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This text includes the following chapters and appendices: • Elementary Signals • The Laplace Transformation • The Inverse Laplace Transformation • Circuit Analysis with Laplace Transfor

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Signals and Systems

with MATLAB® Applications

Second Edition

Steven T Karris

Students and working professionals will find

Signals and Systems with MATLAB® Applications,

Second Edition, to be a concise and easy-to-learn text It provides complete, clear, and detailed expla- nations of the principal analog and digital signal processing concepts and analog and digital filter design illustrated with numerous practical examples.

This text includes the following chapters and appendices:

• Elementary Signals • The Laplace Transformation • The Inverse Laplace Transformation

• Circuit Analysis with Laplace Transforms • State Variables and State Equations • The Impulse Response and Convolution • Fourier Series • The Fourier Transform • Discrete Time Systems and the Z Transform • The DFT and The FFT Algorithm • Analog and Digital Filters

• Introduction to MATLAB • Review of Complex Numbers • Review of Matrices and Determinants Each chapter contains numerous practical applications supplemented with detailed instructions for using MATLAB to obtain quick solutions.

Steven T Karris is the president and founder of Orchard Publications He earned a bachelors degree in electrical engineering at Christian Brothers University, Memphis, Tennessee, a mas- ters degree in electrical engineering at Florida Institute of Technology, Melbourne, Florida, and has done post-master work at the latter He is a registered professional engineer in California and Florida He has over 30 years of professional engineering experience in industry In addi- tion, he has over 25 years of teaching experience that he acquired at several educational insti- tutions as an adjunct professor He is currently with UC Berkeley Extension

ISBN 0-9709511-8-3

$39.95 U.S.A

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 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 Orchard Publications, 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

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 01760-2098

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 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

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Fremont, California

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info@orchardpublications.com

Chapter 2

The Laplace Transformation

Definition 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 Transformation

The 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

Exercises 3-20Solutions to Exercises 3-22

Chapter 4

Circuit Analysis with Laplace Transforms

Circuit 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 Equations

Expressing 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 Convolution

The 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( )

Exercises 6-22Solutions to Exercises 6-24

Chapter 7

Fourier Series

Wave 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 Transform

Definition 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

Chapter 9

Discrete Time Systems and the Z Transform

Definition 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 Algorithm

The 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 Filters

Filter 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

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

Appendix A

Introduction to MATLAB®

MATLAB® and Simulink® A-1

Command 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 Numbers

Definition 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 Determinants

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

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

Chapter 1

Elementary Signals

his chapter begins with a discussion of elementary signals that may be applied to electric 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

net-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 nient to divide the time interval into two parts, , and

conve-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

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

dis-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, we will reserve the designation for any input when we discuss state variables in a later chapter.

u 0(t–t 0)

t

u 0(t–t 0)

The Unit Step Function

Consider the network of Figure 1.6, where the switch is closed at time

Figure 1.6 Network for Example 1.1

Express the output voltage as a function of the unit step function, and sketch the appropriatewaveform

Solution:

For this example, the output voltage for , and for Therefore,

(1.7)and the waveform is shown in Figure 1.7

Chapter 1 Elementary Signals

Figure 1.7 Waveform for Example 1.1

Other forms of the unit step function are shown in Figure 1.8

Figure 1.8 Other forms of the unit step function

Unit step functions can be used to represent other time-varying functions such as the rectangularpulse shown in Figure 1.9

Figure 1.9 A rectangular pulse expressed as the sum of two unit step functions

Thus, the pulse of Figure 1.9(a) is the sum of the unit step functions of Figures 1.9(b) and 1.9(c) is

−Τ

−Τ Τ

a

u 0 ( ) u t – 0(t–1)

The Unit Step Function

The unit step function offers a convenient method of describing the sudden application of a voltage

or current source For example, a constant voltage source of applied at , can be denoted

as Likewise, a sinusoidal voltage source that is applied to a circuit at, can be described as Also, if the excitation in a circuit is a rect-angular, or triangular, or sawtooth, or any other recurring pulse, it can be represented as a sum (dif-ference) of unit step functions

as

(1.9)Line segment } has height , starts at and terminates at This segment is expressed as

(1.10)Line segment ~ has height ,starts at , and terminates at It is expressed as

(1.11)Thus, the square waveform of Figure 1.10 can be expressed as the summation of (1.8) through (1.11),that is,

24 V t = 0 24u 0 ( ) V t v t( ) = V m cos ωt V

0 A

Chapter 1 Elementary Signals

(1.12)

Combining like terms, we get

(1.13)

Example 1.3

Express the symmetric rectangular pulse of Figure 1.11 as a sum of unit step functions

Figure 1.11 Symmetric rectangular pulse for Example 1.3

Express the symmetric triangular waveform of Figure 1.12 as a sum of unit step functions

Figure 1.12 Symmetric triangular waveform for Example 1.4

0 t T 2

–

0 t T 2

–

0

T 2⁄ –

v t( )

T 2⁄

The Unit Step Function

Figure 1.13 Equations for the linear segments of Figure 1.12

For line segment {,

(1.15)and for line segment |,

(1.16)Combining (1.15) and (1.16), we get

(1.17)

Example 1.5

Express the waveform of Figure 1.14 as a sum of unit step functions

Figure 1.14 Waveform for Example 1.5.

Solution:

As in the previous example, we first find the equations of the linear segments { and | shown in ure 1.15

Fig-t 1

0

T 2⁄ –

v t( )

T 2⁄

|

2 T

{

2 T

+

- t+1

0 t T 2

+

– +

=

1 2

Chapter 1 Elementary Signals

Figure 1.15 Equations for the linear segments of Figure 1.14

Following the same procedure as in the previous examples, we get

Multiplying the values in parentheses by the values in the brackets, we get

or

and combining terms inside the brackets, we get

(1.18)Two other functions of interest are the unit ramp function, and the unit impulse or delta function Wewill introduce them with the examples that follow

Example 1.6

In the network of Figure 1.16 is a constant current source and the switch is closed at time

Figure 1.16 Network for Example 1.6

1 2 3