2011 • 822 Pages • 15.78 MB • English

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␦(t) = 0 , t ≠ 0 δ(t) δ[n] ⎧1 , n = 0 t2 ⎧1 , t1 < 0 < t2 1 ␦[n] = ⎨ 1 t∫1 ␦(t) dt = ⎩⎨0 , otherwise t ⎩0 , n ≠ 0 n u(t) ⎧1 , t > 0 ⎪ u(t) = ⎨1/2 , t = 0 1 ⎪ ⎩0 , t < 0 t u [ n ] sgn(t) ⎧1 , n ≥ 0 u n = ⎨ [ ] 1 ⎧1 , t > 0 1 ⎩0 , n < 0 . . . . . . n ⎪ sgn(t) = ⎨0 , t = 0 t ⎪ ⎩−1 , t < 0 -1 ramp(t) ⎧t , t ≥ 0 1 ramp(t) = ⎨ ⎩0 , t < 0 sgn[n] t ⎧1 , n > 0 1 ⎪ 1 sgn[n] = ⎨0 , n = 0 ... ⎪ n ⎩−1 , n < 0 ... δ (t) T -1 ∞ 1 ␦T (t) = ∑ ␦(t − nT) ... ... n=−∞ t -2T -T T 2T rect(t) ⎧1 , | t | < 1/2 ⎪ 1 rect(t) = ⎨1/2 , | t | = 1/2 ramp[n] ⎪ ⎩0 , | t | > 1/2 1 1 t ⎧n , n ≥ 0⎫ 2 2 ramp[n] = ⎨ ⎬ = nu[n] 8 ⎩0 , n < 0⎭ ... 4 ... tri(t) n 4 8 ⎧1 − | t | , | t | < 1 1 tri(t) = ⎨ ⎩0 , | t | ≥ 1 t −1 1 sinc(t) 1 sin(t) sinc(t) = δ [n] t N ∞ t ␦N [n] = ∑ ␦[n − mN] 1 −5 −4 −3 −2 −1 1 2 3 4 5 m = −∞ ... ... n -N N 2N drcl(t,7) 1 sin( Nt) drcl(t, N) = N sin(t) ... ... t -1 1 rob80687_infc.indd 1 12/8/10 1:55:29 PM ISBN: 0073380681 Front Inside Cover Author: Roberts Color: 1 Title: Signals & System, Second Pages: 1,2 Edition

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␦(t) = 0 , t ≠ 0 δ(t) δ[n] ⎧1 , n = 0 t2 ⎧1 , t1 < 0 < t2 1 ␦[n] = ⎨ 1 t∫1 ␦(t) dt = ⎩⎨0 , otherwise t ⎩0 , n ≠ 0 n u(t) ⎧1 , t > 0 ⎪ u(t) = ⎨1/2 , t = 0 1 ⎪ ⎩0 , t < 0 t u [ n ] sgn(t) ⎧1 , n ≥ 0 u n = ⎨ [ ] 1 ⎧1 , t > 0 1 ⎩0 , n < 0 . . . . . . n ⎪ sgn(t) = ⎨0 , t = 0 t ⎪ ⎩−1 , t < 0 -1 ramp(t) ⎧t , t ≥ 0 1 ramp(t) = ⎨ ⎩0 , t < 0 sgn[n] t ⎧1 , n > 0 1 ⎪ 1 sgn[n] = ⎨0 , n = 0 ... ⎪ n ⎩−1 , n < 0 ... δ (t) T -1 ∞ 1 ␦T (t) = ∑ ␦(t − nT) ... ... n=−∞ t -2T -T T 2T rect(t) ⎧1 , | t | < 1/2 ⎪ 1 rect(t) = ⎨1/2 , | t | = 1/2 ramp[n] ⎪ ⎩0 , | t | > 1/2 1 1 t ⎧n , n ≥ 0⎫ 2 2 ramp[n] = ⎨ ⎬ = nu[n] 8 ⎩0 , n < 0⎭ ... 4 ... tri(t) n 4 8 ⎧1 − | t | , | t | < 1 1 tri(t) = ⎨ ⎩0 , | t | ≥ 1 t −1 1 sinc(t) 1 sin(t) sinc(t) = δ [n] t N ∞ t ␦N [n] = ∑ ␦[n − mN] 1 −5 −4 −3 −2 −1 1 2 3 4 5 m = −∞ ... ... n -N N 2N drcl(t,7) 1 sin( Nt) drcl(t, N) = N sin(t) ... ... t -1 1 rob80687_infc.indd 1 12/8/10 1:55:29 PM ISBN: 0073380681 Front Inside Cover Author: Roberts Color: 1 Title: Signals & System, Second Pages: 1,2 Edition

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Signals and Systems ® Analysis Using Transform Methods and MATLAB Second Edition Michael J. Roberts Professor, Department of Electrical and Computer Engineering University of Tennessee rob80687_fm_i-xx.indd i 1/3/11 4:12:46 PM

