This document provides an overview of a textbook on analog and digital signals and systems. It discusses the book's contents, which cover topics such as Fourier analysis, linear systems, analog filter design, sampling theory, discrete-time signals and systems, and an introduction to analog communications. The textbook is intended for courses in signals and systems, systems and filters, and digital signal processing. It emphasizes basic concepts and includes numerous examples and homework problems.

2. Analog and Digital Signals and Systems

4. R.K. Rao Yarlagadda
Analog and Digital Signals
and Systems
1 3

5. R.K. Rao Yarlagadda
School of Electrical & Computer
Engineering
Oklahoma State University
Stillwater OK 74078-6028
202 Engineering South
USA
rao.yarlagadda@okstate.edu
ISBN 978-1-4419-0033-3 e-ISBN 978-1-4419-0034-0
DOI 10.1007/978-1-4419-0034-0
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2009929744
# Springer ScienceþBusiness Media, LLC 2010
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission 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 in
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The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, is not to be taken as an expression of opinion as to whether or not they are
subject to proprietary rights.
Printed on acid-free paper
Springer is part of Springer ScienceþBusiness Media (www.springer.com)

6. This book is dedicated to my wife
Marceil, children, Tammy Bardwell, Ryan Yarlagadda
and Travis Yarlagadda and their families

8. 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

10. 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

11. 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 on MATLAB; 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

12. Topical Title Related topics in chapters
One semester
(Fundamentals of analog
signals and systems ) Chapters 1–4, 6
One semester Systems and analog filters Chapters 4, 5*, 6, 7
One semester
(Introduction to digital
signal processing ) Chapters 4*, 6*, 8, 9
Two semesters
(Signals and an introduction to
analog communications ) Chapters 1–4, 5*, 6, 8*, 10
*Partial coverage
Preface xi

14. 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 from me. 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

16. 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

17. 1.6.2 Complex Periodic Functions . . . . . . . . . . . . . . . . . . . . .. 27
1.6.3 Functions of Periodic Functions . . . . . . . . . . . . . . . . . .. 27
1.6.4 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

18. 3.4.4 Harmonic Form of Trigonometric Fourier Series. . . .. 83
3.4.5 Parseval’s Theorem Revisited . . . . . . . . . . . . . . . . . . .. 84
3.4.6 Advantages and Disadvantages of the Three Forms
of Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5 Fourier Series of Functions with Simple Symmetries. . . . . . . . 85
3.5.1 Simplification of the Fourier Series Coefficient Integral . 86
3.6 Operational Properties of Fourier Series . . . . . . . . . . . . . . . . . 87
3.6.1 Principle of Superposition . . . . . . . . . . . . . . . . . . . . . .. 87
3.6.2 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87
3.6.3 Time and Frequency Scaling . . . . . . . . . . . . . . . . . . . .. 88
3.6.4 Fourier Series Using Derivatives. . . . . . . . . . . . . . . . .. 89
3.6.5 Bounds and Rates of Fourier Series Convergence
by the Derivative Method . . . . . . . . . . . . . . . . . . . . . . 91
3.6.6 Integral of a Function and Its Fourier Series . . . . . . .. 93
3.6.7 Modulation in Time. . . . . . . . . . . . . . . . . . . . . . . . . . .. 93
3.6.8 Multiplication in Time. . . . . . . . . . . . . . . . . . . . . . . . .. 94
3.6.9 Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . . .. 95
3.6.10 Central Ordinate Theorems. . . . . . . . . . . . . . . . . . . . .. 95
3.6.11 Plancherel’s Relation (or Theorem). . . . . . . . . . . . . . .. 95
3.6.12 Power Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . .. 95
3.7 Convergence of the Fourier Series and the Gibbs
Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.7.1 Fourier’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 96
3.7.2 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97
3.7.3 Spectral Window Smoothing. . . . . . . . . . . . . . . . . . . .. 99
3.8 Fourier Series Expansion of Periodic Functions with
Special Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.8.1 Half-Wave Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.8.2 Quarter-Wave Symmetry. . . . . . . . . . . . . . . . . . . . . . . 102
3.8.3 Even Quarter-Wave Symmetry . . . . . . . . . . . . . . . . . . 102
3.8.4 Odd Quarter-Wave Symmetry. . . . . . . . . . . . . . . . . . . 102
3.8.5 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

19. 4.4 Fourier Transform Theorems, Part 2. . . . . . . . . . . . . . . . . . . . 119
4.4.1 Frequency Translation Theorem. . . . . . . . . . . . . . . . . 120
4.4.2 Modulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4.3 Fourier Transforms of Periodic and Some Special
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.4 Time Differentiation Theorem . . . . . . . . . . . . . . . . . . 124
4.4.5 Times-t Property: Frequency Differentiation
Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.4.6 Initial Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.4.7 Integration Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.5 Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.5.1 Convolution in Time . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.5.2 Proof of the Integration Theorem . . . . . . . . . . . . . . . . 132
4.5.3 Multiplication Theorem (Convolution in
Frequency) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.5.4 Energy Spectral Density . . . . . . . . . . . . . . . . . . . . . . . 135
4.6 Autocorrelation and Cross-Correlation . . . . . . . . . . . . . . . . . . 136
4.6.1 Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . 138
4.7 Bandwidth of a Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.7.1 Measures Based on Areas of the Time and Frequency
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.7.2 Measures Based on Moments . . . . . . . . . . . . . . . . . . . 140
4.7.3 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. . 145
4.11.1 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

20. 5.6 One-Sided Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.6.1 Properties of the One-Sided Laplace Transform. . . . . 167
5.6.2 Comments on the Properties (or Theorems)
of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . 167
5.7 Rational Transform Functions and Inverse Laplace
Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.7.1 Rational Functions, Poles, and Zeros . . . . . . . . . . . . . 175
5.7.2 Return to the Initial and Final Value Theorems
and Their Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.8 Solutions of Constant Coefficient Differential Equations
Using Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.8.1 Inverse Laplace Transforms . . . . . . . . . . . . . . . . . . . . 179
5.8.2 Partial Fraction Expansions . . . . . . . . . . . . . . . . . . . . 179
5.9 Relationship Between Laplace Transforms and Other
Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.9.1 Laplace Transforms and Fourier Transforms. . . . . . . 184
5.9.2 Hartley Transforms and Laplace Transforms. . . . . . . 185
5.10 Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.10.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.10.2 Hilbert Transform of Signals with Non-overlapping
Spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5.10.3 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

21. 6.9.2 Output Fourier-Transform of a Nonlinear System. . . 220
6.9.3 Linearization of Nonlinear System Functions . . . . . . 221
6.10 Ideal Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.10.1 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.11.1 Relationship Between Real and Imaginary Parts
of the Fourier Transform of a Causal Function
Using Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . 228
6.11.2 Amplitude Spectrum Hðj!Þ
j j to a Minimum Phase
Function HðsÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.12 More on Filters: Source and Load Impedances . . . . . . . . . . . . 229
6.12.1 Simple Low-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . 231
6.12.2 Simple High-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . 231
6.12.3 Simple Band-Pass Filters. . . . . . . . . . . . . . . . . . . . . . . 233
6.12.4 Simple Band-Elimination or Band-Reject
or Notch Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
6.12.5 Maximum Power Transfer. . . . . . . . . . . . . . . . . . . . . . 238
6.12.6 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

22. 7.7.2 Inverting Operational Amplifier Circuits . . . . . . . . . . . 280
7.7.3 Non-inverting Operational Amplifier Circuits . . . . . . . 282
7.7.4 Simple Second-Order Low-Pass and All-Pass Circuits. .. 284
7.8 Gain Constant Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
7.9 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
7.9.1 Amplitude (or Magnitude) Scaling, RLC Circuits . . . . 287
7.9.2 Frequency Scaling, RLC Circuits . . . . . . . . . . . . . . . . . 288
7.9.3 Amplitude and Frequency Scaling in Active Filters . . . 288
7.9.4 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

