Fundamentals of Signals and Systems Using the Web and MATLAB.pdf

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Cover Page

PROPERTIES OF THE FOURIER TRANSFORM

COMMON FOURIER TRANSFORM PAIRS

Contents

Preface

CHAPTER 1 Fundamental Concepts

1.1 Continuous-Time Signals

1.2 Discrete-Time Signals

1.3 Systems

1.4 Examples of Systems

1.5 Basic System Properties

1.6 Chapter Summary

Problems

CHAPTER 2 Time-Domain Models of Systems

2.1 Input/Output Representation of Discrete-Time Systems

2.2 Convolution of Discrete-Time Signals

2.3 Difference Equation Models

2.4 Differential Equation Models

2.5 Solution of Differential Equations

2.6 Convolution Representation of Continuous-Time Systems

2.7 Chapter Summary

Problems

CHAPTER 3 The Fourier Series and Fourier Transform

3.1 Representation of Signals in Terms of Frequency Components

3.2 Trigonometric Fourier Series

3.3 Complex Exponential Series

3.4 Fourier Transform

3.5 Spectral Content of Common Signals

3.6 Properties of the Fourier Transform

3.7 Generalized Fourier Transform

3.8 Application to Signal Modulation and Demodulation

3.9 Chapter Summary

Problems

CHAPTER 4 Fourier Analysis of Discrete-Time Signals

4.1 Discrete-Time Fourier Transform

4.2 Discrete Fourier Transform

4.3 DFT of Truncated Signals

4.4 FFT Algorithm

4.5 Application to Data Analysis

4.6 Chapter Summary

Problems

CHAPTER 5 Fourier Analysis of Systems

5.1 Fourier Analysis of Continuous-Time Systems

5.2 Response to Periodic and Nonperiodic Inputs

5.3 Analysis of Ideal Filters

5.4 Sampling

5.5 Fourier Analysis of Discrete-Time Systems

5.6 Application to Lowpass Digital Filtering

5.7 Chapter Summary

Problems

CHAPTER 6 The Laplace Transform and the Transfer Function Representation

6.1 Laplace Transform of a Signal

6.2 Properties of the Laplace Transform

6.3 Computation of the Inverse Laplace Transform

6.4 Transform of the Input/Output Differential Equation

6.5 Transform of the Input/Output Convolution Integral

6.6 Direct Construction of the Transfer Function

6.7 Chapter Summary

Problems

CHAPTER 7 The z-Transform and Discrete-Time Systems

7.1 z-Transform of a Discrete-Time Signal

7.2 Properties of the z-Transform

7.3 Computation of the Inverse z-Transform

7.4 Transfer Function Representation

7.5 System Analysis Using the Transfer Function Representation

7.6 Chapter Summary

Problems

CHAPTER 8 Analysis of Continuous-Time Systems by Use of the Transfer Function Representation

