Analog and Digital Signals and Systems

R.K. Rao Yarlagadda

Analog and Digital Signalsand Systems

1 3

R.K. Rao YarlagaddaSchool of Electrical & ComputerEngineering

Oklahoma State UniversityStillwater OK 74078-6028202 Engineering SouthUSArao.yarlagadda@okstate.edu

ISBN 978-1-4419-0033-3 e-ISBN 978-1-4419-0034-0DOI 10.1007/978-1-4419-0034-0Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2009929744

# Springer ScienceþBusiness Media, LLC 2010All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer ScienceþBusiness Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer ScienceþBusiness Media (www.springer.com)

This book is dedicated to my wifeMarceil, children, Tammy Bardwell, Ryan Yarlagaddaand Travis Yarlagadda and their families

Note to Instructors

The solutions manual can be located on the book’s webpage http://www/

springer.com/engineering/cirucitsþ%26þsystems/bok/978-1-4419-0033-3

vii

Preface

This book presents a systematic, comprehensive treatment of analog and discrete

signal analysis and synthesis and an introduction to analog communication

theory. This evolved from my 40 years of teaching at Oklahoma State University

(OSU). It is based on three courses, Signal Analysis (a second semester junior

level course), Active Filters (a first semester senior level course), and Digital

signal processing (a second semester senior level course). I have taught these

courses a number of times using this material along with existing texts. The

references for the books and journals (over 160 references) are listed in the

bibliography section. At the undergraduate level, most signal analysis courses

do not require probability theory. Only, a very small portion of this topic is

included here.

I emphasized the basics in the book with simple mathematics and the sophis-

tication is minimal. Theorem-proof type of material is not emphasized. The book

uses the following model:

1. Learn basics

2. Check the work using bench marks

3. Use software to see if the results are accurate

The book provides detailed examples (over 400) with applications. A three-

number system is used consisting of chapter number – section number –

example or problem number, thus allowing the student to quickly identify

the related material in the appropriate section of the book. The book

includes well over 400 homework problems. Problem numbers are identified

using the above three-number system. Hints are provided wherever addi-

tional details may be needed and may not have been given in the main part

of the text. A detailed solution manual will be available from the publisher

for the instructors.

Summary of the Chapters

This book starts with an introductory chapter that includes most of the basic

material that a junior in electrical engineering had in the beginning classes. For

those who have forgotten, or have not seen the material recently, it gives enough

ix

background to follow the text. The topics in this chapter include singularity

functions, periodic functions, and others. Chapter 2 deals with convolution

and correlation of periodic and aperiodic functions. Chapter 3 deals with

approximating a function by using a set of basis functions, referred to as the

generalized Fourier series expansion. From these concepts, the three basic

Fourier series expansions are derived. The discussion includes detailed dis-

cussion on the operational properties of the Fourier series and their

convergence.

Chapter 4 deals with Fourier transform theory derived from the Fourier

series. Fourier series and transforms are the bases to this text. Considerable

material in the book is based on these topics. Chapter 5 deals with the relatives

of the Fourier transforms, including Laplace, cosine and sine, Hartley and

Hilbert transforms.

Chapter 6 deals with basic systems analysis that includes linear time-

invariant systems, stability concepts, impulse response, transfer functions,

linear and nonlinear systems, and very simple filter circuits and concepts.

Chapter 7 starts with the Bode plots and later deals with approximations

using classical analog Butterworth, Chebyshev, and Bessel filter functions.

Design techniques, based on both amplitude and phase based, are discussed.

Last part of this chapter deals with analysis and synthesis of active filter

circuits. Examples of basic low-pass, high-pass, band-pass, band elimina-

tion, and delay line filters are included.

Chapter 8 builds a bridge to go from the continuous-time to discrete-time

analysis by starting with sampling theory and the Fourier transform of the

ideally sampled signals. Bulk of this chapter deals with discrete basis func-

tions, discrete-time Fourier series, discrete-time Fourier transform (DTFT),

and the discrete Fourier transform (DFT). Chapter 9 deals with fast

implementations of the DFT, discrete convolution, and correlation. Second

part of the chapter deals the z-transforms and their use in the design of

discrete-data systems. Digital filter designs based on impulse invariance and

bilinear transformations are presented. The chapter ends with digital filter

realizations.

Chapter 10 presents an introduction to analog communication theory,

which includes basic material on analog modulation, such as AM and FM,

demodulation, and multiplexing. Pulse modulation methods are

introduced.

Appendix A reviews the basics on matrices; Appendix B gives a brief intro-

duction onMATLAB; and Appendix C gives a list of useful formulae. The book

concludes with a list of references and Author and Subject indexes.

Suggested Course Content

Instructor is the final judge of what topics will best suit his or her class and in

what depth. The suggestions given below are intended to serve as a guide only.

