Basic Transforms for Electrical Engineering

643 Pages·2022·17.286 MB·English

by  Orhan Özhan

Orhan Özhan
Basic
Transforms
for Electrical
Engineering
Basic Transforms for Electrical Engineering
Orhan Özhan
Basic Transforms for
Electrical Engineering
OrhanÖzhan
FatihSultanVakifUniversity
Istanbul,Turkey
ISBN978-3-030-98845-6 ISBN978-3-030-98846-3 (eBook)
https://doi.org/10.1007/978-3-030-98846-3
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TomyMaster,TheSultanofKnowledgeand
Wisdom...
...AndtoTheCherishedMemoryofmy
Parents.
Foreword
Physics and mathematics courses in the freshman and sophomore curriculum of
electricalengineeringserveastheinfra-structureforengineeringstudies.Calculus
and linear algebra are important subjects towards this end. Besides these, signals
and systems concepts in electrical engineering need a sound understanding of
mathematical tools such as Laplace transform and the like. These tools, in turn,
are closely related to study of mathematical complex analysis. This book started
out as lecture notes of an introductory math course I was teaching at FSMVU.1
First,themainfocuswassolelyoncomplexanalysis;thescopeandbreadthofthe
noteswerejustenoughtocoveraone-semestercourse.Then,itwasdeemedthatit
wouldbeappropriatetolaythebookoutsothatitcouldbereferredtobystudents
taking signals and systems courses. As more and more material accumulated, the
work eventually evolved into a multiple-semester reference that includes various
transformsinsignalprocessing.
The main objective of the book is to teach the basic transforms which provide
the theoretical background for circuit analysis and synthesis, filter theory, signal
processing, and control theory. To this end, the book is organized into two parts.
In Part I, the mathematical background is introduced. Calculus is an essential
prerequisite for this part, as many novel concepts are built on calculus and
some “old” ones are refined or generalized. Part I is divided into three chapters.
Chapter 1 deals with complex numbers and operations on complex numbers.
Chapter 2 continues with functions of complex numbers and analyticity, complex
differentiation, and conformal mapping. Chapter 3 is about complex integration
andresiduetheoremwhichisneededtounderstandtransforminversionviacontour
integration,complexconvolutiontheorem,andParsevaltheoremindiscrete-time.If
the reader is already knowledgeable in these topics, they can skip Part I and dive
into Part II directly. Otherwise, the concepts introduced in these chapters must be
mastered. The organization is such that Chap.1 is a prerequisite for Chap.2, and
Chap.2isaprerequisiteforChap.3.Ifthereaderfeelsuncomfortablewithtopicsof
1FatihSultanMehmetVakifUniversity,Istanbul.
vii
viii Foreword
Chap.1whilereadingChap.2,thentheyshouldreturntoChap.1.Thesamething
appliestoChap.3.
InPartII,weassumethatthereaderisfamiliarwithdifferentialequationsinorder
tolinktheLaplacetransformtolinearsystems.ThechaptersinPartIIdealwiththe
Laplace,Fourier,andz-transformsaswellasFourierseries,fastFouriertransform,
short-time Fourier transform, and discrete cosine transform. Laplace, Fourier, and
z-transformcanbestudiedindependentlyandinanyorder.EspeciallyiftheLaplace
transform has been taken in another course, it can be skipped entirely. Chapters 8
and 10 cover the fast Fourier transform and the discrete cosine transform and are
ratheradvanced.TheFouriertransformmustbestudiedbeforeChaps.7,8,and10.
Althoughwehaveliberallyresortedtoelectriccircuitsinexamplestimeandagain,
it is not our intention in this course to teach electric circuits; we just use them for
pedagogicalreasonsasavehicleandmotivation.
Real CEPSTRUM has been briefly mentioned in Chap.6; the complex CEP-
STRUM,alsoknownasthehomomorphicanalysis,hasbeenreservedforapossible
futureedition.Likewise,othertransformsliketheHilberttransformarenotincluded
in this edition. The wavelet transform on the other hand is a very sophisticated
subjectwhichdeservestobededicatedabookonitsownright.
Finally, a word is in order about the software we have used while writing the
book. We enjoy running extremely powerful math software on our computers. In
lieu of the slide rule, you can use SCILAB, MATHEMATICA, MAXIMA, etc.
