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1、CHAPTERTheDiscreteFourierTransform8Fourieranalysisisafamilyofmathematicaltechniques,allbasedondecomposingsignalsintosinusoids.ThediscreteFouriertransform(DFT)isthefamilymemberusedwithdigitizedsignals.ThisisthefirstoffourchaptersontherealDFT,aversionofthediscreteFouriertransformthatusesrealnum
2、berstorepresenttheinputandoutputsignals.ThecomplexDFT,amoreadvancedtechniquethatusescomplexnumbers,willbediscussedinChapter31.InthischapterwelookatthemathematicsandalgorithmsoftheFourierdecomposition,theheartoftheDFT.TheFamilyofFourierTransformFourieranalysisisnamedafterJeanBaptisteJosephFour
3、ier(1768-1830),aFrenchmathematicianandphysicist.(Fourierispronounced:for@¯e@¯a,andisalwayscapitalized).Whilemanycontributedtothefield,Fourierishonoredforhismathematicaldiscoveriesandinsightintothepracticalusefulnessofthetechniques.Fourierwasinterestedinheatpropagation,andpresentedapaperin1807
4、totheInstitutdeFranceontheuseofsinusoidstorepresenttemperaturedistributions.Thepapercontainedthecontroversialclaimthatanycontinuousperiodicsignalcouldberepresentedasthesumofproperlychosensinusoidalwaves.Amongthereviewersweretwoofhistory'smostfamousmathematicians,JosephLouisLagrange(1736-1813)
5、,andPierreSimondeLaplace(1749-1827).WhileLaplaceandtheotherreviewersvotedtopublishthepaper,Lagrangeadamantlyprotested.Fornearly50years,Lagrangehadinsistedthatsuchanapproachcouldnotbeusedtorepresentsignalswithcorners,i.e.,discontinuousslopes,suchasinsquarewaves.TheInstitutdeFrancebowedtothepre
6、stigeofLagrange,andrejectedFourier'swork.ItwasonlyafterLagrangediedthatthepaperwasfinallypublished,some15yearslater.Luckily,Fourierhadotherthingstokeephimbusy,politicalactivities,expeditionstoEgyptwithNapoleon,andtryingtoavoidtheguillotineaftertheFrenchRevolution(literally!).141142TheScientis
7、tandEngineer'sGuidetoDigitalSignalProcessingWhowasright?It'sasplitdecision.Lagrangewascorrectinhisassertionthatasummationofsinusoidscannotformasignalwithacorner.However,youcangetveryclose.Soclosethatthedifferencebetweenthetwohaszeroenergy.Int