The book permits flexibility in teaching analysis, synthesis of continuous-time

and discrete-time systems, analog filters, digital signal processing, and an intro-

duction to analog communications. The following table gives suggestions for

courses.

x Preface

Topical Title Related topics in chapters

One semester (Fundamentals of analogsignals and systems ) Chapters 1–4, 6

One semester Systems and analog filters Chapters 4, 5*, 6, 7

One semester (Introduction to digitalsignal processing ) Chapters 4*, 6*, 8, 9

Two semesters (Signals and an introduction toanalog communications ) Chapters 1–4, 5*, 6, 8*, 10

*Partial coverage

Preface xi

Acknowledgements

The process of writing this book has taken me several years. I am indebted to all

the students who have studied with me and taken classes fromme. Education is a

two-way street. The teachers learn from the students, as well as the students learn

from the teachers. Writing a book is a learning process.

Dr. Jack Cartinhour went through the material in the early stages of the text

and helped me in completing the solution manual. His suggestions made the text

better. I am deeply indebted to him. Dr. George Scheets used an earlier version of

this book in his signal analysis and communications theory class. Dr. Martin

Hagan has reviewed a chapter. Their comments were incorporated into the

manuscript. Beau Lacefield did most of the artwork in the manuscript. Vijay

Venkataraman and Wen Fung Leong have gone through some of the chapters

and their suggestions have been incorporated. In addition, Vijay and Wen have

provided some of the MATLAB programs and artwork. I appreciated Vijay’s

help in formatting the final version of the manuscript.

An old adage of the uncertainty principle is, no matter how many times the

author goes through the text, mistakes will remain. I sincerely appreciate all the

support provided by Springer. Thanks to Alex Greene. He believed in me to

complete this project. I appreciated the patience and support of Katie Chen.

Thanks to Shanty Jaganathan and her associates of Integra-India. They have

been helpful and gracious in the editorial process.

Dr. Keith Teague, Head, School of Electrical and Computer Engineering at

Oklahoma State University has been very supportive of this project and I

appreciated his encouragement.

Finally, the time spent on this book is the time taken away from my wife

Marceil, children Tammy, Ryan and Travis and my grandchildren. Without my

family’s understanding, I could not have completed this book.

Oklahoma, USA R.K. Rao Yarlagadda

xiii

Contents

1 Basic Concepts in Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction to the Book and Signals . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Different Ways of Looking at a Signal . . . . . . . . . . . . . . 1

1.1.2 Continuous-Time and Discrete-Time Signals . . . . . . . . . 3

1.1.3 Analog Versus Digital Signal Processing . . . . . . . . . . . . 5

1.1.4 Examples of Simple Functions . . . . . . . . . . . . . . . . . . . . 6

1.2 Useful Signal Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.4 Amplitude Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.5 Simple Symmetries: Even and Odd Functions . . . . . . . . 9

1.2.6 Products of Even and Odd Functions . . . . . . . . . . . . . . . 9

1.2.7 Signum (or sgn) Function . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.8 Sinc and Sinc2 Functions. . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.9 Sine Integral Function. . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Derivatives and Integrals of Functions . . . . . . . . . . . . . . . . . . . 11

1.3.1 Integrals of Functions with Symmetries . . . . . . . . . . . . . 12

1.3.2 Useful Functions from Unit Step Function . . . . . . . . . . 12

1.3.3 Leibniz’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.4 Interchange of a Derivative and an Integral . . . . . . . . . . 13

1.3.5 Interchange of Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Singularity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.1 Unit Impulse as the Limit of a Sequence. . . . . . . . . . . . . 15

1.4.2 Step Function and the Impulse Function . . . . . . . . . . . . 16

1.4.3 Functions of Generalized Functions . . . . . . . . . . . . . . . . 17

1.4.4 Functions of Impulse Functions . . . . . . . . . . . . . . . . . . . 18

1.4.5 Functions of Step Functions . . . . . . . . . . . . . . . . . . . . . . 19

1.5 Signal Classification Based on Integrals . . . . . . . . . . . . . . . . . . 19

1.5.1 Effects of Operations on Signals . . . . . . . . . . . . . . . . . . . 21

1.5.2 Periodic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5.3 Sum of Two Periodic Functions . . . . . . . . . . . . . . . . . . . 23

1.6 Complex Numbers, Periodic, and Symmetric Periodic

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.6.1 Complex Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

xv

1.6.2 Complex Periodic Functions . . . . . . . . . . . . . . . . . . . . . . 271.6.3 Functions of Periodic Functions . . . . . . . . . . . . . . . . . . . 271.6.4 Periodic Functions with Additional Symmetries. . . . . . . 28

1.7 Examples of Probability Density Functions and their

Moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.8 Generation of Periodic Functions from Aperiodic

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.9 Decibel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.10 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1.1 Scalar Product and Norm . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2.1 Properties of the Convolution Integral . . . . . . . . . . . . . . 41

2.2.2 Existence of the Convolution Integral. . . . . . . . . . . . . . . 44

2.3 Interesting Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4 Convolution and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4.1 Repeated Convolution and the Central Limit

Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4.2 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.5 Convolution Involving Periodic and Aperiodic Functions . . . . 54

2.5.1 Convolution of a Periodic Function with an

Aperiodic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.5.2 Convolution of Two Periodic Functions. . . . . . . . . . . . . 55

2.6 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6.1 Basic Properties of Cross-Correlation Functions . . . . . . 57

