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Applicationofmathematicalmethodsofengineeringtotransientanalysisofr-l-cseriescircuitwithsinusoidalvoltage

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MathematicalModelingandSimulation-IntroductionforScientistsandEngineers

ElementaryMathematicalandComputationalToolsforElectricalandComputerEngineersUsingMATLAB

Alinearsystem'scharacteristicsarecompletelyspecifiedbythesystem'simpulseresponse,asgovernedbythemathematicsofconvolution.Thisisthebasisofmanysignalprocessingtechniques.Forexample:Digitalfiltersarecreatedbydesigninganappropriateimpulseresponse.Enemyaircraftaredetectedwithradarbyanalyzingameasuredimpulseresponse.Echosuppressioninlongdistancetelephonecallsisaccomplishedbycreatinganimpulseresponsethatcounteractstheimpulseresponseofthereverberation.Thelistgoesonandon.Thischapterexpandsonthepropertiesandusageofconvolutioninseveralareas.First,severalcommonimpulseresponsesarediscussed.Second,methodsarepresentedfordealingwithcascadeandparallelcombinationsoflinearsystems.Third,thetechniqueofcorrelationisintroduced.Fourth,anastyproblemwithconvolutionisexamined,thecomputationtimecanbeunacceptablylongusingconventionalalgorithmsandcomputers.CHAPTERPropertiesofConvolution7Alinearsystem'scharacteristicsarecompletelyspecifiedbythesystem'simpulseresponse,asgovernedbythemathematicsofconvolution.Thisisthebasisofmanysignalprocessingtechniques.Forexample:Digitalfiltersarecreatedbydesigninganappropriateimpulseresponse.Enemyaircraftaredetectedwithradarbyanalyzingameasuredimpulseresponse.Echosuppressioninlongdistancetelephonecallsisaccomplishedbycreatinganimpulseresponsethatcounteractstheimpulseresponseofthereverberation.Thelistgoesonandon.Thischapterexpandsonthepropertiesandusageofconvolutioninseveralareas.First,severalcommonimpulseresponsesarediscussed.Second,methodsarepresentedfor……

Althoughcomplexnumbersarefundamentallydisconnectedfromourreality,theycanbeusedtosolvescienceandengineeringproblemsintwoways.First,theparametersfromarealworldproblemcanbesubstitutedintoacomplexform,aspresentedinthelastchapter.Thesecondmethodismuchmoreelegantandpowerful,awayofmakingthecomplexnumbersmathematicallyequivalenttothephysicalproblem.ThisapproachleadstothecomplexFouriertransform,amoresophisticatedversionoftherealFouriertransformdiscussedinChapter8.ThecomplexFouriertransformisimportantinitself,butalsoasasteppingstonetomorepowerfulcomplextechniques,suchastheLaplaceandz-transforms.ThesecomplextransformsarethefoundationoftheoreticalDSP.TheRealDFTAllCHAPTERTheComplexFourierTransform31Althoughcomplexnumbersarefundamentallydisconnectedfromourreality,theycanbeusedtosolvescienceandengineeringproblemsintwoways.First,theparametersfromarealworldproblemcanbesubstitutedintoacomplexform,aspresentedinthelastchapter.Thesecondmethodismuchmoreelegantandpowerful,awayofmakingthecomplexnumbersmathematicallyequivalenttothephysicalproblem.ThisapproachleadstothecomplexFouriertransform,amoresophisticatedversionoftherealFouriertransformdiscussedinChapter8.ThecomplexFouriertransformisimportantinitself,butalsoasasteppingstonetomorepowerfulcomplextechniques,suchastheLaplaceandz-transforms.Thesecomplextr……