23. 8.6.5 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
8.6.6 Differentiation in Frequency . . . . . . . . . . . . . . . . . . . 342
8.6.7 Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
8.6.8 Summation or Accumulation . . . . . . . . . . . . . . . . . . 344
8.6.9 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
8.6.10 Multiplication in Time. . . . . . . . . . . . . . . . . . . . . . . . 345
8.6.11 Parseval’s Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 346
8.6.12 Central Ordinate Theorems. . . . . . . . . . . . . . . . . . . . 346
8.6.13 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). . . . . . . . . . . . . . . . . . . . 350
8.9.1 Matrix Representations of the DFT and
the IDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
8.9.2 Requirements for Direct Computation of
the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
8.10 Discrete Fourier Transform Properties . . . . . . . . . . . . . . . . . . 354
8.10.1 DFTs and IDFTs of Real Sequences. . . . . . . . . . . . . 354
8.10.2 Linearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
8.10.3 Duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
8.10.4 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
8.10.5 Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
8.10.6 Even Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
8.10.7 Odd Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
8.10.8 Discrete-Time Convolution Theorem . . . . . . . . . . . . 357
8.10.9 Discrete-Frequency Convolution Theorem. . . . . . . . 358
8.10.10 Discrete-Time Correlation Theorem . . . . . . . . . . . . . 359
8.10.11 Parseval’s Identity or Theorem . . . . . . . . . . . . . . . . . 359
8.10.12 Zero Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
8.10.13 Signal Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . 360
8.10.14 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

24. 9.4.2 z-Transform and the Discrete-Time Fourier
Transform (DTFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
9.5 Properties of the z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . 384
9.5.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
9.5.2 Time-Shifted Sequences. . . . . . . . . . . . . . . . . . . . . . . . 385
9.5.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
9.5.4 Multiplication by an Exponential . . . . . . . . . . . . . . . . 385
9.5.5 Multiplication by n . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.5.6 Difference and Accumulation . . . . . . . . . . . . . . . . . . . 386
9.5.7 Convolution Theorem and the z-Transform . . . . . . . . 386
9.5.8 Correlation Theorem and the z-Transform. . . . . . . . . 387
9.5.9 Initial Value Theorem in the Discrete Domain . . . . . . 388
9.5.10 Final Value Theorem in the Discrete Domain . . . . . . 388
9.6 Tables of z-Transform Properties and Pairs. . . . . . . . . . . . . . . 389
9.7 Inverse z-Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
9.7.1 Inversion Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
9.7.2 Use of Transform Tables (Partial Fraction
Expansion Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
9.7.3 Inverse z-Transforms by Power Series Expansion. . . . 394
9.8 The Unilateral or the One-Sided z-Transform . . . . . . . . . . . . . 395
9.8.1 Time-Shifting Property . . . . . . . . . . . . . . . . . . . . . . . . 395
9.9 Discrete-Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
9.9.1 Discrete-Time Transfer Functions. . . . . . . . . . . . . . . . 400
9.9.2 Schur–Cohn Stability Test. . . . . . . . . . . . . . . . . . . . . . 401
9.9.3 Bilinear Transformations. . . . . . . . . . . . . . . . . . . . . . . 401
9.10 Designs by the Time and Frequency Domain Criteria. . . . . . . 403
9.10.1 Impulse Invariance Method by Using the Time
Domain Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
9.10.2 Bilinear Transformation Method by Using the
Frequency Domain Criterion. . . . . . . . . . . . . . . . . . . . 407
9.11 Finite Impulse Response (FIR) Filter Design . . . . . . . . . . . . . 410
9.11.1 Low-Pass FIR Filter Design . . . . . . . . . . . . . . . . . . . . 411
9.11.2 High-Pass, Band-Pass, and Band-Elimination
FIR Filter Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
9.11.3 Windows in Fourier Design. . . . . . . . . . . . . . . . . . . . . . . . . .
1416
9.12 Digital Filter Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
9.12.1 Cascade Form of Realization . . . . . . . . . . . . . . . . . . . 422
9.12.2 Parallel Form of Realization . . . . . . . . . . . . . . . . . . . . 422
9.12.3 All-Pass Filter Realization. . . . . . . . . . . . . . . . . . . . . . 423
9.12.4 Digital Filter Transposed Structures. . . . . . . . . . . . . . 423
9.12.5 FIR Filter Realizations . . . . . . . . . . . . . . . . . . . . . . . . 423
9.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
10 Analog Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
10.2 Limiters and Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
10.2.1 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
10.3 Linear Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
Contents xxiii

25. 10.3.1 Double-Sideband (DSB) Modulation . . . . . . . . . . . 432
10.3.2 Demodulation of DSB Signals. . . . . . . . . . . . . . . . . 433
10.4 Frequency Multipliers and Dividers. . . . . . . . . . . . . . . . . . . . 435
10.5 Amplitude Modulation (AM). . . . . . . . . . . . . . . . . . . . . . . . . 437
10.5.1 Percentage Modulation . . . . . . . . . . . . . . . . . . . . . . 438
10.5.2 Bandwidth Requirements . . . . . . . . . . . . . . . . . . . . 438
10.5.3 Power and Efficiency of an Amplitude
Modulated Signal. . . . . . . . . . . . . . . . . . . . . . . . . . . 439
10.5.4 Average Power Contained in an AM Signal . . . . . . 440
10.6 Generation of AM Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
10.6.1 Square-Law Modulators . . . . . . . . . . . . . . . . . . . . . 441
10.6.2 Switching Modulators . . . . . . . . . . . . . . . . . . . . . . . 441
10.6.3 Balanced Modulators. . . . . . . . . . . . . . . . . . . . . . . . 442
10.7 Demodulation of AM Signals . . . . . . . . . . . . . . . . . . . . . . . . 443
10.7.1 Rectifier Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . 443
10.7.2 Coherent or a Synchronous Detector . . . . . . . . . . . 443
10.7.3 Square-Law Detector. . . . . . . . . . . . . . . . . . . . . . . . 444
10.7.4 Envelope Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 444
10.8 Asymmetric Sideband Signals . . . . . . . . . . . . . . . . . . . . . . . . 446
10.8.1 Single-Sideband Signals . . . . . . . . . . . . . . . . . . . . . . 446
10.8.2 Vestigial Sideband Modulated Signals . . . . . . . . . . 447
10.8.3 Demodulation of SSB and VSB Signals . . . . . . . . . 448
10.8.4 Non-coherent Demodulation of SSB. . . . . . . . . . . . 449
10.8.5 Phase-Shift Modulators and Demodulators . . . . . . 449
10.9 Frequency Translation and Mixing . . . . . . . . . . . . . . . . . . . . 450
10.10 Superheterodyne AM Receiver. . . . . . . . . . . . . . . . . . . . . . . . 453
10.11 Angle Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
10.11.1 Narrowband (NB) Angle Modulation. . . . . . . . . . . 458
10.11.2 Generation of Angle Modulated Signals . . . . . . . . . 459
10.12 Spectrum of an Angle Modulated Signal . . . . . . . . . . . . . . . . 460
10.12.1 Properties of Bessel Functions. . . . . . . . . . . . . . . . . 461
10.12.2 Power Content in an Angle Modulated Signal . . . . 463
10.13 Demodulation of Angle Modulated Signals. . . . . . . . . . . . . . 465
10.13.1 Frequency Discriminators . . . . . . . . . . . . . . . . . . . . 465
10.13.2 Delay Lines as Differentiators . . . . . . . . . . . . . . . . . 467
10.14 FM Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
10.14.1 Distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
10.14.2 Pre-emphasis and De-emphasis . . . . . . . . . . . . . . . . 469
10.14.3 Distortions Caused by Multipath Effect . . . . . . . . . 470
10.15 Frequency-Division Multiplexing (FDM) . . . . . . . . . . . . . . . 471
10.15.1 Quadrature Amplitude Modulation (QAM)
or Quadrature Multiplexing (QM). . . . . . . . . . . . . . 472
10.15.2 FM Stereo Multiplexing and the FM Radio . . . . . . 473
10.16 Pulse Modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
10.16.1 Pulse Amplitude Modulation (PAM) . . . . . . . . . . . 475
10.16.2 Problems with Pulse Modulations . . . . . . . . . . . . . . 475
10.16.3 Time-Division Multiplexing (TDM) . . . . . . . . . . . . 477
10.17 Pulse Code Modulation (PCM) . . . . . . . . . . . . . . . . . . . . . . . 478
10.17.1 Quantization Process . . . . . . . . . . . . . . . . . . . . . . . . 478
10.17.2 More on Coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
xxiv Contents

26. 10.17.3 Tradeoffs Between Channel Bandwidth and
Signal-to-Quantization Noise Ratio . . . . . . . . . . . . 481
10.17.4 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) . . . . . . . . . . . . . . . . . . . . 502
A.8.2 Least-Squares Solution of Underdetermined
System of Equations (m n) . . . . . . . . . . . . . . . . . . . . 504
A.9 Numerical-Based Interpolations: Polynomial and Lagrange
Interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
A.9.1 Polynomial Approximations. . . . . . . . . . . . . . . . . . . . . . . . . .
505
A.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,
PoleZero 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

