º January, 2013 Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj Industrial Estate, Delhi-110092 and Printed by Rajkamal Electric Press, Plot No. 2, Phase IV, HSIDC, Kundli-131028, Sonepat, Haryana.

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Contents Preface 1 xiii BASIC PRINCIPLES OF SIGNAL PROCESSING 1 13 1.1 Introduction 1 1.2 Principles of Analog Signal Processing 2 1.3 Principles of Optical Signal Processing 5 1.3.1 Basic Principles of OSP 5 1.3.2 Explanation of the Observed Spectrum 8 1.3.3 Principles of an Optical Signal Processor 10 1.3.4 Optical Signal Processor (or Optical Computer) 11 Review Questions 12 2 BASIC PRINCIPLES OF DIGITAL SIGNAL PROCESSING 14 34 2.1 Introduction 14 2.2 Why Digital 15 2.3 Building Blocks of a Digital Signal Processor 15 2.4 Comparison between OSP and DSP 16 2.5 Analog-to-digital Conversion 17 2.5.1 The Sampling Process 17 2.5.2 Sampling and Holding 18 2.5.3 The Nyquist Sampling Theorem 19 2.5.4 Digital-to-analog Converters 23 2.5.5 Analog-to-digital Converters 25 2.5.6 The Counter-ramp-type ADC 27 2.5.7 The Successive-approximation A/D Converter (SAADC) 30 2.5.8 The Flash Converter 31 2.6 The z-transformer and the Discrete Fourier-transformer 33 Review Questions 34 Problems 34 v

vi Contents 3 SIGNALS AND FUNCTIONS ENCOUNTERED IN SIGNAL PROCESSING 35 62 3.1 Introduction 35 3.2 The Delta (Impulse) Function 35 3.3 The Unit-step Function 38 3.4 The Unit-ramp Function 40 3.5 The Parabolic and Exponential Functions 41 3.6 The Sinusoidal Function 42 3.7 Methods for Plotting Various Types of Signals 44 3.8 Other Classification of Functions 50 3.8.1 Even and Odd Functions 50 3.8.2 Periodic and Nonperiodic Functions 56 Review Questions 60 Problems 61 4 SYSTEMS AND THEIR PROPERTIES 63 99 4.1 Introduction 63 4.2 Properties of Systems 65 4.2.1 Linearity 65 4.2.2 Causality 71 4.2.3 Time-variance 73 4.2.4 Convolution 74 4.2.5 Stability 85 4.2.6 Memory 88 4.3 Worked-out Problems 89 4.3.1 Problems on Linearity 89 4.3.2 Problems on Causality 92 4.3.3 Problems on Time-variance 93 4.3.4 Problems on Stability 95 4.3.5 Problems on Memory 96 Review Questions 97 Problems 97 5 TRANSFORMATIONS 100 134 5.1 Introduction 100 5.2 The Fourier, the Laplace and Other Transformations 102 5.2.1 The Fourier Transformation 102 5.2.2 The Laplace Transformation 104 5.2.3 z-transforms and Discrete-time Fourier Transforms of Various Functions 105 5.2.4 Region of Convergence (ROC), Causality and Stability 107

5.3 Properties of z-transforms and Discrete-time Fourier Transforms 113 5.4 Inverse z-transformation (IZT) 128 Review Questions 131 Problems 131 Contents vii 6 DISCRETE FOURIER TRANSFORMATION 135 167 7 6.1 Introduction 135 6.2 Computation of DFT 136 6.3 Inverse Discrete Fourier Transformation (IDFT) 141 6.4 Periodicity and Symmetry Properties of DFT 142 6.4.1 Periodicity in DFTs 142 6.4.2 Reason for Choosing 0 and (N 1) as the Limits to Evaluate DFTs 143 6.4.3 Conjugate Symmetry of DFTs 143 6.5 Comparison between DTFT and DFT 145 6.6 Circular Convolution Property of DFT 149 6.6.1 Circular (Periodic) Shift 149 6.6.2 Circular or Periodic Convolution 150 6.7 Solving Convolutional Problems using Various Methods 151 6.8 Additional Properties of DFT 157 6.8.1 Linearity 157 6.8.2 Time-shift 158 6.9 Block Convolution 158 6.9.1 Overlap-and-add Method 159 6.9.2 Overlap-and-save Method 163 Review Questions 165 Problems 166 FAST-FOURIER TRANSFORMATION 168 211 7.1 Introduction 168 7.2 Decimation-in-time Radix-2 FFT Algorithm 169 7.3 Saving Obtained in the Decimation-in-time FFT Algorithm 175 7.4 Butterfly Diagrams 181 7.4.1 Further Decimations 183 7.4.2 In-place Computation 188 7.5 Decimation-in-frequency FFT Algorithm 191 7.5.1 Further Decimations in DIF Algorithm 194 7.5.2 More about Butterflies in DIF Algorithm 198 7.6 Composite-N FFT Algorithms 200 7.7 Radix-4 DIT FFT Algorithm 201

