Signals systems and transforms 4e by philips

SIGNALS, SYSTEMS, AND TRANSFORMS FOURTH EDITION This page intentionally left blank SIGNALS, SYSTEMS, AND TRANSFORMS FOURTH EDITION CHARLES L PHILLIPS Emeritus Auburn University Auburn, Alabama JOHN M PARR University of Evansville Evansville, Indiana EVE A RISKIN University of Washington Seattle, Washington Upper Saddle River, NJ 07458 Library of Congress Cataloging-in-Publication Data Phillips, Charles L Signals, systems, and transforms / Charles L Phillips, John M Parr, Eve A Riskin.—4th ed p cm Includes bibliographical references and index ISBN-13: 978-0-13-198923-8 ISBN-10: 0-13-198923-5 Signal processing–Mathematical models Transformations (Mathematics) System analysis I Parr, John M II Riskin, Eve A (Eve Ann) III Title TK5102.9.P47 2008 621.382'2—dc22 2007021144 Vice President and Editorial Director, ECS: Marcia J Horton Associate Editor: Alice Dworkin Acquisitions Editor: Michael McDonald Director of Team-Based Project Management: Vince O’Brien Senior Managing Editor: Scott Disanno Production Editor: Karen Ettinger Director of Creative Services: Christy Mahon Associate Director of Creative Services: Leslie Osher Art Director, Cover: Jayne Conte Cover Designer: Bruce Kenselaar Art Editor: Gregory Dulles Director, Image Resource Center: Melinda Reo Manager, Rights and Permissions: Zina Arabia Manager, Visual Research: Beth Brenzel Manager, Cover Visual Research and Permissions: Karen Sanatar Manufacturing Buyer: Lisa McDowell Marketing Assistant: Mack Patterson © 2008 Pearson Education, Inc Pearson Education, Inc Upper Saddle River, NJ 07458 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 Pearson Prentice Hall® is a trademark of Pearson Education, Inc The author and publisher of this book have used their best efforts in preparing this book These efforts include the development, research, and testing of the theories and programs to determine their effectiveness The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs Printed in the United States of America All other trademark or product names are the property of their respective owners TRADEMARK INFORMATION MATLAB is a registered trademark of the MathWorks, Inc The MathWorks, Inc., Apple Hill Drive, Natick, MA 01760-2098 10 ISBN-13: 978-0-13-198923-8 ISBN-10: 0-13-198923-5 Pearson Education Ltd., London Pearson Education Australia Pty Ltd., Sydney Pearson Education Singapore, Pte Ltd Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Inc., Toronto Pearson Educación de Mexico, S.A de C.V Pearson Education—Japan, Tokyo Pearson Education Malaysia, Pte Ltd Pearson Education, Inc., Upper Saddle River, New Jersey To Taylor, Justin, Jackson, Rebecca, and Alex Michaela, Cadence, Miriam, and Connor Duncan, Gary, Noah, and Aden This page intentionally left blank CONTENTS PREFACE xvii INTRODUCTION 1.1 Modeling 1.2 Continuous-Time Physical Systems Electric Circuits, Operational Amplifier Circuits, Simple Pendulum, DC Power Supplies, 10 Analogous Systems, 12 1.3 Samplers and Discrete-Time Physical Systems 14 Analog-to-Digital Converter, 14 Numerical Integration, 16 Picture in a Picture, 17 Compact Disks, 18 Sampling in Telephone Systems, 19 Data-Acquisition System, 21 1.4 2.1 MATLAB and SIMULINK 22 CONTINUOUS-TIME SIGNALS AND SYSTEMS Transformations of Continuous-Time Signals Time Transformations, 24 Amplitude Transformations, 2.2 Signal Characteristics Even and Odd Signals, Periodic Signals, 34 23 24 30 32 32 vii viii Contents 2.3 Common Signals in Engineering 2.4 Singularity Functions 39 45 Unit Step Function, 45 Unit Impulse Function, 49 2.5 Mathematical Functions for Signals 2.6 Continuous-Time Systems Interconnecting Systems, Feedback System, 64 2.7 54 59 61 Properties of Continuous-Time Systems 65 Stability, 69 Linearity, 74 Summary 76 Problems 78 CONTINUOUS-TIME LINEAR TIME-INVARIANT SYSTEMS 3.1 Impulse Representation of