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Transforms and applications primer for engineers with examples and MATLAB PDF

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Transforms 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 CRC Press is an imprint of the Taylor & Francis Group, an informa business MATLAB® and Simulink® are trademarks of The MathWorks, Inc. and are used with permission. The Math- Works does not warrant the accuracy of the text of exercises in this book. This book’s use or discussion of MATLAB® and Simulink® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® and Simulink® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140512 International Standard Book Number-13: 978-1-4200-8932-5 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com 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. 3AppleHillDriveNatick,MA01760-2098USA Tel:5086477000 Fax:508-647-7001 E-mail:[email protected] Web:www.mathworks.com ix

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