2.6.2 Cross-Correlation and Convolution . . . . . . . . . . . . . . . . 57

2.6.3 Bounds on the Cross-Correlation Functions . . . . . . . . . 58

2.6.4 Quantitative Measures of Cross-Correlation . . . . . . . . . 59

2.7 Autocorrelation Functions of Energy Signals . . . . . . . . . . . . . . 63

2.8 Cross- and Autocorrelation of Periodic Functions . . . . . . . . . . 65

2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2 Orthogonal Basis Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.1 Gram–Schmidt Orthogonalization . . . . . . . . . . . . . . . . . 74

3.3 Approximation Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.1 Computation of c[k] Based on Partials . . . . . . . . . . . . . . 77

3.3.2 Computation of c[k] Using the Method of Perfect

Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.3 Parseval’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4.1 Complex Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4.2 Trigonometric Fourier Series . . . . . . . . . . . . . . . . . . . . . 83

3.4.3 Complex F-series and the Trigonometric F-series

Coefficients-Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

xvi Contents

3.4.4 Harmonic Form of Trigonometric Fourier Series. . . . . 833.4.5 Parseval’s Theorem Revisited . . . . . . . . . . . . . . . . . . . . 843.4.6 Advantages and Disadvantages of the Three Forms

of Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5 Fourier Series of Functions with Simple Symmetries. . . . . . . . 853.5.1 Simplification of the Fourier Series Coefficient Integral . 86

3.6 Operational Properties of Fourier Series . . . . . . . . . . . . . . . . . 873.6.1 Principle of Superposition . . . . . . . . . . . . . . . . . . . . . . . 873.6.2 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.6.3 Time and Frequency Scaling . . . . . . . . . . . . . . . . . . . . . 883.6.4 Fourier Series Using Derivatives . . . . . . . . . . . . . . . . . . 893.6.5 Bounds and Rates of Fourier Series Convergence

by the Derivative Method . . . . . . . . . . . . . . . . . . . . . . 913.6.6 Integral of a Function and Its Fourier Series . . . . . . . . 933.6.7 Modulation in Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.6.8 Multiplication in Time. . . . . . . . . . . . . . . . . . . . . . . . . . 943.6.9 Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . . . . 953.6.10 Central Ordinate Theorems. . . . . . . . . . . . . . . . . . . . . . 953.6.11 Plancherel’s Relation (or Theorem). . . . . . . . . . . . . . . . 953.6.12 Power Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . 95

3.7 Convergence of the Fourier Series and the Gibbs

Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.7.1 Fourier’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.7.2 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.7.3 Spectral Window Smoothing. . . . . . . . . . . . . . . . . . . . . 99

3.8 Fourier Series Expansion of Periodic Functions with

Special Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.8.1 Half-Wave Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 1003.8.2 Quarter-Wave Symmetry. . . . . . . . . . . . . . . . . . . . . . . 1023.8.3 Even Quarter-Wave Symmetry . . . . . . . . . . . . . . . . . . 1023.8.4 Odd Quarter-Wave Symmetry. . . . . . . . . . . . . . . . . . . 1023.8.5 Hidden Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.9 Half-Range Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.10 Fourier Series Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.11 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4 Fourier Transform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.2 Fourier Series to Fourier Integral . . . . . . . . . . . . . . . . . . . . . . . 109

4.2.1 Amplitude and Phase Spectra . . . . . . . . . . . . . . . . . . . . . 112

4.2.2 Bandwidth-Simplistic Ideas . . . . . . . . . . . . . . . . . . . . . . . 114

4.3 Fourier Transform Theorems, Part 1. . . . . . . . . . . . . . . . . . . . . 114

4.3.1 Rayleigh’s Energy Theorem . . . . . . . . . . . . . . . . . . . . . . 114

4.3.2 Superposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3.3 Time Delay Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3.4 Scale Change Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3.5 Symmetry or Duality Theorem . . . . . . . . . . . . . . . . . . . . 118

4.3.6 Fourier Central Ordinate Theorems . . . . . . . . . . . . . . . . 119

Contents xvii

4.4 Fourier Transform Theorems, Part 2 . . . . . . . . . . . . . . . . . . . . 1194.4.1 Frequency Translation Theorem. . . . . . . . . . . . . . . . . 1204.4.2 Modulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1204.4.3 Fourier Transforms of Periodic and Some Special

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.4.4 Time Differentiation Theorem . . . . . . . . . . . . . . . . . . 1244.4.5 Times-t Property: Frequency Differentiation

Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.4.6 Initial Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4.7 Integration Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.5 Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.5.1 Convolution in Time . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.5.2 Proof of the Integration Theorem . . . . . . . . . . . . . . . . 1324.5.3 Multiplication Theorem (Convolution in

Frequency) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.5.4 Energy Spectral Density . . . . . . . . . . . . . . . . . . . . . . . 135

4.6 Autocorrelation and Cross-Correlation . . . . . . . . . . . . . . . . . . 1364.6.1 Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . 138

4.7 Bandwidth of a Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.7.1 Measures Based on Areas of the Time and Frequency