Continuous-Time Signals 3.2 Convolution for Continuous-Time LTI Systems 3.3 Properties of Convolution 3.4 Properties of Continuous-Time LTI Systems 90 91 104 107 Memoryless Systems, 108 Invertibility, 108 Causality, 109 Stability, 110 Unit Step Response, 111 3.5 Differential-Equation Models 112 Solution of Differential Equations, 114 General Case, 116 Relation to Physical Systems, 118 3.6 Terms in the Natural Response Stability, 3.7 System Response for Complex-Exponential Inputs Linearity, 123 Complex Inputs for LTI Systems, Impulse Response, 128 3.8 119 120 Block Diagrams 129 Direct Form I, 133 Direct Form II, 133 nth-Order Realizations, 133 Practical Considerations, 135 124 123 89 Contents ix Summary 137 Problems 139 4.1 FOURIER SERIES 150 Approximating Periodic Functions Periodic Functions, 152 Approximating Periodic Functions, 4.2 Fourier Series 152 156 Fourier Series, 157 Fourier Coefficients, 4.3 151 158 Fourier Series and Frequency Spectra Frequency Spectra, 161 162 4.4 Properties of Fourier Series 171 4.5 System Analysis 4.6 Fourier Series Transformations 174 181 Amplitude Transformations, 182 Time Transformations, 184 Summary 186 Problems 187 THE FOURIER TRANSFORM 197 5.1 Definition of the Fourier Transform 197 5.2 Properties of the Fourier Transform 206 Linearity, 206 Time Scaling, 208 Time Shifting, 211 Time Transformation, 212 Duality, 213 Convolution, 216 Frequency Shifting, 217 Time Differentiation, 219 Time Integration, 224 Frequency Differentiation, 227 Summary, 227 5.3 Fourier Transforms of Time Functions DC Level, 228 Unit Step Function, 228 Switched Cosine, 229 228 760 12.9 12.10 12.13 12.17 12.19 12.22 12.24 12.27 Answers to Selected Problems [2, 0, 2, 0] 1 c , 0, - , d 2 (a) [3, -j, 1, j] N 1d[k - p] + d[k + p]2 (a) A (b) C (c) D (d) B (a) 2p (b) 1024p rad>s (a) [2, 5, 4, 5, -1, -2, -6] (b) [1, 3, -2, 5] (c) [-2, -4, -1, -2, 5, 5, 6] (d) [6, 5, 5, -2, -1, -4, -2] (e) [-4, -2, 2, 9, 2, -2, -4] (a) 30 (b) 38 (c) 12 (d) 30 (e) 60 (f) [121, 5, 9, 5] CHAPTER 13 13.1 13.4 13.5 13.6 13.8 (b) x[n + 1] = c d x[n] + c d u[n] -0.8 y[n] = [1 0] x[n] (b) x[n + 1] = 0.9x[n] + u[n] y[n] = 1.35x[n] + 1.5u[n] (b) x[n + 1] = 0.8x[n] + u[n] y[n] = -0.4x[n] + 4u[n] (c) y[n] = 0.8y[n - 1] + 4u[n] - 3.6u[n - 1] 6.08z + 6.69 (b) H1z2 = 1z - 0.821z - 0.92 (b) H1z2 = 2z2 - 5.035z + 5.0865 z2 - 1.9z + 0.8 Appen H Appen H 13.10 13.12 Answers to Selected Problems (a) x1[n + 1] = 11 - a2x1[n] + au[n] y[n] = 11 - a2x1[n] + au[n] az (b) H1z2 = z - 11 - a2 (a) x[n + 1] = B 0.8 6.08 R x[n] + B R u[n] 0.9 3.2 y[n] = [0 (b) £[n] = C (d) (e) (f) (a) (b) (c) 1.9] x[n] 0.8n 6.08[0.9n - 0.8n] S 0.9n 0.8n R 62.8(0.9) - 60.8(0.8)n y[n] = 119.310.92n - 115.510.82n, n Ú y[n] = 638.4 + 577.610.82n - 121610.92n, n Ú y[n] = 638.5 + 577.710.82n - 1216.210.92n, n Ú y[n] = 638.5 + 462.210.82n - 1096.910.92n n Ú not stable 11.272n, 10.632n a = [1.9 0.8; -1 0]; eig1a2 (c) x[n] = B 13.19 761 n I SIGNALS AND SYSTEMS REFERENCES Some references for particular topics are given throughout this book Many good books in the general area of signals and systems are available An incomplete list of these books is given as Chapter Refs 12 through 20 Any omission of books from this list is inadvertent CHAPTER 1 W A Gardner, Introduction to Random Processes New York: Macmillan, 1986 C L Phillips and R D Harbor, Feedback Control Systems, 4th ed Upper Saddle River, NJ: Prentice Hall, 1999 J Millman, Microelectronics, 2d ed New York: McGraw–Hill, 1999 C L Phillips and H T Nagle, Digital Control System Analysis and Design, 3d ed Upper Saddle River, NJ: Prentice Hall, 1996 S D Conte and C deBoor, Elementary Numerical Analysis: An Algorithmic Approach New York: McGraw-Hill, 1982 M Burkert et al., “IC Set for a Picture-in-Picture System with On-Chip Memory,” IEEE Transactions on Consumer Electronics, February 1990 L Buddine and E Young, The Brady Guide to CD-ROM Englewood