Discrete-Time
`Signal Processing
`
`Alan V. Oppenheim
`Ronald W. Schafer
`
`--
`
`-+-+
`
`PRENTICE HALL, Englewood Clifls, New Jersey 07632
`
`PETITIONERS EXHIBIT 1011
`Page 1 of 896
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`Library of Congress Cataloging-in-Publication Data
`
`Oppenheim, Alan V.
`Discrete-time signal processing I Alan V. Oppenheim, Ronald W.
`Schafer.
`p.
`cm.-(Prentice Hall signal processing series)
`Bibliography: p.
`Includes index.
`rsBN 0-13-216292-X
`1. Signal processing-Math€matics. 2. Discrete-time systems.
`L Schafer, Ronald W. II. Title. III. Series.
`TK5102.5.024s2 1989
`621.38'043 dc 19
`
`88-25562
`CIP
`
`Editorial/production supervision: Barbara G. Flanagan
`Interior design: Roger Brower
`Cover design: Vivian Berman
`Manufacturing buyer: Mary Noonan
`
`@) 1989 by Prentice-Hall, Inc.
`A Division of Simon & Schuster
`Englewood Cliffs, New Jersey 07632
`
`=
`
`All rights reserved. No part of this book may be
`reproduced, in any form or by any means,
`without permission in witing from the publisher.
`
`Printed in the United States of America
`1098765432
`
`ISBN 0-1,3-El,bele-x
`
`Prentice-Hall International (UK) Limited, Lonilon
`Prentice-Hall of Australia Pty. Limited, Sydney
`Prentice-Hall Canada lnc., Toronto
`Prentice-Hall Hispanoamericana, S.A., M exico
`Prentice-Hall of India Private Limited, New Delhi
`Prentice-Hall of Japan, lnc., Tokyo
`Simon & Schuster Asia Pte. Ltd., Singapore
`Editora Prentice-Hall do Brasil, Ltda., Rio ile Janeiro
`
`PETITIONERS EXHIBIT 1011
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`To Phyllis, Jason, and Justine
`
`To Dorothy, Bill, Kate, anil Barbara
`and in memory of John
`
`PETITIONERS EXHIBIT 1011
`Page 3 of 896
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`Alan V. Oppenheim received the S.B. and S.M. degrees in 1961 and the Sc.D.
`degree in 1964, all in electrical engineering, from the Massachusetts Institute
`of Technology. ln 1964 he joined the faculty at MIT, where he is currently
`Professor of Electrical Engineering and computer Science. Since 1967 hi
`has also been affiliated with MIT Lincoln Laboratory and since 1977 with
`woods Hole oceanographic Institution. His researCh interests are in the
`general area of signal processing and its applications to speech, image, and
`seismic data processing. He is coauthor of the widely usedtextbook sbigitat
`signal Processing and signals and systems. He is also editor of several
`advanced books on signal processing.
`Oppenheim is a member of the National Academy of Engineering,
`a Fellow of the IEEE, and a member of Sigma Xi and eta fappa Nu. Hi
`has been a Guggenheim Fellow and a Sackler Fellow. He has ilio received
`a number of awards for outstanding research and teaching including the
`IEEE Education Medal, the IEEE Centennial Award, and the Soliety
`Award and Technical Achievement Award of the IEEE Societv on
`Acoustics, Speech, and Signal Processing.
`
`Ronald W. Schafer received the B.S.E.E. and M.S.E.E. degrees from the
`University of Nebraska in 1961 and, 1962, respectively, and thi ph.D. degree
`from the Massachusetts Institute of rechnblogy in teos. From 196[ to
`1974 he was a member of the Acoustics Research Department at Bell
`Laboratories, Murray Hill, New Jersey, and since 1974 he has been on the
`faculty of the Georgia Institute of rechnology as Regents' professor and
`holder of the John o. Mccarty/Audichron chair in thJ School of Electrical
`Engineering. His research interests are in discrete-time signal processing
`and_ its application to problems in speech communication and imagi
`analysis. He is coauthor of Digital signal processing and Digital processiig
`of Speech Signals.