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.7.2 Measures Based on Moments . . . . . . . . . . . . . . . . . . . 1404.7.3 Uncertainty Principle in Fourier Analysis. . . . . . . . . . 141

4.8 Moments and the Fourier Transform . . . . . . . . . . . . . . . . . . . 143

4.9 Bounds on the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 144

4.10 Poisson’s Summation Formula . . . . . . . . . . . . . . . . . . . . . . . . 145

4.11 Interesting Examples and a Short Fourier Transform Table . . 1454.11.1 Raised-Cosine Pulse Function. . . . . . . . . . . . . . . . . . . 146

4.12 Tables of Fourier Transforms Properties and Pairs. . . . . . . . . 147

4.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5 Relatives of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.2 Fourier Cosine and Sine Transforms . . . . . . . . . . . . . . . . . . . . . 156

5.3 Hartley Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.4 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.4.1 Region of Convergence (ROC) . . . . . . . . . . . . . . . . . . . . 163

5.4.2 Inverse Transform of Two-Sided Laplace Transform. . . 164

5.4.3 Region of Convergence (ROC) of Rational

Functions – Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5 Basic Two-Sided Laplace Transform Theorems . . . . . . . . . . . . 165

5.5.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5.2 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5.3 Shift in s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5.4 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5.5 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.5.6 Differentiation in Time . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.5.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.5.8 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

xviii Contents

5.6 One-Sided Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . 1665.6.1 Properties of the One-Sided Laplace Transform. . . . . 1675.6.2 Comments on the Properties (or Theorems)

of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . 167

5.7 Rational Transform Functions and Inverse Laplace

Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.7.1 Rational Functions, Poles, and Zeros . . . . . . . . . . . . . 1755.7.2 Return to the Initial and Final Value Theorems

and Their Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.8 Solutions of Constant Coefficient Differential Equations

Using Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.8.1 Inverse Laplace Transforms . . . . . . . . . . . . . . . . . . . . 1795.8.2 Partial Fraction Expansions . . . . . . . . . . . . . . . . . . . . 179

5.9 Relationship Between Laplace Transforms and Other

Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.9.1 Laplace Transforms and Fourier Transforms . . . . . . . 1845.9.2 Hartley Transforms and Laplace Transforms . . . . . . . 185

5.10 Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.10.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.10.2 Hilbert Transform of Signals with Non-overlapping

Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.10.3 Analytic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.11 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6 Systems and Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.2 Linear Systems, an Introduction . . . . . . . . . . . . . . . . . . . . . . . . 193

6.3 Ideal Two-Terminal Circuit Components and Kirchhoff’s

Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.3.1 Two-Terminal Component Equations . . . . . . . . . . . . . . 195

6.3.2 Kirchhoff’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.4 Time-Invariant and Time-Varying Systems . . . . . . . . . . . . . . . . 198

6.5 Impulse Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.5.1 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.5.2 Bounded-Input/Bounded-Output (BIBO) Stability . . . . 202

6.5.3 Routh–Hurwitz Criterion (R–H criterion) . . . . . . . . . . . 203

6.5.4 Eigenfunctions in the Fourier Domain . . . . . . . . . . . . . . 206

6.6 Step Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

6.7 Distortionless Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6.7.1 Group Delay and Phase Delay . . . . . . . . . . . . . . . . . . . . 213

6.8 System Bandwidth Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.8.1 Bandwidth Measures Using the Impulse Response

hðtÞ and Its Transform Hðj!Þ . . . . . . . . . . . . . . . . . . . . . 216

6.8.2 Half-Power or 3 dB Bandwidth. . . . . . . . . . . . . . . . . . . . 217

6.8.3 Equivalent Bandwidth or Noise Bandwidth . . . . . . . . . . 217

6.8.4 Root Mean-Squared (RMS) Bandwidth . . . . . . . . . . . . . 218

6.9 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.9.1 Distortion Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Contents xix

6.9.2 Output Fourier-Transform of a Nonlinear System. . . 2206.9.3 Linearization of Nonlinear System Functions . . . . . . 221

6.10 Ideal Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216.10.1 Low-Pass, High-Pass, Band-Pass, and

Band-Elimination Filters . . . . . . . . . . . . . . . . . . . . . . . 222

6.11 Real and Imaginary Parts of the Fourier Transform

of a Causal Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276.11.1 Relationship Between Real and Imaginary Parts

of the Fourier Transform of a Causal Function

Using Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . 2286.11.2 Amplitude Spectrum Hðj!Þj j to a Minimum Phase

Function HðsÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

6.12 More on Filters: Source and Load Impedances . . . . . . . . . . . . 2296.12.1 Simple Low-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . 2316.12.2 Simple High-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . 2316.12.3 Simple Band-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . 2336.12.4 Simple Band-Elimination or Band-Reject

or Notch Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2356.12.5 Maximum Power Transfer. . . . . . . . . . . . . . . . . . . . . . 2386.12.6 A Simple Delay Line Circuit . . . . . . . . . . . . . . . . . . . . 239