27. 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

28. 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 2f;
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

29. Table 9.2.1 Properties of the function WN ¼ ejð2=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

30. Chapter 1
Basic Concepts in Signals
1.1 Introduction to the Book and Signals
The primary goal of this book is to introduce the
reader on the basic principles of signals and to
provide tools thereby to deal with the analysis of
analog and digital signals, either obtained natu-
rally or by sampling analog signals, study the
concepts of various transforming techniques, fil-
tering analog and digital signals, and finally
introduce the concepts of communicating analog
signals using simple modulation techniques. The
basic material in this book can be found in several
books. See references at the end of the book.
A signal is a pattern of some kind used to convey
a message. Examples include smoke signals, a set of
flags, traffic lights, speech, image, seismic signals,
and many others. Smoke signals were used for con-
veying information that goes back before recorded
history. Greeks and Romans used light beacons in
the pre-Christian era. England employed a long
chain of beacons to warn that Spanish Armada is
approaching in the late sixteenth century. Around
this time, the word signal came into use perceptible
by sight, hearing, etc., conveying information. The
present day signaling started with the invention of
the Morse code in 1838. Since then, a variety of
signals have been studied. These include the follow-
ing inventions: Facsimile by Alexander Bain in
1843; telephone by Alexander Bell in 1876; wireless
telegraph system by Gugliemo Marconi in 1897;
transmission of speech signals via radio by Reginald
Fessenden in 1905, invention and demonstration of
television, the birth of television by Vladimir Zwor-
ykin in the 1920 s, and many others. In addition, the
development of radar and television systems during
World War II, proposition of satellite communica-
tion systems, demonstration of a laser in 1955, and
the research and developments of many signal pro-
cessing techniques and their use in communication
systems. Since the early stages of communications,
research has exploded into several areas connected
directly, or indirectly, to signal analysis and com-
munications. Signal analysis has taken a signifi-
cant role in medicine, for example, monitoring
the heart beat, blood pressure and temperature of
a patient, and vital signs of patients. Others include
the study of weather phenomenon, the geological
formations below the surface and deep in the
ground and under the ocean floors for oil and gas
exploration, mapping the underground surface
using seismometers, and others. Researchers have
concluded that computers are powerful and neces-
sary that they need to be an integral part of any
communication system, thus generating significant
research in digital signal processing, development
of Internet, research on HDTV, mobile and cellu-
lar telephone systems, and others. Defense indus-
try has been one of the major organizations in
advancing research in signal processing, coding,
and transmission of data. Several research areas
have surfaced in signals that include processing of
speech, image, radar, seismic, medical, and other
signals.
1.1.1 Different Ways of Looking
at a Signal
Consider a signal xðtÞ, a function representing a
physical quantity, such as voltage, current, pres-
sure, or any other variable with respect to a second
variable t, such as time. The terms of interest are the
time t and the signal xðtÞ. One of the main topics of
R.K.R. Yarlagadda, Analog and Digital Signals and Systems, DOI 10.1007/978-1-4419-0034-0_1,
Ó Springer ScienceþBusiness Media, LLC 2010
1

31. this book is the analysis of signals. Webster’s dic-
tionary defines the analysis as
1. Separation of a thing into the parts or elements
of which it is composed.
2. An examination of a thing to determine its parts
or elements.
3. A statement showing the results of such an
examination.
There are other definitions. In the following the
three parts are considered using simple examples.
Consider the sinusoidal function and its expansion
using Euler’s formula:
xðtÞ ¼ A0 cosðo0t þ y0Þ
¼
A0
2
ejy0
ejo0t
þ
A0
2
ejy0
ejo0t
¼ ReðA0ejo0t
ejy0
Þ:
(1:1:1)
In (1.1.1) A0 is assumed to be positive and real and
A0ejy0
is a complex number carrying the amplitude
and phase angle of the sinusoidal function and is by
definition the phasor representation of the given
sinusoidal function. Some authors refer to this as
phasor transform of the sinusoidal signal, as it trans-
forms the time domain sinusoidal function to the
complex frequency domain. A brief discussion on
complex numbers is included later in Section 1.6.
This signal can be described in another domain, i.e.,
such as the frequency domain. The amplitude is
ðA0=2Þ and the phase angles of y0 corresponding
to the frequencies f0 ¼ o0=2p Hz. In reality,
only positive frequencies are available, but Euler’s
formula in (1.1.1) dictates that both the positive and
negative frequencies need to be identified as illu-
strated in Fig. 1.1.1a. This description is the two-
sided amplitude and phase line spectra of xðtÞ.
Amplitudes are always positive and are located at
f ¼ o0=2p ¼ f0 Hz, symmetrically located
around the zero frequency, i.e., with even symme-
try. The phase spectrum consists of two angles
y ¼ y0 corresponding to the positive and negative
frequencies, respectively, with odd symmetry. Since
ðtÞ is real, we can pictorially describe it by one- or
two-sided amplitude and phase line spectra as shown
in Fig. 1.1.1a,b,c,d. The following example illus-
trates the three steps.
Example 1.1.1 Express the following function in
terms of a sum of cosine functions:
xðtÞ ¼ A0 þ A1 cosðo1t þ y1Þ
A2 cosðo2tÞ A3 sinðo3t þ y3Þ; Ai 0:
(1:1:2)
Solution: Using trigonometric relations to express
each term in (1.1.2) in the form of Ai cosðoit þ yiÞ
results in
xðtÞ ¼ A0 cosðð0Þt 180
Þ
þ A1 cosðo1t þ y1Þ þ A2 cosðo2t 180
Þ
þ A3 cosðo3t þ y3 þ 90
Þ:
(1:1:3)
In the first and the third terms either 1808 or 1808
could be used, as the end result is the same. The
two-sided line spectra of the function in (1.1.2) are
shown in Fig. 1.1.2. How would one get the func-
tions of the type shown in (1.1.3) for an arbitrary
Fig. 1.1.1 xðtÞ ¼ A0 cosðo0t þ y0Þ. (a) Two-sided amplitude
spectrum, (b) two-sided phase spectrum, (c) one-sided ampli-
tude spectrum, and (d) one-sided phase spectrum
2 1 Basic Concepts in Signals

32. function? The sign and cosine functions are the
building blocks of the Fourier series in Chapter 3
and later the Fourier transforms in Chapter 4. The
function xðtÞ has four frequencies:
f1ð¼0Þ; f2; f3; f4 with amplitudes A0; A1; A2; A3
and phases 180o
; y1; 180o
; y3 þ 90o
:
Figure 1.1.2 illustrates pictorially the discrete loca-
tions of the frequencies, their amplitudes, and
phases. The signal in (1.1.2) can be described by
using the time domain function or in terms of fre-
quencies. In the figures, o0
ð¼ 2pf Þ0
s in radians per
second could have been used rather than f 0
s in Hz.
1.1.2 Continuous-Time and Discrete-Time
Signals
A signal xðtÞ is a continuous-time signal if t is a
continuous variable. It can take on any value in
the continuous interval ða; bÞ. Continuous-time sig-
nal is an analog signal. If a function y½n is defined at
discrete times, then it is a discrete-time signal, where
n takes integer values. In Chapter 8 discrete-time
signals will be studied by sampling the continuous
signals at equal sampling intervals of ts seconds and
write xðntsÞ; where n an integer. This is expressed by
x½n xðntsÞ: (1:1:4)
Example(s) 1.1.2 In this example several specific
examples of interest are considered. In the first one,
part of the time signal illustrating a male voice of
speech in the sentence ‘‘. . .Show the rich lady out’’ is
shown in Fig. 1.1.3. The speech signal is sampled at
8000 samples per second. There are three portions
of the speech ‘‘/. . ./, /sh/, /o/’’ shown in the figure.
The first part of the signal does not have any speech
in it and the small amplitudes of the signal represent
the noise in the tape recorder and/or in the room
where the speech was recorded. It represents a ran-
dom signal and can be described only by statistical
means. Random signal analysis is not discussed in
any detail in this book, as it requires knowledge of
probability theory. The second part represents the
phoneme ‘‘sh’’ that does not show any observable
pattern. It is a time signal for a very short time and
has finite energy. Power and energy signals are stu-
died in Section 1.5. Third part of the figure repre-
sents the vowel ‘‘o,’’ showing a structure of (almost)
periodic pulses for a short time. In this book, aper-
iodic or non-periodic signals with finite energy and
periodic signals with finite average power will be
studied. One goal is to come up with a model for
each portion of a signal that can be transmitted and
reconstructed at the receiver.
Next three examples are from food industry.
Small businesses are sprouting that use signal pro-
cessing. For example, when we go to a grocery store
we may like to buy a watermelon. It may not always
be possible to judge the ripeness of the watermelon
Fig. 1.1.3 Speech . . .sho in . . .show ._male 2000 Samples @
8000 samples per second. Printed with the permission from
Hassan et al. (1994)
Fig. 1.1.2 (a) Two-sided amplitude spectra and (b) two-
sided phase spectra
1.1 Introduction to the Book and Signals 3