`Dr. Schafer is a Fellow of the IEEE and the Acoustical Society of
`America and a member of Sigma Xi, Eta Kappa Nu, and phi Kappa
`Phi. He was awarded the Achievement Award and the Society Award of t'he
`IEEE Society on Acoustics, Speech, and Signal processing, th; IEEE Region
`III Outstanding Engineer Award, and the IEEE Centennial Award, u,i h.
`shared the 1980 Emanuel R. Piore Award with L. R. Rabiner for their work
`i1 speech processing. He has received several awards for teaching, including
`the 1985 Class of 1934 Distinguished professor Award at Georgia Tech.
`
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`Contents
`
`8
`
`Preface xi
`I lntroductlon
`I
`2 Discrete-Time Signals and Slrtems
`2.0
`Introduction 8
`2.1 Discrete-Time Signals: Sequences 9
`2.2 Discrete-Time SYstems 17
`2.3
`Linear Time-Invariant Systems 2l
`2.4 Properties of Linear Time-Invariant Systems 21
`2.5
`Linear Constant-Coefficient Difference Equations 33
`2.6
`Frequency-Domain Representation ofDiscrete-Time
`Signals and SYstems 39
`2.7 Representation of Sequences by Fourier Transforms 45
`2.8 Symmetry Properties of the Fourier Transform 52
`2.9 Fourier Transform Theorems 56
`2.lO Discrete-Time Random Signals 63
`2.ll SummarY 67
`Problems 68
`3 Sampllng of Contlnuous-Time Signals
`3.0
`Introduction 80
`3.1 Periodic SamPling 80
`3.2
`Frequency-Domain Representation of Sampling 82
`3.3 Reconstruction of a Bandlimited Signal from Its Samples 87
`3.4 Discrete-Time Processing of Continuous-Time Signals 9l
`3.5 Continuous-Time Processing of Discrete-Time Signals 99
`3.6 Changing the Sampling Rate Using Discrete-Time
`Processing 101
`3.7 Practical Considerations I 12
`3.8 Summary 130
`Problems 131
`4 The z-Transform 149
`4.0
`Introduction 149
`4.1
`The z-Transform 149
`4.2 Properties of the Region of Convergence for the
`z-Transform 160
`
`80
`
`vll
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`vil
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`Contents
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`4.3
`4.4
`4.5
`
`4.6
`4.7
`4.8
`4.9
`
`The Inverse z-Transform 165
`z-Transform Properties 172
`The Inverse z-Transform Using Contour
`Integration I 8 I
`The Complex Convolution Theorem lg4
`Parseval's Relation 186
`The Unilateral z-Transform 188
`Summary 19l
`Problems 192
`5 Transform Ana[ris of Linear Time-tnvariant
`Slzstems 2O2
`5.0
`Introduction 202
`5.1 The Frequency Response of LTI Systems 203
`5.2 System Functions for Systems Characterized bv Linear
`Constant-Coefficient Difference Equations 206
`Frequency Response for Rational System Functions 213
`Relationship Between Magnitude and phase 230
`Allpass Systems 234
`Minimum-Phase Systems 240
`Linear Systems with Generalized Linear phase 250
`Summary 270
`Problems 270
`
`5.3
`5.4
`5.5
`5.6
`5.7
`5.8
`
`290
`
`6.1
`
`6.2
`
`6.3
`6.4
`6.5
`6.6
`6.7
`6.8
`6.9
`6.10
`
`6.tl
`
`6 Structures for Discrete-Tlme Slntems
`6.0
`Introduction 290
`Block Diagram Representation of Linear
`Constant-Coefficient Difference Equations 291
`Signal Flow Graph Representation of Linear
`Constant-Coefficient Difference Equations 297
`Basic Structures for IIR Systems 300
`Transposed Forms 309
`Basic Network Structures for FIR Systems 313
`Lattice Structures 317
`Overview of Finite-Precision Numerical Effects 325
`The Effects of Coefficient euantization 335
`Effects of Roundoff Noise in Digital Filters 351
`Zero-Input Limit Cycles in Fixed-point Realizations
`of IIR Digital Filters 373
`Summary 378
`Problems 379
`7 Filter Design Techniques 4O3
`7.0
