Signals and Systems

1093 Pages·2021·8.39 MB·English

by  S. Palani

S. Palani
Signals and
Systems
Second Edition
Signals and Systems
S. Palani
Signals and Systems
Second Edition
S.Palani
Professor(Retired)
NationalInstituteofTechnology
Tiruchirappalli,India
ISBN978-3-030-75741-0 ISBN978-3-030-75742-7 (eBook)
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Preface to Second Edition
IhaveventuredtobringoutthesecondeditionofthebookSignalsandSystemsinits
newformduetothesuccessandwidepatronageextendedtothepreviouseditionand
reprintsbythemembersoftheteachingfacultyandstudentcommunity.Thepresent
editionasinthepreviouseditioncoverstheundergraduatesyllabusinSignalsand
Systems fortheB.E.degreecourses.Athoroughrevisionofallthechaptersinthe
previouseditionhasbeenundertaken.Fewerrorsnoticedinthepreviouseditionhave
beenremovedandappropriatecorrectionshavebeenmade.Signalrepresentationis
a vital topic to understand the importance of the theoretical concepts in Signals
andSystems.AlargenumberofnumericalproblemshavebeenincludedinChap.1
which describes signal representation (both continuous and discrete time signals).
Similarly,theclassificationofsystemsiswellexplainedinChap.2withgraphical
illustrationwhereverpossible.Morenumberofnumericalproblemshavebeenadded
inChap.4whichdescribesFourierSeriesAnalysis.Further,thepropertiesofFSare
well explained and applied in solving many FS problems by cutting short lengthy
procedures.Similarly,inChap.6,explanationisprovidedfortheFourierTransform
methodofAnalysisandforthepropertiesofFTwhicharefrequentlyusedtosolve
numericalproblemsinaneasierway.However,inFSandFT,conventionalmethods
ofsolvingthenumericalproblemsarealsoretained.InChap.8,numericalproblems
using LT properties have been solved. I hope that the readers of this book would
appreciate the above attempts. Since every theoretical concept is explained by a
varietyofnumericalexampleswhicharepresentedinagradedmanner,thebookis
voluminous.ItakethisopportunitytothankAneBooksPvt.Ltdandthepublisher
fortakingupthisdifficultjob.
Pudukkottai,India S.Palani
v
Preface to First Edition
ThebookSIGNALSANDSYSTEMSpresentsacomprehensivetreatmentofsignals
and linear systems for the undergraduate level study. It is a rich subject with
diverse applications such as signal processing, control systems and communica-
tion systems. It provides an integrated treatment of continuous-time and discrete-
timeformsofsignalsandsystems.Thesetwoformsaretreatedsidebyside.Even
thoughcontinuous-timeanddiscrete-timetheoryhavemanymathematicalproperties
commonbetweenthem,thephysicalprocessesthataremodelledbycontinuous-time
systemsareverymuchdifferentfromthediscrete-timesystemscounterpart.
IhavewrittenthisbookwiththematerialIhavecollectedduringmylongexperi-
enceofteachingsignalsandsystemstotheundergraduatelevelstudentsinnational
levelreputedinstitutions.Thebookinthepresentformiswrittentomeettherequire-
mentsofundergraduatesyllabusofIndianUniversitiesingeneralandAnnaUniver-
sityinparticularforB.E./B.Tech.degreecourses.Theorganizationofthechapters
isasfollows.
Chapter1dealswiththerepresentationofsignalsandsystems.Itmotivatesthe
reader as to what signals and systems are and how they are related to other areas
such as communication systems, control systems and digital signal processing. In
thischapter,variousterminologiesrelatedtosignalsandsystemsaredefined.Further,
mathematicaldescription,representationandclassificationsofsignalsandsystems
areexplained.
Chapter2presentsadetaileddescriptionsofsystemclassifications.Underbroader
category,systemsareclassifiedascontinuous-timeanddiscrete-timesystems.Each
ofthemisfurtherclassifiedaslinearandnon-linear,timeinvariantandtimevarying,
static and dynamic, causal and non-causal, stable and unstable and invertible and
non-invertible.Systemsareidentifiedaccordingly.
