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<ul><li><p>Start of Citation[PU]McGraw Hill[/PU][DP]1999[/DP]End of Citation</p><p>Schaum's Outline of Theory and Problems of </p><p>Digital Signal Processing </p><p>Monson H. Hayes </p><p>Professor of Electrical and Computer Engineering Georgia Institute of Technology </p><p>SCHAUM'S OUTLINE SERIES </p></li><li><p>Start of Citation[PU]McGraw Hill[/PU][DP]1999[/DP]End of Citation</p><p>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 &amp; 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." </p><p>Schaum's Outline of Theory and Problems of DIGITAL SIGNAL PROCESSING </p><p>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. </p><p>2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 PRS PRS 9 0 2 10 9 </p><p>ISBN 0070273898 </p><p>Sponsoring Editor: Barbara Gilson Production Supervisor: Pamela Pelton Editing Supervisor: Maureen B. Walker </p><p>Library of Congress Cataloging-in-Publication Data </p><p>Hayes, M. H. (Monson H.), date. </p><p>Schaum's outline of theory and problems of digital signal </p><p>processing / Monson H. Hayes. </p><p>p. cm. (Schaum's outline series) </p><p>Includes index. </p><p>ISBN 0070273898 </p><p>1. Signal processingDigital techniquesProblems, exercises, </p><p>etc. 2. Signal processingDigital techniquesOutlines, syllabi, </p><p>etc. I. Title. II. Title: Theory and problems of digital signal </p><p>processing. </p><p>TK5102.H39 1999 </p><p>621.382'2dc21 9843324 </p><p>CIP </p></li><li><p>Start of Citation[PU]McGraw Hill[/PU][DP]1999[/DP]End of Citation</p><p>For Sandy </p></li><li><p>Preface </p><p>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. </p><p>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. </p><p>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. </p><p>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 </p></li><li><p>Start of Citation[PU]McGraw Hill[/PU][DP]1999[/DP]End of Citation</p><p>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. </p><p>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 </p><p>http://www.ee.gatech.edu/users/mhayes/schaum </p><p>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. </p></li><li><p>Contents </p><p>Chapter 1. Signals and Systems 1 </p><p>1.1 Introduction 1 </p><p>1.2 Discrete-Time Signals 1 </p><p>1.2.1 Complex Sequences 2 </p><p>1.2.2 Some Fundamental Sequences 2 </p><p>1.2.3 Signal Duration 3 </p><p>1.2.4 Periodic and Aperiodic Sequences 3 </p><p>1.2.5 Symmetric Sequences 4 </p><p>1.2.6 Signal Manipulations 4 </p><p>1.2.7 Signal Decomposition 6 </p><p>1.3 Discrete-Time Systems 7 </p><p>1.3.1 Systems Properties 7 </p><p>1.4 Convolution 11 </p><p>1.4.1 Convolution Properties 11 </p><p>1.4.2 Performing Convolutions 12 </p><p>1.5 Difference Equations 15 </p><p>Solved Problems 18 </p><p>Chapter 2. Fourier Analysis 55 </p><p>2.1 Introduction 55 </p><p>2.2 Frequency Response 55 </p><p>2.3 Filters 58 </p><p>2.4 Interconnection of Systems 59 </p><p>2.5 The Discrete-Time Fourier Transform 61 </p><p>2.6 DTFT Properties 62 </p><p>2.7 Applications 64 </p><p>2.7.1 LSI Systems and LCCDEs 64 </p><p>2.7.2 Performing Convolutions 65 </p><p>2.7.3 Solving Difference Equations 66 </p><p>2.7.4 Inverse Systems 66 </p><p>Solved Problems 67 </p><p>Chapter 3. Sampling 101 </p><p>3.1 Introduction 101 </p><p>3.2 Analog-to-Digital Conversion 101 </p><p>3.2.1 Periodic Sampling 101 </p><p>3.2.2 Quantization and Encoding 104 </p><p>3.3 Digital-to-Analog Conversion 106 </p><p>3.4 Discrete-Time Processing of Analog Signals 108 </p><p>3.5 Sample Rate Conversion 110 </p><p>3.5.1 Sample Rate Reduction by an Integer