Signals and Systems: A Practical Approach

479 Pages·2022·6.899 MB·English

by  D. Sundararajan

D. Sundararajan
Signals
and Systems
A Practical Approach
Second Edition
Signals and Systems
D. Sundararajan
Signals and Systems
A Practical Approach
Second Edition
D.Sundararajan
FormerlyatConcordiaUniversity
Montreal,QC,Canada
ISBN978-3-031-19376-7 ISBN978-3-031-19377-4 (eBook)
https://doi.org/10.1007/978-3-031-19377-4
1stedition:©JohnWiley&SonsSingaporePteLtd.2008
2ndedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
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Preface to the Second Edition
A thorough revision of all the chapters has been carried out. Some sections have
been significantly expanded. Some sections have been interchanged for a better
sequenceofpresentation.Minornecessarycorrectionshavebeenmadethroughout
thetext.
The new topics included are: (i) Bode plots, (ii) Nyquist diagrams, (iii) down-
sampling, (iv) bilinear transformation, (v) diagonalization, (vi) similarity trans-
formation, (vii) controllability, (viii) observability, (ix) complex numbers, and (x)
moreapplications.ExpandedsetsofMATLABprogramsareavailableatthebook’s
website. Further, simulation programs are introduced to make the learning of
mathematicalconceptseasier.
I am grateful to my editor and his team at Springer for their help and encour-
agementincompletingthesecondeditionofthebook.Ithankmyfamilyfortheir
supportduringthisendeavor.
D.Sundararajan
v
Preface to the First Edition
Theincreasingnumberofapplications,requiringaknowledgeofthetheoryofsig-
nalsandsystems,andtherapiddevelopmentsindigitalsystemstechnologyandfast
numericalalgorithmscallforachangeinthecontentandapproachusedinteaching
thesubject.Ibelievethatamodernsignalsandsystemscourseshouldemphasizethe
practical and computational aspects in presenting the basic theory. This approach
ofteachingthesubjectmakesthestudentmoreeffectiveinsubsequentcourses.In
addition,thestudentisexposedtopracticalandcomputationalsolutionshewillbe
using in his professional career. This book is my attempt to adapt the theory of
signalsandsystemstousethedigitalcomputerefficientlyasananalysistool.
A good knowledge of the fundamentals of the analysis of signals and systems
is required to specialize in such areas as signal processing, communication, and
control. As most of the practical signals are continuous functions of time and
digital systems are mostly used to process them due to several advantages, the
studyofbothcontinuousanddiscretesignalsandsystemsisrequired.Theprimary
objective of writing this book is to present the fundamentals of time-domain and
frequency-domainmethodsofsignalandlineartime-invariantsystemanalysisfrom
a practical viewpoint. As the discrete signals and systems are more often used in
practice and their concepts are relatively easier to understand, for each topic, the
discreteversionispresentedfirstfollowedbythecorrespondingcontinuousversion.
Typical applications of the methods of analysis are also provided. Comprehensive
coverageofthetransformmethods,andemphasisonpracticalmethodsofanalysis
and physical interpretation of the concepts are the key features of this book. The
well documented software, which is a supplement to this book and available on
the Internet, further greatly reduces much of the difficulty in understanding the
concepts. Based on this software, a laboratory course can be tailored to suit the
individualcourserequirements.
This book is intended to be a textbook for junior undergraduate level one-
semester signals and systems course. This book will also be useful for self-study.
Answers to selected exercises, marked ∗, are given at the end of the book. I
assume the responsibility for all the errors in this book and in the accompanying
supplements, and would very much appreciate receiving readers’ suggestions and
vii
viii PrefacetotheFirstEdition
pointing of any errors (email address: [email protected]). I am grateful
tomyeditorandhisteamatWileyfortheirhelpandencouragementincompleting
thisproject.IthankmyfamilyandmyfriendDr.A.Pedarfortheirsupportduring
thisendeavor.
D.Sundararajan
Contents
1 DiscreteSignals............................................................. 1
1.1 Introduction........................................................... 1
1.2 BasicSignals ......................................................... 3
1.2.1 Unit-ImpulseSignal....................................... 4
1.2.2 Unit-StepSignal........................................... 5
1.2.3 Unit-RampSignal......................................... 6
1.2.4 SinusoidsandExponentials............................... 6
1.3 ClassificationofSignals ............................................. 16
1.3.1 Continuous,Discrete,andDigitalSignals ............... 16
1.3.2 PeriodicandAperiodicSignals........................... 17
1.3.3 EnergyandPowerSignals................................ 19
1.3.4 Even-andOdd-SymmetricSignals....................... 20
1.3.5 CausalandNoncausalSignals............................ 23
1.3.6 DeterministicandRandomSignals....................... 24
1.4 SignalOperations .................................................... 24
1.4.1 TimeShifting.............................................. 24
1.4.2 TimeReversal ............................................. 25
1.4.3 TimeScaling............................................... 26
1.4.4 ZeroPadding .............................................. 28
1.5 NumericalIntegration................................................ 28
1.6 TheOrganizationofthisBook....................................... 29
1.7 Summary.............................................................. 32
Exercises ..................................................................... 32
2 ContinuousSignals......................................................... 37
2.1 BasicSignals ......................................................... 37
2.1.1 TheUnit-StepSignal...................................... 38
