Schaum's Outline of Theory and Problems of Digital Signal Processing Monson H Hayes Professor of Electrical and Computer Engineering Georgia Institute of Technology SCHAUM'S OUTLINE SERIES Start of Citation[PU]McGraw Hill[/PU][DP]1999[/DP]End of Citation MONSON H HAYES is a Professor of Electrical and Computer Engineering at the Georgia Institute of Technology in Atlanta, Georgia He received his B.A degree in Physics from the University of California, Berkeley, and his M.S.E.E and Sc.D degrees in Electrical Engineering and Computer Science from M.I.T His research interests are in digital signal processing with applications in image and video processing He has contributed more than 100 articles to journals and conference proceedings, and is the author of the textbook Statistical Digital Signal Processing and Modeling, John Wiley & Sons, 1996 He received the IEEE Senior Award for the author of a paper of exceptional merit from the ASSP Society of the IEEE in 1983, the Presidential Young Investigator Award in 1984, and was elected to the grade of Fellow of the IEEE in 1992 for his "contributions to signal modeling including the development of algorithms for signal restoration from Fourier transform phase or magnitude." Schaum's Outline of Theory and Problems of DIGITAL SIGNAL PROCESSING Copyright © 1999 by The McGraw-Hill Companies, Inc All rights reserved Printed in the United States of America Except as permitted under the Copyright Act of 1976, no part of this publication may be reproduced or distributed in any forms or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher 10 11 12 13 14 15 16 17 18 19 20 PRS PRS 10 ISBN 0–07–027389–8 Sponsoring Editor: Barbara Gilson Production Supervisor: Pamela Pelton Editing Supervisor: Maureen B Walker Library of Congress Cataloging-in-Publication Data Hayes, M H (Monson H.), date Schaum's outline of theory and problems of digital signal processing / Monson H Hayes p cm — (Schaum's outline series) Includes index ISBN 0–07–027389–8 Signal processing—Digital techniques—Problems, exercises, etc Signal processing—Digital techniques—Outlines, syllabi, etc I Title II Title: Theory and problems of digital signal processing TK5102.H39 1999 621.382'2—dc21 98–43324 CIP Start of Citation[PU]McGraw Hill[/PU][DP]1999[/DP]End of Citation For Sandy Preface Digital signal processing (DSP) is concerned with the representation of signals in digital form, and with the processing of these signals and the information that they carry Although DSP, as we know it today, began to flourish in the 1960's, some of the important and powerful processing techniques that are in use today may be traced back to numerical algorithms that were proposed and studied centuries ago Since the early 1970's, when the first DSP chips were introduced, the field of digital signal processing has evolved dramatically With a tremendously rapid increase in the speed of DSP processors, along with a corresponding increase in their sophistication and computational power, digital signal processing has become an integral part of many commercial products and applications, and is becoming a commonplace term This book is concerned with the fundamentals of digital signal processing, and there are two ways that the reader may use this book to learn about DSP First, it may be used as a supplement to any one of a number of excellent DSP textbooks by providing the reader with a rich source of worked problems and examples Alternatively, it may be used as a self-study guide to DSP, using the method of learning by example With either approach, this book has been written with the goal of providing the reader with a broad range of problems having different levels of difficulty In addition to problems that may be considered drill, the reader will find more challenging problems that require some creativity in their solution, as well as problems that explore practical applications such as computing the payments on a home mortgage When possible, a problem is worked in several different ways, or alternative methods of solution are suggested The nine chapters in this book cover what is typically considered to be the core material for an introductory course in DSP The first chapter introduces the basics of digital signal processing, and lays the foundation for the material in the following chapters The topics covered in this chapter include the description and characterization of discrete-type signals and systems, convolution, and linear constant coefficient difference equations The second chapter considers the represention of discrete-time signals in the frequency domain Specifically, we introduce the discrete-time Fourier transform (DTFT), develop a number of DTFT properties, and see how the DTFT may be used to solve difference equations and perform convolutions Chapter covers the important issues associated with sampling continuous-time signals Of primary importance in this chapter is the sampling theorem, and the notion of aliasing In Chapter 4, the z-transform is developed, which is the discrete-time equivalent of the Laplace transform for continuous-time signals Then, in Chapter 5, we look at the system function, which is the z-transform of the unit sample response of a linear shift-invariant system, and introduce a number of different types of systems, such as allpass, linear phase, and minimum phase filters, and feedback systems The next two chapters are concerned with the Discrete Fourier Transform (DFT) In Chapter 6, we introduce the DFT, and develop a number of DFT properties The key idea in this chapter is that multiplying the DFTs of two sequences corresponds to circular convolution in the