Optimal Design of Partial-Band Time-Varying System

OptimalDesignofPartial-BandTime-VaryingSystemsW.M.CampbellMotorolaGSTG,H11758201E.McDowellRoadScottsdale,AZ85252(602)[e-mail]ngineeringCornellUniversityIthaca,NY14853(607)[e-mail]25,1996Abstract.Thedesignofpartial-bandlinearperiodicallytime-varyingsystems(PBTV)imitatinglineartime-invariant(LTI)systemsisapproachedusingarelative‘2errorcriterion.ThiserrorcriterionresultsfromanaturalextensionoftheChebysheverrorcriterionforthedesignofLTIsystems.Acompleteanalysisoftheerrorcriterionispresented.Analgorithmisintroducedwhichndsalocallyoptimalsolution.Thisalgorithmisbasedonrecentresultsinnonsmoothoptimization.ThemethodsintroducedallowthedesignertocompareaswellasdesignPBTVsystemsinasystematicandmeaningfulmanner.ThisworkwassupportedbytheNationalScienceFoundationunderGrantMIP9224424.121.IntroductionThedesignofmultiratesystemshasbeenextensivelyconsidered;see[1,2]andtherefer-encelisttherein.Inthispaper,thedesignofpartial-bandlinearperiodicallytime-varying(PBTV)systemswhichimitatelineartime-invariant(LTI)systemsisconsidered.Awellknownsystem[1]forpartial-bandlteringisshowninFigure1.(Inthegure,HiandPdenotelteringbyHi(z)andP(z),respectively.)Themainadvantageofthissystemiscomputationaleciency[1].Sinceonlyasectionofthefrequencybandoftheinputisprocessed,thesamplingrateoftheinputcanbereducedbydecimation.ThesignalisthenprocessedbyPandrestoredtotheoriginalsamplingratebyinterpolation.IfH1,P,andH2arenotideal,thenthesystemintroducesaliasing.Thegoalofthispaperistodealwiththeerrorarisingfromthisapproximationprobleminapreciseform.OurnewresultsincludetheanalysisofthiscriterionforthePBTVsystem,andtheformulationandanalysisoftheproblemusingnonsmoothoptimizationmethods.Figure1.Systemforpartial-bandltering.InSection2,theproblemisdescribedindetail.Notationisintroducedtosimplifyanalysis.Thesystemisthenrearrangedtoacommutatorform[1].InSections3and4,therelative‘2errorcriterionisintroduced.ThiserrorcriterionismotivatedbyseveralprinciplesincludingtheneedtohaveageneraldeterministiccriterionwhichisanaturalextensionoftheChebyshevcriterionforLTIsystemdesign[3].Analysisoftheerrorcriterionrevealsseveralaspectsofthedesignproblem.Theerrorcriterionisnonsmoothandleadstoapproximationofamatrix-valuedfunction.Also,structuralconstraintsinthesystemmaketheproblemnontrivial.InSections5thru6,methodsarepresentedforthedesignofPBTVsystemsusingtheproposederrorcriterion.Amethodwhichndsalocallyoptimalsolutionisdiscussed,and3anexampleshowsthefeasibilityofthismethod.Adesignexampleintroducesseveralnewideasincludingtransitionregions.1.1.Notation.Afewconventionsareusedunlessotherwisenoted.Weuselower-caseletters,e.g.x,toindicatefunctionsontheintegers,Z.Capitalletters,e.g.X,indicateeitherthez-transformortheFouriertransformofxwhereambiguityisresolvedbythevariable,X(z)vs.X(f),orthecontext.TheFourier(-Plancherel)transformofxisgivenbyX(f)=[F(x)](f)=1Xk=1x(n)ej2knf:(1.1)wherej=p1.Forconvenience,wespecifyfrequencyresponseson[0:5;0:5]or[0;1].Scriptlettersdenoteoperators.Theoperatorgivenbyconvolutionwithhisalsodenotedbyascriptletter,[H(x)](n)=[hx](n).Allmatricesandvectorshavestartingindex0.2.ProblemStructureWebeginbystatingsomesimplifyingassumptions.First,inFigure1,P(z)canbeelimi-natedbycombiningitwithH1(z)togiveH1(z)P(zL)usingthenobleidentities[2].Second,weassumeH1andH2areunconstrainedcausalFIRlterswithlengthsN1andN2.Werefertothesystemwiththeseassumptionsasapartial-bandtime-varyingsystem(PBTVsystem).Toachieveadditionalcomputationalimprovement,structuralconstraintsmaybeintroducedforH1andH2.Forinstance,theinputdecimator(thecombinationofH1anddownsamplingbyL)couldbedesignedasacascadestructure.ThisintroducesaconstraintonH1.Formoreinformationreferto[1]andrelatedreferences.AtypicalidealPBTVsystemwouldhaveH1andH2inFigure1withfrequencyresponsesHi(f)=8:ej2Diff2[B+;B]transitionfunctionf2(B;B+)[(B;B)0f2[0:5;B][[B;0:5](2.1)4where0B12L,andtheDiarethedesiredgroupdelays.ThiswouldpreventaliasingandmaketheoverallsystemLTIwithfrequencyresponseHideal(f)=1LH1(f)H2(f).Notethatthetransitionfunctionhasnotbeenspecied.WewillimplicitlydesignthetransitionfunctioninSection6usingatransitionregion.TwoformsofthesystemthatprovideinsightintothePBTVsystemarethematrixformandthecommutatorform.Thematrixform[4,5]providesapossibleimplementationformforthesystemandprovidesaninitialformfortheanalysisoftheerrorcriterioninSection4.Thecommutatorform[1]providesanaturalinterpretationforPBTVsystemsandshowsstructuralconstraints.Thecommutatorformwillbeusedtoanalyzetheerrorcriterionindetailandanalyzealiasingerror.Figure2.MatrixFormofthePBTVSystem.ThematrixformofthesystemisshowninFigure2.InFigure2,\)indicatesavectorsignal;PListheL-polyphasedecompositionoperatordenedforarbitraryxbyxL(n)=PL(x)(n)=26666664xL;0(n)...xL;L1(n)37777775(2.2)wherexL;k(n)=x(Ln+k).PyLindicatestheadjointofPL.ThematrixT(z)isanouterproductofthepolyphasecomponentsofH1andH2T(z)=26666664H2;0(z)...H2;L1(z)37777775H1;0(z)zH1;L1(z):::zH1;1(z):(2.3)The