6.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

7 Approximations and Filter Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

7.2 Bode Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

7.2.1 Gain and Phase Margins . . . . . . . . . . . . . . . . . . . . . . . . . 252

7.3 Classical Analog Filter Functions . . . . . . . . . . . . . . . . . . . . . . . 254

7.3.1 Amplitude-Based Design. . . . . . . . . . . . . . . . . . . . . . . . . 254

7.3.2 Butterworth Approximations . . . . . . . . . . . . . . . . . . . . . 255

7.3.3 Chebyshev (Tschebyscheff) Approximations . . . . . . . . . 257

7.4 Phase-Based Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

7.4.1 Maximally Flat Delay Approximation . . . . . . . . . . . . . . 263

7.4.2 Group Delay of Bessel Functions . . . . . . . . . . . . . . . . . . 264

7.5 Frequency Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

7.5.1 Normalized Low-Pass to High-Pass Transformation . . . 266

7.5.2 Normalized Low-Pass to Band-Pass Transformation. . . 268

7.5.3 Normalized Low-Pass to Band-Elimination

Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

7.5.4 Conversions of Specifications from Low-Pass,

High-Pass, Band-Pass, and Band Elimination Filters

to Normalized Low-Pass Filters . . . . . . . . . . . . . . . . . . . 270

7.6 Multi-terminal Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

7.6.1 Two-Port Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

7.6.2 Circuit Analysis Involving Multi-terminal Components

and Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

7.6.3 Controlled Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

7.7 Active Filter Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7.7.1 Operational Amplifiers, an Introduction . . . . . . . . . . . . 279

xx Contents

7.7.2 Inverting Operational Amplifier Circuits . . . . . . . . . . . 2807.7.3 Non-inverting Operational Amplifier Circuits . . . . . . . 2827.7.4 Simple Second-Order Low-Pass and All-Pass Circuits. .. 284

7.8 Gain Constant Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

7.9 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2877.9.1 Amplitude (or Magnitude) Scaling, RLC Circuits . . . . 2877.9.2 Frequency Scaling, RLC Circuits . . . . . . . . . . . . . . . . . 2887.9.3 Amplitude and Frequency Scaling in Active Filters . . . 2887.9.4 Delay Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

7.10 RC–CR Transformations: Low-Pass to High-Pass Circuits . . 292

7.11 Band-Pass, Band-Elimination and Biquad Filters . . . . . . . . . . 294

7.12 Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

7.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

8 Discrete-Time Signals and Their Fourier Transforms . . . . . . . . . . . . . 311

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

8.2 Sampling of a Signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.2.1 Ideal Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.2.2 Uniform Low-Pass Sampling or the Nyquist

Low-Pass Sampling Theorem . . . . . . . . . . . . . . . . . . . . . 314

8.2.3 Interpolation Formula and the Generalized Fourier

Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

8.2.4 Problems Associated with Sampling Below

the Nyquist Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

8.2.5 Flat Top Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

8.2.6 Uniform Band-Pass Sampling Theorem . . . . . . . . . . . . . 324

8.2.7 Equivalent continuous-time and discrete-time

systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

8.3 Basic Discrete-Time (DT) Signals . . . . . . . . . . . . . . . . . . . . . . . 325

8.3.1 Operations on a Discrete Signal . . . . . . . . . . . . . . . . . . . 327

8.3.2 Discrete-Time Convolution and Correlation . . . . . . . . . 329

8.3.3 Finite duration, right-sided, left-sided, two-sided,

and causal sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

8.3.4 Discrete-Time Energy and Power Signals . . . . . . . . . . . . 330

8.4 Discrete-Time Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

8.4.1 Periodic Convolution of Two Sequences with the

Same Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

8.4.2 Parseval’s Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

8.5 Discrete-Time Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . 335

8.5.1 Discrete-Time Fourier Transforms (DTFTs) . . . . . . . . . 335

8.5.2 Discrete-Time Fourier Transforms of Real Signals

with Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

8.6 Properties of the Discrete-Time Fourier Transforms. . . . . . . . . 339

8.6.1 Periodic Nature of the Discrete-Time Fourier

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

8.6.2 Superposition or Linearity . . . . . . . . . . . . . . . . . . . . . . . 340

8.6.3 Time Shift or Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

8.6.4 Modulation or Frequency Shifting . . . . . . . . . . . . . . . . . 341

Contents xxi

8.6.5 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3418.6.6 Differentiation in Frequency . . . . . . . . . . . . . . . . . . . 3428.6.7 Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3428.6.8 Summation or Accumulation . . . . . . . . . . . . . . . . . . 3448.6.9 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3448.6.10 Multiplication in Time. . . . . . . . . . . . . . . . . . . . . . . . 3458.6.11 Parseval’s Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 3468.6.12 Central Ordinate Theorems . . . . . . . . . . . . . . . . . . . . 3468.6.13 Simple Digital Encryption . . . . . . . . . . . . . . . . . . . . . 346