`Introduction 403
`7.1 Design of Discrete-Time IIR Filters from Continuous_
`Time Filters 406
`7.2
`Frequency Transformations of Lowpass IIR Filters 430
`7.3 Computer-Aided Design ol Discrete-Time IIR Filters 43g
`7.4 Design of FIR Filters by Windowing 444
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`Contents
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`'7.5 Examples of FIR Filter Design by the Kaiser Window
`Method 458
`7.6 Optimum Approximations of FIR Filters 464
`7.7
`Examples of FIR Equiripple Approximation 481
`Comments on IIR and FIR Digital Filters 488
`7.8
`1.9 Summary 489
`Problems 490
`8 The Discrete Fourier Transform 514
`8.0
`Introduction 514
`8.1 Representation of Periodic Sequences: The Discrete
`Fourier Series 515
`8.2 Properties of the Discrete Fourier Series 520
`8.3 Summary of Properties of the DFS Representation of
`Periodic Sequences 525
`8.4 The Fourier Transform of Periodic Signals 526
`8.5 Sampling the Fourier Transform 527
`8.6 FourierRepresentationofFinite-DurationSequences:
`The Discrete Fourier Transform 530
`8.7 Properties of the Discrete Fourier Transform 535
`8.8 Summary of Properties of the Discrete Fourier
`Transform 54'7
`8.9
`Linear Convolution Using the Discrete Fourier
`Transform 548
`8.10 Summary 560
`Problems 561
`I Computation of the Discrete Fourier
`Transform 581
`9.0
`Introduction 581
`g.l
`Efficient Computation of the Discrete Fourier Transform 582
`9.2
`The Goertzel Algorithm 585
`9.3 Decimation-in-Time FFT Algorithms 587
`9.4 Decimation-in-FrequencyFFTAlgorithms 599
`9.5
`Implementation of FFT Algorithms 605
`9.6 FFT Algorithms for Composite N 610
`9.7
`Implementation of the DFT Using Convolution 622
`9.8
`Effects of Finite Register Length in Discrete Fourier
`Transform Computations 628
`9.9 Summary 641
`Problems 642
`
`lO Discrete Hilbert Transforms 662
`10.0 Introduction 662
`10.1 Real and Imaginary Part Sufficiency of the Fourier
`Transform for Causal Sequences 664
`10.2 Sufficiency Theorems for Finite-Length Sequences 670
`10.3 Relationships Between Magnitude and Phase 674
`10.4 Hilbert Transform Relations for Complex Sequences 676
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`10.5 Summary 689
`Problems 689
`ll Fourier Ana[rsis of Signals Using the Discrete
`Fourier Transform 695
`1 1.0 Introduction 695
`11.1 Fourier Analysis of Signals Using the DFT 696
`ll.2 DFT Analysis of Sinusoidal Signals 699
`11.3 The Time-Dependent Fourier Transform 713
`11.4 Block Convolution Using the Time-Dependent Fourier
`Transform 721
`11.5 Fourier Analysis of Nonstationary Signals 723
`11.6 Fourier Analysis of Stationary Random Signals:
`The Periodogram 730
`ll.7
`Spectrum Analysis of Random Signals Using Estimates
`of the Autocorrelation Sequence 742
`I 1 .8 Summary 7 55
`Problems 756
`l2 Cepstrum Anafzsis and Homomorphic
`Deconvolution 76A
`12.0 Introduction 768
`l2.l Definition of the Complex Cepstrum 769
`12.2 HomomorphicDeconvolution 771
`12.3 Properties of the Complex Logarithm 775
`12.4 Alternative Expressions for the Complex
`Cepstrum 718
`12.5 The Complex Cepstrum of Exponential Sequences 779
`12.6 Minimum-Phase and Maximum-Phase Sequences 781
`12.7 Realizations of the Characteristic System D*[.] 787
`12.8 Examples of Homomorphic Filtering 797
`12.9 Applications to Speech Processing 815
`12.10 Summary 825
`Problems 826
`Appendix A Random Signals
`835
`A.1 Discrete-Time Random Processes 835
`4.2
`Averages 837
`A.3
`Properties of Correlation and Covariance
`Sequences 841
`4,.4 Transform Representations of Random Signals 843
