傅里叶变换综述

ASurveyofFourierTransformAbstract:Asalinearintegraltransformtool,Fouriertransformisbecomingmoreandmorewidelyused.Inthispaper,theoriginofFouriertransformandclassificationanddiscreteFouriertransformandfastFouriertransformareexplainedandfoundthatbothinthefieldofmathematicsorengineeringapplicationsareofgreatsignificance,isaverycommonToolstohelppeoplesolveproblems.Keywords:FouriertransformFFTDFTapplicationsONE.FouriertransformoriginFouriertransformisalinearintegraltransform.[1-3]BecauseitsbasicideafirstbytheFrenchscholarJosephFouriersystematicallyputforward,sonamedbyitsnametocommemorate.Fouriertransformisakindoflinearintegraltransformwhichiswidelyusedinthefieldsofphysics,acoustics,optics,structuraldynamics,numbertheory,combinatorialmathematics,probabilitytheory,statistics,signalprocessing,cryptography,oceanographyandcommunication.[4-7]Applications.Fouriertransformscanexpresscertainfunctionssatisfyingcertainconditionsasalinearcombinationoftrigonometricfunctions(sineand/orcosinefunctions)ortheirintegrals.Indifferentresearchfields,Fouriertransformhasmanydifferentvariants,suchascontinuousFouriertransformanddiscreteFouriertransform.TheinitialFourieranalysiswasproposedasatoolforanalyticalanalysisofthermalprocesses.Thesinebasisfunctionistheeigenfunctionofthedifferentialoperator,sothatthesolutionofthelineardifferentialequationcanbetransformedintoalgebraicequationwithconstantcoefficients.Inalineartime-invariantphysicalsystem,thefrequencyisinvariantinnature,sothattheresponseofthesystemtocomplexstimulicanbeobtainedbycombiningitsresponsetosinusoidalsignalsofdifferentfrequencies.TWO.FouriertransformclassificationAccordingtothedifferenttypesoftheoriginalsignal,wecandividetheFouriertransformintofourcategories:[8-9]1non-periodiccontinuoussignalFouriertransform(FourierTransform)2periodiccontinuoussignalFourierseries(FourierSeries)3DiscreteTimeDiscreteTimeFourierTransform(DiscreteTimeFourierTransform)4periodicdiscretesignalDiscreteFourierTransform(DiscreteFourierTransform)ThesefourFouriertransformsareforinfiniteandnegativeinfinityofthesignal,thatis,thelengthofthesignalisinfinite,weknowthatthisisnotpossibleforcomputerprocessing,thenthereisnolimitedforthelengthoftheFouriertransformit?No.Sinceasine-cosinewaveisdefinedfromnegativeinfinitytopositiveinfinity,wecannotcombineasignalofinfinitelengthintoasignaloffinitelength.Facedwiththisdifficulty,themethodistolimitthelengthofthesignalisexpressedasaninfinitelengthofthesignal,thesignalcanbeinfinitelyextendedfromlefttoright,theextensionoftheparttozero,sothatthesignalcanbeseenasaperiodicDissociationofthesignal,wecanusethediscrete-timeFouriertransformmethod.Also,thesignalcanbeextendedwiththereplicationmethod,sothatthesignalbecomesaperiodicdiscretesignal,thenwecanusediscreteFouriertransformmethodtotransform.Herewewanttolearnisadiscretesignal,forcontinuoussignalwedonotdiscuss,becausethecomputercanonlydealwithdiscretenumericalsignals,ourultimategoalistousethecomputertoprocessthesignal.Butforaperiodicsignals,weneedtouseaninfinitenumberofdifferentfrequencysinecurvetorepresent,whichforthecomputerisimpossibletoachieve.Therefore,onlythediscreteFouriertransform(DFT)canbeappliedtothetransformationofdiscretesignals.Forthecomputer,onlydiscreteandfinitelengthdatacanbeprocessed.Forothertransformations,onlymathematicalcalculuscanbeused,infrontofthecomputerWecanonlyusetheDFTmethod,weneedtounderstandistheDFTmethod.Itistobeunderstoodherethatweuseperiodicsignalsinordertobeabletosolveproblemsmathematically,asitismeaninglesstoconsiderwheretheperiodicsignalsarederivedorhowtheyareobtained.EachFouriertransformisdividedintorealandcomplextwomethods,therealmethodisthebestunderstanding,butthecomplexmethodisrelativelycomplex,andneedtounderstandthetheoreticalknowledgeofcomplexnumbers,butifyouunderstandtherealnumberdiscreteFouriertransform(realDFT),andthentounderstandthecomplexFourieriseasier,sowefirstputtheFouriersideofthecomplextofirstunderstandtherealFouriertransform,wewillfirsttalkaboutthebasictheoryofcomplexnumbers,andthenunderstandtherealnumberFouriertransformonthebasisofunderstandingthecomplexFouriertransform.THERE.FastFourierTransformandDiscreteFourierTransform1.FastFouriertransformFastFouriertransform(FastFouriertransform),thatis,theuseofcomputer-baseddiscreteFouriertransform(DFT)efficient,fastcalculationmethodcollectively,referredtoasFFT.[10-13]ThefastFouriertransformwasproposedin1965byJ.W.CooleyandT.W.Tooki.Withthisalgorithm,thenumberofmultiplicationsrequiredforthecomputationofthediscreteFouriertransformcanbegreatlyreduced.Inparticular,themorethenumberofsampledpointsNisconverted,themoresignificantthecomputationalcomplexityoftheFFTalgorithmis.Thefinite-lengthsequencescanalsodiscretizetheirfrequencydomainintofinite-lengthsequencesbydiscreteFouriertransform(DFT).(FFT).In1965,CooleyandTukeyproposedafastalgorithmforcalculatingthediscreteFouriertransform(DFT),andtheDFTalgorithmwasproposedtosolvetheproblemofthefastFouriertransformTheamountofcomputationisreducedbyseveralordersofmagnitude.Sincethen,thefastFouriertransfor