33. by outward characteristics such as external color,
stem conditions, or just the way it looks. A sure way
of looking at the quality is to cut the watermelon
open and taste it before we buy it. This implies we
break it first, which is destructive testing. Instead,
we can use our grandmother’s procedure in select-
ing a watermelon. She uses her knuckles to send a
signal into the watermelon. From the audio
response of the watermelon she decides whether it
is good or not based on her prior experience. We can
simulate this by putting the watermelon on a stand,
use a small hammer like device, give a slight tap on
the watermelon, and record the response. A simplis-
tic model of this is shown in Fig. 1.1.4. The
responses can be categorized by studying the out-
puts of tasty watermelons. For an interesting
research work on this topic, see Stone et al (1996).
Image processing can be used to check for
burned crusts, topping amount distribution, such
as the location of pepperoni pizza slices, and others.
For an interesting article on this subject, see Wagner
(1983), which has several applications in the food
industry.
The next two examples are from the surface seis-
mic signal analysis. In the first one, we use a source
in the form of dynamite sticks representing a source,
dig a small hole, and blow them in the hole. The
ground responds to this input and the response is
recorded using a seismometer and a tape recorder.
The analysis of the recorded waveform can provide
information about the underground cavities and
pockets of oil and other important measures.
Geologists drill holes into the ground and a small
slice of the core sample is used to measure the oil
content by looking at the percentage of the area
with dark spots on the slice, which is image
processing.
Another example of interest is measuring the
distance from a ground station to an airplane.
Send a signal with square wave pulses toward the
airplane and when the signal hits it, a return signal is
received at the ground station. A simple model is
shown in Fig. 1.1.5. If we can measure the time
between the time the signal left from the ground
station and the time it returned, identified as T in
the figure, we can determine the distance between
the ground station and the target by the formula
x ¼ 3ð108
Þ ðm=sÞ
Tðsignal round trip time in secondsÞ=2:
(1:1:5)
The constant c ¼ 3ð108
Þ m/s is the speed of light.
Radar and sonar signal processing are two impor-
tant areas of signal processing applications.
An exciting field of study is the biomedical
area. We are well aware of a healthy heart that
beats periodically, which can be seen from a record
of an electrocardiogram (ECG). The ECG
represents changes in the voltage potential due
(a)
(b)
Fig. 1.1.5 (a) Radar range measurement and (b) transmitted
and received filtered signals
Fig. 1.1.4 Watermelon responses to a tap
4 1 Basic Concepts in Signals

34. to electrochemical processes in the heart cells.
Inferences can be made about the health of the
heart under observation from the ECG. Another
important example is the electroencephalogram
(EEG), which measures the electrical activity in
brain.
Signal processing is an important area that inter-
ests every engineer. Pattern recognition and classifi-
cation is almost on top of the list. See, for example,
O’Shaughnessy (1987) and Tou and Gonzalez
(1974). For example, how do we distinguish two
phonemes, one is a vowel and the other one is a
consonant. A rough measure of frequency of a
waveform with zero average value is the number of
zero crossings per unit time. We will study in much
more detail the frequency content in a signal later in
terms of Fourier transforms in Chapter 4. Vowel
sounds have lower frequency content than the con-
sonants. A simple procedure to measure frequency
in a speech segment is by computing the number of
zero crossings in that segment. To differentiate a
vowel from a consonant, set a threshold level for
the frequency content for vowels and consonants
that differentiate between vowels and consonants.
If the frequency content is higher than this thresh-
old, then the phoneme is a consonant. Otherwise, it
is a vowel. If we like to distinguish one vowel from
another we may need more than one measure. Vocal
tract can be modeled as an acoustic tube with reso-
nances, called formants. Two formant frequencies
can be used to distinguish two vowels, say /u/ and /
a/. See Problem 1.1.1. Two formant frequencies may
not be enough to distinguish all the phonemes, espe-
cially if the signal is corrupted by noise.
Consider a simple pattern classification problem
with M prototype patterns z1; z2; . . . ; zM, where zi
is a vector representing an ith pattern. For simpli-
city we assume that each pattern can be represented
by a pair of numbers, say zi ¼ zi1; zi2
ð Þ; i = 1,2 . . . M
and classify an arbitrary pattern x ¼ x1; x2
ð Þ to
represent one of the prototype patterns. The Eucli-
dean distance between a pattern x and the ith pro-
totype pattern is defined by
Di ¼ x zi
k k ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x1 zi1
ð Þ2
þ x2 zi2
ð Þ2
q
: (1:1:6)
A simple classifier is a minimum distance classifier
that computes the distance from a pattern x of the
signal to be classified to the prototype of each class
and assigns the unknown pattern to the class which it
is closest to. That is, if Di5Dj; for all i 6¼ j, then we
make the decision that x belongs to the ith prototype
pattern. Ties are rare and if there are, they are
resolved arbitrarily. In the above discussion two
measures are assumed for each pattern. More mea-
sures give a better separation between classes.
There are several issues that would interest a
biomedical signal processor. These include removal
of any noise present in the signals, such as 60-Hz
interference picked up by the instruments, interference
of the tools or meters that measure a parameter, and
other signals that interfere with the desired signal.
Finding the important facets in a signal, such as the
frequency content, and many others is of interest.
1.1.3 Analog Versus Digital Signal
Processing
Most signals are analog signals. Analog signal pro-
cessing uses analog circuit elements, such as resistors,
capacitors, inductors, and active components, such
as operational amplifiers and non-linear devices.
Since the inductors are made from magnetic mate-
rial, they have inherent resistance and capacitance.
This brings the quality of the components low. They
tend to be bulky and their effectiveness is reduced. To
alleviate this problem, active RC networks have been
popular. Analog processing is a natural way to solve
differential equations that describe physical systems,
without having to resort to approximate solutions.
Solutions are obtained in real time. In Chapter 10 we
will see an example of analog encryption of a signal,
wherein the analog speech is scrambled by the use of
modulation techniques.
Digital signal processing makes use of a special
purpose computer, which has three basic elements,
namely adders, multipliers, and memory for sto-
rage. Digital signal processing consists of numerical
computations and there is no guarantee that the
processing can be done in real time. To encrypt a
set of numbers, these need to be converted into
another set of numbers in the digital encryption
scheme, for example. The complete encrypted signal
is needed before it can be decrypted. In addition, if
the input and the output signals are analog, then an
1.1 Introduction to the Book and Signals 5