`Appendix B Continuous-Time Filters
`B.1 Butterworth Lowpass Filters 845
`8.2 Chebyshev Filters 847
`B.3 Elliptic Filters 849
`Bibliography 851
`lndex
`A69
`
`845
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`Preface
`
`This text has its origins in our initial thought several years ago of revising and
`updating our first text, Digital Signal Processing, which was published in 1975. The
`vitality of that book attests to the tremendous interest in and influence of signal
`processing, and it is clear that the field continues to grow in importance as the
`available technologies for implementing signal processing continue to develop.
`Shortly after beginning the revision, we realized that it would be more appropriate to
`develop a new textbook strongly based on our first one and at the same time continue
`to have the original text also available.
`The title Discrete-Time Signal Processing was chosen for this new book for
`several reasons. In the mid-1960s digital signal processing emerged as a new branch
`of signal processing, driven by the potential and feasibility of implementing real-time
`signal processing using digital computers. The term digital signal processing refers
`specifically to processing based on digital technology, which inherently involves both
`time and amplitude quantization. However, the principal focus in essentially all texts
`on digital signal processing is on time quantization, i.e., the discrete-time nature of the
`signals. Furthermore, many signal processing technologies (e.g., charge transport
`devices and switched capacitor filters) are discrete time but not digital; i.e., signal
`values are clocked so that time is quantized but the signal amplitudes are represented
`in analog form.
`In the mid-1970s, when the original text was published, courses on digital and
`discrete-time signal processing were available in only a few schools, and only at the
`graduate level. Now the basic principles are often taught at the undergraduate level,
`sometimes even as part of a first course on linear systems, or at a somewhat more
`advanced level in third-year, fourth-year, or beginning graduate subjects. Much of
`our thinking in planning this new text is in recognition of the importance of this
`material at the undergraduate level. In particular, we have considerably expanded the
`treatment of a number of topics, including linear systems, sampling, multirate signal
`processing, applications, and spectral analysis. In addition, a large number of
`examples are included to emphasize and illustrate important concepts. We have also
`removed and condensed some topics. This new text contains a rich set of more than
`400 problems, and a solutions manual is available for course instructors.
`It is assumed that the reader has a background of advanced calculus, including
`an introduction to complex variables, and an exposure to linear system theory for
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`xii
`
`preFace
`
`continuous-time signals, including Laplace and Fourier transforms, as taught in most
`undergraduate electrical and mechanical engineering curricula. With this back-
`ground, the book is self-contained. In particular, no prior experience with discrete-
`time signals, z-transforms, discrete Fourier transforms, and the like is assumed. In
`later sections of some chapters, some topics such as quantization noise are included
`that assume a basic background in stochastic signals. A brief review of the back-
`ground for these sections is included as Appendix A.