viii Contents 8 7.8 The Generalized Twiddle-factor Table for the N-point Radix-2 DIT FFT Algorithm 206 7.9 Evaluation of 16-point DFT using Radix-2 DIT Algorithm 207 Review Questions 211 Problems 211 BASIC PRINCIPLES OF FILTERS AND FILTERING 212 229 8.1 Introduction 212 8.2 Low-pass Filters 214 8.2.1 A Practical First-order RC Low-pass Filter Circuit 214 8.2.2 A Practical Second-order RLC Low-pass Filter Circuit 216 8.2.3 A Practical Second-order RC Low-pass Filter Circuit 217 8.3 High-pass Filters 218 8.4 Band-pass Filters 219 8.5 Band-reject Filters 220 8.6 Problems Associated with Passive Filters 221 8.7 Theory of Analog Filter Design 221 8.7.1 Introduction 221 8.7.2 Filters using Operational Amplifiers 224 8.7.3 Another Way of Representing the Structure of Analog Computers 227 Review Questions 228 Problems 229 9 STEP-BY-STEP DESIGN OF ANALOG FILTERS 230 256 9.1 Introduction 230 9.2 The Butterworth Approximation Function 231 9.3 Step-by-step Design of Butterworth Filters 232 9.3.1 Introduction 232 9.3.2 Design Procedure for Butterworth Filter 233 9.3.3 Transfer Function of a Fourth-order Butterworth Filter 246 9.3.4 Transfer Function of a Fifth-order Butterworth Filter 247 9.4 Chebyshev Approximation 247 9.4.1 Introduction 247 9.4.2 Step-by-step Design of Chebyshev Low-pass Filter 249 9.5 Comparison between Butterworth and Chebyshev Filters 252 Review Questions 254 Problems 254

Contents ix 10 STEP-BY-STEP DESIGN OF DIGITAL IIR FILTERS 257 295 11 10.1 Introduction 257 10.2 Theory of Infinite-impulse-response (IIR) Filters 258 10.3 Finding the Expression for H(s) in the Digital Domain 259 10.4 Step-by-step Design of Butterworth Digital IIR Filters 267 10.5 Chebyshev Digital IIR Filters 272 10.5.1 Introduction 272 10.5.2 Step-by-step Design of Type-I Chebyshev Digital IIR Filters 273 10.6 The Inverse (Type-II) Chebyshev IIR Filter Design 277 10.7 Digital Filter Design using Bilinear Transformation 282 10.7.1 Introduction 282 10.7.2 Frequency Warping and Prewarping 283 10.7.3 Design of Digital Filters using Bilinear Transformation 284 10.8 Frequency Transformation 287 10.8.1 Low-pass Frequency Transformation 288 10.8.2 High-pass Frequency Transformation 288 10.8.3 Band-pass Transformation 290 10.8.4 Band-reject (Band-elimination or Notch-filter) Transformation 291 10.9 Why the Name Infinite-impulse-response Filter 292 Review Questions 293 Problems 293 STRUCTURES FOR REALIZING DIGITAL IIR FILTERS 296 318 11.1 Introduction 296 11.2 Direct-form I Structure of IIR Filters 297 11.3 Direct-form II (Canonical-form) Structure for IIR Filters 301 11.4 Transposed-form Structure for IIR Filters 305 11.5 Parallel Structure for IIR Filters 306 11.6 Cascade-form Structures for IIR Filters 309 11.7 Signal-flow Graphs in Filter-structure Realization 312 11.7.1 Introduction 312 11.7.2 Transposed-form Structure of Digital Filters 314 11.8 Lattice Structure of IIR Filters 315 Review Questions 317 Problems 317

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