Cliffs, NJ: Prentice Hall, 1988 B E Keiser and E Strange, Digital Telephony and Network Integration New York: Van Nostrand Reinhold, 1995 J D Irwin, Basic Engineering Circuit Analysis, 6th ed New York: Macmillan, 1999 10 Learning MATLAB 6, Natick, MA: The Mathworks, Inc., 2001 11 Learning Simulink 4, Natick, MA: The Mathworks, Inc., 2001 12 R A Gabel and R A Roberts, Signals and Linear Systems New York: Wiley, 1987 13 L B Jackson, Signals, Systems, and Transforms Reading, MA: Addison–Wesley, 1991 14 B P Lathi, Linear Systems and Signals, New York: Berkeley-Cambridge, 1992 15 R J Mayhan, Discrete-Time and Continuous-Time Linear Systems, 2d ed Reading, MA: Addison–Wesley, 1998 762 Appen I Signals and Systems References 763 16 C D McGillem and G R Cooper, Continuous and Discrete Signal and System Analysis, 3d ed New York: Holt, Rinehart and Winston, 1995 17 M O’Flynn and E Moriarty, Linear Systems Time Domain and Transform Analysis New York: Harper & Row, 1987 18 A V Oppenheim and A S Willsky, Signals and Systems, 2d ed Upper Saddle River, NJ: Prentice Hall, 1996 19 S S Soliman and M D Srinath, Continuous and Discrete Signals and Systems, 2d ed Upper Saddle River, NJ: Prentice Hall, 1997 20 R E Ziemer, W H Tranter, and S R Fannin, Signals and Systems Continuous and Discrete, 4th ed New York: Macmillan, 1998 CHAPTER G Carlson, Signal and Linear System Analysis, 2d ed New York, John Wiley & Sons, 1998 G Doetsch, Guide to the Applications of the Laplace and z-Transforms London: Van Nostrand Reinhold, 1971 W Kaplan, Operational Methods for Linear Systems Reading, MA: Addison–Wesley, 1962 R V Churchill, Operational Mathematics, 3d ed New York: McGraw–Hill, 1977 G Doetsch, Guide to the Applications of Laplace Transforms London: Van Nostrand Reinhold, 1961 G Doetsch, Introduction to the Theory and Application of the Laplace Transform New York: Springer–Verlag, 1974 R F Wigginton, Evaluation of OPS-II Operational Program for the Automatic Carrier Landing System Saint Inigoes, MD: Naval Electronic Systems Test and Evaluation Facility, 1971 CHAPTER F B Hildebrand, Advanced Calculus and Applications, 2d ed Englewood Cliffs, NJ: Prentice-Hall, 1976 G Birkhoff and G.-C Rota, Ordinary Differential Equations, 4th ed New York: Wiley, 1994 C L Phillips and R D Harbor, Feedback Control Systems, 4th ed Upper Saddle River, NJ: Prentice Hall, 1999 G Doetsch, Guide to the Applications of the Laplace and z-Transforms London: Van Nostrand Reinhold, 1971 CHAPTER C L Phillips and R D Harbor, Feedback Control Systems, 4th ed Upper Saddle River, NJ: Prentice Hall, 1999 D Jackson, Fourier Series and Orthogonal Polynomials Menosha, WI: Collegiate Press, 1981 764 Signals and Systems References Appen I A V Oppenheim and A S Willsky, Signals and Systems Upper Saddle River, NJ: Prentice Hall, 1996 W Kaplan, Operational Methods for Linear Systems Reading, MA: Addison–Wesley, 1962 R V Churchill, Operational Mathematics, 2d ed New York: McGraw–Hill, 1972 CHAPTER R Bracewell, The Fourier Transform and Its Applications, 2d ed New York: McGraw–Hill, 1986 M J Lighthill, Fourier Analysis and Generalised Functions Cambridge: Cambridge University Press, 1958 A Papoulis, The Fourier Integral and Its Applications New York: McGraw–Hill, 1962 R A Gabel and R A Roberts, Signals and Linear Systems New York: Wiley, 1987 H P Hsu, Fourier Analysis New York: Simon & Schuster, 1970 B P Lathi, Signals, Systems and Communication New York: Wiley, 1965 N K Sinha, Linear Systems New York: Wiley, 1991 S S Soliman and M D Srinath, Continuous and Discrete Signals and Systems, 2d ed Upper Saddle River, NJ: Prentice Hall, 1997 A V Oppenheim and A S Willsky, Signals and Systems, 2d ed Upper Saddle River, NJ: Prentice Hall, 1996 CHAPTER L W Couch II, Modern Communication Systems, Upper Saddle River, NJ: Prentice Hall, 1995 G E Carlson, Signal and Linear System Analysis Boston: Houghton Mifflin, 