An-economical-class-of-digital-filters-for-decimat

IEEETRANSACTIONSONACOUSTICS,SPEECH,ANDSIGNALPROCESSING,VOL.ASSP-29,NO.2,APRIL1981155AnEconomicalClassofDigitalFiltersforDecimationandInterpolationAbstruct-Aclassofdigitalhearphasefiiiteimpulseresponse(FIR)filtersfordecimation(samplingratedecrease)andinterpolation(sam-plingrateincrease)arepresented.Theyrequirenomultipliersanduselimitedstoragemakingthemaneconomicalalternativetoconventionalimplementationsforcertainapplications.Adigitalfiiterinthisclassconsistsofcascadedidealintegratorstagesoperatingatahighsamplingrateandanequalnumberofcombstagesoperatingatalowsamplingrate.Together,asingleintegrator-combpairproducesauniformFIR.Thenumberofcascadedintegrator-combpairsischosentomeetdesignrequirementsforaliasingorimagingerror.Designproceduresandexamplesaregivenforbothdecimationandinterpolationfilterswiththeemphasisonfrequencyresponseandregis-terwidth.I.INTRODUCTIONINrecentliterature,CrochiereandRabiner[l]-[3]havepresentedageneraltheoryforFIRmultistagedecimatorsandinterpolatorswithemphasisonoptimaldesignsintermsofminimizingthenumberofmultiplicationspersecondortherequiredamountofstorage.GoodmanandCarey[4]havetakentheapproachthatacarefulchoiceoffiltercoefficientsforhalf-banddecimatorsandinterpolatorscanleadtoefficienthardwaredesigns.Inthefieldofefficientdigitalfilters,PeledandLiu[SIhaveintroducedthe“coefficientslicing”approachtofilterdesign.Forthesefilte,rs,multipliersarereplacedwithaddersandROMlook-uptables.Thisapproachcanbeappliedprofitablytodecimatorandinterpolatordesigns.Theessentialfunctionofadecimationorinterpolationfilteristodecreaseorincreasethesamplingrateandtokeepthepassbandaliasingorimagingerrorwithinprescribedbounds.Inthispaper,aclassoflinearphaseFIRfiltersfordecimationandinterpolationthatfulfillthisbasicrequirementareintro-duced.Thefiltersrequirenomultipliersanduselimitedstor-agetherebyleadingtomoreeconomicalhardwareimplemen-tations.Theyaredesignatedcascadedintegrator-comb(CIC)filtersbecausetheirstructureconsistsofanintegratorsectionoperatingatthehighsamplingrateandacombsectionoper-atingatthelowsamplingrate.UsingCICfilters,theamountofpassbandaliasingorimagingerrorcanbebroughtwithinprescribedboundsbyincreasingthenumberofstagesinthefilter.However,thewidthofthepassbandandthefrequencycharacteristicsoutsidethepass-bandareseverelylimited.ForcriticalapplicationstheseManuscriptreceivedMarch17,1980;revisedSeptember22,1980.TheauthoriswithESL,Inc.Sunnyvale,CA94086.limitationscanbeovercomebyusingCICfilterstomakethetransitionbetweenhighandlowsamplingrates,andtouseconventionalfiltersatthelowsamplingrateto“shape”or“clean-up”thefrequencyresponse.Inthismanner,CICfiltersareusedathighsamplingrateswhereeconomyiscritical,andconventionalfiltersareusedatlowsamplingrateswherethenumberofmultipliespersecondislow.LikeCICfilters,someofthefiltersdescribedin[4]donotrequiremultipliers;however,thesefiltersarerestrictedtoaratechangefactoroftwo,andhavelimitedattenuationinthealiasing/imagingbands.ThenextsectiondescribesCICfiltersintermsoftheirfunc-tionalbuildingblocks,relatingtheirz-transformstothez-transformofthecompositefilter.,SectionI11discussesthefrequencyresponseofCICfiltersgivinganapproximationthatisusableforawiderangeofdesignproblems.Tablesarepro-videdfordeterminingfilterparametersasafunctionofthedesiredbandwidthandaliasing/imagingerror.InSectionIV,CICdecimationfiltersaredescribedwithparticularattentiongiventotheeffectsoftruncationandroundingonthefilter’serrorstatistics.Designequationsaregivenandareappliedtoaspecificdesignproblem.InSectionVasimilartreatmentisgivenforCICinterpolationfilterswiththemajorconsiderationgiventoregistergrowthanditsrela-tiontothefilterdesign.11.CICFILTERDESCRIPTIONFig.1showsthebasicstructureoftheCICdecimationfilter.AnanalogousstructurefortheCICinterpolationfilterispre-sentedinFig.2.TheintegratorsectionofCICfiltersconsistsofNidealdigitalintegratorstagesoperatingatthehighsamplingrate,f,.Eachstageisimplementedasaone-polefilterwithaunityfeedbackcoefficient.ThesystemfunctionforasingleintegratorisThecombsectionoperatesatthelowsamplingratefJRwhereRistheintegerratechangefactor.ThissectionconsistsofNcombstageswithadifferentialdelayofMsamplesperstage.Thedifferentialdelayisafilterdesignparameterusedtocontrolthefilter’sfrequencyresponse.Inpractice,thedifferentialdelayisusuallyheldtoM=1or2.Thesystemfunctionforasinglecombstagereferencedtothehighsam-plingrateisH&)=1-Z-RM.(2)0096-3518/81/0400-01S5$00.7S01981IEEE156IEEETRANSACTIONSONACOUSTICS,SPEECH,ANDSIGNALPROCESSING,VOL.ASSP-29,NO,2,APRIL1981INTEQRATORSECTIONCOMBSECTIONSTAQE10.0STAQENSTAGEN+1STAGE2N0.0.Fig.1.CICdecimationfilter.x.+fJRCOMBSECTIONINTEGRATORSECTIONSTAGE1STAGENSTAGEN+1eSTAGE2NFig.2.CICinterpolationfilter.Thereisaratechangeswitchbetweenthetwofiltersections.Fordecimation,theswitchsubsamplestheoutputofthelastintegratorstage,reducingthesamplingratefromf,tof,/R;andforinterpolation,theswitchcausesarateincreasebyafactorofRbyinsertingR-1zerovaluedsamplesbetweenconsecutivesamplesofthecombsectionoutput.Itfollowsfrom(1)and(2)thatthesystemfunctionforthecompositeCICfilte