Factor 110 </p><p>3.5.2 Sample Rate Increase by an Integer Factor 111 </p><p>3.5.3 Sample Rate Conversion by a Rational Factor 113 </p><p>Solved Problems 114 </p><p>Chapter 4. The Z-Transform 142 </p><p>4.1 Introduction 142 </p><p>4.2 Definition of the z-Transform 142 </p><p>4.3 Properties 146 </p><p>vii </p></li><li><p>4.4 The Inverse z-Transform 149 </p><p>4.4.1 Partial Fraction Expansion 149 </p><p>4.4.2 Power Series 150 </p><p>4.4.3 Contour Integration 151 </p><p>4.5 The One-Sided z-Transform 151 </p><p>Solved Problems 152 </p><p>Chapter 5. Transform Analysis of Systems 183 </p><p>5.1 Introduction 183 </p><p>5.2 System Function 183 </p><p>5.2.1 Stability and Causality 184 </p><p>5.2.2 Inverse Systems 186 </p><p>5.2.3 Unit Sample Response for Rational System Functions 187 </p><p>5.2.4 Frequency Response for Rational System Functions 188 </p><p>5.3 Systems with Linear Phase 189 </p><p>5.4 Allpass Filters 193 </p><p>5.5 Minimum Phase Systems 194 </p><p>5.6 Feedback Systems 195 </p><p>Solved Problems 196 </p><p>Chapter 6. The DFT 223 </p><p>6.1 Introduction 223 </p><p>6.2 Discrete Fourier Series 223 </p><p>6.3 Discrete Fourier Transform 226 </p><p>6.4 DFT Properties 227 </p><p>6.5 Sampling the DTFT 231 </p><p>6.6 Linear Convolution Using the DFT 232 </p><p>Solved Problems 235 </p><p>Chapter 7. The Fast Fourier Transform 262 </p><p>7.1 Introduction 262 </p><p>7.2 Radix-2 FFT Algorithms 262 </p><p>7.2.1 Decimation-in-Time FFT 262 </p><p>7.2.2 Decimation-in-Frequency FFT 266 </p><p>7.3 FFT Algorithms for Composite N 267 </p><p>7.4 Prime Factor FFT 271 </p><p>Solved Problems 273 </p><p>Chapter 8. Implementation of Discrete-Time Systems 287 </p><p>8.1 Introduction 287 </p><p>8.2 Digital Networks 287 </p><p>8.3 Structures for FIR Systems 289 </p><p>8.3.1 Direct Form 289 </p><p>8.3.2 Cascade Form 289 </p><p>8.3.3 Linear Phase Filters 289 </p><p>8.3.4 Frequency Sampling 291 </p><p>8.4 Structures for IIR Systems 291 </p><p>8.4.1 Direct Form 292 </p><p>8.4.2 Cascade Form 294 </p><p>8.4.3 Parallel Structure 295 </p><p>8.4.4 Transposed Structures 296 </p><p>8.4.5 Allpass Filters 296 </p><p>8.5 Lattice Filters 298 </p><p>8.5.1 FIR Lattice Filters 298 </p><p>8.5.2 All-Pole Lattice Filters 300 </p><p>viii </p></li><li><p>8.5.3 IIR Lattice Filters 301 </p><p>8.6 Finite Word-Length Effects 302 </p><p>8.6.1 Binary Representation of Numbers 302 </p><p>8.6.2 Quantization of Filter Coefficients 304 </p><p>8.6.3 Round-Off Noise 306 </p><p>8.6.4 Pairing and Ordering 309 </p><p>8.6.5 Overflow 309 </p><p>Solved Problems 310 </p><p>Chapter 9. Filter Design 358 </p><p>9.1 Introduction 358 </p><p>9.2 Filter Specifications 358 </p><p>9.3 FIR Filter Design 359 </p><p>9.3.1 Linear Phase FIR Design Using Windows 359 </p><p>9.3.2 Frequency Sampling Filter Design 363 </p><p>9.3.3 Equiripple Linear Phase Filters 363 </p><p>9.4 IIR Filter Design 366 </p><p>9.4.1 Analog Low-Pass Filter Prototypes 367 </p><p>9.4.2 Design of IIR Filters from Analog Filters 373 </p><p>9.4.3 Frequency Transformations 376 </p><p>9.5 Filter Design Based on a Least Squares Approach 376 </p><p>9.5.1 Pade Approximation 377 </p><p>9.5.2 Prony's Method 378 </p><p>9.5.3 FIR Least-Squares Inverse 379 </p><p>Solved Problems 380 </p><p>Index 429 </p><p>ix </p></li><li><p>Chapter 1 Signals and Systems </p><p>1.1 INTRODUCTION In this chapter we begin our study of digital signal processing by developing the notion of a discrete-time signal and a discrete-time system. We will concentrate on solving problems related to signal representations, signal manipulations, properties of signals, system classification, and system