A comprehensive treatment of time domain analysis of continuous-time and
discrete-timesystemsaregiveninChapter3.Itdevelopsconvolutionfromtherepre-
sentationofaninputsignalsasasuperpositionofimpulses.Tofindtheconvolution
oftwotimesignals,bothanalyticalaswellasgraphicalmethodsareexplained.
Chapter 4 deals with the Fourier representation of continuous-time signals.
Continuous time periodic signals are represented by trigonometric Fourier series,
polarFourierseriesandexponentialFourierseries.
vii
viii PrefacetoFirstEdition
In Chapter 5, discrete-time signals is represented by exponential Fourier series
andtheirpropertiesarederived.TheFourierspectraofdiscrete-timesignalisalso
determinedinthischapter.
ItisnotpossibletofindFourierseriesrepresentationofnon-periodicsignals.In
Chapter6,Fouriertransformisintroducedwhichcanrepresentperiodicaswellas
non-periodicsignals.InthischaptertheFouriertransformforcontinuous-timesignal
isexplained.
InChapter7,therepresentationofdiscrete-timesignalusingdiscretetimeFourier
transformisexplained.Further,discreteFouriertransformandFastFourierTrans-
form algorithm are also explained here. The Laplace transform is a very powerful
toolintheanalysisofcontinuoustimesignalsandsystems.
InChapter8,theLaplacetransformmethodisexplainedanditspropertiesderived.
The use of Laplace transform to solve differential. equation is described. Finally
differentformsofstructurerealizationofcontinuous-timesystemsarediscussed.
Chapter9isdevotedtothez-transformanditsapplicationtodiscretetimesignals
and systems. The properties of z-transform and techniques for inversion are intro-
duced in this chapter. The use of z-transform for solving difference equation is
explained. Different forms of structure realization of discrete-time system is also
explainedinthischapter.
In Chapter 10, the sampling theorem is explained. The necessary condition to
avoidaliasingisalsoexplainedhere.
Thenotablefeaturesofthisbookincludesthefollowing:
1. Thesyllabuscontentofsignalsandsystemsforundergraduatelevelofmostof
the Indian Universities in general and Anna University in particular has been
covered.
2. Theorganizationofthechapteraresequentialinnature.
3. Largenumberofnumericalexampleshavebeenworkedout.
4. Chapterobjectivesandsummaryaregivenineachchapter.
5. Forthestudentstopractice,shortandlongquestionswithanswersaregivenat
theendofeachchapter.
ItakethisopportunitytothankShri.Sunil,ManagingDirectorAneBooksIndia,for
coming forward to publish the book. I would like to express my sincere thanks to
Shri.R.Krishnamoorthi,salesmanagerAneBooksIndiawhotooktheinitiativesto
publishthebookinashortspanoftime.Iwouldliketoexpressmysincerethanksto
Mr.V.Ashokwhohasdoneawonderfuljobtokeythevoluminousbooklikethisin
averyshorttimeandbeautifullytoo.Mysincerethanksarealsoduetomycolleague
PrefacetoFirstEdition ix
Mr.N.Sathurappanwhogavesomeusefulsuggestions.Iwouldalsoliketothankmy
wifeDr.S.Manimegalai,M.B.B.S.,M.D.,whowasthesourceofinspirationwhile
preparingthisbook.
