3multidimensional filter banks and wavelets

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1、TOAPPEARASANINVITEDARTICLEINASPECIALISSUEONSIGNALPROCESSINGINTHEJOURNALOFFRANKLININSTITUTE,1998Multidimensionalfilterbanksandwavelets-asystemtheoreticperspectiveSankarBasu*IBMThomasJ.WatsonResearchCenterYorktownHeights,NY10598email:sankar@wats0n.ibm.comNov

2、ember10,1997AbstractWereviewthecurrentstatusofmultidimensionalfiltersbankandwaveletdesignfromtheperspectiveofsignalandsystemtheory.Thestudyofwaveletsandperfectreconstructionfilterbanksareknowntohaverootsintraditionalfilterdesigntechniques.Ontheotherhand,th

3、efieldofmultidimensionalsystemsandsignalprocessinghasdevelopedasetoftoolsintrinsictoitself,andhasattainedacertainlevelofmaturityoverthelasttwodecades.Wehaverecentlynotedadegreeofsynergybetweenthetwofieldsofwaveletsandmultidimensionalsystems.ThisarisesfIomt

4、hefactthatmanyideascrucialtothewaveletdesignareinherentlysystemtheoreticinnature.Whiletherearemanyexamplesofthissynergymanifestedinrecentpublications,weprovideaflavouroftechniquesgermanetothisdevelopmentbyconsideringafewspecificproblemsindetail.Theconstruc

5、tionoforthogonalwaveletscanbeessen-tiallyviewedasacircuitandsystemtheoreticproblemofdesignofenergydissipative(passive)filters,themultidimensionalversionofwhichhasveryclosetieswithclassicproblemoflumped-distributedpassivenetworksyn-thesis.Groebnerbasistechn

6、iques,matrixcompletionproblemsoverringsofpolynomialsorringsofstablerationalfunctionsi.e.,Quillen-Suslin[31]typeproblemsarestillotherexamples,whichfeatureinourdiscussioninimportantmanner.Anumberofopenproblemsarealsocited.'TheauthorrecievedsupportfromNSFgran

7、tMIP969617611Introduction:Webeginbyconsideringamultidimensional(k-D)subbandcodingschemeshowninfigure1.Weshallfollowthenotationfromstandardliteratureasin[42].AssociatedwitheachsuchschemethereisadecimationmatrixDofsizekxk.Weshallalsoneedtoconsideranumberofpa

8、rametersasscoiatedwithD.First,notethatdetDistheratioofsamplingdensitiesoftheoriginalsignalandthesubsampledsignal,andthus,forcriticallysampledsignals,mustbeequaltothenumberofchannelsminthesubbandcodingschemein