Signals and Systems - DedicatedTeacher.com · Signals and Systems with MATLAB® Applications Second Edition Steven T. Karris Orchard Publications

Signals and Systems with MATLAB® Applications

Second Edition

Steven T. Karris

Orchard Publicationswww.orchardpublications.com

Signals and Systems with MATLAB Applications, Second Edition

Copyright © 2003 Orchard Publications. All rights reserved. Printed in the United States of America. No part of thispublication 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 Orchard Publications, 39510 Paseo Padre Parkway, Fremont, California 94538

Product and corporate names are trademarks or registered trademarks of the Microsoft™ Corporation and TheMathWorks™ 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 systemswith 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 andintegral calculus, and basic electric circuit theory.

This book can be used in a two-quarter, or one semester course. This author has taught the subjectmaterial for many years at San Jose State University, San Jose, California, and was able to cover allmaterial 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 familiarwith it. The Student Edition of MATLAB is an inexpensive, and yet a very powerful softwarepackage; it can be found in many college bookstores, or can be obtained directly from

The MathWorks™ Inc., 3 Apple Hill Drive , Natick, MA 01760-2098Phone: 508 647-7000, Fax: 508 647-7001http://www.mathworks.come-mail: [email protected]

The elementary signals are reviewed in Chapter 1 and several examples are presented. The intent ofthis chapter is to enable the reader to express any waveform in terms of the unit step function, andsubsequently the derivation of the Laplace transform of it. Chapters 2 through 4 are devoted toLaplace transformation and circuit analysis using this transform. Chapter 5 discusses the statevariable method, and Chapter 6 the impulse response. Chapters 7 and 8 are devoted to Fourier seriesand 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 withthe simplest possible explanations. Chapter 11 contains a thorough discussion to analog and digitalfilters analysis and design procedures. As mentioned above, Appendix A is an introduction toMATLAB. 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 workingprofessionals 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 thoseinstructors who adopt this text for their class.

The chapter-end summaries will undoubtedly be a valuable aid to instructors for the preparation ofpresentation material.

The last major change is the improvement of the plots generated by the latest revisions of theMATLAB® Student Version, Release 13.

Orchard PublicationsFremont, Californiawww.orchardpublications.cominfo@orchardpublications.com

2

Table of Contents

Chapter 1

Elementary SignalsSignals Described in Math Form.................................................................................................................1-1The Unit Step Function ................................................................................................................................1-2The Unit Ramp Function ...........................................................................................................................1-10The Delta Function .....................................................................................................................................1-12Sampling Property of the Delta Function................................................................................................1-12Sifting Property of the Delta Function.....................................................................................................1-13Higher Order Delta Functions...................................................................................................................1-15Summary........................................................................................................................................................1-19Exercises........................................................................................................................................................1-20Solutions to Exercises .................................................................................................................................1-21

Chapter 2

The Laplace TransformationDefinition of the Laplace Transformation................................................................................................. 2-1Properties of the Laplace Transform.......................................................................................................... 2-2The Laplace Transform of Common Functions of Time .....................................................................2-12The Laplace Transform of Common Waveforms..................................................................................2-23Summary........................................................................................................................................................2-29Exercises........................................................................................................................................................2-34Solutions to Exercises .................................................................................................................................2-37

Chapter 3

The Inverse Laplace TransformationThe Inverse Laplace Transform Integral....................................................................................................3-1Partial Fraction Expansion ...........................................................................................................................3-1Case where is Improper Rational Function ( )................................................................... 3-13Alternate Method of Partial Fraction Expansion................................................................................... 3-15Summary....................................................................................................................................................... 3-18

F s( ) m n≥

Signals and Systems with MATLAB Applications, Second Edition iOrchard Publications

Exercises .......................................................................................................................................................3-20Solutions to Exercises .................................................................................................................................3-22

Chapter 4

Circuit Analysis with Laplace TransformsCircuit Transformation from Time to Complex Frequency................................................................................................................... 4-1Complex Impedance ...........................................................................................................................4-8Complex Admittance ........................................................................................................................4-10Transfer Functions ......................................................................................................................................4-13Summary .......................................................................................................................................................4-16Exercises .......................................................................................................................................................4-18Solutions to Exercises .................................................................................................................................4-21

Chapter 5

State Variables and State EquationsExpressing Differential Equations in State Equation Form...................................................................5-1Solution of Single State Equations..............................................................................................................5-7The State Transition Matrix .........................................................................................................................5-9Computation of the State Transition Matrix ...........................................................................................5-11Eigenvectors .................................................................................................................................................5-18Circuit Analysis with State Variables ........................................................................................................5-22Relationship between State Equations and Laplace Transform...........................................................5-28Summary .......................................................................................................................................................5-35Exercises .......................................................................................................................................................5-39Solutions to Exercises .................................................................................................................................5-41