1992 International Telephone and Telegraph Corporation, Reference Data for Radio Engineers, 5th ed Indianapolis, IN: Howard W Sams, 1973 C J Savant, Jr., M S Roden, and G L Carpenter, Electronic Design: Circuits and Systems, 2d ed Redwood City, CA: Benjamin/Cummings, 1991 A B Williams, Electronic Filter Design Handbook New York: McGraw–Hill, 1981 S Haykin, An Introduction to Analog and Digital Communications New York: Wiley, 1989 A J Jerri, “The Shannon Sampling Theorem—Its Various Extensions and Applications: A Tutorial Review,” Proceedings of IEEE, vol 65, pp 1565–1596, 1977 CHAPTER R V Churchill, Operational Mathematics, 3d ed New York: McGraw–Hill, 1977 G Doetsch, Introduction to the Theory and Application of the Laplace Transform New York: Springer–Verlag, 1970 G Doetsch, Guide to the Applications of the Laplace and z-Transforms New York: Van Nostrand Reinhold, 1971 W Kaplan, Operational Methods for Linear Systems Reading, MA: Addison–Wesley, 1962 Appen I Signals and Systems References 765 CHAPTER D Graupe, Identification of Systems Huntington, NY: Robert E Kreiger, 1976 C L Phillips and R D Harbor, Feedback Control Systems, 4th ed Upper Saddle River, NJ: Prentice Hall, 1999 B Friedlander, Control System Design New York: McGraw–Hill, 1986 G H Golub and C F Van Loan, Matrix Computations, 2d ed Baltimore, MD: Johns Hopkins University Press, 1996 C L Phillips and H T Nagle, Digital Control System Analysis and Design, 3d ed Upper Saddle River, NJ: Prentice Hall, 1996 G F Franklin and J D Powell, Digital Control of Dynamic Systems 3d ed Reading, MA: Addison–Wesley, 1997 W L Brogan, Modern Control Theory, 3d ed Upper Saddle River, NJ: Prentice Hall, 1991 CHAPTER C L Phillips and H T Nagle, Digital Control System Analysis and Design, 3d ed Upper Saddle River, NJ: Prentice Hall, 1996 CHAPTER 10 L A Pipes, Applied Mathematics for Engineers New York: McGraw–Hill, 1946 CHAPTER 11 G Doetsch, Guide to the Applications of the Laplace and z-Transforms New York: Van Nostrand Reinhold, 1971 E I Jury, Theory and Application of the z-Transform Method New York: Krieger, 1973 C L Phillips and H T Nagle, Digital Control System Analysis and Design, 3d ed Upper Saddle River, NJ: Prentice Hall, 1996 Martin Vetterli and Jelena Kovaˇcevi´c, Wavelets and Subband Coding, Upper Saddle River, NJ: Prentice Hall, 1995 CHAPTER 12 H Nyquist, “Certain Topics in Telegraph Transmission Theory,” Transactions of AIEE, vol 47, April 1928 C E Shannon, “Communication in the Presence of Noise,” Proceedings of the IRE, vol 37, January 1949 G E Carlson, Signal and Linear System Analysis, 2d ed New York: John S Wiley & Sons, 1998 766 Signals and Systems References Appen I S S Soliman and M D Srinath, Continuous and Discrete Signals and Systems, 2d ed Upper Saddle River, NJ: Prentice Hall, 1997 L B Jackson, Signals, Systems and Transforms, Reading, MA: Addison–Wesley, 1991 R D Strum and D E Kirk, First Principles of Discrete Systems and Digital Signal Processing Reading, MA: Addison–Wesley, 1988 K Sayood, Introduction to Data Compression, 2d ed San Francisco: Morgan Kaufmann Publishers, 2000 http://ftp.math.hkbu.edu.hk/help/toolbox/images/transfo6.html CHAPTER 13 C L Phillips and H T Nagle, Digital Control Systems, 3d ed Upper Saddle River, NJ: Prentice Hall, 1996 G F Franklin and J D Powell, Digital Control of Dynamic Systems, 3d ed Reading, MA: Addison–Wesley, 1997 B Friedlander, Control System Design New York: McGraw–Hill, 1986 APPENDIX D R V Churchill, J W Brown, and R F Verkey, Complex Variables and Applications, 4th ed New York: McGraw–Hill, 1989 R E Larson and R P Hostetler, Algebra and Trigonometry, 2d ed Lexington, MA: D.C Heath, 1993 APPENDIX E W E Boyce and R C DePrima, Elementary Differential Equations and Boundary Value Problems, 2d ed New York: Wiley, 1992 APPENDIX F R V Churchill, Operations Mathematics, 2d ed New York: McGraw-Hill, 1972 APPENDIX G F R