Table Of ContentTransforms and
Applications Primer
for Engineers
with Examples
and MATLAB®
Electrical Engineering Primer Series
Series Editor
Alexander D. Poularikas
University of Alabama
Huntsville, Alabama
Transforms and Applications Primer for Engineers with Examples and MATLAB®,
Alexander D. Poularikas
Discrete Random Signal Processing and Filtering Primer with MATLAB®,
Alexander D. Poularikas
Signals and Systems Primer with MATLAB®, Alexander D. Poularikas
Adaptive Filtering Primer with MATLAB®, Alexander D. Poularikas
and Zayed M. Ramadan
Transforms and
Applications Primer
for Engineers
with Examples
and MATLAB®
Alexander D. Poularikas
Boca Raton London New York
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Contents
Preface..........................................................................................................................ix
Author..........................................................................................................................xi
1 Signals and Systems..................................................................................... 1-1
1.1 Introduction.............................................................................................................1-1
1.2 Signals.......................................................................................................................1-1
1.3 CircuitElementsandEquation........................................................................1-13
1.4 LinearMechanicalandRotationalMechanicalElements..........................1-21
1.4.1 LinearMechanicalSystems.................................................................. 1-21
1.4.2 RotationalMechanicalSystems........................................................... 1-22
1.5 DiscreteEquationsandSystems......................................................................1-23
1.6 DigitalSimulationofAnalogSystems............................................................1-26
1.7 ConvolutionofAnalogSignals ........................................................................1-26
1.8 ConvolutionofDiscreteSignals.......................................................................1-29
2 Fourier Series................................................................................................. 2-1
2.1 Introduction.............................................................................................................2-1
2.2 FourierSeriesinaComplexExponentialForm..............................................2-1
2.3 FourierSeriesinTrigonometricForm..............................................................2-2
2.3.1 DifferentiationoftheFourierSeries.....................................................2-2
2.3.2 IntegrationoftheFourierSeries............................................................2-3
2.4 WaveformSymmetries..........................................................................................2-3
2.5 SomeAdditionalFeaturesofPeriodicContinuousFunctions.....................2-3
2.5.1 PowerContent:Parseval’sTheorem.....................................................2-3
2.5.2 OutputofanLTISystemWhentheInput
IsaPeriodicFunction..............................................................................2-4
2.5.3 TransmissionwithoutDistortion..........................................................2-4
2.5.4 Band-LimitedPeriodicSignals...............................................................2-5
2.5.5 SumandDifferenceofFunctions..........................................................2-5
2.5.6 ProductofTwoFunctions......................................................................2-5
v
vi Contents
2.5.7 ConvolutionofTwoFunctions.........................................................2-6
2.5.8 Gibbs’Phenomenon.............................................................................2-6
2.5.9 FourierSeriesoftheCombFunction..............................................2-7
3 Fourier Transforms...................................................................................... 3-1
3.1 Introduction—FourierTransform....................................................................3-1
3.2 OtherFormsofFourierTransform.................................................................3-1
3.2.1 f(t)IsaComplexFunction.................................................................3-1
3.2.2 RealTimeFunctions............................................................................3-2
3.2.3 ImaginaryTimeFunctions.................................................................3-2
3.2.4 f(t)IsEven..............................................................................................3-3
3.2.5 f(t)Odd .................................................................................................3-3
3.2.6 OddandEvenRepresentations.........................................................3-3
3.2.7 Causal-TimeFunctions.......................................................................3-4
3.3 FourierTransformExamples............................................................................3-5
3.4 FourierTransformProperties...........................................................................3-8
3.5 ExamplesonFourierProperties.......................................................................3-8
3.6 FTExamplesofSingularFunctions..............................................................3-12
3.7 DurationofaSignalandtheUncertaintyPrinciple.................................3-37
3.8 ApplicationstoLinear-TimeInvariantSystems........................................3-38
3.9 ApplicationstoCommunicationSignals.....................................................3-47
