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1、《数字信号处理》研究型作业一——离散余弦变换DCT的基本介绍学生：黄琬茹学号：2011019160026指导老师：潘晔二零一三年11月10日No.1TheDefinitionofDCTAdiscretecosinetransform(DCT)expressesafinitesequenceofdatapointsintermsofasumofcosinefunctionsoscillatingatdifferentfrequencies.DCTsareimportanttonumerousapplic
2、ationsinscienceandengineering,Theuseofcosineratherthansinefunctionsiscriticalintheseapplications:forcompression,itturnsoutthatcosinefunctionsaremuchmoreefficient(asdescribedbelow,fewerfunctionsareneededtoapproximateatypicalsignal),whereasfordifferential
3、equationsthecosinesexpressaparticularchoiceofboundaryconditions.Inparticular,aDCTisaFourier-relatedtransformsimilartothediscreteFouriertransform(DFT),butusingonlyrealnumbers.DCTsareequivalenttoDFTsofroughlytwicethelength,operatingonrealdatawithevensymme
4、try(sincetheFouriertransformofarealandevenfunctionisrealandeven),whereinsomevariantstheinputand/oroutputdataareshiftedbyhalfasample.ThereareeightstandardDCTvariants,ofwhichfourarecommon.Arealsymmetricorantisymmetricfinite-lengthsequenceisaproductofaline
5、ar-phasetermandarealamplitudefunction.Sincethephasetermisknownforgivenlengthsequence,theamplitudefunctionuniquelydescribesthetime-domainsequenceinthetransformdomain.Oneclassofrealorthogonaltransformisbasedonconvertingtherealarbitrarysequenceintoeitheras
6、ymmetricoranantisymmetricsequenceandthenextractingtherealorthogonaltransformcoefficientsfromtheDFTofthegeneratedsequencewithgeometricsymmetry.Thetransformsdevelopedviathisapproacharecalledthediscretecosinetransform,oftenabbreviatedasDCT,anddiscretesinet
7、ransform,oftenabbreviatedasDST.No.2TheengineeringbackgroundofDCTTheDCT,andinparticulartheDCT-II,isoftenusedinsignalandimageprocessing,especiallyforlossydatacompression,becauseithasastrong"energycompaction"property:[1][2]mostofthesignalinformationtendsto
8、beconcentratedinafewlow-frequencycomponentsoftheDCT,approachingtheKarhunen-Loèvetransform(whichisoptimalinthedecorrelationsense)forsignalsbasedoncertainlimitsofMarkovprocesses.Asexplainedbelow,thisstemsfromtheboundaryconditionsim