Gantmacher, Theory of Matrices, Vols I and II New York: Chelsea, 1959 G Strang, Linear Algebra and Its Applications, 2d ed New York: Academic Press, 1988 G H Golub and C F Van Loan, Matrix Computations, 2d ed Baltimore, MD: Johns Hopkins University Press, 1996 INDEX A B Abscissa of absolute convergence, 344 Absolutely integrable, 110, 143, 202, 203, 264 Absolutely summable, 508, 510, 603 Accuracy, adequate, 3, 126, 151, 152 Aliasing, 242, 301 Amplifiers: differentiating, integrating, 8, 66, 96 operational, 6, 16, 96, 225 voltage, Amplitude modulation: pulse, 19, 167, 317–324 sinusoidal, 306–317 Amplitude sensitivity, 312 AM radio, 312, 315 Analog computer, 131 Analogous systems, 12–14 Analog signal, 2, 23, 618 Analog simulation, 131 Analog-to-digital converter (A/D, ADC), 14–16, 20, 21, 52, 237, 443, 444 Aperiodic signal, 35, 171, 181 Autocorrelation, 660, 666–667 Automatic control, 64, 74, 148 Auxiliary equation (See Characteristic equation) Average power, normalized, 203, 260–261, 283 Averaging periodogram method, 667 Bandpass signal, 296 Bandwidth: 3-dB, 296, absolute, 296, 298 first null, 297 half-power, 296, 298 null-to-null, 297, 298 zero-crossing, 297 Baseband signal, 296, 297 BIBO stability (See Stability) Bilateral Laplace transform, 129, 336–337, 380–389 Bilateral z-transform, 534, 547–549, 551, 555, 559, 579–590 Block diagrams, 129–137, 472, 521–527 (See also Simulation diagrams) Block filtering, 655–659 Bode plots, 246, 248–250, 379 Bounded-input–bounded-output (BIBO) stability, 69, 110 Bounded signals, 69 Butterfly diagrams, 628 Butterworth filters, 284–290 C Canonical form, 406 Carrier signals, 238, 307, 317 Cathode ray tube, 36, 57 767 768 Causality, 68, 109, 374–375, 477–478, 506–507, 575 Causal system, 68, 477, 575 Characteristic equation, 117, 376, 424, 513, 577, 703–704, 731 Characteristic values (See Eigenvalues) Chebyschev filters, 290–295, 324 Circuits, 4, 6–9, 11–16, 19–22 Circular convolution, 609–610, 646–651 Compact disk, 18–19 Complementary function: difference equations, 512, 513, 518 differential equations, 115, 730–733 Complex exponential functions (See Signals and also Response.) Complex inversion integral, 337–338, 547 Complex numbers, 723–729 Complex poles, 370–374, 573–575, 736 Continuous-time signal (See Signal) Continuous-time system (See System) Control canonical form, 406 Convergence: discrete-time Fourier transform, 603 Laplace transform, 382 z-transform, 555, 559, 577, 580, 586 Converters: analog-to-digital, 14–16, 18, 20–22, 52, 237, 443–444 digital-to-analog, 14, 20–21, 24, 100, 303–306, 369, 466 Convolution: continuous time, 91–104, 239 properties, 104–107 discrete time, 493–505 properties, 502–505 Correlation, 660–666 Cost function, 153–154 Cramer’s rule, 742–743 Cross correlation, 660–666 Cutoff frequency, 282–284 D Data reconstruction, 240, 299–306 DC power supplies, 10–12, 37 DC value (See Signals) Decomposition-in-frequency FFT, 632–635 Decomposition-in-time FFT, 627–632 Index Demodulation, synchronous, 310–311, 315–316 Demultiplex, 320 Detection (See Demodulation) DFT shorthand notation, 620 Difference equations, 446, 510–518, 682–692 solution, 512, 692–699 Differential equations, 112–119, 398 solution, 114–116 Differentiator, 132 Digital filters, 510, 525–526, 552 Digital signal processing, 3, 24, 443 Digital simulation, 131 Diodes, 10, 11 Dirac delta function, 49 Dirichlet conditions, 171, 202 Discrete cosine transform (DCT), 667–672 Discrete Fourier transform (DFT), 617–627, 635–667 Discrete frequency variable, 600 Discrete-time Fourier transform (DTFT): 600–605 definition, 600 properties: (See Tables) convolution in frequency, 609 convolution in time, 609 frequency shift, 607–608 linearity, 606 multiplication by n, 610 periodicity, 605–606 symmetry, 608 time reversal, 608 time shift, 606–607 z-transform relationship, 602–605 Discrete-time impulse function, 448 Discrete-time