3.10 Signals,Noise,andCorrelation......................................................................3-50
3.11 AveragePowerSpectra,RandomSignals,Input–OutputRelations......3-51
3.12 FTinProbabilityTheory.................................................................................3-53
3.12.1 CharacteristicFunction.................................................................... 3-55
3.12.2 JointCumulativeDistributionFunction...................................... 3-55
3.12.3 CharacteristicFunctionofTwoVariables................................... 3-56
4 Relatives to the Fourier Transform.......................................................... 4-1
4.1 InfiniteFourierSineTransform .......................................................................4-1
4.2 InfiniteFourierCosineTransform...................................................................4-1
4.3 ApplicationstoBoundary-ValueProblems...................................................4-9
4.4 FiniteSineFourierTransformandFiniteCosine
FourierTransform ........................................................................................... 4-15
4.5 Two-DimensionalFourierTransform..........................................................4-18
4.5.1 Two-DimensionalConvolution...................................................... 4-21
4.5.2 Two-DimensionalCorrelation........................................................ 4-21
4.5.3 TheoremsofTwo-DimensionalFunctions.................................. 4-22
5 Sampling of Continuous Signals............................................................... 5-1
5.1 FundamentalsofSampling................................................................................5-1
5.2 TheSamplingTheorem......................................................................................5-6
Contents vii
6 Discrete-Time Transforms......................................................................... 6-1
6.1 Discrete-TimeFourierTransform....................................................................6-1
6.1.1 ApproximatingtheFourierTransform...........................................6-1
6.1.2 SymmetryPropertiesoftheDTFT...................................................6-5
6.2 SummaryofDTFTProperties..........................................................................6-5
6.3 DTFTofFiniteTimeSequences.......................................................................6-7
6.3.1 Windowing.............................................................................................6-9
6.4 FrequencyResponseofLTIDiscreteSystems............................................6-11
6.5 DiscreteFourierTransform............................................................................6-13
6.6 SummaryofDFTProperties..........................................................................6-15
6.7 MultirateDigitalSignalProcessingandSpectra........................................6-27
6.7.1 DownSampling(orDecimation).................................................. 6-28
6.7.2 FrequencyDomainofDown-SampledSignals........................... 6-30
6.7.3 Interpolation(Up-Sampling)byaFactorU................................ 6-34
6.7.4 FrequencyDomainCharacterization
ofUp-SampledSignals..................................................................... 6-35
Appendix.........................................................................................................................6-38
6.A.1 ProofsofDTFTProperties...........................................................................6-38
6.A.2 ProofsofDFTProperties..............................................................................6-40
6.A.3 FastFourierTransform..................................................................................6-43
6.A.3.1 DecimationinTimeProcedure.................................................... 6-43
7 Laplace Transform ....................................................................................... 7-1
7.1 One-SidedLaplaceTransform..........................................................................7-1
7.2 SummaryoftheLaplaceTransformProperties ...........................................7-4
7.3 SystemsAnalysis:TransferFunctionsofLTISystems................................7-8
7.4 InverseLaplaceTransform.............................................................................7-19
7.5 ProblemSolvingwithLaplaceTransform...................................................7-26
7.5.1 OrdinaryDifferentialEquations..................................................... 7-26
7.5.2 PartialDifferentialEquations.......................................................... 7-39
7.6 FrequencyResponseofLTISystems............................................................7-49
7.7 PoleLocationandtheStabilityofLTISystems.........................................7-57
7.8 FeedbackforLinearSystems..........................................................................7-60
7.9 BodePlots...........................................................................................................7-71
7.10 *InversionIntegral............................................................................................7-75
7.11 *ComplexIntegrationandtheBilateralLaplaceTransform...................7-86
7.12 *StateSpaceandStateEquations ..................................................................7-88
7.12.1 StateEquationsinPhaseVariableForm...................................... 7-90
7.12.2 TimeResponseUsingStateSpaceRepresentation .................... 7-98
7.12.3 SolutionUsingtheLaplaceTransform.......................................7-102
7.12.4 StateSpaceTransferFunction......................................................7-105
