Table Of ContentSchaum'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.
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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
1. Signal processing—Digital techniques—Problems, exercises,
etc. 2. 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
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 3 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 finite-
length 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 8 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 9 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.
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Contents
Chapter 1. Signals and Systems 1
1.1 Introduction 1
1.2 Discrete-Time Signals 1
1.2.1 Complex Sequences 2
1.2.2 Some Fundamental Sequences 2
1.2.3 Signal Duration 3
1.2.4 Periodic and Aperiodic Sequences 3
1.2.5 Symmetric Sequences 4
1.2.6 Signal Manipulations 4
1.2.7 Signal Decomposition 6
1.3 Discrete-Time Systems 7
1.3.1 Systems Properties 7
1.4 Convolution 11
1.4.1 Convolution Properties 11
1.4.2 Performing Convolutions 12
1.5 Difference Equations 15
Solved Problems 18
Chapter 2. Fourier Analysis 55
2.1 Introduction 55
2.2 Frequency Response 55
2.3 Filters 58
2.4 Interconnection of Systems 59
2.5 The Discrete-Time Fourier Transform 61
2.6 DTFT Properties 62
2.7 Applications 64
2.7.1 LSI Systems and LCCDEs 64
2.7.2 Performing Convolutions 65
2.7.3 Solving Difference Equations 66
2.7.4 Inverse Systems 66
Solved Problems 67
Chapter 3. Sampling 101
3.1 Introduction 101
3.2 Analog-to-Digital Conversion 101
3.2.1 Periodic Sampling 101
3.2.2 Quantization and Encoding 104
3.3 Digital-to-Analog Conversion 106
3.4 Discrete-Time Processing of Analog Signals 108
3.5 Sample Rate Conversion 110
3.5.1 Sample Rate Reduction by an Integer Factor 110
3.5.2 Sample Rate Increase by an Integer Factor 111
3.5.3 Sample Rate Conversion by a Rational Factor 113
Solved Problems 114
Chapter 4. The Z-Transform 142
4.1 Introduction 142
4.2 Definition of the z-Transform 142
4.3 Properties 146
vii
4.4 The Inverse z-Transform 149
4.4.1 Partial Fraction Expansion 149
4.4.2 Power Series 150
4.4.3 Contour Integration 151
4.5 The One-Sided z-Transform 151
Solved Problems 152
Chapter 5. Transform Analysis of Systems 183
5.1 Introduction 183
5.2 System Function 183
5.2.1 Stability and Causality 184
5.2.2 Inverse Systems 186
5.2.3 Unit Sample Response for Rational System Functions 187
5.2.4 Frequency Response for Rational System Functions 188
5.3 Systems with Linear Phase 189
5.4 Allpass Filters 193
5.5 Minimum Phase Systems 194
5.6 Feedback Systems 195
Solved Problems 196
Chapter 6. The DFT 223
6.1 Introduction 223
6.2 Discrete Fourier Series 223
6.3 Discrete Fourier Transform 226
6.4 DFT Properties 227
6.5 Sampling the DTFT 231
6.6 Linear Convolution Using the DFT 232
Solved Problems 235
Chapter 7. The Fast Fourier Transform 262
7.1 Introduction 262
7.2 Radix-2 FFT Algorithms 262
7.2.1 Decimation-in-Time FFT 262
7.2.2 Decimation-in-Frequency FFT 266
7.3 FFT Algorithms for Composite N 267
7.4 Prime Factor FFT 271
Solved Problems 273
Chapter 8. Implementation of Discrete-Time Systems 287
8.1 Introduction 287
8.2 Digital Networks 287
8.3 Structures for FIR Systems 289
8.3.1 Direct Form 289
8.3.2 Cascade Form 289
8.3.3 Linear Phase Filters 289
8.3.4 Frequency Sampling 291
8.4 Structures for IIR Systems 291
8.4.1 Direct Form 292
8.4.2 Cascade Form 294
8.4.3 Parallel Structure 295
8.4.4 Transposed Structures 296
8.4.5 Allpass Filters 296
8.5 Lattice Filters 298
8.5.1 FIR Lattice Filters 298
8.5.2 All-Pole Lattice Filters 300
viii
8.5.3 IIR Lattice Filters 301
8.6 Finite Word-Length Effects 302
8.6.1 Binary Representation of Numbers 302
8.6.2 Quantization of Filter Coefficients 304
8.6.3 Round-Off Noise 306
8.6.4 Pairing and Ordering 309
8.6.5 Overflow 309
Solved Problems 310
Chapter 9. Filter Design 358
9.1 Introduction 358
9.2 Filter Specifications 358
9.3 FIR Filter Design 359
9.3.1 Linear Phase FIR Design Using Windows 359
9.3.2 Frequency Sampling Filter Design 363
9.3.3 Equiripple Linear Phase Filters 363
9.4 IIR Filter Design 366
9.4.1 Analog Low-Pass Filter Prototypes 367
9.4.2 Design of IIR Filters from Analog Filters 373
9.4.3 Frequency Transformations 376
9.5 Filter Design Based on a Least Squares Approach 376
9.5.1 Pade Approximation 377
9.5.2 Prony's Method 378
9.5.3 FIR Least-Squares Inverse 379
Solved Problems 380
Index 429
ix
