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Alan V. Oppenheim/ Ronald W. Schafer DIGITAL SIGNAL PROCESSING Alan V. Oppenheim Ronald W. Schafer Department of Electrical Engineering School of Electrical Engineering Massachusetts Institute of Technology Georgia Institute of Technology (Formerly with Bell Laboratories) PRENTICE-HALL, INC., Englewood Cliffs, New Jersey. Library of Congress Cataloging in Publication Data Oppenheim, Alan V. Digital Signal Processing. Includes bibliographical references. 1. Signal theory (Telecommunication) 2. Digital electronics. I. Schafer, Ronald W. joint author. II. Title. TK5102.5.0245 621.3819'58'2 74-17280 ISBN 0-13-214635-5 © 1975 by Alan V. Oppenheim and Bell Telephone Laboratories, Inc. All rights reserved. No part of this book may be reproduced in any form or by any means without permission in writing from the publisher. 18 Printed in the United States of America Prentice-Hall International, Inc., London Prentice-Hall of Australia, Pty. Ltd., Sydney Prentice-Hall of Canada, Ltd., Toronto Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo To Phyllis and Dorothy Contents PREFACE XI* INTRODUCTION 1 1 DISCRETE-TIME SIGNALS AND SYSTEMS 6 1.0 Introduction 6 1.1 Discrete-Time Signals-Sequences 8 1.2 Linear Shift-Invariant Systems 11 1.3 Stability and Causality 13 1.4 Linear Constant-Coefficient Difference Equations 16 1.5 Frequency-Domain Representation of Discrete-Time Systems and Signals 18 1.6 Some Symmetry Properties of the Fourier Transform 24 1.7 Sampling of Continuous-Time Signals 26 1.8 Two-Dimensional Sequences and Systems 30 Summary 34 Problems 35 2 THE Z-TRANSFORM 45 2.0 Introduction 45 2.1 z-Transform 45 2.2 Inverse z-Transform 52 2.3 z-Transform Theorems and Properties 58 2.4 System Function 67 2.5 Two-Dimensional Z-Transform 73 Summary 77 Problems 78 viii Contents 3 THE DISCRETE FOURIER TRANSFORM 3.0 Introduction 87 3.1 Representation of Periodic Sequences-The Discrete Fourier Series 3.2 Properties of the Discrete Fourier Series 91 3.3 Summary of Properties of the DFS Representation of Periodic Sequences 95 3.4 Sampling the z-Transform 96 3.5 Fourier Representation of Finite-Duration Sequences - The Discrete Fourier Transform 99 3.6 Properties of the Discrete Fourier Transform 101 3.7 Summary of Properties of the Discrete Fourier Transform 110 3.8 Linear Convolution Using the Discrete Fourier Transform 110 3.9 Two-Dimensional Discrete Fourier Transform 115 Summary 121 Problems 121 4 FLOW GRAPH AND MATRIX REPRESENTA¬ TION OF DIGITAL FILTERS 4.0 Introduction 136 4.1 Signal Flow Graph Representation of Digital Networks 137 4.2 Matrix Representation of Digital Networks 143 4.3 Basic Network Structures for HR Systems 148 4.4 Transposed Forms 153 4.5 Basic Network Structures for FIR Systems 155 4.6 Parameter Quantization Effects 165 4.7 Tellegen’s Theorem for Digital Filters and Its Applications 173 Summary 181 Problems 182 5 DIGITAL FILTER DESIGN TECHNIQUES 5.0 Introduction 195 5.1 Design of HR Digital Filters from Analog Filters 197 5.2 Design Examples: Analog-Digital Transformation 211 5.3 Computer-Aided Design of HR Digital Filters 230 5.4 Properties of FIR Digital Filters 237 5.5 Design of FIR Filters Using Windows 239 5.6 Computer-Aided Design of FIR Filters 250 5.7 A Comparison of HR and FIR Digital Filters 268 Summary 269 Problems 271 Contents ix 6 COMPUTATION OF THE DISCRETE FOURIER TRANSFORM 284 6.0 Introduction 284 6.1 Goertzel Algorithm 287 6.2 Decimation-in-Time FFT Algorithms 291 6.3 Decimation-in-Frequency FFT Algorithms 302 6.4 FFT Algorithms for N a Composite Number 307 6.5 General Computational