7.12.5 ImpulseandStepResponse...........................................................7-106
viii Preface
8 The z-Transform .......................................................................................... 8-1
8.1 Thez-Transform..................................................................................................8-1
8.2 Convergenceofthez-Transform .....................................................................8-5
8.3 Propertiesofthez-Transform........................................................................8-11
8.4 z-TransformPairs............................................................................................. 8-20
8.5 Inversez-Transform......................................................................................... 8-21
8.5.1 PartialFractionExpansion.............................................................. 8-21
8.5.2 *InverseTransformbyIntegration................................................ 8-28
8.5.3 *ResiduesforSimplePoles.............................................................. 8-28
8.5.4 *ResiduesforMultiplePoles........................................................... 8-28
8.5.5 *ResiduesforSimplePolesNotFactorable................................. 8-29
8.6 TransferFunction............................................................................................. 8-31
8.6.1 HigherOrderTransferFunctions.................................................. 8-37
8.6.1.1 Stability................................................................................ 8-39
8.6.1.2 Causality.............................................................................. 8-39
8.7 FrequencyResponseofFirst-OrderDiscreteSystems..............................8-39
8.7.1 PhaseShiftinDiscreteSystems...................................................... 8-45
8.8 FrequencyResponseofHigherOrderDigitalSystems............................8-46
8.9 z-TransformSolutionofFirst-OrderDifferenceEquations....................8-49
8.10 HigherOrderDifferenceEquations..............................................................8-53
8.10.1 MethodofUndeterminedCoefficients......................................... 8-59
8.11 *LTIDiscrete-TimeDynamicalSystems......................................................8-64
8.12 *z-TransformandRandomProcesses..........................................................8-69
8.12.1 PowerSpectralDensities.................................................................. 8-69
8.12.2 LinearDiscrete-TimeFilters........................................................... 8-71
8.12.3 OptimumLinearFiltering............................................................... 8-72
8.13 *RelationshipbetweentheLaplaceandz-Transforms.............................8-74
8.14 *RelationshiptotheFourierTransform......................................................8-78
Appendix.........................................................................................................................8-79
9 *Hilbert Transforms..................................................................................... 9-1
9.1 Definition...............................................................................................................9-1
9.2 HilbertTransforms,Properties,andtheAnalyticSignal............................9-2
9.3 HilbertTransformPropertiesandHilbertPairs........................................9-15
Appendix A: Functions of a Complex Variable.............................................A-1
Appendix B: Series and Summations................................................................B-1
Appendix C: Definite Integrals...........................................................................C-1
Appendix D: Suggestions and Explanations for MATLAB1 Use ............ D-1
Index.......................................................................................................................IN-1
Preface
Thisbookpresentsthemostcommonandusefulmathematicaltransformsforstudents
andpracticingengineers.Itcanbeconsideredasacompanionforstudentsandahandy
referenceforpracticingengineerswhowillneedtousetransformsintheirwork.
TheLaplacetransform,whichundoubtedlyisthemostfamiliarexample,isbasictothe
solutionofinitialvalueproblems.TheFouriertransform,beingsuitedtosolvingbound-
ary-value problems, isbasic to the frequency spectrum analysisof time-varying signals.
Fordiscretesignals,wedevelopthez-transformanditsuses.Thepurposeofthisbookis
to develop the most important integral transforms and present numerous examples
elucidating their use. Laplace and Fourier transforms are by far the most widely and
most useful of all integral transforms. For this reason, they have been given a more
extensivetreatmentinthisbookwhencomparedtootherbooksonthesamesubject.
Thisbookisprimarilywrittenforseniors,first-yeargraduatestudents,andpracticing
engineers and scientists. To comprehend some of the topics, the reader should have a
basicknowledgeofcomplexvariabletheory.Advancedtopicsareindicatedbyastar(*).
Thebookcontainsseveralappendicestocomplementthemainsubjects.Theextensive
tables of the transforms are the most important contributions in this book. Another
important contribution is the inclusion of an ample number of examples drawn from
severaldisciplines.Theincludedexampleshelpthereadersunderstandanyofthetrans-
formsandgivethemtheconfidencetouseit.Furthermore,itincludes,whereverneeded,
1
MATLAB functionsandBookMATLABfunctionsdevelopedbytheauthor,whichare
includedinthetext.
MATLABisaregisteredtrademarkofTheMathWorks,Inc.Forproductinformation,
pleasecontact:
TheMathWorks,Inc.
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