Signals and Systems with MATLAB Applications Second Edition Steven T. Karris www.orchardpublications.com
Signals and Systems with MATLAB Applications, Second Edition Copyright 2003. All rights reserved. Printed in the United States of America. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. Direct all inquiries to, 39510 Paseo Padre Parkway, Fremont, California 94538 Product and corporate names are trademarks or registered trademarks of the Microsoft Corporation and The MathWorks Inc. They are used only for identification and explanation, without intent to infringe. Library of Congress Cataloging-in-Publication Data Library of Congress Control Number: 2003091595 ISBN 0-9709511-8-3 Copyright TX 5-471-562
Preface This text contains a comprehensive discussion on continuous and discrete time signals and systems with many MATLAB examples. It is written for junior and senior electrical engineering students, and for self-study by working professionals. The prerequisites are a basic course in differential and integral calculus, and basic electric circuit theory. This book can be used in a two-quarter, or one semester course. This author has taught the subject material for many years at San Jose State University, San Jose, California, and was able to cover all material in 16 weeks, with 2½ lecture hours per week. To get the most out of this text, it is highly recommended that Appendix A is thoroughly reviewed. This appendix serves as an introduction to MATLAB, and is intended for those who are not familiar with it. The Student Edition of MATLAB is an inexpensive, and yet a very powerful software package; it can be found in many college bookstores, or can be obtained directly from The MathWorks Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 Phone: 508 647-7000, Fax: 508 647-7001 http://www.mathworks.com e-mail: info@mathwork.com The elementary signals are reviewed in Chapter 1 and several examples are presented. The intent of this chapter is to enable the reader to express any waveform in terms of the unit step function, and subsequently the derivation of the Laplace transform of it. Chapters 2 through 4 are devoted to Laplace transformation and circuit analysis using this transform. Chapter 5 discusses the state variable method, and Chapter 6 the impulse response. Chapters 7 and 8 are devoted to Fourier series and transform respectively. Chapter 9 introduces discrete-time signals and the Z transform. Considerable time was spent on Chapter 10 to present the Discrete Fourier transform and FFT with the simplest possible explanations. Chapter 11 contains a thorough discussion to analog and digital filters analysis and design procedures. As mentioned above, Appendix A is an introduction to MATLAB. Appendix B contains a review of complex numbers, and Appendix C discusses matrices. New to the Second Edition This is an refined revision of the first edition. The most notable changes are chapter-end summaries, and detailed solutions to all exercises. The latter is in response to many students and working professionals who expressed a desire to obtain the author s solutions for comparison with their own. The author has prepared more exercises and they are available with their solutions to those instructors who adopt this text for their class. The chapter-end summaries will undoubtedly be a valuable aid to instructors for the preparation of presentation material.
The last major change is the improvement of the plots generated by the latest revisions of the MATLAB Student Version, Release 13. Fremont, California www.orchardpublications.com info@orchardpublications.com 2