`It has become common in many signal processing courses to include exercises to
`be done on a computer, and many of the homework problems in this book are easily
`turned into problems to be solved with the aid of a computer. with one or two
`exceptions, we have purposely avoided providing software to implement algorithms
`described in this book, for a variety of reasons. Foremost u-ong them is that there
`are readily available a variety of inexpensive signal processing software packages for
`demonstrating and implementing signal processing on any of the popular peisonal
`computers and workstations. These packages are well documented and have excel-
`lent technical support, and many of them have excellent user interfaces that make
`them easily accessible to students. Furthermore, they are in a constant state of
`evolution, which strongly suggests that available software for classroom use should be
`constantly reviewed and updated. While we may have current favorites, these will no
`doubt change over time, and consequently our preference is for computer-based
`exercises to be independent of any specific software system or vendor. we have on
`occasion in this text illustrated points through the use of FoRTRAN programs. we
`chose FORTRAN specifically for its general readability rather than with the
`implication that these specific programs are recommended for use in research or
`practical applications. Even though FORTRAN is often inefficient as an implementa-
`tion of an algorithm, it can be a convenient language for communicating the structure
`of an algorithm.
`The material in this book is organized in a way that provides considerable
`flexibility in its use at both the undergraduate and graduatqlevel. A typical one-
`semester undergraduate elective might cover in depth chapter 2, Sections 2.0-2.9;
`chapter 3, Sections 3.0-3.6; chapter 4; chapter 5, Sections 5.0 5.3; chapter 6,
`Sections 6.0 6.5; chapter 7, sections 7.0-7.2 and 7.4-7.5 and a brief overview of
`Sections 7.6-7.7. If students have studied discrete-time signals and systems in a
`general signals and systems course, it may be possible to move more quickly through
`the material of chapters 2,3, and 4, thus freeing time for covering chapter g. A first-
`year graduate course could augment the above topics with the remaining topics in
`Chapter 5, a brief exposure to the practical considerations in Section 3.7 andSections
`6.7 -6.10, a discussion of optimal FIR filters as incorporated in Sectio ns 7.6 and 7 .7,
`and a thorough treatment of the discrete Fourier transform (chapter g) and its
`computation using the FFT (Chapter 9). The discussion of the DFT can be effectively
`augmented with many of the examples in chapter 11. In a two-semester graduate
`course, the entire text together with a number of current advanced topics can be
`covered.
`In Chapter 2, we introduce the basic class of discrete-time signals and systems
`and define basic system properties such as linearity, time invariance, stabiliiy, and
`causality. The primary focus of the book is on linear time-invariant systems because
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`Preliace
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`xiii
`
`of the rich set of tools available for designing and analyzing this class of systems. In
`particular, in Chapter 2 we develop the time-domain representation of linear time-
`invariant systems through the convolution sum and introduce the class of linear time-
`invariant systems represented by linear constant-coefficient difference equations.
`In Chapter 6, we develop this class of systems in considerably more detail. Also in
`Chapter 2, we introduce the frequency-domain representation of signals and systems
`through the Fourier transform. The primary focus in Chapter 2 is on the representa-
`tion of sequences in terms of the Fourier transform, i.e., as a linear combination of
`complex exponentials, and the development of the basic properties of the Fourier
`transform. We defer until Chapter 5 a detailed discussion of the analysis of linear
`time-invariant systems using the Fourier transform.
`In Chapter 3, we carry out a detailed discussion of the relationship between
`continuous-time and discrete-time signals when the discrete-time signals are obtained
`through periodic sampling of continuous-time signals. This includes a development
`of the Nyquist sampling theorem. In addition, we discuss upsampling and downsam-
`pling of discrete-time signals, as used, for example, in multirate signal processing
`systems and for sampling rate conversion. The chapter concludes with a discussion of
`some of the practical issues encountered in conversion from continuous time to
`discrete time including prefiltering to avoid aliasing and modeling the effects of
`amplitude quantization when the discrete-time signals are represented digitally.