8.1 Stability and the Impulse Response

8.2 Routh-Hurwitz Stability Test

8.3 Analysis of the Step Response

8.4 Response to Sinusoids and Arbitrary Inputs

8.5 Frequency Response Function

8.6 Causal Filters

8.7 Chapter Summary

Problems

CHAPTER 9 Application to Control

9.1 Introduction to Control

9.2 Tracking Control

9.3 Root Locus

9.4 Application to Control System Design

9.5 Chapter Summary

Problems

CHAPTER 10 Design of Digital Filters and Controllers

10.1 Discretization

10.2 Design of IIR Filters

10.3 Design of IIR Filters Using MATLAB

10.4 Design of FIR Filters

10.5 Design of Digital Controllers

10.6 Chapter Summary

Problems

CHAPTER 11 State Representation

11.1 State Model

11.2 Construction of State Models

11.3 Solution of State Equations

11.4 Discrete-Time Systems

11.5 Equivalent State Representations

11.6 Discretization of State Model

11.7 Chapter Summary

Problems

Appendix A Brief Review of Complex Variables

Appendix B Brief Review of Matrices

Bibliography

Index

COMMON LAPLACE TRANSFORM PAIRS

PROPERTIES OF THE LAPLACE TRANSFORM

Back Cover

,_~-__ '_'_· __ *P -·f '1' ( 1hnsform Pair/Property ax(t) + bv(t) H aX(w) + bV(w) x(t- c) HX(w)e-Jwc x(at) H 1 (w) X - a a a > 0 x(-t) H X(-w) = X(w) t"x(I)Hj" lc X(w) n=1,2, ... l" cw" x(t)ei"'"1 H X(w - w0) w0 real x(t) sin w0t H f [X(w + w0) - X(w - w 0)] x(t) cos w 0t H ~[X(w + w0) + X(w - w 0)] d" -111 x(t)H(jw)"X(w) n = 1,2, ... ct r L, x(A-) dA H fw X(w) + .n:X(O)o(w) 1 PROPERTIES OF THE FOURIER TRANSFORM Property Linearity Right or left shift in time Time scaling Time reversal Multiplication by a power oft Multiplication by a complex exponential Multiplication by sin wot Multiplication by cos wot Differentiation in the time domain Integration Convolution in the time domain x(t) * v(t) HX(w)V(w) Multiplication in the time domain Parseval's theorem Special case of Parseval's theorem Duality x(t)v(t) H - - X(w) * V(w) 1 2.n: r, x(t)v(t) dt r, x2(t) dt = 1 Loo X(t) H 2.n:x( -w) l r X(w)V(w) dw n )_'lJ 2 {% iXCwW dw 2.n: Loo

COMMON FOURIER TRANSFORM PAIRS l, -oo 0 1 JW + b ejwot H 2nb(w - w 0), w 0 any real number '[OJ Pr t H rsn1C 2n ( ) • . ( ) r smc- H 2npr w rt 2n ( r 1 - 21tl)p (t) H _'£ sinc2 ('w) ~ sinc2 ( ;~) H 2n (1 - 21:1)Pr(w) 4n 2 r cos w0t H n[c)(w + w0) + b(w - w0)] cos (w(l + 0) H n[e-j0b(w + uJ0) + ej0b(w - w0)] sin w0t H jn[c)(w + w0) - <)(w sin (w(/ + 0) H jn[e-jl)b(w + w0) - ej0b(o! - w0)] w0)]

- Library of' Congress Cataloging-in-Publication Data on File Vice President and Editorial Director, ECS: Marcia J. Horton Acquisitions Editor: Michael McDonald Associate Editor: Alice Dworkin Executive Managing Editor: Vince O'Brien Managing Editor: David A. George Production Editor: Scott Disanno Director of Creative Services: Paul Be!fanti Art Director: Jayne Conte Cover Design: Bruce Kenselaar Art Editor: Greg Dulles Manufacturing Manager: Alexis Heydt-Long Manufacturing Buyer: Lisa McDowell Marketing Manager: Tim Galligan © 2007 by Pearson Education, Inc. Pearson Prentice Hall Upper Saddle River, New Jersey 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 HallTM is a trademark of Pearson Education, Inc. MATLAB is a registered trademark of The Math Works, Inc., 3 Apple Hill Drive Natick, MA. LEGO, MINDSTORMS, and the LEGO logo are trademarks of The LEGO Group, DK-7190, Billund, Denmark. 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 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 with, or arising out of, the furnishing, performance, or use of these programs. Printed in the United States of America 1098765432 ISBN 0-13-168737-9 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 Educaei6n de Mexico, S.A. de C.V. Pearson Education-Japan, Tokyo Pearson Education Malaysia, Pte. Ltd. Pearson Education, Inc., Upper Saddle River, New Jersey