Chapter 6

The Impulse Response and ConvolutionThe Impulse Response in Time Domain...................................................................................................6-1Even and Odd Functions of Time..............................................................................................................6-5Convolution....................................................................................................................................................6-7Graphical Evaluation of the Convolution Integral ..................................................................................6-8Circuit Analysis with the Convolution Integral...................................................................................... 6-18Summary ...................................................................................................................................................... 6-20

Z s( )Y s( )

ii Signals and Systems with MATLAB Applications, Second EditionOrchard Publications

Exercises....................................................................................................................................................... 6-22Solutions to Exercises ................................................................................................................................ 6-24

Chapter 7

Fourier SeriesWave Analysis.................................................................................................................................................7-1Evaluation of the Coefficients .....................................................................................................................7-2Symmetry.........................................................................................................................................................7-7Waveforms in Trigonometric Form of Fourier Series .......................................................................... 7-11Gibbs Phenomenon.................................................................................................................................... 7-24Alternate Forms of the Trigonometric Fourier Series .......................................................................... 7-25Circuit Analysis with Trigonometric Fourier Series .............................................................................. 7-29The Exponential Form of the Fourier Series ......................................................................................... 7-31Line Spectra ................................................................................................................................................. 7-35Computation of RMS Values from Fourier Series ................................................................................ 7-40Computation of Average Power from Fourier Series ........................................................................... 7-42Numerical Evaluation of Fourier Coefficients ....................................................................................... 7-44Summary....................................................................................................................................................... 7-48Exercises....................................................................................................................................................... 7-51Solutions to Exercises ................................................................................................................................ 7-53

Chapter 8

The Fourier TransformDefinition and Special Forms ...................................................................................................................... 8-1Special Forms of the Fourier Transform ................................................................................................... 8-2Properties and Theorems of the Fourier Transform................................................................................ 8-9Fourier Transform Pairs of Common Functions ...................................................................................8-17Finding the Fourier Transform from Laplace Transform.....................................................................8-25Fourier Transforms of Common Waveforms.........................................................................................8-27Using MATLAB to Compute the Fourier Transform ...........................................................................8-33The System Function and Applications to Circuit Analysis..................................................................8-34Summary........................................................................................................................................................8-41Exercises........................................................................................................................................................8-47Solutions to Exercises .................................................................................................................................8-49

Signals and Systems with MATLAB Applications, Second Edition iiiOrchard Publications

iv Signals and Systems with MATLAB Applications, Second EditionOrchard Publications

Chapter 9

Discrete Time Systems and the Z TransformDefinition and Special Forms ......................................................................................................................9-1Properties and Theorems of the Z Tranform ..........................................................................................9-3The Z Transform of Common Discrete Time Functions....................................................................9-11Computation of the Z transform with Contour Integration ...............................................................9-20Transformation Between and Domains...........................................................................................9-22The Inverse Z Transform..........................................................................................................................9-24The Transfer Function of Discrete Time Systems .................................................................................9-38State Equations for Discrete Time Systems ............................................................................................9-43Summary .......................................................................................................................................................9-47Exercises .......................................................................................................................................................9-52Solutions to Exercises .................................................................................................................................9-54

Chapter 10

The DFT and the FFT AlgorithmThe Discrete Fourier Transform (DFT) ..................................................................................................10-1Even and Odd Properties of the DFT.....................................................................................................10-8Properties and Theorems of the DFT................................................................................................... 10-10The Sampling Theorem ........................................................................................................................... 10-13Number of Operations Required to Compute the DFT.................................................................... 10-16The Fast Fourier Transform (FFT) ....................................................................................................... 10-17Summary .................................................................................................................................................... 10-28Exercises .................................................................................................................................................... 10-31Solutions to Exercises .............................................................................................................................. 10-33

Chapter 11

Analog and Digital FiltersFilter Types and Classifications ................................................................................................................ 11-1Basic Analog Filters.................................................................................................................................... 11-2Low-Pass Analog Filters............................................................................................................................ 11-7Design of Butterworth Analog Low-Pass Filters ...............................................................................11-11Design of Type I Chebyshev Analog Low-Pass Filters......................................................................11-22Other Low-Pass Filter Approximations................................................................................................11-34High-Pass, Band-Pass, and Band-Elimination Filters.........................................................................11-39

s z

Digital Filters ............................................................................................................................................. 11-49Summary..................................................................................................................................................... 11-69Exercises..................................................................................................................................................... 11-73Solutions to Exercises .............................................................................................................................. 11-79