8.7 Tables of Discrete-Time Fourier Transform (DTFT)

Properties and Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

8.8 Discrete-Time Fourier-transforms from Samples of the

Continuous-Time Fourier-Transforms . . . . . . . . . . . . . . . . . . 348

8.9 Discrete Fourier Transforms (DFTs) . . . . . . . . . . . . . . . . . . . . 3508.9.1 Matrix Representations of the DFT and

the IDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3528.9.2 Requirements for Direct Computation of

the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

8.10 Discrete Fourier Transform Properties . . . . . . . . . . . . . . . . . . 3548.10.1 DFTs and IDFTs of Real Sequences. . . . . . . . . . . . . 3548.10.2 Linearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3548.10.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3558.10.4 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3558.10.5 Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3568.10.6 Even Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3568.10.7 Odd Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3568.10.8 Discrete-Time Convolution Theorem . . . . . . . . . . . . 3578.10.9 Discrete-Frequency Convolution Theorem. . . . . . . . 3588.10.10 Discrete-Time Correlation Theorem . . . . . . . . . . . . . 3598.10.11 Parseval’s Identity or Theorem . . . . . . . . . . . . . . . . . 3598.10.12 Zero Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3598.10.13 Signal Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 3608.10.14 Decimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

8.11 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

9 Discrete Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

9.2 Computation of Discrete Fourier Transforms (DFTs) . . . . . . . 368

9.2.1 Symbolic Diagrams in Discrete-Time

Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

9.2.2 Fast Fourier Transforms (FFTs). . . . . . . . . . . . . . . . . . . 369

9.3 DFT (FFT) Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

9.3.1 Hidden Periodicity in a Signal. . . . . . . . . . . . . . . . . . . . . 372

9.3.2 Convolution of Time-Limited Sequences . . . . . . . . . . . . 374

9.3.3 Correlation of Discrete Signals . . . . . . . . . . . . . . . . . . . . 377

9.3.4 Discrete Deconvolution. . . . . . . . . . . . . . . . . . . . . . . . . . 378

9.4 z-Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

9.4.1 Region of Convergence (ROC) . . . . . . . . . . . . . . . . . . . . 381

xxii Contents

9.4.2 z-Transform and the Discrete-Time Fourier

Transform (DTFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

9.5 Properties of the z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . 3849.5.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3849.5.2 Time-Shifted Sequences. . . . . . . . . . . . . . . . . . . . . . . . 3859.5.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3859.5.4 Multiplication by an Exponential . . . . . . . . . . . . . . . . 3859.5.5 Multiplication by n . . . . . . . . . . . . . . . . . . . . . . . . . . . 3869.5.6 Difference and Accumulation . . . . . . . . . . . . . . . . . . . 3869.5.7 Convolution Theorem and the z-Transform . . . . . . . . 3869.5.8 Correlation Theorem and the z-Transform. . . . . . . . . 3879.5.9 Initial Value Theorem in the Discrete Domain . . . . . . 3889.5.10 Final Value Theorem in the Discrete Domain . . . . . . 388

9.6 Tables of z-Transform Properties and Pairs. . . . . . . . . . . . . . . 389

9.7 Inverse z-Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3909.7.1 Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3909.7.2 Use of Transform Tables (Partial Fraction

Expansion Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.7.3 Inverse z-Transforms by Power Series Expansion. . . . 394

9.8 The Unilateral or the One-Sided z-Transform . . . . . . . . . . . . . 3959.8.1 Time-Shifting Property . . . . . . . . . . . . . . . . . . . . . . . . 395

9.9 Discrete-Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3979.9.1 Discrete-Time Transfer Functions. . . . . . . . . . . . . . . . 4009.9.2 Schur–Cohn Stability Test. . . . . . . . . . . . . . . . . . . . . . 4019.9.3 Bilinear Transformations. . . . . . . . . . . . . . . . . . . . . . . 401

9.10 Designs by the Time and Frequency Domain Criteria. . . . . . . 4039.10.1 Impulse Invariance Method by Using the Time

Domain Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4059.10.2 Bilinear Transformation Method by Using the

Frequency Domain Criterion. . . . . . . . . . . . . . . . . . . . 407

9.11 Finite Impulse Response (FIR) Filter Design . . . . . . . . . . . . . 4109.11.1 Low-Pass FIR Filter Design . . . . . . . . . . . . . . . . . . . . 4119.11.2 High-Pass, Band-Pass, and Band-Elimination

FIR Filter Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4139.11.3 Windows in Fourier Design. . . . . . . . . . . . . . . . . . . . . . . . . .1416

9.12 Digital Filter Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4199.12.1 Cascade Form of Realization . . . . . . . . . . . . . . . . . . . 4229.12.2 Parallel Form of Realization . . . . . . . . . . . . . . . . . . . . 4229.12.3 All-Pass Filter Realization. . . . . . . . . . . . . . . . . . . . . . 4239.12.4 Digital Filter Transposed Structures . . . . . . . . . . . . . . 4239.12.5 FIR Filter Realizations . . . . . . . . . . . . . . . . . . . . . . . . 423

9.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

10 Analog Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

10.2 Limiters and Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43110.2.1 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

10.3 Linear Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Contents xxiii

10.3.1 Double-Sideband (DSB) Modulation . . . . . . . . . . . 43210.3.2 Demodulation of DSB Signals. . . . . . . . . . . . . . . . . 433