`In Chapter 4, we develop the z-transform as a generalization of the Fourier
`transform. In Chapter 5, we carry out an extensive and detailed discussion of the use
`of the Fourier transform and the z-transform for the representation and analysis of
`linear time-invariant systems. In particular, in Chapter 5 we define the class of ideal,
`frequency-selective filters and develop the system function and pole-zero representa-
`tion lor systems described by linear constant-coefficient difference equations, a class of
`systems whose implementation is considered in detail in Chapter 6. Also in Chapter 5,
`we define and discuss group delay, phase response and phase distortion, and the
`relationships between the magnitude response and the phase response of systems,
`including a discussion of minimum-phase, allpass, and generalized linear phase
`systems.
`In Chapter 6, we focus specifically on systems described by linear constant-
`coefficient difference equations and develop their representation in terms of block
`diagrams and linear signal flow graphs. Much of this chapter is concerned with
`developing a variety of the important system structures and comparing some of their
`properties. The importance of this discussion and the variety of filter structures relate
`to the fact that in a practical implementation of a discrete-time system, the effects of
`coefficient inaccuracies and arithmetic error can be very dependent on the specific
`structure used. While these basic issues are similar whether the technology used for
`implementation is digital or discrete-time analog, we illustrate them in this chapter in
`the context of a digital implementation through a discussion of the effects of coefficient
`quantization and arithmetic roundoff noise for digital filters.
`While Chapter 6 is concerned with the representation and implementation of
`linear constant-coefficient difference equations, Chapter 7 is a discussion of the
`procedures for obtaining the coefficients of this class of difference equations to
`approximate a desired system response. The design techniques separate into those
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`xiv
`
`preface
`
`(IIR) filters and those used for finite impulse
`
`used for infinite impulse response
`response (FIR) filters.
`In continuous-time linear system theory, the Fourier transform is primarily an
`analytical tool for representing signals and systems. In contrast, in the discrete-time
`case' many signal processing systems and algorithms involve the explicit computation
`of the Fourier transform. While the Fourier transform itself canntt be computed, a
`sampled version of it, the discrete Fourier transform (DFT), can be computed, and for
`finiteJength signals the DFT is a complete Fourier representation of ihe signal. In
`Chapter 8, the discrete Fourier transform is introduced and its properties and
`relationship to the discrete-time Fourier transform are developed in aetail. In
`Chapter 9, the rich and important variety of algorithms for compuiing or generating
`the discrete Fourier transform is introduced and discussed, including the Goertze-i
`algorithm, the fast Fourier transform (FFT) algorithms, and the chirp transform.
`In Chapter 10, we introduce the discrete Hitbert transform. This transform
`arises in a variety of practical applications, including inverse filtering, complex
`representations for real bandpass signals, single-sideband modulation techniques, and
`many others. It also has particular significance for the class of signal piocessing
`techniques referred to as cepstral analysis and homomorphic signal processing, ai
`discussed in Chapter 12.
`With the background developed in the earlier chapters and particularly
`chapters 2,3,5, and 8, we focus in chapter 1 1 on Fourier analysis of signals using thl
`discrete Fourier transform. Without a careful understanding of the issues involved
`and the relationship between the DFT and the Fourier transform, using the DFT for
`practical signal analysis can often lead to confusions and misinterpretations. We
`address a number of these issues in Chapter 11. We also consider in some detail the
`Fourier analysis of signals with time-varying characteristics by means of the time-
`dependent Fourier transform.
`In Chapter 12, we introduce a class ofsignal processing techniques referred to as
`cepstral analysis and homomorphic signal processing. This class of techniques,
`although nonlinear, is based on a generalization of the linear techniques that weri the
`focus of the earlier chapters of the book.
`In writing this book, we have been fortunate to receive valuable assistance,
`suggestions, and support from numerous colleagues, students, and friends. over the
`years, a number of our colleagues at MIT and Georgia Institute of Technology have
`taught the material with us, and we have benefited greatly from their perspectives and
`input. These colleagues include Professors Jae Lim, Bruce Musicus, and Victor Zte at
`MIT and Professors Tom Barnwell, Mark clements, Monty Hayes, Jim Mcclellan,
`Russ Mersereau, David Schwartz, and Mark Smith at Georgia Tech. professors
`McClellan andZue along with Jim Glass of MIT were also generous with their time in
`helping us to prepare several of the figures in the book.