Considerations in FFT Algorithms 315 6.6 Chirp Z-Transform Algorithm 321 Summary 326 Problems 328 7 DISCRETE HILBERT TRANSFORMS 337 7.0 Introduction 337 7.1 Real- and Imaginary-part Sufficiency for Causal Sequences 339 7.2 Minimum-Phase Condition 345 7.3 Hilbert Transform Relations for the DFT 353 7.4 Hilbert Transform Relations for Complex Sequences 358 Summary 365 Problems 367 8 DISCRETE RANDOM SIGNALS 376 8.0 Introduction 376 8.1 A Discrete-Time Random Process 377 8.2 Averages 382 8.3 Spectrum Representations of Infinite-Energy Signals 388 8.4 Response of Linear Systems to Random Signals 391 Summary 395 Problems 395 9 EFFECTS OF FINITE REGISTER LENGTH IN DIGITAL SIGNAL PROCESSING 404 9.0 Introduction 404 9.1 Effect of Number Representation on Quantization 406 9.2 Quantization in Sampling Analog Signals 413 9.3 Finite-Register-Length Effects in Realizations of IIR Digital Filters 418 9.4 Finite-Register-Length Effects in Realizations of FIR Digital Filters 438 9.5 Effects of Finite Register Length in Discrete Fourier Transform Computations 444 Summary 462 Problems 464 x Contents 10 HOMOMORPHIC SIGNAL PROCESSING 480 10.0 Introduction 480 10.1 Generalized Superposition 481 10.2 Multiplicative Homomorphic System 484 10.3 Homomorphic Image Processing 487 10.4 Homomorphic Systems for Convolution 490 10.5 Properties of the Complex Cepstrum 500 10.6 Computational Realizations of the Characteristic System D* 507 10.7 Applications of Homomorphic Deconvolution 511 Summary 527 - Problems 529 11 POWER SPECTRUM ESTIMATION 532 11.0 Introduction 532 11.1 Basic Principles of Estimation Theory 533 11.2 Estimates of the Autocovariance 539 11.3 The Periodogram as an Estimate of the Power Spectrum 541 11.4 Smoothed Spectrum Estimators 548 11.5 Estimates of the Cross Covariance and Cross Spectrum 554 11.6 Application of the FFT in Spectrum Estimation 555 11.7 Example of Spectrum Estimation 562 Summary 571 Problems 571 INDEX 577 Preface This book has grown out of our teaching and research activities in the field of digital-signal processing. It is designed, primarily, as a text for a senior or first-year graduate level course. The notes on which this book is based have been used for a one-semester introductory course in the M.I.T. Department of Electrical Engineering as well as for a continuing education course at Bell Laboratories. The notes were also used for three years in a condensed two- week summer course at M.I.T. and, in latter stages of development, at a number of universities around the country for one-semester courses. A typical one-semester course would cover Chapters 1, 2, and 3, in depth, and selected fundamental topics from Chapters 4, 5, 6, and 7. The remainder of the text, in conjunction with supplemental reading, forms the basis for a second-semester course on advanced topics and applications. Essential to learning a subject of this nature is the actual practice in working out new results and the application of the results to the solutions of real problems. Therefore, an important part of the text is a collection of approximately 250 homework problems. These problems are designed to extend results developed in the text, to develop some results that were referred to in the text, and to illustrate applications to practical problems. Solutions to the problems are available to instructors through the publisher. A self- study course consisting of a set of video-tape lectures and a study guide have been developed to accompany this text. Further information about the lecture tapes and study guide may be obtained through the M.I.T. Center for Advanced Engineering Study.