samples, 237 Discrete-time signal, 3, 443–482 Discrete-time system, 3, 443–482 Double-sideband modulation (DSB) (See Modulation: double-sideband modulation) Duration-bandwidth relationship, 211, 324 Dynamic system (See Systems) E Eigenvalue, 739 Eigenvector, 739 Index Elliptic filters, 290–294 Energy compaction, 669 Energy signal, 203, 255 Energy spectral density, 255–258 Energy spectral density estimate, 666–667 Energy transmission, 261–263 Equivalent operations, 449–450 Euler’s relation, 40, 43, 77, 723–729 Euler’s rule, 16, 445, 447 Even signal: continuous-time, 33, 34 discrete-time, 459–460 Exponential order, 339 F Fast Fourier transform, 627–635, 672 Federal Communications Commission (FCC), 307 Feedback systems (See Systems) Filters: bandpass, 274, 294–295, 324 bandstop, 274, 324 Butterworth (See Butterworth filters) Chebyschev (See Chebyschev filters) digital (See Digital filters) high-pass, 274, 277, 281, 324 ideal (See Ideal filters) low-pass, 274, 277, 278, 282–284, 324 noncausal, 276, 295 RC-lowpass, 282–284 real (See Real filters) transformation of, 294, 295–296 Filtering with DFT, 653–659 Final value theorem: Laplace transform, 358–359 z-transform, 562–264 Finite impulse response (FIR), 496 First difference, 449 Flat-top PAM (See Modulation: flat-top PAM) Forced response (See Response) Fourier series, 150–196 coefficients, 158–160 combined trigonometric form, 158 common signals, 168 exponential form, 158 periodic functions, 151–156 769 properties, 171–174 system analysis, 174–181 transformations, 181–186 trigonometric form, 158 Fourier transform: 197–273 definition, 197 energy density spectrum, 255–258 existence, 197 power and energy transmission, 261–264 power density spectrum, 258–261 properties (See Tables) convolution, 216–217 duality, 213–215 frequency differentiation, 227 frequency shifting, 217–219 linearity, 206–208, 218, 225 multiplication, 216 time differentiation, 219–224 time integration, 224–227 time scaling, 208–211 time shifting, 211–212 time transformation, 212–213 sampling, 237–243 time functions, 228–237 Frequency band assignments (See Federal Communications Commission) Frequency content of signal (See Frequency spectra) Frequency-division multiplexing, 315–317 Frequency response, 174, 243–252, 274, 282, 284, 285–286, 291–295, 378–379 Frequency-response function, 282 Frequency spectra, 161–171, 252–254 Functions (See also Signals) finite duration, 385, 584 left sided, 384, 385, 583 rational, 126, 253, 264, 531, 734 right sided, 384, 583 two sided, 384, 385, 584 Fundamental frequency, 36, 77, 152, 157 Fundamental matrix (See State transition matrix) Fundamental period, 35, 39, 152 G Generating function, 233 Gibbs phenomenon, 173 770 H Hamming window, 640 Hanning window, 640 Harmonic series, 157 Homogeneous equation, 115 Hybrid system (See Systems: hybrid) I Ideal filters, 274–281 Ideal sampling, 238, 299 Ideal time delay, 68,100, 466 Identity matrix (See Matrix: identity) Impulse function: continuous-time, 49–54 properties, 52 discrete time, 447–448, 613–617 Impulse response (See Response) Impulse sampling, 238–240, 303 Infinite impulse response (IIR), 500 Integrator (See Systems) Intermediate frequency, 315 Interpolating function, 301–303 Inverse transforms (See the Transform) Invertibility (See Systems) Iterative solution, 518 J JPEG, 599, 668 L Laplace transform, 335–397 bilateral, 336, 380 initial conditions, 362 periodic functions, 361 properties: (See Tables) convolution, 368 differentiation, 342, 353 final value, 358 initial value, 357 integration, 355 linearity, 348 Index multiplication by t, 356 time scaling, 359, 360 time-shifting, 350, 360 time transformation, 359 variable transformation, 359 region of convergence (ROC), 381 system response, 364 unilateral, 337 Leibnitz’s rule, 720 L’Hôpital’s rule, 721 Linear convolution with DFT, 646 Linear differential equations, 4, 9, 112, 