2.1.2 TheUnit-ImpulseSignal.................................. 39
2.1.3 TheUnit-RampSignal.................................... 47
2.1.4 Sinusoids .................................................. 48
ix
x Contents
2.2 ClassificationofSignals ............................................. 52
2.2.1 ContinuousSignals........................................ 52
2.2.2 PeriodicandAperiodicSignals........................... 53
2.2.3 EnergyandPowerSignals................................ 53
2.2.4 Even-andOdd-SymmetricSignals....................... 54
2.2.5 CausalandNoncausalSignals............................ 56
2.3 SignalOperations .................................................... 56
2.3.1 TimeShifting.............................................. 56
2.3.2 TimeReversal ............................................. 57
2.3.3 TimeScaling............................................... 58
2.4 Summary.............................................................. 59
Exercises ..................................................................... 60
3 Time-DomainAnalysisofDiscreteSystems ............................. 65
3.1 DifferenceEquationModel.......................................... 65
3.1.1 SystemResponse.......................................... 67
3.1.2 ImpulseResponse......................................... 71
3.1.3 Characterization of Systems by Their
ResponsestoImpulseandUnit-StepSignals............ 73
3.2 ClassificationofSystems ............................................ 74
3.2.1 LinearandNonlinearSystems............................ 74
3.2.2 Time-InvariantandTime-VaryingSystems.............. 75
3.2.3 CausalandNoncausalSystems........................... 76
3.2.4 InstantaneousandDynamicSystems..................... 77
3.2.5 InverseSystems ........................................... 77
3.2.6 ContinuousandDiscreteSystems........................ 77
3.3 Convolution-SummationModel..................................... 78
3.3.1 PropertiesofConvolution-Summation................... 83
3.3.2 The Difference Equation and the
Convolution-Summation.................................. 84
3.3.3 ResponsetoComplexExponentialInput ................ 85
3.4 SystemStability...................................................... 86
3.5 RealizationofDiscreteSystems..................................... 87
3.5.1 DecompositionofHigher-OrderSystems................ 88
3.5.2 FeedbackSystems......................................... 90
3.6 Summary.............................................................. 90
Exercises ..................................................................... 91
4 Time-DomainAnalysisofContinuousSystems ......................... 97
4.1 ClassificationofSystems ............................................ 98
4.1.1 LinearandNonlinearSystems............................ 98
4.1.2 Time-InvariantandTime-VaryingSystems.............. 99
4.1.3 CausalandNoncausalSystems........................... 100
4.1.4 InstantaneousandDynamicSystems..................... 101
4.1.5 Lumped-ParameterandDistributed-ParameterSystems 101
4.1.6 InverseSystems ........................................... 101
Contents xi
4.2 DifferentialEquationModel......................................... 102
4.3 Convolution-IntegralModel ......................................... 106
4.3.1 PropertiesofConvolution-Integral ....................... 108
4.3.2 ConvolutionofaFunctionwithaNarrowUnit
AreaPulse ................................................. 109
4.4 SystemResponse..................................................... 110
4.4.1 ImpulseResponse......................................... 111
4.4.2 ResponsetoUnit-StepInput.............................. 112
4.4.3 Characterization of Systems by Their
ResponsestoImpulseandUnit-StepSignals............ 114
4.4.4 ResponsetoComplexExponentialInput ................ 115
4.5 SystemStability...................................................... 116
4.6 RealizationofContinuousSystems ................................. 117
4.6.1 DecompositionofHigher-OrderSystems................ 118
4.6.2 FeedbackSystems......................................... 119
4.7 Summary.............................................................. 119
Exercises ..................................................................... 120
5 TheDiscreteFourierTransform.......................................... 125
5.1 TheTime-DomainandtheFrequency-Domain..................... 125
5.2 TheFourierAnalysis................................................. 127
5.2.1 TheFourVersionsofFourierAnalysis................... 129
5.3 TheDiscreteFourierTransform..................................... 129
5.3.1 TheApproximationofArbitraryWaveforms
withFiniteNumberofSamples .......................... 129
5.3.2 TheDFTandtheIDFT ................................... 130
5.3.3 DFTofSomeBasicSignals .............................. 134
5.4 PropertiesoftheDiscreteFourierTransform....................... 138
5.4.1 Linearity................................................... 139
5.4.2 Periodicity................................................. 139
5.4.3 CircularTimeReversal ................................... 140
5.4.4 Duality..................................................... 140
5.4.5 SumandDifferenceofSequences........................ 140
5.4.6 UpsamplingofaSequence................................ 141
5.4.7 ZeroPaddingtheData .................................... 142
5.4.8 CircularShiftofaSequence.............................. 143
5.4.9 CircularShiftofaSpectrum.............................. 143
5.4.10 Symmetry.................................................. 144
5.4.11 CircularConvolutionofTime-DomainSequences...... 148
5.4.12 CircularConvolutionofFrequency-DomainSequences 150
5.4.13 Parseval’sTheorem........................................ 150
5.5 ApplicationsoftheDiscreteFourierTransform.................... 151
5.5.1 ComputationoftheLinearConvolutionUsing
theDFT.................................................... 151
5.5.2 InterpolationandDecimation............................. 152