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SIGNALS AND SYSTEMS: ANALYSIS USING TRANSFORM METHODS AND MATLAB®, SECOND EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Previous edition © 2004. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on recycled, acid-free paper containing 10% postconsumer waste. 1 2 3 4 5 6 7 8 9 0 QDQ/QDQ 1 0 9 8 7 6 5 4 3 2 1 ISBN 978-0-07-338068-1 MHID 0-07-338068-7 Vice President & Editor-in-Chief: Marty Lange Vice President EDP/Central Publishing Services: Kimberly Meriwether David Publisher: Raghothaman Srinivasan Senior Sponsoring Editor: Peter E. Massar Senior Marketing Manager: Curt Reynolds Development Editor: Darlene M. Schueller Project Manager: Melissa M. Leick Cover Credit: © Digital Vision/Getty Images Buyer: Sandy Ludovissy Design Coordinator: Margarite Reynolds Media Project Manager: Balaji Sundararaman Compositor: Glyph International Typeface: 10.5/12 Times Roman Printer: Quad/Graphics Library of Congress Cataloging-in-Publication Data Roberts, Michael J., Dr. Signals and systems: analysis using transform methods and MATLAB / Michael J. Roberts.—2nd ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-07-338068-1 (alk. paper) ISBN-10: 0-07-338068-7 (alk. paper) 1. Signal processing. 2. System analysis. 3. MATLAB. I. Title. TK5102.9.R63 2012 621.382’2–dc22 2010048334 www.mhhe.com rob80687_fm_i-xx.indd i 1/3/11 5:29:41 PM

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To my wife Barbara for giving me the time and space to complete this effort and to the memory of my parents, Bertie Ellen Pinkerton and Jesse Watts Roberts, for their early emphasis on the importance of education. rob80687_fm_i-xx.indd i 1/3/11 4:12:47 PM

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CONTENTS Preface, xii Time Scaling, 39 Simultaneous Shifting and Scaling, 43 Chapter 1 2.6 Differentiation and Integration, 47 Introduction, 1 2.7 Even and Odd Signals, 49 Combinations of Even and Odd Signals, 51 1.1 Signals and Systems Deﬁ ned, 1 Derivatives and Integrals of Even and Odd Signals, 53 1.2 Types of Signals, 3 2.8 Periodic Signals, 53 1.3 Examples of Systems, 8 2.9 Signal Energy and Power, 56 A Mechanical System, 9 Signal Energy, 56 A Fluid System, 9 Signal Power, 57 A Discrete-Time System, 11 2.10 Summary of Important Points, 60 Feedback Systems, 12 Exercises, 60 1.4 A Familiar Signal and System Example, 14 ® Exercises with Answers, 60 1.5 Use of MATLAB , 18 Signal Functions, 60 Scaling and Shifting, 61 Chapter 2 Derivatives and Integrals, 65 Even and Odd Signals, 66 Mathematical Description of Continuous-Time Signals, 19 Periodic Signals, 68 Signal Energy and Power, 69 2.1 Introduction and Goals, 19 Exercises without Answers, 70 2.2 Functional Notation, 20 Signal Functions, 70 2.3 Continuous-Time Signal Functions, 20 Scaling and Shifting, 71 Complex Exponentials and Sinusoids, 21 Generalized Derivative, 74 Functions with Discontinuities, 23 Derivatives and Integrals, 74 The Signum Function, 24 Even and Odd Signals, 75 The Unit-Step Function, 24 Periodic Signals, 75 The Unit-Ramp Function, 26 Signal Energy and Power, 76 The Unit Impulse, 27 The Impulse, the Unit Step and Generalized Derivatives, 29 Chapter 3 The Equivalence Property of the Impulse, 30 Discrete-Time Signal Description, 77 The Sampling Property of the Impulse, 31 The Scaling Property of the Impulse, 31 3.1 Introduction and Goals, 77 The Unit Periodic Impulse or Impulse Train, 32 3.2 Sampling and Discrete Time, 78 A Coordinated Notation for Singularity 3.3 Sinusoids and Exponentials, 80 Functions, 33 Sinusoids, 80 The Unit-Rectangle Function, 33 Exponentials, 83 2.4 Combinations of Functions, 34 3.4 Singularity Functions, 84 2.5 Shifting and Scaling, 36 The Unit-Impulse Function, 84 Amplitude Scaling, 36 The Unit-Sequence Function, 85 Time Shifting, 37 The Signum Function, 85 iv rob80687_fm_i-xx.indd iv 1/3/11 4:12:47 PM