730–733 Linearity: continuous-time systems, 75 discrete-time systems, 491 Linearization of systems, 9–10 Local oscillator, 310 LTI system (See Systems) M Magnitude frequency spectrum, 260, 274, 282 MATLAB, 22 Matrix, 737–743 addition, 741 adjoint, 740 cofactor, 740 determinants, 739 diagonal, 738 differentiation, 743 eigenvalues, 739 eigenvectors, 739 identity, 738 integration, 743 inverse, 740 minor, 740 multiplication, 741 properties, 739 symmetric, 738 trace, 739 transpose, 738 Matrix exponential, 417 Mean-square minimization, 154 Modeling, 1–4 difference equations, 17, 446 differential equations, 1, 9, 60, 112–119 Index Modes (See Systems) Modulation: delta modulation, 21 double-sideband modulation, 307, 312 DSB/SC-AM, 307 DSB/WC-AM, 312 flat-top PAM, 321 natural-top PAM, 317 pulse-amplitude modulation (PAM), 19, 167, 317–324 pulse-code modulation, 20 pulse-width modulation, 21 MPEG, 668 Multiplexing, 19, 315 Multivariable systems, 402, 681 N Natural response (See Response) Natural-top PAM, 317 Newton’s law, 1, 74 Nonanticipatory system, 68 Noncausal system, 276, 281 Nonlinear differential equations, Normalized average power, 203, 260, 283 Normalized frequency, 290 Numerical integration, 16, 445 Nyquist rate (Nyquist frequency), 240, 303 771 Particular solutions: difference equations, 510 differential equations, 114, 730–733 Passband, 274 Pendulum, 9, 10, 13 clock, 36, 44, 164 Periodic convolution (See Circular convolution) Periodic functions (See Signals) Periodic in frequency, 465 Periodogram spectrum estimate, 666 Phase angle, 205, 213 Phase delay, 284 Phase shift, 212, 284 Phase spectrum, 213 Phasor, rotating, 206, 208, 212 Physical systems (See Systems) Picture in a picture, 17–18 Pixel, 17 Pixel block, 668–669 Power series, 547, 572 Power signal, 203, 255 Power spectral density (PSD), 255, 259–265 Programming forms (See Simulation diagrams) Pulse-amplitude modulation (PAM) (See Modulation: pulse-amplitude modulation) Pulsed cosine, 229 R O Odd signal: continuous-time, 32 discrete-time, 459 Order of system, 113 Orthogonal functions, 160 Oscillators, 161 Overlap-add technique, 655 P Parseval’s theorem, 256 Partial-fraction expansions, 343, 344, 734–736 Ramp function, 50 Rational functions (See Functions) RC low-pass filter, 282 Real filters, 281–295 Recomposition equations, FFT, 629 Reconstruction, 299 Rectangular pulse (rect), 46, 47, 202 Region of convergence, 382, 586 discrete-time Fourier transform, 603 Laplace transform, 341 z-transform, 555, 559, 577, 580 Repeated poles, 373, 735–736 Resolvant, 410, 692–693 Response: complex-exponential inputs, 123, 527 forced, 115, 415, 512, 514, 576 772 Response—contd frequency, 378 impulse, 128, 496 initial conditions, 115, 362, 414 natural, 115, 118, 119, 512, 516, 576 steady-state, 118, 124, 174, 517, 529 transient, 118, 517 unforced, 118, 516 unit impulse, 91, 205, 345, 494 unit step, 96, 111, 509 zero-input, 118, 414, 516 zero-state, 118, 414, 517 Rotating phasor (See Phasor) S Sample-and-hold circuit, 322 Sampled-data signal (See Signals) Sampled-data system (See Systems) Sample estimate of autocorrelation, 666 Sample period, 237, 301 Sampling, 237, 301, 443, 533 ideal (See Ideal sampling) Sampling theorem, 240, 303 Sensors, 64, 68 Servomotor, 126 Sifting property, impulse function, 52 Signals: advanced, 69, 455 analog, 2, 6, 8, 23 aperiodic, 34 causal, 109, 477 complex exponential, 40, 123, 157, 463, 527 continuous-amplitude, 23, 444 continuous-time, 18, 23, 443 dc, 30 delayed, 27 digital, 443 discrete amplitude, 24, 444 discrete time, 3, 24, 443 discrete time exponential, 468 envelope, 44, 169 even and odd, 32, 459 exponential, 39, 468, 519 impulse representation, 90, 492 mathematical functions, 23, 54 periodic, 34–39, 151, 462 physical, 1–22, 23 Index sampled-data, 237, 299 sinusoidal, 43, 466 transformations, 24, 181, 450 Signum function (sgn), 203 Simulation, 131 