- Contents Preface ix CHAPTER 1 Fundamental Concepts 1 1.1 1.2 1.3 1.4 1.5 1.6 21 Continuous-Time Signals Discrete-Time Signals Systems Examples of Systems Basic System Properties Chapter Summary Problems 37 37 1 11 24 31 CHAPTER 2 Time-Domain Models of Systems 44 Input/Output Representation of Discrete-Time Systems 2.1 2.2 Convolution of Discrete-Time Signals 2.3 Difference Equation Models 2.4 Differential Equation Models 2.5 Solution of Differential Equations 2.6 Convolution Representation of Continuous-Time Systems 2.7 Chapter Summary 55 64 49 69 84 44 75 Problems 85 CHAPTER 3 The Fourier Series and Fourier Transform 96 3.1 Representation of Signals in Terms of Frequency Components 3.2 Trigonometric Fourier Series 3.3 Complex Exponential Series 3.4 Fourier Transform 101 108 114 96 y

r vi Contents 3.5 Spectral Content of Common Signals 3.6 Properties of the Fourier Transform 3.7 Generalized Fourier Transform 3.8 Application to Signal Modulation and Demodulation 3.9 Chapter Summary 154 Problems 152 126 141 122 144 CHAPTER 4 Fourier Analysis of Discrete-Time Signals 166 166 4.1 Discrete-Time Fourier Transform 4.2 Discrete Fourier Transform 4.3 DFT of Truncated Signals 4.4 FFT Algorithm 4.5 Application to Data Analysis 4.6 Chapter Summary 217 Problems 216 195 179 188 206 CHAPTER 5 Fourier Analysis of Systems 222 5.1 Fourier Analysis of Continuous-Time Systems 5.2 Response to Periodic and Nonperiodic Inputs 5.3 Analysis of Ideal Filters 5.4 Sampling 5.5 Fourier Analysis of Discrete-Time Systems 5.6 Application to Lowpass Digital Filtering 5.7 Chapter Summary 263 Problems 262 242 237 249 254 222 228 CHAPTER 6 The Laplace Transform and the Transfer Function 6.1 6.2 6.3 6.4 6.5 6.6 6.7 279 Representation 279 Laplace Transform of a Signal Properties of the Laplace Transform Computation of the Inverse Laplace Transform Transform of the Input/Output Differential Equation Transform of the Input/Output Convolution Integral 330 Direct Construction of the Transfer Function Chapter Summary 342 Problems 285 341 296 316 323 CHAPTER 7 The z-Transform and Discrete-Time Systems 352 z-Transform of a Discrete-Time Signal 357 7.1 7.2 Properties of the z-Transform 7.3 Computation of the Inverse z-Transform 352 367

Contents vii 377 7.4 ll"ansfer Function Representation 7.5 System Analysis Using the Transfer 388 Function Representation 400 7.6 Chapter Summary 401 Problems CHAPTER 8 Analysis of Continuous-Time Systems by Use of the Transfer Function Representation 411 411 Stability and the Impulse Response 8.1 8.2 Routh-Hurwitz Stability Test 8.3 Analysis of the Step Response 8.4 Response to Sinusoids and Arbitrary Inputs 8.5 Frequency Response Function 8.6 Causal Filters 8.7 Chapter Summary 479 414 419 Problems 478 443 463 437 CHAPTER 9 Application to Control 489 489 Introduction to Control 9.1 9.2 Tracking Control 9.3 Root Locus 9.4 Application to Control System Design 9.5 Chapter Summary 526 Problems 508 497 525 516 CHAPTER 10 Design of Digital Filters and Controllers 536 536 543 10.1 Discretization 10.2 Design of IIR Filters 10.3 Design of IIR Filters Using MATLAB 556 10.4 Design of FIR Filters 10.5 Design of Digital Controllers 10.6 Chapter Summary 578 Problems 568 577 549 CHAPTER 11 State Representation 584 584 11.1 State Model 11.2 Construction of State Models 11.3 Solution of State Equations 11.4 Discrete-Time Systems 604 587 595

viii Contents 11.5 Equivalent State Representations 11.6 Discretization of State Model 11.7 Chapter Summary 623 Problems 622 612 618 Appendix A Appendix B Bibliography Index 645 Brief Review of Complex Variables 633 Brief Review of Matrices 638 644

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