10.4 Frequency Multipliers and Dividers. . . . . . . . . . . . . . . . . . . . 435

10.5 Amplitude Modulation (AM). . . . . . . . . . . . . . . . . . . . . . . . . 43710.5.1 Percentage Modulation . . . . . . . . . . . . . . . . . . . . . . 43810.5.2 Bandwidth Requirements . . . . . . . . . . . . . . . . . . . . 43810.5.3 Power and Efficiency of an Amplitude

Modulated Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 43910.5.4 Average Power Contained in an AM Signal . . . . . . 440

10.6 Generation of AM Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 44110.6.1 Square-Law Modulators . . . . . . . . . . . . . . . . . . . . . 44110.6.2 Switching Modulators . . . . . . . . . . . . . . . . . . . . . . . 44110.6.3 Balanced Modulators. . . . . . . . . . . . . . . . . . . . . . . . 442

10.7 Demodulation of AM Signals . . . . . . . . . . . . . . . . . . . . . . . . 44310.7.1 Rectifier Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . 44310.7.2 Coherent or a Synchronous Detector . . . . . . . . . . . 44310.7.3 Square-Law Detector. . . . . . . . . . . . . . . . . . . . . . . . 44410.7.4 Envelope Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 444

10.8 Asymmetric Sideband Signals . . . . . . . . . . . . . . . . . . . . . . . . 44610.8.1 Single-Sideband Signals . . . . . . . . . . . . . . . . . . . . . . 44610.8.2 Vestigial Sideband Modulated Signals . . . . . . . . . . 44710.8.3 Demodulation of SSB and VSB Signals . . . . . . . . . 44810.8.4 Non-coherent Demodulation of SSB. . . . . . . . . . . . 44910.8.5 Phase-Shift Modulators and Demodulators . . . . . . 449

10.9 Frequency Translation and Mixing . . . . . . . . . . . . . . . . . . . . 450

10.10 Superheterodyne AM Receiver. . . . . . . . . . . . . . . . . . . . . . . . 453

10.11 Angle Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45510.11.1 Narrowband (NB) Angle Modulation. . . . . . . . . . . 45810.11.2 Generation of Angle Modulated Signals . . . . . . . . . 459

10.12 Spectrum of an Angle Modulated Signal . . . . . . . . . . . . . . . . 46010.12.1 Properties of Bessel Functions . . . . . . . . . . . . . . . . . 46110.12.2 Power Content in an Angle Modulated Signal . . . . 463

10.13 Demodulation of Angle Modulated Signals. . . . . . . . . . . . . . 46510.13.1 Frequency Discriminators . . . . . . . . . . . . . . . . . . . . 46510.13.2 Delay Lines as Differentiators . . . . . . . . . . . . . . . . . 467

10.14 FM Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46810.14.1 Distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46810.14.2 Pre-emphasis and De-emphasis . . . . . . . . . . . . . . . . 46910.14.3 Distortions Caused by Multipath Effect . . . . . . . . . 470

10.15 Frequency-Division Multiplexing (FDM) . . . . . . . . . . . . . . . 47110.15.1 Quadrature Amplitude Modulation (QAM)

or Quadrature Multiplexing (QM). . . . . . . . . . . . . . 47210.15.2 FM Stereo Multiplexing and the FM Radio . . . . . . 473

10.16 Pulse Modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47410.16.1 Pulse Amplitude Modulation (PAM) . . . . . . . . . . . 47510.16.2 Problems with Pulse Modulations . . . . . . . . . . . . . . 47510.16.3 Time-Division Multiplexing (TDM) . . . . . . . . . . . . 477

10.17 Pulse Code Modulation (PCM) . . . . . . . . . . . . . . . . . . . . . . . 47810.17.1 Quantization Process . . . . . . . . . . . . . . . . . . . . . . . . 47810.17.2 More on Coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

xxiv Contents

10.17.3 Tradeoffs Between Channel Bandwidth and

Signal-to-Quantization Noise Ratio . . . . . . . . . . . . 48110.17.4 Digital Carrier Modulation . . . . . . . . . . . . . . . . . . . 482

10.18 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Appendix A: Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

A.1 Matrix Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

A.2 Elements of Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

A.2.1 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

A.3 Solutions of Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . 492

A.3.1 Determinants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

A.3.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

A.3.3 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

A.4 Inverses of Matrices and Their Use in Determining

the Solutions of a Set of Equations . . . . . . . . . . . . . . . . . . . . . . 495

A.5 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 496

A.6 Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . . . . 500

A.7 Generalized Inverses of Matrices . . . . . . . . . . . . . . . . . . . . . . . 501

A.8 Over- and Underdetermined System of Equations . . . . . . . . . . 502

A.8.1 Least-Squares Solutions of Overdetermined

System of Equations (m > n) . . . . . . . . . . . . . . . . . . . . 502A.8.2 Least-Squares Solution of Underdetermined

System of Equations (m � n) . . . . . . . . . . . . . . . . . . . . 504

A.9 Numerical-Based Interpolations: Polynomial and Lagrange

Interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505A.9.1 Polynomial Approximations . . . . . . . . . . . . . . . . . . . . . . . . . .505A.9.2 Lagrange Interpolation Formula . . . . . . . . . . . . . . . . . . . . . .506

