Fractional Fourier domain analysis of cyclic mult

ScienceinChinaSeriesE:TechnologicalSciences©2008SCIENCEINCHINAPRESSReceivedDecember18,2007;acceptedMarch24,2008doi:10.1007/s11431-008-0092-y†Correspondingauthor(email:[e-mail])SupportedpartiallybytheNationalNaturalScienceFoundationofChina(GrantsNos.60232010and60572094),theNationalNaturalScienceFoundationofChinaforDistinguishedYoungScholars(GrantNo.60625104),aswellastheDoctorshipFoundationofChinaEducationMinistry(GrantsNo.1010036620602)SciChinaSerE-TechSci|Jun.2008|vol.51|no.6|803-819†&WANGYueDepartmentofElectronicEngineering,BeijingInstituteofTechnology,Beijing100081,ChinaThecyclicfilterbanks,whichareusedwidelyintheimagesubbandcoding,refertosignalprocessingonthefinitefield.ThisstudyinvestigatesthefractionalFourierdomain(FRFD)analysisofcyclicmultiratesystemsbasedonthefractionalcircularconvolutionandchirpperiod.TheproposedtheoremsincludethefractionalFourierdomainanalysisofcyclicdecimationandcyclicinterpolation,thenobleidentitiesofcyclicdecimationandcyclicinterpolationintheFRFD,thepolyphaserepresen-tationofcyclicsignalintheFRFD,andtheperfectreconstructionconditionforthecyclicfilterbanksintheFRFD.Furthermore,thispaperproposesthedesignmethodsforperfectreconstructioncyclicfilterbankandcyclicfilterbankwithchirpmodulationintheFRFD.TheproposedtheoremsextendthemultiratesignalprocessingintheFRFD,whichalsoadvancetheapplicationsofthetheoremsoffilterbankintheFRFDonthefinitesignalfield,suchasdigitalimageprocessing.Atlast,theproposeddesignmethodsforthecyclicfilterbanksintheFRFDarevalidatedbysimulations.discretefractionalFouriertransform,fractionalcircularconvolution,cyclicfilterbanks,perfectreconstruction1IntroductionWiththedevelopmentofdigitalsignalprocessing,computationalamountandstorageloadhavegraduallyincreased.Thus,samplingrateconversionisalwaysmadeinordertodecreasethecomputationalamountandstorageloadinthesystem,andmultiratesignalprocessinghasbeenestablishedtomeetthatneed[1,2].TheZ-transformandcorrespondinglinearconvolutionhavebeenthecenterpieceoftheclassicalmultiratesignalprocessing.However,circularconvolutionandDFTblocksarealwaysutilizedinthepracticalDSPsystem.Thus,Smithetal.[3,4]proposedtheimagesubbandcodingbasedonthecircularconvolutionandthewavelettransformsassoci-atedwithcyclicfilterbanks[5].Then,thetheoremsforcyclicmultiratesystemswereproposedby804MENGXiangYietal.SciChinaSerE-TechSci|Jun.2008|vol.51|no.6|803-819Vaidyananthanetal.[6,7],whicharethegeneralizationsoftheclassicalmultiratesignalprocessingonthefinitefield.ThefractionalFouriertransform(FRFT)wasfirstlyproposedbyWienerin1929[8].Ithasbeenfoundmanyapplicationsinopticsfromthe1980’s,andhasgraduallybeenappliedtosignalproc-essingfromthe1990’s.AsageneralizationofFourierTransform(FT),theFRFTismoresuitabletodealwithanonstationarysignalthanFTforitstime-frequencyanalysischaracter.Also,theFRFTofasignalcanbeconsideredasthedecompositionofthesignalintermsofchirpsets,whichhavethesamesweepingrateanddifferentinitialfrequencies.ComparedwiththeclassicalFT,theFRFThasanadditionalparameterforthetransformorder[9,10].Thus,theapplicationsoftheFRFTonthefinitesignalfield,suchasimageprocessing,focusontheimagewatermarkandimageencryption[10—12],becausetheadditionalparametercanbeutilizedtoincreasethesecuritylevel.However,thepresentanalysisofanonstationarysignalisunderthesituationwithsinglesamplingrate,whichlacksthecombinationwiththemulti-resolutionanalysisofsignals.ThefractionalFourierdomainanalysisofcyclicfilterbanksis,therefore,requiredtosupportthemulti-resolutionanalysisoffinitesignalinthefractionalFourierdomain(FRFD).Thisstudyin-vestigatesthefractionalFourierdomainanalysisofcyclicfilterbanksbasedonthefractionalcircularconvolutiontheorem[13,14].Theproposedtheoremsarethegeneralizationsofthefrac-tionalFourierdomainanalysisofmultiratesignalprocessinginrefs.[15—17]onthefinitefield.Therestofthispaperisorganizedasfollows.ThealgorithmsforthediscretefractionalFouriertransform(DFRFT)andthefractionalcircularconvolutiontheoremarereviewedinsection2.ThefractionalFourierdomainanalysisofcyclicdecimationandcyclicinterpolation,andtheno-bleidentitiesforcyclicmultiratesignalprocessingarededucedinsection3.Insection4,thedefinitionofcyclicfilterbankswithchirpmodulationandthedesignmethodsforperfectrecon-struction(PR)cyclicfilterbanksareintroduced.Insection5,thesimulationsaregiventoverifythedesignmethodinthispaper.2DiscretefractionalFouriertransformThep-thorderFRFTofsignalx(t)isdefinedas[9,10](){}()()()(,)d,pppXuFxtuxtKutt+∞−∞==⎡⎤⎣⎦∫(1)wherep=2α/πindicatesthetransformorderoftheFRFT,αindicatestherotationangleofthetransformedsignalfortheFRFT,Fp[·]indicatestheFRFToperator,andKp(u,t)theFRFTkernel,whichisdefinedas()221cotexpcotcsc,,22(,)(),2,(),21.pjtujjutnKuttuntunααααππδαπδαπ⎧⎛⎞−+−≠⎪⎜⎟⎪⎝⎠=⎨−=⎪⎪+=±⎩(2)TheinversetransformoftheFRFTisdefinedas(){}()()()(,)d.ppppxtFXutXuKtuu+∞−−−∞⎡⎤==⎣⎦∫(3)Inpractice,thesignalinthestudyisinthediscreteform.Thus,theDFRFTisutilized.Theal-gorithmsfortheDFRFTcomput