35. analog-to-digital converter (A/D), a digital proces-
sor, and a digital-to-analog converter (D/A) are
needed to implement analog processing by digital
means. Special purpose processor with A/D and
D/A converters can be expensive.
Digital approach has distinct advantages over
analog approaches. Digital processor can be used
to implement different versions of a system by chan-
ging the software on the processor. It has flexibility
and repeatability. In the analog case, the system has
to be redesigned every time the specifications are
changed. Design components may not be available
and may have to live with the component values
within some tolerance. Components suffer from
parameter variations due to room temperature,
humidity, supply voltages, and many other aspects,
such as aging, component failure. In a particular
situation, many of the above problems need to be
investigated before a complete decision can be
made. Future appears to be more and more digital.
Many of the digital signal processing filter designs
are based on using analog filter designs. Learning
both analog and digital signal processing is desired.
Deterministic and random signals: Deterministic
signals are specified for any given time. They can be
modeled by a known function of time. There is no
uncertainty with respect to any value at any time.
For example, xðtÞ ¼ sinðtÞ is a deterministic signal.
A random signal yðtÞ ¼ xðtÞ þ nðtÞ can take ran-
dom values at any given time, as there is uncertainty
about the noise signal nðtÞ. We can only describe
such signals through statistical means, and the dis-
cussion on this topic will be minimal.
1.1.4 Examples of Simple Functions
To begin the study we need to look into the concept
of expressing a signal in terms of functions that can
be generated in a laboratory. One such function is
the sinusoidal function xðtÞ ¼ A0 cosðo0t þ y0Þ seen
earlier in (1.1.1), where A0; o0; and y0 are some
constants. A digital signal can be defined as a
sequence in the forms
x½n ¼
an
; n 0
0; n50
; (1:1:7)
fx½ng ¼ f. . . ; 0; 0; 0; 1; a; a2
; . . . ; an
; . . .g; (1:1:8)
fx½ng ¼ f. . . ; 1
#
; a; a2
; . . . ; an
; . . .g or fx½ng
¼ f1
#
; a; a2
; . . . ; an
; . . .g
(1:1:9)
In (1.1.8) reference points are not identified. In
(1.1.9), the arrow below 1 is the 0 index term. The
first term is assumed to be zero index term if there is
no arrow and all the values of the sequence are zero
for n50. We will come back to this in Chapter 8.
A signal xðtÞ is a real signal if its value at some t
is a real number. A complex signal xðtÞ consists of
two real signals, x1ðtÞ and x2ðtÞ such that x(t) = x_1
(t) + jx2ðtÞ; where j ¼
ffiffiffiffiffiffiffiffi
1:
p
The symbol j (or i) is
used to represent the imaginary part.
Interesting functions: a. P Function: The P func-
tion is centered at t0 with a width of t s shown in
Fig. 1.1.6. It is not defined at t ¼ t0 t=2 and is
symbolically expressed by
Y t t0
t
h i 1; t t0
j j5t=2
0; otherwise
(1:1:10)
xðtÞP
t t0
t
h i
¼
xðtÞ; t t0
j j5t=2
0; otherwise
This P function is a deterministic signal. It is even, as
P½t ¼ P½t and P½t0 t ¼ P½t t0. Some use the
symbol‘‘rect’’andð1=tÞrectððt t0Þ=tÞisarectangular
pulse of width t scenteredatt ¼ t0 with a height ð1=tÞ:
b. L Function: The triangular function shown in
Fig. 1.1.7 is defined by
L
t t0
t
h i
¼
1 jtt0j
t ; jt t0j5t
0; otherwise
:
(
(1:1:11)
0
t t
τ
−
Π
Fig. 1.1.6 A P function
6 1 Basic Concepts in Signals

36. Rectangular function defined in (1.1.10) has a width
of t seconds, whereas the triangular function
defined in (1.1.11) has a width of 2t s. The symbol
‘‘tri’’ is also used for a triangular function and
triððt t0Þ=tÞ describes the function in (1.1.11).
c. Unit step function: It is shown in Fig. 1.1.8 and is
u t
ð Þ ¼
1; t40
0; t50
not defined at t ¼ 0
ð Þ: (1:1:12)
The unit step function at t ¼ 0 can be defined expli-
citly as 0 or 1 or ½uð0þ
Þ þ uð0
Þ=2=.5
d. Exponential decaying function: A simple such
function is
x t
ð Þ ¼
X0et=Tc
; t 0; Tc40
0; otherwise
:
(
(1:1:13)
See Fig. 1.1.9. It has a special significance, as xðtÞ is
the solution of a first-order differential equation.
The constants X0 and Tc can take different values and
xðTcÞ
X0
¼ e1
:37 or xðTcÞ ¼ :37X0: (1:1:14)
xðtÞ decreases to about 37% of its initial value in Tc
s and is the time constant. It decreases to 2% in four
time constants and Xð4TcÞ :018X0: A measure
associated with exponential functions is the half
life Th defined by
xðThÞ ¼
1
2
X0; eTh=Tc
¼
1
2
; Th ¼ Tc logeð2Þ ffi :693Tc:
(1:1:15)
e. One-sided and two-sided exponentials: These are
described by
x1ðtÞ ¼
eat
; t 0
0; t50
; a40
x2ðtÞ ¼
0; t 0
eat
; t50
a40; x3ðtÞ ¼ eajtj
; a40:
(1:1:16)
x1ðtÞ is the right-sided exponential, x2ðtÞ is the left-
sided exponential, and x3ðtÞ is the two-sided expo-
nential. These are sketched in Fig. 1.1.10. Using the
unit step function, we have x2ðtÞ ¼ x3ðtÞuðtÞ and
x1ðtÞ ¼ x3ðtÞuðtÞ.
Fig. 1.1.8 Unit step function
– /
0
( ) ( ), 0
c
t T
x t X e u t τ
=
Fig. 1.1.9 Exponential decaying function
0
t t
τ
−
Λ
Fig. 1.1.7 A L function
2 ( )
x t 1( )
x t 3 ( )
x t
(a) (b) (c)
Fig. 1.1.10 Exponential
functions (a) x2 (t) = eat
u(t),a 0,
(b) x1ðtÞ ¼ eat
uðtÞ; a 0;
and (c) x3ðtÞ ¼ ea t
j j
; a 0
1.1 Introduction to the Book and Signals 7

37. 1.2 Useful Signal Operations
1.2.1 Time Shifting
Consider an arbitrary signal starting at t ¼ 0
shown in Fig. 1.2.1a. It can be shifted to the
right as shown in Fig. 1.2.1b. It starts at time
t ¼ a 0, a delayed version of the one in
Fig. 1.2.1a. Similarly it can be shifted to the left
starting at time a shown in Fig. 1.2.1c. It is an
advanced version of the one in Fig. 1.2.1a. We
now have three functions: xðtÞ, xðt aÞ; and
xðt þ aÞ with a 0. The delayed and advanced
unit step functions are
u t a
ð Þ ¼
1; t4a
0; t5a
;
uðt þ aÞ ¼
1; t4 a
0; t5 a
; ða40Þ: (1:2:1)
From (1.1.16), the right-sided delayed exponential
decaying function is
x1ðt tÞ ¼ eaðttÞ
uðt tÞ; a 0; t 0: (1:2:2)
1.2.2 Time Scaling
The compression or expansion of a signal in time
is known as time scaling. It is expanded in time if
a5 1 and compressed in time if a 1 in
f t
ð Þ ¼ x at
ð Þ; a 0: (1:2:3)
Example 1.2.1 Illustrate the rectangular pulse func-
tions P½t; P 2t
½ ; and P t=2
½ .
Solution: These are shown in Fig. 1.2.2 and are of
widths 1, (1/2), and 2, respectively. The pulse P 2t
½
is a compressed version and the pulse P½t=2 is an
expanded version of the pulse function P t
½ .
1.2.3 Time Reversal
If a ¼ 1 in (1.2.3), that is, fðtÞ ¼ xðtÞ, then the
signal is time reversed (or folded).
Example 1.2.2 Let x1ðtÞ ¼ eat
uðtÞ: Give its time-
reversed signal.
Solution: The time-reversed signal of x1ðtÞ is
x2ðtÞ ¼ eat
uðtÞ.
1.2.4 Amplitude Shift
The amplitude shift of xðtÞ by a constant K is
fðtÞ ¼ K þ xðtÞ.
Combined operations: Some of the above signal
operations can be combined into a general form.
The signal yðtÞ ¼ xðat t0Þ may be described by
one of the two ways, namely:
Π[t] Π[2t] Π[t /2]
(a) (b) (c)
Fig. 1.2.2 Pulse functions
x(t) x(t–a) x(t+a)
(a) (b) (c)
Fig. 1.2.1 (a) x(t), (b) x(t a), (c) x(t + a), a 0
8 1 Basic Concepts in Signals