Thesesoftwareprovidecomplexnumberoperationsaswellasverypowerfulsignal
processingtoolboxes.Theyjustmakeourjobeasier,morefun,andmoreenjoyable.
LabVIEWisjustanotherfantasticsoftwareforstudents.Itisagraphicalprogram-
mingplatformwhichcanbeusedinreal-timeornonreal-timeapplications.Inorder
to make the subject matter more interesting and attractive, we have mostly used
LabVIEWinthetext,examples,chapterproblems,andprojects.Youcandownload
a trial version of LabVIEW from National Instruments website for free. With
LabVIEW or similar tools, you can derive more pleasure and satisfaction learning
the transforms than the older generations did using the archaic tools. LabVIEW
motivatesthestudenttoexperimentwithideasandconceptsburiedintheexamples
andproblemsandthuscontributestotheirunderstanding.AlthoughLabVIEWisour
favoritesoftware,wehaveoccasionallygivenexamplesusingMATLAB,SCILAB,
orLTSPICE.SCILABandLTSPICEarealsofreetodownload.However,awarning
is in order for the reader of wit. All these super software are no good unless a
soundknowledgeandanunderstandingoftheunderlyingmathematicsareattained.
Convinceyourselfthat,withoutproperunderstanding,evenwiththissoftwareyou
can’t go very far. Don’t settle for a trial-and-error strategy with software because
only the knowledge inspires you and gives you insight. Hopefully, the software
of our choice will help you like the topics, and make it a fun to learn through its
number-crunchingpowerandbreath-takinggraphics.
Istanbul,Turkey OrhanÖzhan
Contents
PartI Background
1 ComplexNumbers ......................................................... 3
1.1 RepresentationofComplexNumbers............................... 5
1.2 Euler’sIdentity ....................................................... 6
1.2.1 ComplexExponential ...................................... 6
1.2.2 ConjugateofaComplexNumber.......................... 7
1.3 MathematicalOperations ............................................ 8
1.3.1 Identity...................................................... 8
1.3.2 AdditionandSubtraction................................... 8
1.3.3 MultiplicationandDivision................................ 10
1.3.4 RotatingaNumberinComplexPlane..................... 14
1.4 RootsofaComplexNumber ........................................ 16
1.5 ApplicationsofComplexNumbers.................................. 18
1.5.1 ComplexNumbersVersusTrigonometry.................. 19
1.5.2 Integration................................................... 22
1.5.3 Phasors...................................................... 23
1.5.4 3-PhaseElectricCircuits................................... 31
1.5.5 NegativeFrequency ........................................ 33
1.5.6 ComplexNumbersinMathematicsSoftware ............. 34
1.5.7 RootsofaPolynomial...................................... 37
2 FunctionsofaComplexVariable ........................................ 55
2.1 LimitofaComplexFunction........................................ 57
2.2 DerivativeofComplexFunctionsandAnalyticity.................. 58
2.3 Cauchy–RiemannConditions ....................................... 60
2.4 RulesofDifferentiation.............................................. 67
2.5 HarmonicFunctions.................................................. 69
2.6 ApplicationsofComplexFunctionsandAnalyticity............... 72
2.6.1 ElementaryFunctions ...................................... 72
2.6.2 ConformalMapping........................................ 82
2.6.3 Fractals...................................................... 90
ix
x Contents
3 ComplexIntegration ....................................................... 99
3.1 IntegratingComplexFunctionsofaRealVariable ................. 101
3.2 Contours .............................................................. 102
3.3 IntegratingFunctionsofaComplexVariable....................... 105
3.4 NumericalComputationoftheComplexIntegral .................. 108
3.5 PropertiesoftheComplexIntegral.................................. 111
3.6 TheCauchy–GoursatTheorem...................................... 121
3.6.1 IntegratingDifferentiableFunctions....................... 122
3.6.2 ThePrincipleofContourDeformation ................... 126
3.6.3 Cauchy’sIntegralforMultiplyConnectedDomains...... 127
3.7 Cauchy’sIntegralFormula........................................... 129
3.8 Higher-OrderDerivativesofAnalyticFunctions ................... 134
3.9 ComplexSequencesandSeries...................................... 138
3.10 PowerSeriesExpansionsofFunctions.............................. 145
3.10.1 TaylorandMaclaurinSeries ............................... 146
3.10.2 DifferentiationandIntegrationofPowerSeries........... 156
3.11 LaurentSeries ........................................................ 157
3.12 Residues .............................................................. 161