time domain Then, in Chapter 7, we develop a number of efficient algorithms for computing the DFT of a finitelength sequence These algorithms are referred to, generically, as fast Fourier transforms (FFTs) Finally, the last two chapters consider the design and implementation of discrete-time systems In Chapter we look at different ways to implement a linear shift-invariant discrete-time system, and look at the sensitivity of these implementations to filter coefficient quantization In addition, we analyze the propagation of round-off noise in fixed-point implementations of these systems Then, in Chapter we look at techniques for designing FIR and IIR linear shiftinvariant filters Although the primary focus is on the design of low-pass filters, techniques for designing other frequency selective filters, such as high-pass, bandpass, and bandstop filters are also considered It is hoped that this book will be a valuable tool in learning DSP Feedback and comments are welcomed through the web site for this book, which may be found at http://www.ee.gatech.edu/users/mhayes/schaum Also available at this site will be important information, such as corrections or amplifications to problems in this book, additional reading and problems, and reader comments Start of Citation[PU]McGraw Hill[/PU][DP]1999[/DP]End of Citation Contents Chapter Signals and Systems 1.1 Introduction 1.2 Discrete-Time Signals 1.2.1 Complex Sequences 1.2.2 Some Fundamental Sequences 1.2.3 Signal Duration 1.2.4 Periodic and Aperiodic Sequences 1.2.5 Symmetric Sequences 1.2.6 Signal Manipulations 1.2.7 Signal Decomposition 1.3 Discrete-Time Systems 1.3.1 Systems Properties 1.4 Convolution 1.4.1 Convolution Properties 1.4.2 Performing Convolutions 1.5 Difference Equations Solved Problems 1 2 3 4 7 11 11 12 15 18 Chapter Fourier Analysis 2.1 Introduction 2.2 Frequency Response 2.3 Filters 2.4 Interconnection of Systems 2.5 The Discrete-Time Fourier Transform 2.6 DTFT Properties 2.7 Applications 2.7.1 LSI Systems and LCCDEs 2.7.2 Performing Convolutions 2.7.3 Solving Difference Equations 2.7.4 Inverse Systems Solved Problems 55 55 55 58 59 61 62 64 64 65 66 66 67 Chapter Sampling 3.1 Introduction 3.2 Analog-to-Digital Conversion 3.2.1 Periodic Sampling 3.2.2 Quantization and Encoding 3.3 Digital-to-Analog Conversion 3.4 Discrete-Time Processing of Analog Signals 3.5 Sample Rate Conversion 3.5.1 Sample Rate Reduction by an Integer Factor 3.5.2 Sample Rate Increase by an Integer Factor 3.5.3 Sample Rate Conversion by a Rational Factor Solved Problems 101 101 101 101 104 106 108 110 110 111 113 114 Chapter The Z-Transform 4.1 Introduction 4.2 Definition of the z-Transform 4.3 Properties 142 142 142 146 vii 4.4 The Inverse z-Transform 4.4.1 Partial Fraction Expansion 4.4.2 Power Series 4.4.3 Contour Integration 4.5 The One-Sided z-Transform Solved Problems 149 149 150 151 151 152 Chapter Transform Analysis of Systems 5.1 Introduction 5.2 System Function 5.2.1 Stability and Causality 5.2.2 Inverse Systems 5.2.3 Unit Sample Response for Rational System Functions 5.2.4 Frequency Response for Rational System Functions 5.3 Systems with Linear Phase 5.4 Allpass Filters 5.5 Minimum Phase Systems 5.6 Feedback Systems Solved Problems 183 183 183 184 186 187 188 189 193 194 195 196 Chapter The DFT 6.1 Introduction 6.2 Discrete Fourier Series 6.3 Discrete Fourier Transform 6.4 DFT Properties 6.5 Sampling the DTFT 6.6 Linear Convolution Using the DFT Solved Problems 223 223 223 226 227 231 232 235 Chapter The Fast Fourier Transform 7.1 Introduction 7.2 Radix-2 FFT Algorithms 7.2.1 Decimation-in-Time FFT 7.2.2 Decimation-in-Frequency FFT 7.3 FFT Algorithms for Composite N 7.4 Prime Factor FFT Solved Problems 262 262 262 262 266 267 271 273 Chapter Implementation of Discrete-Time Systems 8.1 Introduction 8.2 Digital Networks 8.3 Structures for FIR Systems 8.3.1 Direct Form 8.3.2 Cascade Form 8.3.3 Linear Phase Filters 8.3.4 Frequency Sampling 8.4 Structures for IIR Systems 8.4.1 Direct Form 8.4.2 Cascade Form 8.4.3 Parallel Structure 8.4.4 Transposed Structures 8.4.5 Allpass Filters 8.5 Lattice Filters 8.5.1 FIR Lattice Filters 8.5.2 All-Pole Lattice Filters 287 287 287 289 289 289 289 291 291 292 294 295 296 296 298 298 300 viii 8.5.3 IIR Lattice Filters 8.6 Finite Word-Length Effects 8.6.1 Binary Representation of Numbers 8.6.2 Quantization of Filter Coefficients 8.6.3 Round-Off Noise 8.6.4 Pairing and Ordering 8.6.5 Overflow Solved Problems 301 302 302 304 306 309 309 310 Chapter Filter Design 9.1 Introduction 9.2 Filter Specifications 9.3 FIR Filter Design 9.3.1 Linear Phase FIR Design Using Windows 9.3.2 Frequency Sampling Filter Design 9.3.3 Equiripple Linear Phase Filters 9.4 IIR Filter Design 9.4.1 Analog Low-Pass Filter Prototypes 9.4.2 Design of IIR Filters from Analog Filters 9.4.3 Frequency Transformations 9.5 Filter Design Based on a Least Squares Approach 9.5.1 Pade Approximation 9.5.2 Prony's Method 9.5.3 FIR Least-Squares Inverse Solved Problems 358 358 358 359 359 363 363 366 367 373 376 376 377 378 379 380 Index 429 ix ... digital signal processing / Monson H Hayes p cm — (Schaum's outline series) Includes index ISBN 0–07–027389–8 Signal processing? ? ?Digital techniques—Problems, exercises, etc Signal processing? ? ?Digital. .. problems of digital signal processing TK5102.H39 1999 621.382'2—dc21 98–43324 CIP Start of Citation[PU]McGraw Hill[/PU][DP]1999[/DP]End of Citation For Sandy Preface Digital signal processing. .. analog-to -digital converter that is converting an Analog-to -digital conversion will be discussed in Chap SIGNALS AND SYSTEMS [CHAP analog signal into a discrete-time signal Examples of signals