properties. First, in Sec. 1.2 we define precisely what is meant by a discrete-time signal and then develop some basic, yet important, operations that may be performed on these signals. Then, in Sec. 1.3 we consider discrete-time systems. Of special importance will be the notions of linearity, shift-invariance, causality, stability, and invertibility. It will be shown that for systems that are linear and shift-invariant, the input and output are related by a convolution sum. Properties of the convolution sum and methods for performing convolutions are then discussed in Sec. 1.4. Finally, in Sec. 1.5 we look at discrete-time systems that are described in terms of a difference equation. </p><p>1.2 DISCRETE-TIME SIGNALS A discrete-time signal is an indexed sequence of real or complex numbers. Thus, a discrete-time signal is a function of an integer-valued variable, n, that is denoted by x(n). Although the independent variable n need not necessarily represent "time" (n may, for example, correspond to a spatial coordinate or distance), x(n) is generally referred to as a function of time. A discrete-time signal is undefined for noninteger values of n. Therefore, a real-valued signal x(n) will be represented graphically in the form of a lollipop plot as shown in Fig. 1-I. In </p><p>A Fig. 1-1. The graphical representation of a discrete-time signal x(n) . </p><p>some problems and applications it is convenient to view x(n) as a vector. Thus, the sequence values x(0) to x(N - 1) may often be considered to be the elements of a column vector as follows: </p><p>Discrete-time signals are often derived by sampling a continuous-time signal, such as speech, with an analog- to-digital (AID) converter.' For example, a continuous-time signal x,(t) that is sampled at a rate of fs = l/Ts samples per second produces the sampled signal x(n), which is related to xa(t) as follows: </p><p>Not all discrete-time signals, however, are obtained in this manner. Some signals may be considered to be naturally occurring discrete-time sequences because there is no physical analog-to-digital converter that is converting an </p><p>Analog-to-digital conversion will be discussed in Chap. 3. </p><p>1 </p></li><li><p>2 SIGNALS AND SYSTEMS [CHAP. 1 </p><p>analog signal into a discrete-time signal. Examples of signals that fall into this category include daily stock market prices, population statistics, warehouse inventories, and the Wolfer sunspot number^.^ </p><p>1.2.1 Complex Sequences In general, a discrete-time signal may be complex-valued. In fact, in a number of important applications such as digital communications, complex signals arise naturally. A complex signal may be expressed either in terms of its real and imaginary parts, </p><p>or in polar form in terms of its magnitude and phase, </p><p>The magnitude may be derived from the real and imaginary parts as follows: </p><p>whereas the phase may be found using </p><p>ImMn))arg{z(n)) = tan-' -</p><p>Re(z(n)) If z(n) is a complex sequence, the complex conjugate, denoted by z*(n), is formed by changing the sign on the imaginary part of z(n): </p><p>1.2.2 Some Fundamental Sequences Although most information-bearing signals of practical interest are complicated functions of time, there are three simple, yet important, discrete-time signals that are frequently used in the representation and description of more complicated signals. These are the unit sample, the unit step, and the exponential. The unit sample, denoted by S(n), is defined by </p><p>1 n = OS(n) = 0 otherwise </p><p>and plays the same role in discrete-time signal pr...</p></li></ul>