Signals and systems

434 Pages·2018·3.86 MB·English

by  Rao

K. Deergha Rao
Signals
and Systems

K. Deergha Rao
Signals and Systems
K.DeerghaRao
DepartmentofElectronicsandCommunicationEngineering
VasaviCollegeofEngineering(AffiliatedtoOsmaniaUniversity)
Hyderabad,Telangana,India
ISBN978-3-319-68674-5 ISBN978-3-319-68675-2 (eBook)
https://doi.org/10.1007/978-3-319-68675-2
LibraryofCongressControlNumber:2017958547
MathematicsSubjectClassification(2010):94A12;94A05;93C55;93C20;35Q93
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To
My Parents Dalamma and Boddu,
My Beloved Wife Sarojini,
and
My Mentor Prof. M.N.S. Swamy
Preface
Thesignalsandsystemscourseisnotonlyanimportantelementforundergraduate
electrical engineering students but the fundamentals and techniques of the subject
areessentialinallthedisciplinesofengineering.Signalsandsystemsanalysishasa
longhistory,withitstechniquesandfundamentalsfoundinbroadareasofapplica-
tions.Thesignalsandsystemsiscontinuouslyevolvinganddevelopinginresponse
tonewproblems,suchasthedevelopmentofintegratedcircuitstechnologyandits
applications.
In this book, many illustrative examples are included in each chapter for easy
understanding of the fundamentals and methodologies of signals and systems. An
attractive feature of this book is the inclusion of MATLAB-based examples with
codes to encourage readers to implement exercises on their personal computers in
order to become confident with the fundamentals and to gain more insight into
signals and systems. In addition to the problems that require analytical solutions,
MATLABexercisesareintroducedtothereaderattheendofsomechapters.
Thisbookisdividedinto8chapters.Chapter1presentsanintroductiontosignals
and systems with basic classification of signals, elementary operations on signals,
andsomereal-worldexamplesofsignalsandsystems.Chapter2givestime-domain
analysis of continuous time signals and systems, and state-space representation of
continuous-time LTI systems. Fourier analysis of continuous-time signals and sys-
tems is covered in Chapter 3. Chapter 4 deals with the Laplace transform and
analysis of continuous-time signals and systems, and solution of state-space equa-
tions of continuous-time LTI systems using Laplace transform. Ideal continuous-
time (analog) filters, practical analog filter approximations and design methodolo-
gies,anddesignofspecialclassfiltersbasedonpole-zeroplacementarediscussedin
Chapter 5. Chapter 6 discusses the time-domain representation of discrete-time
signals and systems, linear time-invariant (LTI) discrete-time systems and their
properties,characterizationofdiscrete-timesystems,andstate-spacerepresentation
ofdiscrete-timeLTIsystems.Representationofdiscrete-timesignalsandsystemsin
frequencydomain,representation ofsamplinginfrequencydomain,reconstruction
ofaband-limitedsignalfromitssamples,andsamplingofdiscrete-timesignalsare
vii
viii Preface
detailed in Chapter 7. Chapter 8 describes the z-transform and analysis of LTI
discrete-time systems, the solution of state-space equations of discrete-time LTI
systems using z-transform, and transformations between the continuous-time sys-
temsanddiscrete-timesystems.
Thesalientfeaturesofthisbookareasfollows:
(cid:129) Provides introductory and comprehensive exposure to all aspects of signal and
systemswithclarityandinaneasywaytounderstand.
(cid:129) Provides an integrated treatment of continuous-time signals and systems and
discrete-timesignalsandsystems.
(cid:129) Several fully worked numerical examples are provided to help students under-
standthefundamentalsofsignalsandsystems.
(cid:129) PC-based MATLAB m-files for the illustrative examples are included in
thisbook.
This book is written at introductory level for undergraduate classes in electrical
engineering and applied sciences that are the prerequisite for upper level courses,
suchascommunicationsystems,digitalsignalprocessing,andcontrolsystems.
Hyderabad,India K.DeerghaRao
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 WhatisaSignal?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 WhatisaSystem?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 ElementaryOperationsonSignals. . . . . . . . . . . . . . . . . . . . . . . 1
1.3.1 TimeShifting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.2 TimeScaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3.3 TimeReversal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 ClassificationofSignals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Continuous-TimeandDiscrete-TimeSignals. . . . . . . . . 5
1.4.2 AnalogandDigitalSignals. . . . . . . . . . . . . . . . . . . . . . 5
1.4.3 PeriodicandAperiodicSignals. . . . . . . . . . . . . . . . . . . 6
1.4.4 EvenandOddSignals. . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4.5 Causal,Noncausal,andAnticausalSignal. . . . . . . . . . . 12
1.4.6 EnergyandPowerSignals. . . . . . . . . . . . . . . . . . . . . . 13
1.4.7 DeterministicandRandomSignals. . . . . . . . . . . . . . . . 20
1.5 BasicContinuous-TimeSignals. . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 TheUnitStepFunction. . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 TheUnitImpulseFunction. . . . . . . . . . . . . . . . . . . . . . 21
1.5.3 TheRampFunction. . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.4 TheRectangularPulseFunction. . . . . . . . . . . . . . . . . . 22
1.5.5 TheSignumFunction. . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.6 TheRealExponentialFunction. . . . . . . . . . . . . . . . . . . 23
1.5.7 TheComplexExponentialFunction. . . . . . . . . . . . . . . 24
1.5.8 TheSincFunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 GenerationofContinuous-TimeSignals
UsingMATLAB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.7 TypicalSignalProcessingOperations. . . . . . . . . . . . . . . . . . . . . 30