3.12.1 ResidueTheorem........................................... 162
3.12.2 ResidueatInfinity .......................................... 165
3.12.3 FindingResidues ........................................... 169
3.13 ResidueIntegrationofRealIntegrals................................ 171
3.14 FourierIntegrals...................................................... 179
PartII Transforms
4 TheLaplaceTransform ................................................... 191
4.1 MotivationtoUseLaplaceTransform .............................. 192
4.2 DefinitionoftheLaplaceTransform ................................ 195
4.3 PropertiesoftheLaplaceTransform ................................ 201
4.3.1 Linearity..................................................... 201
4.3.2 RealDifferentiation ........................................ 202
4.3.3 RealIntegration............................................. 203
4.3.4 Differentiationbys......................................... 204
4.3.5 RealTranslation ............................................ 205
4.3.6 ComplexTranslation ....................................... 205
4.3.7 PeriodicFunctions.......................................... 206
4.3.8 LaplaceTransformofConvolution ........................ 207
4.3.9 InitialValueTheorem ...................................... 210
4.3.10 FinalValueTheorem ....................................... 212
4.4 TheInverseLaplaceTransform...................................... 214
4.4.1 RealPoles................................................... 215
4.4.2 ComplexPoles.............................................. 217
4.4.3 MultiplePoles .............................................. 218
Contents xi
4.5 MoreonPolesandZeros............................................. 221
4.5.1 FactoringPolynomials...................................... 222
4.5.2 PolesandTimeResponse .................................. 224
4.5.3 AnAlternativeWaytoSolveDifferentialEquations ..... 229
4.6 InverseLaplaceTransformbyContourIntegration................. 234
4.7 ApplicationsofLaplaceTransform ................................. 239
4.7.1 ElectricalSystems.......................................... 239
4.7.2 InverseLTISystems........................................ 244
4.7.3 EvaluationofDefiniteIntegrals............................ 249
5 TheFourierSeries.......................................................... 257
5.1 VectorsandSignals................................................... 258
5.2 TheFourierSeries.................................................... 261
5.3 CalculatingFourierSeriesCoefficients ............................. 264
5.4 PropertiesoftheFourierSeries...................................... 272
5.4.1 Linearity..................................................... 273
5.4.2 SymmetryProperties ....................................... 273
5.4.3 ShiftinginTime ............................................ 281
5.4.4 TimeReversal............................................... 281
5.4.5 Differentiation .............................................. 283
5.4.6 Integration................................................... 286
5.5 Parseval’sRelation ................................................... 287
5.6 ConvergenceofFourierSeries....................................... 288
5.7 GibbsPhenomenon................................................... 290
5.8 Discrete-TimeFourierSeries ........................................ 291
5.8.1 PeriodicConvolution....................................... 303
5.8.2 Parseval’sRelationforDiscrete-TimeSignals............ 314
5.9 ApplicationsofFourierSeries....................................... 316
6 TheFourierTransform .................................................... 335
6.1 Introduction........................................................... 335
6.2 DefinitionoftheFourierTransform................................. 336
6.3 FourierTransformVersusFourierSeries............................ 338
6.4 ConvergenceoftheFourierTransform.............................. 341
6.5 PropertiesoftheFourierTransform................................. 345
6.5.1 SymmetryIssues............................................ 345
6.5.2 Linearity..................................................... 350
6.5.3 TimeScaling................................................ 351
6.5.4 TimeReversal............................................... 352
6.5.5 TimeShift................................................... 352
6.5.6 FrequencyShift(AmplitudeModulation) ................ 353
6.5.7 DifferentiationwithRespecttoTime...................... 353
6.5.8 IntegrationwithRespecttoTime .......................... 354
6.5.9 Duality ...................................................... 355
6.5.10 Convolution................................................. 355
6.5.11 MultiplicationinTimeDomain............................ 356