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Contents v The Unit-Ramp Function, 86 Additivity, 128 The Unit Periodic Impulse Function or Impulse Linearity and Superposition, 129 Train, 86 LTI Systems, 129 3.5 Shifting and Scaling, 87 Stability, 133 Amplitude Scaling, 87 Causality, 134 Time Shifting, 87 Memory, 134 Time Scaling, 87 Static Nonlinearity, 135 Time Compression, 88 Invertibility, 137 Time Expansion, 88 Dynamics of Second-Order Systems, 138 3.6 Differencing and Accumulation, 92 Complex Sinusoid Excitation, 140 3.7 Even and Odd Signals, 96 4.3 Discrete-Time Systems, 140 Combinations of Even and Odd Signals, 97 System Modeling, 140 Symmetrical Finite Summation of Even and Odd Block Diagrams, 140 Signals, 97 Difference Equations, 141 3.8 Periodic Signals, 98 System Properties, 147 3.9 Signal Energy and Power, 99 4.4 Summary of Important Points, 150 Signal Energy, 99 Exercises, 151 Signal Power, 100 Exercises with Answers, 151 3.10 Summary of Important Points, 102 System Models, 151 Exercises, 102 System Properties, 153 Exercises with Answers, 102 Exercises without Answers, 155 Signal Functions, 102 System Models, 155 Scaling and Shifting, 104 System Properties, 157 Differencing and Accumulation, 105 Even and Odd Signals, 106 Chapter 5 Periodic Signals, 107 Time-Domain System Analysis, 159 Signal Energy and Power, 108 Exercises without Answers, 108 5.1 Introduction and Goals, 159 Signal Functions, 108 5.2 Continuous Time, 159 Shifting and Scaling, 109 Impulse Response, 159 Differencing and Accumulation, 111 Continuous-Time Convolution, 164 Even and Odd Signals, 111 Derivation, 164 Periodic Signals, 112 Graphical and Analytical Examples of Signal Energy and Power, 112 Convolution, 168 Convolution Properties, 173 System Connections, 176 Chapter 4 Step Response and Impulse Response, 176 Description of Systems, 113 Stability and Impulse Response, 176 4.1 Introduction and Goals, 113 Complex Exponential Excitation and the Transfer 4.2 Continuous-Time Systems, 114 Function, 177 System Modeling, 114 Frequency Response, 179 Differential Equations, 115 5.3 Discrete Time, 181 Block Diagrams, 119 Impulse Response, 181 System Properties, 122 Discrete-Time Convolution, 184 Introductory Example, 122 Derivation, 184 Homogeneity, 126 Graphical and Analytical Examples of Time Invariance, 127 Convolution, 187 rob80687_fm_i-xx.indd v 1/3/11 4:12:47 PM

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vi Contents Convolution Properties, 191 6.3 The Continuous-Time Fourier Transform, 241 Numerical Convolution, 191 Extending the Fourier Series to Aperiodic Signals, 241 Discrete-Time Numerical Convolution, 191 The Generalized Fourier Transform, 246 Continuous-Time Numerical Convolution, 193 Fourier Transform Properties, 250 Stability and Impulse Response, 195 Numerical Computation of the Fourier Transform, 259 System Connections, 195 6.4 Summary of Important Points, 267 Unit-Sequence Response and Impulse Response, 196 Exercises, 267 Complex Exponential Excitation and the Exercises with Answers, 267 Transfer Function, 198 Fourier Series, 267 Frequency Response, 199 Orthogonality, 268 5.4 Summary of Important Points, 201 CTFS Harmonic Functions, 268 Exercises, 201 System Response to Periodic Excitation, 271 Exercises with Answers, 201 Forward and Inverse Fourier Transforms, 271 Continuous Time, 201 Relation of CTFS to CTFT, 280 Impulse Response, 201 Numerical CTFT, 281 Convolution, 201 System Response , 282 Stability, 204 Exercises without Answers, 282 Discrete Time, 205 Fourier Series, 282 Impulse Response, 205 Orthogonality, 283 Convolution, 205 Forward and Inverse Fourier Transforms, 283 Stability, 208 Exercises without Answers, 208 Chapter 7 Continuous Time, 208 Discrete-Time Fourier Methods, 290 Impulse Response, 208 Convolution, 209 7.1 Introduction and Goals, 290 Stability, 210 7.2 The Discrete-Time Fourier Series and the Discrete Discrete Time, 212 Fourier Transform, 290 Impulse Response, 212 Linearity and Complex-Exponential Excitation, 290 Convolution, 212 Orthogonality and the Harmonic Function, 294 Stability, 214 Discrete Fourier Transform Properties, 298 The Fast Fourier Transform, 302 7.3 The Discrete-Time Fourier Transform, 304 Chapter 6 Extending the Discrete Fourier Transform to Aperiodic Continuous-Time Fourier Methods, 215 Signals, 304 6.1 Introduction and Goals, 215 Derivation and Deﬁ nition, 305 6.2 The Continuous-Time Fourier Series, 216 The Generalized DTFT, 307 Conceptual Basis, 216 Convergence of the Discrete-Time Fourier Orthogonality and the Harmonic Function, 220 Transform, 308 The Compact Trigonometric Fourier Series, 223 DTFT Properties, 309 Convergence, 225 Numerical Computation of the Discrete-Time Fourier Transform, 315 Continuous Signals, 225 7.4 Fourier Method Comparisons, 321 Discontinuous Signals, 226 7.5 Summary of Important Points, 323 Minimum Error of Fourier-Series Partial Sums, 228 Exercises, 323 The Fourier Series of Even and Odd Periodic Functions, 229 Exercises with Answers, 323 Fourier-Series Tables and Properties, 230 Orthogonality, 323 Numerical Computation of the Fourier Series, 234 Discrete Fourier Transform, 324 rob80687_fm_i-xx.indd vi 1/3/11 4:12:47 PM

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