1.7.1 Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.7.2 Filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.7.3 ModulationandDemodulation. . . . . . . . . . . . . . . . . . . 31
ix
x Contents
1.7.4 Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.7.5 MultiplexingandDemultiplexing. . . . . . . . . . . . . . . . . 32
1.8 SomeExamplesofReal-WorldSignalsandSystems. . . . . . . . . . 32
1.8.1 AudioRecordingSystem. . . . . . . . . . . . . . . . . . . . . . . 32
1.8.2 GlobalPositioningSystem. . . . . . . . . . . . . . . . . . . . . . 33
1.8.3 Location-BasedMobileEmergency
ServicesSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.8.4 HeartMonitoringSystem. . . . . . . . . . . . . . . . . . . . . . . 34
1.8.5 HumanVisualSystem. . . . . . . . . . . . . . . . . . . . . . . . . 36
1.8.6 MagneticResonanceImaging. . . . . . . . . . . . . . . . . . . . 36
1.9 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.10 MATLABExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
FurtherReading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Continuous-TimeSignalsandSystems. . . . . . . . . . . . . . . . . . . . . . . 41
2.1 TheRepresentationofSignalsinTermsofImpulses. . . . . . . . . . 41
2.2 Continuous-TimeSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.1 LinearSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.2 Time-InvariantSystem. . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.3 CausalSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.4 StableSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.5 MemoryandMemorylessSystem. . . . . . . . . . . . . . . . . 49
2.2.6 InvertibleSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.7 StepandImpulseResponses. . . . . . . . . . . . . . . . . . . . . 49
2.3 TheConvolutionIntegral. . .. . . . .. . . .. . . .. . . .. . . . .. . . .. 49
2.3.1 SomePropertiesoftheConvolutionIntegral. . . . . . . . . 50
2.3.2 GraphicalConvolution. . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3.3 ComputationofConvolutionIntegral
UsingMATLAB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3.4 InterconnectedSystems. . . . . . . . . . . . . . . . . . . . . . . . 74
2.3.5 PeriodicConvolution. . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.4 PropertiesofLinearTime-InvariantContinuous-Time
System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.4.1 LTISystemsWithandWithoutMemory. . . . . . . . . . . . 77
2.4.2 CausalityforLTISystems. . . . . . . . . . . . . . . . . . . . . . 77
2.4.3 StabilityforLTISystems. . . . . . . . . . . . . . . . . . . . . . . 77
2.4.4 InvertibleLTISystem. . .. . . .. . . . .. . . .. . . .. . . . .. 79
2.5 SystemsDescribedbyDifferentialEquations. . . . . . . . . . . . . . . 82
2.5.1 LinearConstant-CoefficientDifferentialEquations. . . . . 82
2.5.2 TheGeneralSolutionofDifferentialEquation. . . . . . . . 85
2.5.3 Linearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.5.4 Causality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.5.5 Time-Invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.5.6 ImpulseResponse. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.5.7 SolutionofDifferentialEquationsUsing
MATLAB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Contents xi
2.5.8 DeterminingImpulseResponseandStep
ResponseforaLinearSystemDescribedby
aDifferentialEquationUsingMATLAB. . . . . . . . . . . . 92
2.6 Block-DiagramRepresentationsofLTISystems
DescribedbyDifferentialEquations. . . . . . . . . . . . . . . . . . . . . . 93
2.7 SingularityFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.8 State-SpaceRepresentationofContinuous-Time
LTISystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.8.1 StateandStateVariables. . . . . . . . . . . . . . . . . . . . . . . 98
2.8.2 State-SpaceRepresentationofSingle-Input
Single-OutputContinuous-TimeLTISystems. . . . . . . . 99
2.8.3 State-SpaceRepresentationofMulti-input
Multi-outputContinuous-TimeLTISystems. . . . . . . . . 104
2.9 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
2.10 MATLABExercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
FurtherReading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3 FrequencyDomainAnalysisofContinuous-Time
SignalsandSystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.1 ComplexExponentialFourierSeriesRepresentation
oftheContinuous-TimePeriodicSignals. . . . . . . . . . . . . . . . . . . 111
3.1.1 ConvergenceofFourierSeries. . . . . . . . . . . . . . . . . . . . 113
3.1.2 PropertiesofFourierSeries. .. . . . . . . . . .. . . . . . . . . .. 113
3.2 TrigonometricFourierSeriesRepresentation. . . . . . . . . . . . . . . . 128
3.2.1 SymmetryConditionsinTrigonometric
FourierSeries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.3 TheContinuousFourierTransformforNonperiodic
Signals. . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . . .. 133
3.3.1 ConvergenceofFourierTransforms. . . . . . . . . . . . . . . . 135
3.3.2 FourierTransformsofSomeCommonlyUsed
Continuous-TimeSignals. . . . . . . . . . . . . . . . . . . . . . . . 136
3.3.3 PropertiesoftheContinuous-TimeFourier
Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.4 TheFrequencyResponseofContinuous-TimeSystems. . . . . . . . . 159
3.4.1 DistortionDuringTransmission. . . . . . . . . . . . . . . . . . . 160
3.5 SomeCommunicationApplicationExamples. . . . . . . . . . . . . . . . 162
3.5.1 AmplitudeModulation(AM)andDemodulation
AmplitudeModulation. . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.5.2 Single-Sideband(SSB)AM. . . . . . . . . . . . . . . . . . . . . . 164
3.5.3 FrequencyDivisionMultiplexing(FDM). . . . . . . . . . . . . 164
3.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
FurtherReading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
4 LaplaceTransforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.1 TheLaplaceTransform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
4.1.1 DefinitionofLaplaceTransform. . . . . . . . . . . . . . . . . . 171