analog, 131, 407 digital, 131 Simulation diagrams, 131, 403, 522, 686–691 direct form I, 133, 523, 681 direct form II, 133, 404, 523, 681 SIMULINK, 22, 288, 413, 699 Sinc function, 169, 200–202 Singularity functions, 45 Sinusoidal system response, 244 Spectrum analyzer, 181 Spectrum leakage distortion, 639 Square-law device, 279 Stability (See Systems) State equations: continuous-time, 398–442 solutions, 408 discrete-time, 681–717 solutions, 692 State transition matrix: continuous time, 410 properties, 418 discrete time, 692 properties, 699–701 Steady-state response (See Response) Step function, 45, 201, 446 Step response (See Response) Stopband, 274 Super-heterodyne, 315 Superposition, 74, 207, 479, 491 Switched cosine, 229 Symmetry, 213, 608 Synchronous detection, 310 Systems: analog, 1, 12–14, 18–21, 23, 24 analogous, 12, 13 causal, 68, 109, 368, 477, 506, 575 continuous-time, 1–14, 18, 23–88 data-acquisition, 21 differentiator, definition, 448, 472 discrete-time, 14–22, 443–490, 491–545 dynamic, 66, 475 feedback, 64 hybrid, 3, 24 Index integrator, 17 interconnecting, 61, 473 inverse, 67, 377, 477, 506, 578 linear, 74, 479, 491 LTI: continuous-time, 89, 107–112, 369 discrete time, 480, 491, 505 memory, 66, 475, 506 memoryless, 108 modes, 118, 519, 576 noncausal, 276 physical, 14–22, 112 sampled-data, 3, 24, 237 square-law device, 279 stability, 69, 110, 120, 375, 422, 478, 507 static, 66, 475 telephone, 19, 321 time invariant, 71, 478 transformation notation, 60, 472 System-identification, 403 T Tables: discrete-time Fourier transform, 604 properties, 611 Fourier transforms, 223, 265 properties, 207 Laplace transform, 341, 348 properties, 361 z-transform, bilateral, 558, 584 z-transform, unilateral, 552 properties, 567 Taylor’s series, 417 Telephone systems (See Systems) Thermistor, 67 Thermometer, 68 Time-average value, 224 Time constant, 21, 41, 468 Time-division multiplexing (TDM), 19, 319 Time duration, 227 Time invariant (See Systems) Transfer functions, 106, 125, 217, 244, 363, 420, 530, 568, 681, 697–700 Transforms: discrete cosine, 667–672 discrete Fourier, 617–627, 635–667 discrete-time Fourier, 600–605 773 Laplace, bilateral (See Laplace) Laplace, unilateral (See Laplace) z, bilateral (See z-transform) z, unilateral (See z-transform) Transformations: signal (See signals) similarity, 424–432, 700–705 properties, 430–432, 704–705 systems ( See Systems) Transform, linear, 339 Triangular pulse (tri), 226–227 U Undetermined coefficients, 514, 730 Unforced response (See Response) Unit impulse function (See impulse function) Unit impulse response (See Impulse response) Unit ramp function, 48, 49 Unit rectangular function, 46 Unit sample function, 448, 492 Unit step function (See Step function) V Vectors, 737 Video compression, 668 W Wavelength, 307 Weighting factor, 628, 630 Windowing function, 637 Z Zero-input response (See Response) Zero-order hold, 304 Zero padding, 650 z-transform: bilateral, 547, 580 properties: convolution, 566, 567 final value, 562, 567 frequency scaling, 559 774 z-transform—contd initial value, 562, 567 linearity, 559, 567 multiplication by n, 567 properties tables (See Tables) Index tables (See Tables) time scaling, 564, 567 time shifting, 561 regions of convergence, 580, 586 unilateral, 547–555, 560–561, 572, 580 ... systems and (2) discrete-time signals and systems Chapters through cover continuoustime signals and systems, while Chapters through 13 cover discrete-time signals and systems The material may be... 9.2 DISCRETE-TIME SIGNALS AND SYSTEMS Discrete-Time Signals and Systems 445 Unit Step and Unit Impulse Functions, Equivalent Operations, 449 447 Transformations of Discrete-Time Signals 443 450... Discrete-Time Signals 459 Even and Odd Signals, 459 Signals Periodic in n, 462 Signals Periodic in Ỉ, 465 9.4 Common Discrete-Time Signals 9.5 Discrete-Time Systems Interconnecting Systems, 9.6