`In choosing and developing an effective and complete set of homework
`problems to include in this book, a number of students provided considerable help in
`sorting through, categorizing, and critiquing the large selection of potential home-
`work problems that have accumulated over the years. We would particularly like to
`express our appreciation to Joseph Bondaryk, Dan Cobra, and Rosalind wright for
`their indispensable help with this task as well as their further help with a variety of
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`Preface
`
`other aspects such as figure preparation and proofreading. The later stages of
`production of any text require the time-consuming and often tedious job of proofread-
`ing and scrutinizing the galley proofs and page proofs for errors, omissions, and last-
`minute improvements. We were extremely fortunate to have a long list of "volun-
`teers" to help with this task. At MIT, Hiroshi Miyanaga and Patrick Velardo read a
`large portion of both the galley proofs and page proofs with exceptional care and
`dedication. Our sincere thanks also to MIT students Larry Candell and Avi Lele for
`meticulous reading of many chapters of the page proofs and to Michele Covell, Lee
`Hetherington, Paul Hillner, Tae Joo, Armando Rodriguez, Paul Shen, and Gregory
`Wornell for their help with galley proofs. Similarly, our thanks to Georgia Tech
`students Robert Bamberger, Jae Chung, Larry Heck, and David Pepper for careful
`reading of the page proofs. Cheung Au-Yeung, Beth Carlson, Kate Cummings, Brian
`George, Lois Hertz, David Mazel,Doug Reynolds, Craig Richardson, Janet Rutledge,
`and Kevin Tracy also gave valuable assistance with the galley proofs. We greatly
`appreciate the many valuable and perceptive suggestions made by all our students.
`We would also like to express our thanks to Monica Dove and Deborah Gage at
`MIT and Cherri Dunn, Kayron Gilstrap, Pam Majors, and Stacy Schultz at Georgia
`Tech for their typing of various parts of the manuscript and for their help with the
`incredibly long list of details involved in the teaching and writing associated with the
`development of a textbook. We are particularly indebted to Barbara Flanagan, who
`served as production editor. Barbara's concern for perfection and meticulous atten-
`tion to detail has contributed immeasurably to the quality of the final product. We
`would also like to express our appreciation to Vivian Berman, an artist and a friend,
`who helped us sort through many ideas for the cover design.
`MIT and Georgia Tech have provided us with a stimulating environment for
`reiearch and teaching throughout a major part of our technical careers and have
`provided significant encouragement and support for this project. In addition RWS
`particularly thanks the John and Mary Franklin Foundation for many years of
`support through the John O. McCarty Chair'
`Much of the structure and content of this book was shaped during the summer
`of 1985 when we were both guests in the Ocean Engineering Department at the
`Woods Hole Oceanographic Institution, and we wish to express our gratitude for
`this hospitality. In addition, AVO gives special thanks to the Woods Hole Oceano-
`graphic Institution and to our friends and summer neighbors the Wares and the
`Voses of Gansett Point for providing an exceptionally rejuvenating, productive, and
`enjoyable summer environment since 1978.
`We feel extremely fortunate to have worked with Prentice Hall on this
`project. Our relationship with Prentice Hall spans many years and many writing
`projects. The encouragement and support provided by Tim Bozik, Hank Kennedy,
`and many others at Prentice Hall enhance the enjoyment of writing and completing a
`project such as this one.