S.Palani
Contents
1 RepresentationofSignals ..................................... 1
1.1 Introduction ............................................ 1
1.2 TerminologiesRelatedtoSignalsandSystems ............... 2
1.2.1 Signal ......................................... 2
1.2.2 System ........................................ 2
1.3 ContinuousandDiscreteTimeSignals ..................... 3
1.4 BasicContinuousTimeSignals ............................ 6
1.4.1 UnitImpulseFunction ........................... 6
1.4.2 UnitStepFunction .............................. 7
1.4.3 UnitRampFunction ............................ 8
1.4.4 UnitParabolicFunction ......................... 10
1.4.5 UnitRectangularPulse(orGate)Function ......... 10
1.4.6 UnitAreaTriangularFunction .................... 11
1.4.7 UnitSignumFunction ........................... 12
1.4.8 UnitSincFunction .............................. 12
1.4.9 SinusoidalSignal ............................... 12
1.4.10 RealExponentialSignal ......................... 13
1.4.11 ComplexExponentialSignal ..................... 14
1.5 BasicDiscreteTimeSignals .............................. 15
1.5.1 TheUnitImpulseSequence ...................... 15
1.5.2 TheBasicUnitStepSequence .................... 15
1.5.3 TheBasicUnitRampSequence ................... 16
1.5.4 UnitRectangularSequence ....................... 16
1.5.5 SinusoidalSequence ............................ 17
1.5.6 DiscreteTimeRealExponentialSequence .......... 19
1.6 BasicOperationsonContinuousTimeSignals ............... 19
1.6.1 AdditionofCTSignals .......................... 20
1.6.2 MultiplicationsofCTSignals .................... 21
1.6.3 AmplitudeScalingofCTSignals ................. 21
1.6.4 TimeScalingofCTSignals ...................... 22
1.6.5 TimeShiftingofCTSignals ...................... 22
1.6.6 SignalReflectionorFolding ...................... 24
xi
xii Contents
1.6.7 InvertedCTSignal .............................. 24
1.6.8 MultipleTransformation ......................... 26
1.7 BasicOperationsonDiscreteTimeSignals .................. 77
1.7.1 AdditionofDiscreteTimeSequence ............... 77
1.7.2 MultiplicationofDTSignals ..................... 77
1.7.3 AmplitudeScalingofDTSignal .................. 79
1.7.4 TimeScalingofDTSignal ....................... 79
1.7.5 TimeShiftingofDTSignal ...................... 80
1.7.6 MultipleTransformation ......................... 81
1.8 ClassificationofSignals .................................. 103
1.8.1 Deterministic and Non-deterministic
ContinuousSignals ............................. 103
1.8.2 PeriodicandNon-periodicContinuousSignals ...... 103
1.8.3 FundamentalPeriodofTwoPeriodicSignals ....... 105
1.8.4 OddandEvenFunctionsofContinuousTime
Signals ........................................ 121
1.8.5 EnergyandPowerofContinuousTimeSignals ..... 143
1.9 ClassificationofDiscreteTimeSignals ..................... 162
1.9.1 PeriodicandNon-PeriodicDTSignals ............. 162
1.9.2 OddandEvenDTSignals ........................ 170
1.9.3 EnergyandPowerofDTSignals .................. 178
2 ContinuousandDiscreteTimeSystems ......................... 197
2.1 Introduction ............................................ 197
2.2 LinearTimeInvariantContinuous(LTIC)TimeSystem ....... 198
2.3 LinearTimeInvariantDiscrete(LTID)TimeSystem ......... 199
2.4 Properties(Classification)ofContinuousTimeSystem ....... 199
2.4.1 LinearandNon-linearSystems ................... 200
2.4.2 TimeInvariantandTimeVaryingSystems .......... 212
2.4.3 StaticandDynamicSystems(Memoryless
andSystemwithMemory) ....................... 216
2.4.4 CausalandNon-causalSystems .................. 220
2.4.5 StableandUnstableSystems ..................... 225
2.4.6 InvertibilityandInverseSystem ................... 230
2.5 DiscreteTimeSystem .................................... 241
2.6 PropertiesofDiscreteTimeSystem ........................ 241
2.6.1 LinearandNon-linearSystems ................... 242
2.6.2 TimeInvariantandTimeVaryingDTSystems ...... 246
2.6.3 CausalandNon-causalDTSystems ............... 248
2.6.4 StableandUnstableSystems ..................... 251
2.6.5 StaticandDynamicSystems ..................... 254
2.6.6 InvertibleandInverseDiscreteTimeSystems ....... 256