Table of Contents Chapter 1 Elementary Signals Signals Described in Math Form...1-1 The Unit Step Function...1-2 The Unit Ramp Function...1-10 The Delta Function...1-12 Sampling Property of the Delta Function...1-12 Sifting Property of the Delta Function...1-13 Higher Order Delta Functions...1-15 Summary...1-19 Exercises...1-20 Solutions to Exercises...1-21 Chapter 2 The Laplace Transformation Definition of the Laplace Transformation...2-1 Properties of the Laplace Transform...2-2 The Laplace Transform of Common Functions of Time...2-12 The Laplace Transform of Common Waveforms...2-23 Summary...2-29 Exercises...2-34 Solutions to Exercises...2-37 Chapter 3 The Inverse Laplace Transformation The Inverse Laplace Transform Integral...3-1 Partial Fraction Expansion...3-1 Case where Fs ( ) is Improper Rational Function ( m n)... 3-13 Alternate Method of Partial Fraction Expansion... 3-15 Summary... 3-18 Signals and Systems with MATLAB Applications, Second Edition i
Exercises...3-20 Solutions to Exercises...3-22 Chapter 4 Circuit Analysis with Laplace Transforms Circuit Transformation from Time to Complex Frequency...4-1 Complex Impedance Zs ( )...4-8 Complex Admittance Ys ( )...4-10 Transfer Functions...4-13 Summary...4-16 Exercises...4-18 Solutions to Exercises...4-21 Chapter 5 State Variables and State Equations Expressing Differential Equations in State Equation Form...5-1 Solution of Single State Equations...5-7 The State Transition Matrix...5-9 Computation of the State Transition Matrix...5-11 Eigenvectors...5-18 Circuit Analysis with State Variables...5-22 Relationship between State Equations and Laplace Transform...5-28 Summary...5-35 Exercises...5-39 Solutions to Exercises...5-41 Chapter 6 The Impulse Response and Convolution The Impulse Response in Time Domain...6-1 Even and Odd Functions of Time...6-5 Convolution...6-7 Graphical Evaluation of the Convolution Integral...6-8 Circuit Analysis with the Convolution Integral... 6-18 Summary... 6-20 ii Signals and Systems with MATLAB Applications, Second Edition
Exercises... 6-22 Solutions to Exercises... 6-24 Chapter 7 Fourier Series Wave Analysis...7-1 Evaluation of the Coefficients...7-2 Symmetry...7-7 Waveforms in Trigonometric Form of Fourier Series... 7-11 Gibbs Phenomenon... 7-24 Alternate Forms of the Trigonometric Fourier Series... 7-25 Circuit Analysis with Trigonometric Fourier Series...7-29 The Exponential Form of the Fourier Series... 7-31 Line Spectra... 7-35 Computation of RMS Values from Fourier Series... 7-40 Computation of Average Power from Fourier Series... 7-42 Numerical Evaluation of Fourier Coefficients... 7-44 Summary... 7-48 Exercises... 7-51 Solutions to Exercises... 7-53 Chapter 8 The Fourier Transform Definition and Special Forms...8-1 Special Forms of the Fourier Transform...8-2 Properties and Theorems of the Fourier Transform...8-9 Fourier Transform Pairs of Common Functions...8-17 Finding the Fourier Transform from Laplace Transform...8-25 Fourier Transforms of Common Waveforms...8-27 Using MATLAB to Compute the Fourier Transform...8-33 The System Function and Applications to Circuit Analysis...8-34 Summary...8-41 Exercises...8-47 Solutions to Exercises...8-49 Signals and Systems with MATLAB Applications, Second Edition iii
Chapter 9 Discrete Time Systems and the Z Transform Definition and Special Forms...9-1 Properties and Theorems of the Z Tranform...9-3 The Z Transform of Common Discrete Time Functions...9-11 Computation of the Z transform with Contour Integration...9-20 Transformation Between s and z Domains...9-22 The Inverse Z Transform...9-24 The Transfer Function of Discrete Time Systems...9-38 State Equations for Discrete Time Systems...9-43 Summary...9-47 Exercises...9-52 Solutions to Exercises...9-54 Chapter 10 The DFT and the FFT Algorithm The Discrete Fourier Transform (DFT)...10-1 Even and Odd Properties of the DFT...10-8 Properties and Theorems of the DFT... 10-10 The Sampling Theorem... 10-13 Number of Operations Required to Compute the DFT... 10-16 The Fast Fourier Transform (FFT)... 10-17 Summary... 10-28 Exercises... 10-31 Solutions to Exercises... 10-33 Chapter 11 Analog and Digital Filters Filter Types and Classifications... 11-1 Basic Analog Filters... 11-2 Low-Pass Analog Filters... 11-7 Design of Butterworth Analog Low-Pass Filters...11-11 Design of Type I Chebyshev Analog Low-Pass Filters...11-22 Other Low-Pass Filter Approximations...11-34 High-Pass, Band-Pass, and Band-Elimination Filters...11-39 iv Signals and Systems with MATLAB Applications, Second Edition