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

Appendix B: MATLAB1 for Digital Signal Processing . . . . . . . . . . . . . . . 509

B.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

B.2 Signal Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

B.3 Signal Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

B.4 Fast Fourier Transforms (FFTs) . . . . . . . . . . . . . . . . . . . . . . . 511

B.5 Convolution of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

B.6 Differentiation Using Numerical Methods . . . . . . . . . . . . . . . 515

B.7 Fourier Series Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 515

B.8 Roots of Polynomials, Partial Fraction Expansions,

Pole�Zero Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

B.8.1 Partial Fraction Expansions . . . . . . . . . . . . . . . . . . . . 518

B.9 Bode Plots, Impulse and Step Responses . . . . . . . . . . . . . . . . 518

B.9.1 Bode Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

B.9.2 Impulse and Step Responses . . . . . . . . . . . . . . . . . . . . 518

B.10 Frequency Responses of Digital Filter Transfer

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

B.11 Introduction to the Construction of Simple MATLAB

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

B.12 Additional MATLAB Code. . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Contents xxv

Appendix C: Mathematical Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

C.1 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

C.2 Logarithms, Exponents and Complex Numbers . . . . . . . . . . . . 523

C.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

C.4 Indefinite Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

C.5 Definite Integrals and Useful Identities. . . . . . . . . . . . . . . . . . . 525

C.6 Summation Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

C.7 Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

C.8 Special Constants and Factorials . . . . . . . . . . . . . . . . . . . . . . . 526

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

xxvi Contents

List of Tables

Table 1.4.1 Properties of the impulse function. . . . . . . . . . . . . . . . . . . 18

Table 1.9.1 Sound Power (loudness) Comparison . . . . . . . . . . . . . . . . 33

Table 1.9.2 Power ratios and their corresponding values in dB. . . . . . 33

Table 2.4.1 Properties of aperiodic convolution . . . . . . . . . . . . . . . . . 53

Table 2.6.1 Example 2.6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Table 3.4.1 Summary of the three Fourier series representations . . . . 84

Table 3.10.1 Symmetries of real periodic functions and their

Fourier-series coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 105

Table 3.10.2 Periodic functions and their Trigonometric Fourier Series . . 105

Table 4.12.1 Fourier transform properties. . . . . . . . . . . . . . . . . . . . . . . 148

Table 4.12.2 Fourier Transform Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Table 5.6.1 One-sided Laplace transform properties . . . . . . . . . . . . . . 168

Table 5.6.2 One-sided Laplace tranform pairs . . . . . . . . . . . . . . . . . . . 175

Table 5.8.1 Typical rational replace transforms and their inverses . . . 182

Table 5.9.1 One sided Laplace transforms and Fourier transforms. . . 185

Table 5.10.1 Hilbert transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Table 7.1.1 Formula for computing sensitivities . . . . . . . . . . . . . . . . . 244

Table 7.4.1 Normalized frequencies, ! ¼ !�0. Time delay and a loss

table giving the normalized frequency ! at which the

zero frequency delay and loss values deviate by specified

amounts for Bessel filter functions . . . . . . . . . . . . . . . . . . 265

Table 7.5.1 Frequency transformations . . . . . . . . . . . . . . . . . . . . . . . . 269

Table 7.7.1 Guidelines for passive components . . . . . . . . . . . . . . . . . . 283

Table 8.1.1 Fourier representations of discrete-time and

continuous-time signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Table 8.2.1 Common interpolation functions . . . . . . . . . . . . . . . . . . . 319

Table 8.2.2 Spectral occupancy of Xðjð!� n!sÞÞ; ! ¼ 2�f;

n ¼ 0;�1;�2;�3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Table 8.3.1 Properties of discrete convolution . . . . . . . . . . . . . . . . . . . 329

Table 8.7.1 Discrete-time Fourier transform (DTFT) properties . . . . 347

Table 8.7.2 Discrete-time Fourier transform (DTFT) pairs. . . . . . . . . 348

Table 8.10.1 Discrete Fourier transform (DFT) properties . . . . . . . . . . 36

Table 9.1.1 Discrete-time and continuous-time signals and their

transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

xxvii

1

Table 9.2.1 Properties of the functionWN ¼ e�jð2�=NÞ. . . . . . . . . . . . . 369

Table 9.6.1 Z-transform properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Table 9.6.2 Z-transform pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

Table 9.11.1 Ideal low-pass filter FIR coefficients with �c ¼ �=4 . . . . . 412

Table 9.11.2 FIR Filter Coefficients for the Four Basic Filters. . . . . . . 415

Table 10.9.1 Inputs and outputs of the system in Fig. 10.9.1 . . . . . . . . 453

Table 10.12.1 Bessel function values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

Table 10.17.1 Quantization values and codes corresponding to

Fig. 10.17.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Table 10.17.2 Binary representation of quantized values . . . . . . . . . . . . 480

Table 10.17.3 Normal binary and Gray code representations

for N¼8.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Table B.7.1 Amplitudes and phase angles of the harmonic Fourier

series coefficients (Example B.7.1). . . . . . . . . . . . . . . . . . . 516

xxviii List of Tables