38. 1. Time shift of t0 followed by time scaling by a.
2. Time scaling by (a) followed by time shift of ðt0=aÞ.
These can be visualized by the following:
1: x t
ð Þ
shift
t ! t t0
! v t
ð Þ ¼ x t t0
ð Þ
scale
t ! at
! y t
ð Þ
¼ v at
ð Þ ¼ x at t0
ð Þ:
(1:2:4)
2: x t
ð Þ
scale
t ! at
! g t
ð Þ ¼ x at
ð Þ
shift
t ! t t0=a
ð Þ
! y t
ð Þ
¼ g t t0=a
ð Þ
ð Þ ¼ x at t0
ð Þ:
(1:2:5)
Notation can be simplified by writing
yðtÞ ¼ xðaðt ðt0=aÞÞÞ. Noting that b ¼ at t0 is a
linear equation in terms of two constants a and t0, it
follows:
yð0Þ ¼ xðt0Þ and yðt0=aÞ ¼ xð0Þ: (1:2:6)
These two equations provide checks to verify the
end result of the transformation. Following exam-
ple illustrates some pitfalls in the order of time
shifting and time scaling.
Example 1.2.3 Derive the expression for
yðtÞ ¼ xð3t þ 2Þ assuming xðtÞ ¼ P½t=2.
Solution: Using (1.2.4) with a ¼ 3 and t0 ¼ 2, we
have
vðtÞ ¼xðt t0Þ ¼ P
t þ 2
2
;
yðtÞ ¼vð3tÞ ¼ P
3t þ 2
2
¼ P
t þ ð2=3Þ
2=3
:
Using (1.2.5), we have
gðtÞ ¼xðatÞ ¼ P
3t
2
;
yðtÞ ¼gðt
t0
a
Þ ¼ gðt þ
2
3
Þ
¼P
3ðt þ ð2=3ÞÞ
2
¼ P
t þ ð2=3Þ
2=3
:
It is a rectangular pulse of unit amplitude centered
at t ¼ ð2=3Þ with width (2/3). We can check the
equations in (1.2.6) and yð0Þ ¼ xð3Þ ¼ 0 and
yðt0=aÞ ¼ yð3=2Þ ¼ 0.
1.2.5 Simple Symmetries: Even and Odd
Functions
Continuous-time even and odd functions satisfy
xðtÞ ¼ xðtÞ xeðtÞ, an even function, x ( t) =
xðtÞ x0ðtÞ, an odd function. (1.2.7)
Examples of even and odd functions are shown in
Fig. 1.2.3. The function cosðo0tÞ is an even function
and x0ðtÞ ¼ sinðo0tÞ is an odd function. An arbi-
trary real signal, xðtÞ; can be expressed in terms of
its even and odd parts by
xðtÞ ¼ xeðtÞ þ x0ðtÞ; xeðtÞ
¼ ðxðtÞ þ xðtÞÞ=2
½ ; x0ðtÞ
¼ ðxðtÞ xðtÞÞ=2
½ :
(1:2:8)
1.2.6 Products of Even and Odd Functions
Let xeðtÞ and yeðtÞ be two even functions and x0ðtÞ
and y0ðtÞ be two odd functions and arbitrary. Some
general comments can be made about their products.
xeðtÞyeðtÞ ¼ xeðtÞyeðtÞ; even function: (1:2:9)
xeðtÞy0ðtÞ ¼ xeðtÞy0ðtÞ; odd function: (1:2:10)
x0ðtÞy0ðtÞ ¼ ð1Þ2
x0ðtÞy0ðtÞ
¼ x0ðtÞy0ðtÞ; even function:
(1:2:11)
xe(t) x0(t)
(a) (b)
Fig. 1.2.3 (a) Even function and (b) odd function
1.2 Useful Signal Operations 9

39. Note that the functions P[t], L[t] and P[t]cosðo0tÞ are
even functions and P½t sinðo0tÞ is an odd function.
The even and odd parts of the exponential pulse
x1ðtÞ ¼ et
uðtÞ are shown in Fig. 1.2.4 and are
x1eðtÞ ¼
1
2
ðx1ðtÞ þ x1ðtÞÞ;
x10ðtÞ ¼
1
2
ðx1ðtÞ x1ðtÞÞ:
(1:2:12)
1.2.7 Signum (or sgn) Function
The signum (or sgn) function is an odd function
shown in Fig. 1.2.5:
sgnðtÞ ¼ uðtÞ uðtÞ ¼ 2uðtÞ 1: (1:2:13)
sgnðtÞ ¼ lim
a!0
½eat
uðtÞ eat
uðtÞ; a 0: (1:2:14)
It is not defined at t ¼ 0 and is chosen as 0.
1.2.8 Sinc and Sinc2
Functions
The sinc and sinc2
functions are defined in terms of
an independent variable l by
sincðplÞ ¼
sinðplÞ
pl
; sinc2
ðplÞ ¼
sin2
ðplÞ
ðplÞ2
(1:2:15)
Some authors use sincðlÞ for sincðplÞ in (1.2.15).
Notation in (1.2.15) is common. Sinc ðplÞ is inde-
terminate at t ¼ 0: Using the L’Hospital’s rule,
lim
l!0
sinðplÞ
pl
¼ lim
l!0
d sinðplÞ
dl
dðplÞ
dl
¼ lim
l!0
p cosðplÞ
p
¼ 1:
(1:2:16)
In addition, since sinðplÞ is equal to zero for
l ¼ n; n an integer, it follows that
sincðplÞ ¼ 0; l ¼ n; n 6¼ 0 and an integer:
(1:2:17a)
Interestingly, the function sincðplÞ
j j is bounded by
ð1=plÞ
j j as sinðplÞ
j j 1. The side lobes of
sincðplÞ
j j are larger than the side lobes of
sinc2
ðplÞ, which follows from the fact that the
square of a fraction is less than the fraction we
started with. Both the sinc and the sinc2
functions
are even. That is,
sincð plÞ ¼ sincðplÞ and sinc2
ðplÞ ¼ sinc2
ðplÞ:
(1:2:17b)
These functions can be evaluated easily by a calcu-
lator. For the sketch of a sinc function using
MATLAB, see Fig. B.5.2 in Appendix B.
1.2.9 Sine Integral Function
The sine integral function is an odd function defined
by (Spiegel, 1968)
sgn(t)
t
Fig. 1.2.5 Signum function sgnðtÞ
x1(t)
x2(t)
x3(t)
(a) (b) (c)
Fig. 1.2.4 (a) x1ðtÞ, (b) xie(t)
= x2 (t) even part of x1(t),
and (c) x10ðtÞ = x3 (t) odd
part of x1ðtÞ
10 1 Basic Concepts in Signals

40. SiðyÞ ¼
Zy
0
sinðaÞ
a
da: (1:2:18a)
The values of this function can be computed
numerically using the series expression
SiðyÞ ¼
y
ð1Þ1!
y3
ð3Þ3!
þ
y5
ð5Þ5!
y7
ð7Þ7!
þ . . .
(1:2:18b)
Some of its important properties are
SiðyÞ ¼ SiðyÞ; Sið0Þ ¼ 0;
SiðpÞ ffi 2:0123; Sið1Þ ¼ ðp=2Þ : (1:2:18c)
Si function converges fast and only a few terms in
(1.2.18b) are needed for a good approximation.
1.3 Derivatives and Integrals
of Functions
It will be assumed that the reader is familiar with some
of the basic properties associated with the derivative
and integral operations. We should caution that
derivatives of discontinuous functions do not exist
in the conventional sense. To handle such cases,
generalized functions are defined in the next section.
The three well-known formulas to approximate a
derivative of a function, referred to as forward differ-
ence, central difference, and backward difference, are
x0
ðtÞ ¼
dxðtÞ
dt
:
xðt þ hÞ xðhÞ
h
;
x0
ðtÞ :
xðt þ hÞ xðt hÞ
2 h
;
x0
ðtÞ :
xðtÞ xðt hÞ
h
: (1:3:1)
MATLAB evaluations of the derivatives are given
in Appendix B. If we have a function of two vari-
ables, then we have the possibility of taking the
derivatives one or the other, leading to partial deri-
vatives. Let xðt; aÞ be a function of two variables.
The two partial derivatives of xðt; aÞ with respect to
t, keeping a constant, and with respect to a, keeping
t constant are, respectively, given by
@xðt; aÞ
@t
¼ lim
Dt!0
xðt þ Dt; aÞ xðt; aÞ
Dt
;
@xðt; aÞ
@a
¼ lim
Da!0
xðt; a þ DaÞ xðt; aÞ
Da
: (1:3:2)
Assuming the second (first) variable is not a func-
tion of the first (second) variable, the differential of
xðt; aÞ is
dx ¼
@x
@t
dt þ
@x
@a
da:
The integral of a function over an interval is the area
of the function over that interval.
Example 1.3.1 Compute the value of the integral of
xðtÞ shown in Fig. 1.3.1.
Solution: Divide the area into three parts as identi-
fied in the figure. The three parts are in the intervals
ð0; aÞ, ða; bÞ, and ðb; cÞ, respectively. The areas of the
two triangles are identified by A1 and A2 and the
area of the rectangle by A3. They can be individually
computed and then add the three areas to get the
total area. That is,
A1 ¼
1
2
aB; A2 ¼ ðb aÞC; A3 ¼
1
2
ðc bÞB;
A ¼ A1 þ A2 þ A3 ¼
Zc
0
xðtÞdt:
If the function is arbitrary and cannot be divided
into simple functions like in the above example, we
can approximate the integral by dividing the area
into small rectangular strips and compute the area
by adding the areas in each strip.
Fig. 1.3.1 Computation of Integral of x(t) using areas
1.3 Derivatives and Integrals of Functions 11