`
`Alan V. Oppenheim
`Ronald W. Schafer
`
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`ffiffiffiffiffffifi$ffiSfl*Wffiffi*ffiffiiHfilffiffifl#ffiiaffiffi:
`
`Ilntroduction
`
`ffiffiffiffi ffiffi
`
`Signals are used to communicate between humans and between humans and
`machines; they are used to probe our environment to uncover details of structure and
`state not easily observable; and they are used to control and utilize energy and
`information. Signal processing is concerned with the representation, transformation,
`and manipulation of signals and the information they contain. For example, we may
`wish to separate two or more signals that have somehow been combined or to
`enhance some component or parameter of a signal model. For many decades signal
`processing has played a major role in such diverse fields as speech and data
`communication, biomedical engineering, acoustics, sonar, radar, seismology, oil
`exploration, instrumentation, robotics, consumer electronics, and many others.
`Sophisticated signal processing algorithms and hardware are prevalent in a wide
`range of systems, from highly specialized military systems through industrial applica-
`tions to low-cost, high-volume consumer electronics. Although we routinely take for
`granted the perlormance of home entertainment systems such as television and high-
`fidelity audio, these systems have always relied heavily on state-of-the-art signal
`processing. As another example, speech synthesis, which rapidly found its way into
`automatic voice response systems and consumer items such as learning aids and toys,
`moved at an astonishing pace from research literature to realization in military,
`industrial, and consumer systems.
`The field of signal processing has always benefited from a close coupling
`between the theory, applications, and technologies for implementing signal processing
`systems. Prior to the 1960s, the technology for signal processing was almost
`exclusively continuous-time analog technology.t The rapid evolution of digital
`computers and microprocessors together with some important theoretical develop-
`ments caused a major shift to digital technologies, giving rise to the field of digital
`signal processing. A fundamental aspect of digital signal processing is that it is based
`
`t In a general context, we typically refer to the independent variable as "time" even though in
`specific contexts the independent variable may take on any of a broad range of possible dimensions. Conse-
`quently, continuous time and discrete time should be thought ofas generic terms referring to a continuous
`independent variable and a discrete independent variable, respectively.
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`Page 15 of 896
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`lntroduction Chap. I
`
`on processing of sequences of samples. This discrete-time nature of digital signal
`processing technology is also characteristic of other signal processing technologies
`such as surface acoustic wave (SAW) devices, charge-coupled devices (CCDs), charge
`transport devices (CTDs), and switched-capacitor technologies. In digital signal
`processing, signals are represented by sequences of finite-precision numbers, and
`processing is implemented using digital computation. The more general term dis-
`crete-time signal processfng includes digital signal processing as a special case but also
`includes the possibility that sequences of samples (sampled data) are processed with
`other discrete-time technologies. Often the distinction between the terms discrete-
`time signal processing and digital signal processing is ol minor importance, since both
`are concerned with discrete-time signals.
`While there are many examples in which signals to be processed are inherently
`sequences, most applications involve the use of discrete-time technology for process-
`ing continuous-time signals. In this case, a continuous-time signal is converted into a
`sequence of samples, i.e., a discrete-time signal. After discrete-time processing, the
`output sequence is converted back to a continuous-time signal. Real-time operation
`is often desirable for such systems, meaning that the discrete-time system is imple-
`mented so that samples of the output are computed at the same rate at which the
`continuous-time signal is sampled. Discrete-time processing of continuous-time
`signals is cornmonplace in communication systems, radar and sonar, speech and video
`coding and enhancement, and biomedical engineering, to name just a few.
`Much of traditional signal processing involves processing one signal to obtain
`another signal. Another important class of signal processing problems is signal
`interpretation. In such problems the objective of the processing is not to obtain an
`output signal but to obtain a characterization of the input signal. For example, in a
`speech recognition or understanding system, the objective is to interpret the input
`signal or extract information from it. Typically, such a system will apply prepro-
`cessing (filtering, parameter estimation, etc.) followed by a pattern recognition system
`to produce a symbolic representation such as a phonemic transcript of the speech.
`This symbolic output can in turn be the input to a symbolic processing system, such as
`a rule-based expert system, to provide the final signal interpretation.
`Still another and relatively new category of signal processing involves the
`symbolic manipulation of signal processing expressions.