Appendix A

Introduction to MATLAB®MATLAB® and Simulink® ........................................................................................................................A-1Command Window.......................................................................................................................................A-1Roots of Polynomials ...................................................................................................................................A-3Polynomial Construction from Known Roots.........................................................................................A-4Evaluation of a Polynomial at Specified Values.......................................................................................A-6Rational Polynomials ....................................................................................................................................A-8Using MATLAB to Make Plots ................................................................................................................A-10Subplots ........................................................................................................................................................A-18Multiplication, Division and Exponentiation .........................................................................................A-18Script and Function Files ...........................................................................................................................A-25Display Formats ..........................................................................................................................................A-30

Appendix B

Review of Complex NumbersDefinition of a Complex Number.............................................................................................................. B-1Addition and Subtraction of Complex Numbers..................................................................................... B-2Multiplication of Complex Numbers......................................................................................................... B-3Division of Complex Numbers .................................................................................................................. B-4Exponential and Polar Forms of Complex Numbers ............................................................................. B-4

Appendix C

Matrices and DeterminantsMatrix Definition .......................................................................................................................................... C-1Matrix Operations......................................................................................................................................... C-2Special Forms of Matrices ........................................................................................................................... C-5Determinants ................................................................................................................................................. C-9Minors and Cofactors.................................................................................................................................C-12

Signals and Systems with MATLAB Applications, Second Edition vOrchard Publications

Cramer’s Rule ..............................................................................................................................................C-16Gaussian Elimination Method..................................................................................................................C-19The Adjoint of a Matrix.............................................................................................................................C-20Singular and Non-Singular Matrices ........................................................................................................C-21The Inverse of a Matrix .............................................................................................................................C-21Solution of Simultaneous Equations with Matrices ..............................................................................C-23Exercises ......................................................................................................................................................C-30

vi Signals and Systems with MATLAB Applications, Second EditionOrchard Publications

Chapter 1

Elementary Signals

his chapter begins with a discussion of elementary signals that may be applied to electric net-works. The unit step, unit ramp, and delta functions are introduced. The sampling and siftingproperties of the delta function are defined and derived. Several examples for expressing a vari-

ety 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 .

Figure 1.1. A switched network with open terminals.

We wish to describe in a math form for the time interval . To do this, it is conve-nient to divide the time interval into two parts, , and .

For the time interval , the switch is open and therefore, the output voltage is zero. Inother words,

(1.1)

For the time interval , the switch is closed. Then, the input voltage appears at the output,i.e.,

(1.2)

Combining (1.1) and (1.2) into a single relationship, we get

(1.3)

We can express (1.3) by the waveform shown in Figure 1.2.

T

t 0=

+−+

−vout

vSt 0=

R

open terminals

vout ∞ t +∞< <–

∞ t 0< <– 0 t ∞< <

∞ t 0< <– vout

vout 0 for ∞ t 0 < <–=

0 t ∞< < vS

vout vS for 0 t ∞ < <=

vout0 ∞– t 0< <vS 0 t ∞< <⎩

⎨⎧

=

Signals and Systems with MATLAB Applications, Second Edition 1-1Orchard Publications

Chapter 1 Elementary Signals

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 dis-continuous if it exhibits points of discontinuity, that is, the function jumps from one value to anotherwithout taking on any intermediate values.

1.2 The Unit Step Function

A well-known discontinuous function is the unit step function * that is defined as

(1.4)

It is also represented by the waveform of Figure 1.3.

Figure 1.3. Waveform for

In the waveform of Figure 1.3, the unit step function changes abruptly from to at .But if it changes at instead, it is denoted as . Its waveform and definition are asshown in Figure 1.4 and relation (1.5).

Figure 1.4. Waveform for

* In some books, the unit step function is denoted as , that is, without the subscript 0. In this text, however, wewill reserve the designation for any input when we discuss state variables in a later chapter.

0

voutvS

t

vout

u0 t( )

u0 t( )

u t( )u t( )

u0 t( )0 t 0<1 t 0>⎩

⎨⎧

=

u0 t( )

0

1

t

u0 t( )

u0 t( ) 0 1 t 0=

t t0= u0 t t0–( )

1

t00

u0 t t0–( )t

u0 t t0–( )

1-2 Signals and Systems with MATLAB Applications, Second EditionOrchard Publications