41. Example 1.3.2 Consider the function xðtÞ shown in
Fig. 1.3.2. Find the integral of this function using
the above approximation for a5t5b. Assume the
values of the function are known as xðaÞ; xða þ DtÞ;
xða þ 2DtÞ; :::; and x((a þ N - 1)D t)
Solution: Assuming Dt is small enough that we can
approximate the area in terms of rectangular strips
using the rectangular integration formula and
Zb
a
xðtÞdt
X
N1
n¼0
xða þ nDtÞ
#
Dt; Dt ¼ ðb aÞ=N:
(1:3:3a)
Note that xðaÞDt gives the approximate area in the
first strip. If the width of the strips gets smaller and
the number of strips increases correspondingly, then
the approximation gets better. In the limit, i.e., when
Dt ! 0, approximation approaches the value of the
integral. In computing the area of the kth rectangular
strip, xða þ ðk 1ÞDtÞ was used to approximate the
height of the pulse. Some other value of the function
in the interval, such as the value of the function in the
middle of the strip, could be used.
MATLAB evaluation of integrals is discussed in
Appendix B. In Chapter 8 appropriate values for
Dt will be considered in terms of the frequency con-
tent in the signal. Instead of rectangular integration
formula, there are other formulas that are useful.
One could assume that each strip is a trapezoid and
using the trapezoidal integration formula the integral
is approximated by
Zb
a
xðtÞdt ½xðaÞ þ 2xða þ DtÞ þ 2xða þ 2DtÞ þ
þ 2xða þ ðN 1ÞDtÞ þ xða þ NDtÞðDt=2Þ: (1:3:3b)
If the time interval Dt ¼ ðb aÞ=N is sufficiently
small, then the difference between the two formulas
in (1.3.3a) and (1.3.3b) would be small and (1.3.3a)
is adequate.
1.3.1 Integrals of Functions with
Symmetries
The integrals of functions with even and odd sym-
metries around a symmetric interval are
Za
a
xeðtÞdt ¼ 2
Za
0
xeðtÞdt and
Za
a
x0ðtÞdt ¼ 0;
(1:3:4)
where a is an arbitrary positive number.
Example 1.3.3 Evaluate the integrals of the func-
tions given below:
x1ðtÞ ¼ P t=2a
½ ; x2ðtÞ ¼ tx1ðtÞ: (1:3:5)
Solution: x1ðtÞ is a rectangular pulse with an even
symmetry and x2ðtÞ is an odd function with an odd
symmetry. The integrals are
A1 ¼
Za
a
x1ðtÞdt ¼ 2
Za
0
dt ¼ 2a;
A2 ¼
Za
a
x2ðtÞdt ¼ 0:
1.3.2 Useful Functions from Unit Step
Function
The ramp and the parabolic functions can be
obtained by
xrðtÞ ¼
Zt
0
uðtÞdt ¼ tuðtÞ and
xpðtÞ ¼
Zt
0
xrðtÞdt ¼ðt2
=2ÞuðtÞ: (1:3:6)
Section 1.4 considers the derivatives of the unit step
functions.
x(t)
Fig. 1.3.2 xðtÞ and its approximation using its samples
12 1 Basic Concepts in Signals

42. Now consider Leibniz’s rule, interchange of deri-
vative and integral, and interchange of integrals
without proofs. For a summary, see Peebles (2001).
1.3.3 Leibniz’s Rule
LetgðtÞ ¼
Z
bðtÞ
aðtÞ
zðx; tÞdx: (1:3:7)
aðtÞ and bðtÞ are assumed to be real differentiable
functions of a real parameter t, and zðx; tÞ and its
derivative dzðx; tÞ=dt are both continuous functions
of x and t. The derivative of the integral with respect
to t is Leibniz’s rule Spiegel (1968) and is
dgðtÞ
dt
¼ z½bðtÞ; tÞ
daðtÞ
dt
z½aðtÞ; t
dbðtÞ
dt
þ
Z
bðtÞ
aðtÞ
@zðx; tÞ
@t
dx: (1:3:8)
1.3.4 Interchange of a Derivative
and an Integral
When the limits in (1.3.7) are constants, say aðtÞ ¼ a
and bðtÞ ¼ b, then the derivatives of these limits will
be zero and (1.3.7) and (1.3.8) reduce to
gðtÞ ¼
Zb
a
zðx; tÞdx;
dgðtÞ
dt
¼
d
dt
Zb
a
zðx; tÞdt ¼
Zb
a
@zðx; tÞ
@ t
dx: (1:3:9)
The derivative and the integral operations may be
interchanged.
1.3.5 Interchange of Integrals
If any one of the following conditions,
Z1
1
Z1
1
xðt;aÞ
j jdtda51;
Z1
1
½
Z1
1
xðt;aÞ
j jdadt51;
Z1
1
½
Z1
1
xðt;aÞdt
j jda
51; (1:3:10)
is true, then Fubini’s theorem (see Korn and Korn
(1961) for a proof) states that
Z1
1
Z1
1
xðt; aÞdt
2
4
3
5da ¼
Z1
1
Z1
1
xðt; aÞdt
2
4
3
5da
¼
Z1
1
Z1
1
xðt; aÞdtda
: (1:3:11)
Signals generated in a lab are well behaved and they
are valid.
In Chapter 3 on Fourier series, integrating a
product of a simple function, say hðtÞ, with its nth
derivative goes to zero; such a polynomial, and the
other one is a sinusoidal function gðtÞ, such as
sinðo0tÞ or cosðo0tÞ or ejo0t
is applicable. The gen-
eralized integration by parts formula comes in
handy.
R
hðnÞ
ðtÞgðtÞdt ¼ hðn1Þ
ðtÞgðtÞ hðn2Þ
ðtÞg0
ðtÞ
þhðn3Þ
ðtÞg00
ðtÞ ð1Þn R
hðtÞgðnÞ
ðtÞdt;
gðkÞ
ðtÞ ¼
dk
gðtÞ
dtk
; and hðkÞ
ðtÞ ¼
dk
hðtÞ
dtk
:
(1:3:12)
Using (1.3.12), the following equalities can be seen:
Z
t cosðtÞdt ¼ cosðtÞ þ t sinðtÞ;
Z
t sinðtÞdt ¼ sinðtÞ t cosðtÞ (1:3:13a)
Z
t2
cosðtÞdt ¼ 2t cosðtÞ þ ðt2
2Þ sinðtÞ;
Z
t2
sinðtÞdt ¼ 2t sinðtÞ ðt2
2Þ cosðtÞ: (1:3:13b)
1.3 Derivatives and Integrals of Functions 13

43. 1.4 Singularity Functions
The impulse function, or the Dirac delta function, a
singularity function, is defined by
dðtÞ ¼
0; t 6¼ 0
1; t ¼ 0
with
Z1
1
dðtÞdt ¼ 1: (1:4:1)
dðtÞ takes the value of infinity at t ¼ 0 and is zero
everywhere else. See Fig. 1.4.1b. Impulse function is
a continuous function and the area under this func-
tion is equal to one. Note that a line has a zero area.
Here, a generalized or a distribution function is
defined that is nonzero only at one point and has a
unit area. A delayed or an advanced impulse func-
tion can be defined by dðt t0Þ, where t0 is assumed
to be positive in the expressions. The ideal impulse
function cannot be synthesized. It is useful in the
limit. For example,
dðtÞ ¼ lim
e!0
1
e
P
t
e
h i
; dðtÞ ¼ lim
e!0
1
e
L
t
e
h i
:
(1:4:2)
Figure 1.4.1a illustrates the progression of rectan-
gular pulses of unit area toward the delta function.
As e is reduced, the height increases and, in the limit,
the function approaches infinity at t ¼ 0 and the
area of the rectangle is 1. There are other functions
that approximate the impulse function in the limit.
A nice definition is given in terms of an integral of a
product of an impulse and a test function fðtÞ by
Korn and Korn (1961):
Zt2
t1
fðtÞdðt t0Þdt
¼
0; t05t1 or t25t0
ð1=2Þ fðtþ
0 Þ þ fðt
0 Þ ; t15t05t2
ð1=2Þfðtþ
0 Þ; t0 ¼ t1
ð1=2Þfðt
0 Þ; t0 ¼ t2
8
:
: (1:4:3)
fðtÞ is a testing (or a test) function of t and is
assumed to be continuous and bounded in the
neighborhood of t ¼ t0 and is zero outside a finite
interval. That is fð 1Þ ¼ 0. The integral in (1.4.3)
is not an ordinary (Riemann) integral. In this sense,
dðtÞ is a generalized function. A simpler form of
(1.4.3) is adequate. dðtÞ has the property that
Z1
1
fðtÞdðt t0Þdt ¼ fðt0Þ; (1:4:4a)
where fðtÞ is a test function that is continuous at
t ¼ t0: As a special case, consider t0 ¼ 0 in (1.4.4a)
and fðtÞ ¼ 1. Equation (1.4.4a) can be written as
Z1
1
dðtÞdt ¼
Z0þ
0
dðtÞdt ¼ 1: (1:4:4b)
That is, the area under the impulse function is 1. The
integral in (1.4.4a) sifts the value of fðtÞ at t ¼ t0
and dðtÞ is called a sifting function. In summary, the
impulse dðt t0Þ has unit area (or weight) centered
at the point t ¼ t0 and zero everywhere else. Since
the dðt t0Þ exists only at t ¼ t0, and fðtÞ at t ¼ t0 is
fðt0Þ, we have an important result
fðtÞdðt t0Þ ¼ fðt0Þdðt t0Þ: (1:4:5)
There are many limiting forms of impulse functions.
Some of these are given below.
Notes: In the limit, the following functions can be
used to approximate dðtÞ:
dðtÞ ¼ lim
t!0
xiðtÞ; x1ðtÞ ¼
2
t
e t=t
j j
;
x2ðtÞ ¼
1
t
sincðt=tÞ; x3ðtÞ ¼
1
t
sinc2
ðt=tÞ
(a) (b)
Fig. 1.4.1 (a) Progression toward an impulse as e ! 0 and
(b) symbol for dðtÞ
14 1 Basic Concepts in Signals