Digital Filters... 11-49 Summary... 11-69 Exercises... 11-73 Solutions to Exercises... 11-79 Appendix A Introduction to MATLAB MATLAB and Simulink...A-1 Command Window...A-1 Roots of Polynomials...A-3 Polynomial Construction from Known Roots...A-4 Evaluation of a Polynomial at Specified Values...A-6 Rational Polynomials...A-8 Using MATLAB to Make Plots...A-10 Subplots...A-18 Multiplication, Division and Exponentiation...A-18 Script and Function Files...A-25 Display Formats...A-30 Appendix B Review of Complex Numbers Definition of a Complex Number... B-1 Addition and Subtraction of Complex Numbers...B-2 Multiplication of Complex Numbers... B-3 Division of Complex Numbers... B-4 Exponential and Polar Forms of Complex Numbers... B-4 Appendix C Matrices and Determinants Matrix Definition... C-1 Matrix Operations... C-2 Special Forms of Matrices... C-5 Determinants... C-9 Minors and Cofactors...C-12 Signals and Systems with MATLAB Applications, Second Edition v
Cramer s Rule...C-16 Gaussian Elimination Method...C-19 The Adjoint of a Matrix...C-20 Singular and Non-Singular Matrices...C-21 The Inverse of a Matrix...C-21 Solution of Simultaneous Equations with Matrices...C-23 Exercises...C-30 vi Signals and Systems with MATLAB Applications, Second Edition
Chapter 1 Elementary Signals T his chapter begins with a discussion of elementary signals that may be applied to electric networks. The unit step, unit ramp, and delta functions are introduced. The sampling and sifting properties of the delta function are defined and derived. Several examples for expressing a variety of waveforms in terms of these elementary signals are provided. 1.1 Signals Described in Math Form Consider the network of Figure 1.1 where the switch is closed at time t = 0. R v t = 0 S + + v out open terminals Figure 1.1. A switched network with open terminals. We wish to describe v out in a math form for the time interval < t < +. To do this, it is convenient to divide the time interval into two parts, < t < 0, and 0 < t <. For the time interval < t < 0, the switch is open and therefore, the output voltage v out is zero. In other words, v out = 0 for < t < 0 For the time interval 0< t<, the switch is closed. Then, the input voltage v S appears at the output, i.e., v out = v S for 0< t< Combining (1.1) and (1.2) into a single relationship, we get (1.1) (1.2) 0 < t < 0 v out = v S 0< t< (1.3) We can express (1.3) by the waveform shown in Figure 1.2. Signals and Systems with MATLAB Applications, Second Edition 1-1
Chapter 1 Elementary Signals vs v out 0 t v out Figure 1.2. Waveform for as defined in relation (1.3) The waveform of Figure 1.2 is an example of a discontinuous function. A function is said to be discontinuous if it exhibits points of discontinuity, that is, the function jumps from one value to another without taking on any intermediate values. 1.2 The Unit Step Function u 0 () t A well-known discontinuous function is the unit step function u * 0 () t that is defined as u 0 () t = 0 t < 0 1 t > 0 (1.4) It is also represented by the waveform of Figure 1.3. 1 0 u 0 () t t Figure 1.3. Waveform for u 0 () t In the waveform of Figure 1.3, the unit step function u 0 () t changes abruptly from 0 to 1 at t = 0. But if it changes at t = t 0 instead, it is denoted as u 0 ( t t 0 ). Its waveform and definition are as shown in Figure 1.4 and relation (1.5). 1 0 t 0 u 0 ( t t 0 ) t Figure 1.4. Waveform for u 0 ( t t 0 ) * In some books, the unit step function is denoted as ut (), that is, without the subscript 0. In this text, however, we will reserve the ut () designation for any input when we discuss state variables in a later chapter. 1-2 Signals and Systems with MATLAB Applications, Second Edition