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1、IEEESIGNALPROCESSINGLETTERS,VOL.16,NO.10,OCTOBER2009853AConvolutionandProductTheoremfortheLinearCanonicalTransformDeyunWei,StudentMember,IEEE,QiwenRan,YuanminLi,JingMa,andLiyingTanAbstract—Thelinearcanonicaltransform(LCT)playsanLCTavailableintheliterature,however,donotg
2、eneralizeveryimportantroleinmanyfieldsofopticsandsignalprocessing.nicelytheclassicalresultfortheFT.Tobespecific,thecon-Manypropertiesforthistransformarealreadyknown,however,volutiontheoremoftheFTforthesignalsandwiththeconvolutiontheoremsdon’thavetheeleganceandsimplicityas
3、sociatedFTs,and,respectivelyisgivenby:comparabletothatoftheFouriertransform(FT),whichstatesthattheFouriertransformoftheconvolutionoftwofunctionsistheproductoftheirFouriertransforms.Thepurposeofthis(1)letteristointroduceanewconvolutionstructurefortheLCTthatpreservestheco
4、nvolutiontheoremfortheFouriertransformwheredenotestheconvolutionoperation.Thisconvolutionandisalsoeasytoimplementinthedesigningoffilters.Someoftheoremcanalsobewrittenaswell-knownresultsabouttheconvolutiontheoreminFTdomain,fractionalFouriertransform(FRFT)domainareshowntob
5、especialcasesofourachievedresults.(2)IndexTerms—Convolutionandproducttheorems,linearcanon-icaltransform.ThistheoremstatesthepowerfulresultthattheconvolutionoftwosignalsintimedomainresultsinsimplemultiplicationofI.INTRODUCTIONtheirFTsintheFTdomain.However,theconvolutiont
6、heoremfortheLCTgivenin[8]lacksthissimple,butimportantresultascanbeseenfromequationreproducedhere,wheredenotestheLCTofsignal.HElinearcanonicaltransform(LCT)[1][5]isafourpa-TrameterclassoflinearintegraltransformwhichencompassesanumberoftransformsimportantinDSPandop-ticals
7、ystemmodeling.Thewell-knownsignalprocessingoper-ations,suchastheFouriertransform(FT),thefractionalFourier(3)transform(FRFT),theFresneltransform(FST)andthescalingoperationsareallspecialcasesoftheLCT[5],[6].TheLCThasfoundmanyapplicationsinoptics,radarsystemanalysis,GRINTh
8、econvolutiontheoremfortheLCTderivedin[8]withmediumsystemanalysis,filterdesign,phaseretrieval,patternthemodifiedc