44. x4ðtÞ ¼
1
t
epðt=tÞ2
; x5ðtÞ ¼
t
pðt2 þ t2Þ
: (1:4:6a)
x1ðtÞ is a two-sided exponential function; x2ðtÞ and
x3ðtÞ are sinc functions. Sinc function does not go to
zero in the limit for all t. See the discussion by
Papoulis (1962). x4ðtÞ is a Gaussian function and
x5ðtÞ is a Lorentzian function. To prove these,
approximate the impulse function in the limit,
using (1.4.4a) and (1.4.4b).
Example 1.4.1 Show that the Lorentzian function
x5ðtÞ approaches an impulse function as t ! 0.
Show the result by using the equations: a. (1.4.1)
and b. (1.4.4a and b).
Solution: a. Clearly as t ! 0, x5ðtÞ ! 0 for t not
equal to zero. As t ! 0; x5ðtÞ ! 1. From tables,
1
p
Z1
1
t
t2 þ t2
dt ¼ 1; lim
t!0
x5ðtÞ ¼ dðtÞ: (1:4:6b)
b. First by (1.4.4a), with t0 ¼ 0 and fðtÞ ¼ 1 in the
neighborhood of t ¼ 0 results in
lim
t!0
Z1
1
fðtÞx5ðt t0Þdt ¼ lim
t!0
Z1
1
x5ðtÞdt ¼ 1:
(1:4:6c)
Using (1.4.4b) and the integral tables, it follows that
Z0þ
0
1
p
t
ðt2 þ t2Þ
dt ¼
t
p
1
t
tan1 t
t
0þ
0
¼
1
p
p
2
p
2
h i
¼ 1:
1.4.1 Unit Impulse as the Limit
of a Sequence
Another approach to the above is through a
sequence. See the work of Lighthill (1958). Also
see Baher, (1990). A good function fðtÞ is
differentiable everywhere any number of times,
and, in addition, the function and its derivative
decrease at least as rapidly as (1=tn
) as t ! 1 for
all n. The derivative of a good function is another
good function and the sums and the products of two
good functions are good functions. A sequence of
good functions
xnðtÞ
f g x1ðtÞ; x2ðtÞ; . . . ; xnðtÞ
f g (1:4:7)
is called regular, if for any good function fðtÞ, the
following limit exists:
VxðfÞ ¼ lim
n!1
Z1
1
xnðtÞ
f gfðtÞdt: (1:4:8)
Example 1.4.2 Find the limit in (1.4.8) of the follow-
ing sequence:
xnðtÞ
f g ¼ eðt2
=n2
Þ
n o
: (1:4:9)
Solution: The limit in (1.4.8) is
VxðfÞ ¼
Z1
1
fðtÞdt: (1:4:10)
Two regular sequences of good functions are con-
sidered equivalent if the limit in (1.4.8) is the same
for the two sequences. For example, eðt4
=n4
Þ
and
eðt2
=n2
Þ
are equivalent only in that sense. The func-
tion VxðfÞ defines a distribution xðtÞ and the limit
of the sequence
xðtÞ lim
n!1
xnðtÞ
f g: (1:4:11)
An impulse function can be defined in terms of a
sequence of functions and write
dðtÞ lim
n!1
xnðtÞ
f g if lim
n!1
Z1
1
fðtÞ xnðtÞ
f gdt ¼ fð0Þ
(1:4:12)
¼
)
Z1
1
fðtÞdðtÞdt ¼ fð0Þ: (1:4:13)
Many sequences can be used to approximate an
impulse.
1.4 Singularity Functions 15

45. Notes: A constant can be interpreted as a general-
ized function defined by the regular sequence
xnðtÞ
f g so that, for any good function fðtÞ
lim
n!1
Z1
1
xnðtÞ
f gfðtÞdt ¼ K
Z1
1
fðtÞdt: (1:4:14)
Using the function in (1.4.9), it follows that
xnðtÞ
f gKet2
=n2
;K¼ lim
n!1
fKeðt2
=n2
Þ
g: (1:4:15)
1.4.2 Step Function and the Impulse
Function
Noting the area under the impulse function is one, it
follows that
Zt
1
dðaÞda ¼ uðtÞ: (1:4:16)
Asymmetrical functions ðdþðtÞ and dðtÞÞ and
(uþðtÞ and uðtÞÞ can be defined and are
uþðtÞ ¼
Z1
1
dþðtÞdt ¼
1; t 0
0; t 0
;
uðtÞ ¼
Z1
1
dðtÞdt ¼
1; t 0
0; t50
(1:4:17)
uðtÞ ¼
1; t 0
ð1=2Þ; t ¼ 0
0; t50
8
:
(1:4:18)
These step functions differ in how the value of the
function at t ¼ 0 is assigned and are only of theore-
tical interest. Here the step function is assumed to be
uðtÞ and ignore the subscript: Derivative of the unit
step function is an impulse function, which can be
shown by using the generalized function concept.
Example 1.4.3 Using the property of the test func-
tion, lim
t! 1
fðtÞ ¼ 0, show that
Z1
1
x0
ðtÞfðtÞdt ¼
Z1
1
xðtÞf0
ðtÞdt; f0
ðtÞ ¼
df
dt
(1:4:19)
Solution: Using the integration by parts, we have
the result below and (1.4.19) follows:
Z1
1
x0
ðtÞfðtÞdt ¼ xðtÞfðtÞj1
1
Z1
1
xðtÞf0ðtÞdt:
(1:4:20)
Notes: Let g1ðtÞ and g2ðtÞ be two generalized func-
tions and
Z1
1
fðtÞg1ðtÞdt ¼
Z1
1
fðtÞg2ðtÞdt
: (1:4:21)
The two generalized functions are equal, i.e.,
g1ðtÞ ¼ g2ðtÞ only in the sense of (1.4.21). It is called
the equivalency property.
Example 1.4.4 Show that the derivative of the unit
step function is an impulse function using the
equivalence property.
Solution: First
Z1
1
uðtÞfðtÞdt ¼
Z1
0
fðtÞdt: (1:4:22)
Noting that fð1Þ ¼ 0, it follows that
Z1
1
u0
ðtÞfðtÞdt ¼
Z1
1
uðtÞf0
ðtÞdt
¼
Z1
0
f0
ðtÞdt ¼ ½fð1Þ fð0Þ ¼ fð0Þ; (1:4:23)
Z1
1
u0
ðtÞfðtÞdt
¼
Z1
1
dðtÞfðtÞdtand u0
ðtÞ
¼
duðtÞ
dt
¼ dðtÞ: (1:4:24)
The derivative of a parabolic function results in a
ramp function, the derivative of a ramp function
results in a unit step function, and the derivative of a
unit step function results in an impulse function. All
of these are true only in the generalized sense. What
is the derivative of an impulse function? That is,
d0
ðtÞ ¼
ddðtÞ
dt
: (1:4:25)
16 1 Basic Concepts in Signals