CIC补偿滤波器设计

AlteraCorporation1AN-455-1.0PreliminaryApplicationNote455UnderstandingCICCompensationFiltersIntroductionThecascadedintegrator-comb(CIC)filterisaclassofhardware-efficientlinearphasefiniteimpulseresponse(FIR)digitalfilters.CICfiltersachievesamplingratedecrease(decimation)andsamplingrateincrease(interpolation)withoutusingmultipliers.Altera’sCICCompilerMegaCore®functionimplementsvariousCICfiltersbasedonHogenauer’smethod.fCICfilterswerefirstproposedbyEugeneHogenauerin1981,FormoreinformationaboutCICfilters,refertoEugeneB.Hogenauer,“Aneconomicalclassofdigitalfiltersfordecimationandinterpolation,”IEEETransactionsonAcoustics,SpeechandSignalProcessing,pp.155-162,April1981.ACICfilterconsistsofanequalnumberofstagesofidealintegratorfiltersandcombfilters.Itsfrequencyresponsemaybetunedbyselectingtheappropriatenumberofcascadedintegratorandcombfilterpairs.ThehighlysymmetricstructureofaCICfilterallowsefficientimplementationinhardware.However,thedisadvantageofaCICfilteristhatitspassbandisnotflat,whichisundesirableinmanyapplications.Fortunately,thisproblemcanbealleviatedbyacompensationfilter.ThisapplicationnotepresentstheoryandmethodsfordesigningCICcompensatingfiltersforsamplerateconversionsystems.TheMATLABSignalProcessingToolboxisusedtodesignthecoefficientsofthecompensatingFIRfilters.ThisapplicationnotealsodescribeshowtochooseparametersfordesigningacompensationfilterandthenimplementsanexampledecimationsystemusingtheAltera®CICCompilerMegaCorefunctionandtheFIRCompilerMegaCorefunction.Thefollowingtopicsarediscussedinthisdocument:■“Prerequisites”onpage2■“CICFilterStructure”onpage2■“CICCompensationFilterDesign”onpage4■“DataRateDownConversionExample”onpage11■“Conclusion”onpage17April2007,ver.1.02AlteraCorporationPreliminaryUnderstandingCICCompensationFiltersPrerequisitesThisdocumenttargetsdigitalsignalprocessing(DSP)systemsengineerswhomustdesignCICcompensationfiltersforrateconversionsystems.AbasicknowledgeofDSPanddigitalfilterdesignwillhelpyouunderstandthetrade-offbetweenvariousCICcompensationfilterdesignmethods.Inaddition,tounderstandandduplicatetheexamplesandfiguresusedinthisapplicationnote,youshouldhavethefollowing:■SomeexperiencewithMATLABandSIMULINK■SomeknowledgeofAlteraDSPSolutions,includingDSPBuilder1Thedesignexampleusedinthisapplicationnotecanbefoundat:www.altera.com/support/examples/dsp-builder/exm-digital-down-conv-cic-fir.htmlCICFilterStructureThebasicelementsofaCICfilterareintegratorfiltersandcombfilters,asshowninFigure1.Figure1.BlockDiagramofThree-StageCICDecimationandInterpolationFiltersAlteraCorporation3PreliminaryCICFilterStructureAnintegratorfilterisasinglepoleaccumulatorwithatransferfunctionHI(z)(Equation1):(1)AcombfilterisadifferentiatorwithatransferfunctionHC(z)(Equation2):(2)Inthisequation,Misthedifferentialdelay,andisusuallylimitedto1or2.InaCICfilter,theintegratorsoperateathighsamplingfrequency(fS),andthecombfiltersoperateatlowfrequency(fS/R).UsingtheNobleidentities,theequivalentfrequencyresponseoftheircascadecanbecalculated(Figure2).Figure2.BlockDiagramoftheEquivalentFrequencyResponseofanN-StageCICFilterEquation3showsthetotalresponseofaCICfilterathighfrequency(fS):(3)Inthisequation,Nisthenumberofintegrator-combfilterpairs,andRistheratechangefactor.Equation3impliesthattheequivalenttimedomainimpulseresponseofaCICfiltercanbeviewedasacascadeofNrectangularpulses.EachrectangularpulsehasRMtaps.HIz()11z1––----------------=Hcz()1zM––=Hz()HINz()HcNzR()zk–k0=RM1–∑⎝⎠⎜⎟⎜⎟⎛⎞N==4AlteraCorporationPreliminaryUnderstandingCICCompensationFiltersEquation4showsthemagnituderesponseofanN-stageCICfilterathighfrequency(fS):(4)Figure3showsanexampleofaCICfiltermagnituderesponse:Figure3.MagnitudeResponseofaCICFilterwithN=9,R=8,andM=1fFormoreinformationaboutCICfilters,refertoMatthewDonadio,CascadedIntegrator-Comb(CIC)FilterIntroduction,availableatwww.dspguru.com/info/tutor/cic.htm.CICCompensationFilterDesignFigure3showsthatwhenthenumberofstagesislarge,theCICfilterfrequencyresponsedoesnothaveawide,flatpassband.Toovercomethemagnitudedroop,aFIRfilterthathasamagnituderesponsethatistheinverseoftheCICfiltercanbeappliedtoachievefrequencyresponsecorrection.Suchfiltersarecalled“compensationfilters.”Fordataratedownconversion,thecompensationfilterfollowstheCICfilter.Forupsamplingsystems,thecompensationFIRfilterpre-conditionsthedataandisfollowedbyaCICfilter.Inotherwords,thecompensationfilteralwaysoperatesatthelowerrateinarateconversionHf()πMf()sinπfR-----⎝⎠⎛⎞sin-----------------------N=AlteraCorporation5PreliminaryCICCompensationFilterDesigndesign.Onebenefitofrunningthecompensationfilteratthelowrateistoachieveamoreefficienthardwaresolution,thatis,moretimesharinginthecompensationFIRfilter.Equation4givesthemagnituderesponseofaCICfilter.Toachieveaflatpassband,thecompensationFIRfiltershouldhaveamagnituderesponsethatistheinverseofEquation4,asshowninEquation5:(5)WhenRislarge,thecompensationfilterresponsecanbeapproximatedbytheinversesincfunction,sothecompensationfilterissometimesreferredtoasthe“inversesincfilter.”Insomeratechangesystems,compensationfiltersarealsomultiratefilters.Theycanimplementadditionaldecimationorinterpolationasnecessary,butusuallybyafactorof2orless.GeneratingCompensationFilterCoefficientsUsingMATLABAnyfilterdesigntoolthatgeneratesfiltercoefficientsbasedonthespecifiedfrequencyresponsecanbeusedtodesignaCICcompensationfilter.Inthisapplicationnote,theMATLABSignalProcessingToolboxfunctionfir2isusedtogeneratethecoefficientsforCICcompensationfilters.Thefir2functiondesignsFIRfilterswithanarbitraryfrequencyresponsebasedonthefrequencysamplingmethod.Thegeneratedfiltercoefficientsarerealandsymmetric.YoucanspecifyadigitalFIRfilterorderLwiththefrequencyresponsespecifiedbyvectorsF(frequency)andA(magnituderesponseonbreakpointsF).Thefir2functionreturnsan(L+1)vectoroffiltercoefficients.1TheMATLABscriptprovidedinthisapplicationnoteisautomaticallygeneratedbytheAlteraCICCompilerMegaCorefunction,version7.1.fFormoreinformationaboutthefrequencysamplingmethod,refertoL.R.Rabiner,B.Gold,andC.A.McGonegal,“Anapproachtotheapproximationproblemfornonrecursivedigitalfilters,”IEEETransactionsonAudioandElectroacoustics,pp.83-106,vol.Au-18,No.2,June1970.fFormoreinformationaboutthefir2function,refertotheMATLABHelp.Gf()MRπfR⁄()sinπMf()sin----------------------------⎝⎠⎛⎞NπMfπMf()sin-----------------------N≈sinc1–Mf()N==6AlteraCorporationPreliminaryUnderstandingCICCompensationFiltersExample1showsaMATLABscriptthatgeneratestheCICcompensationfiltercoefficients:Example1.GeneratingtheCICCompensationFilterCoefficients%%%%%%CICfilterparameters%%%%%%R=4;%%DecimationfactorM=1;%%DifferentialdelayN=8;%%NumberofstagesB=18;%%Coeffi.Bit-widthFs=91.392e6;%%(High)SamplingfreqinHzbeforedecimationFc=4.85e6;%%PassbandedgeinHz%%%%%%%fir2.mparameters%%%%%%L=110;%%Filterorder;mustbeevenFo=R*Fc/Fs;%%NormalizedCutofffreq;0<Fo<=0.5/M;%Fo=0.5/M;%%useFo=0.5ifyoudon'tcareresponsesare%%outsidethepassband%%%%%%%CICCompensatorDesignusingfir2.m%%%%%%p=2e3;%%Granularitys=0.25/p;%%Stepsizefp=[0:s:Fo];%%Passbandfrequencysamplesfs=(Fo+s):s:0.5;%%Stopbandfrequencysamplesf=[fpfs]*2;%%Normalizedfrequencysamples;0<=f<=1Mp=ones(1,length(fp));%%Passbandresponse;Mp(1)=1Mp(2:end)=abs(M*R*sin(pi*fp(2:end)/R)./sin(pi*M*fp(2:end))).^N;Mf=[Mpzeros(1,length(fs))];f(end)=1;h=fir2(L,f,Mf);%%FilterlengthL+1h=h/max(h);%%Floatingpointcoefficientshz=round(h*power(2,B-1)-1);%%FixedpointcoefficientsCompensationFilterExampleFigure4showsasimpleexampleofaCICfilterresponseanditscompensationfilterresponse.ThebluedottedlineisthemagnituderesponseofaCICfilterwithratechangefactorR=4,differentialdelayM=1,andnumberofintegrator-combfilterpairsN=4.TheresponseisplottedatlowfrequencyfS/R.Thegreendashedcurveisthesingleratecompensationfilterresponseplottedoveritsoperatingfrequency,fS/R.TheproductofthedottedcurveandthedashedcurveisthetotalresponseoftheCICandcompensationfiltercascade,representedbytheredsolidcurve.Thefiltercascadeclearlyhasareasonablyflatresponse.Inthisexample,a15-tapFIRcompensationfilterisdesignedusingfir2.Its18-bitwidecoefficientsare=[–215,446,–1258,3213,–7586,17668,–44268,131071,–44268,17668,–7586,3213,–1258,446,–215].TheoverallresponseplottedatthehighfrequencyfSisshowninFigure5onpage8.AlteraCorporation7PreliminaryCICCompensationFilterDesignFigure4.CompensationFilterResponsefora4-StageCICFilter,PlottedOverfS/RNotethatthecompensationfilteroperatesatlowfrequency(fS/R).Forsingleratecompensationfilterstoavoidaliasing,thecutofffrequency(fC)is,atmost,halfitsoutputfrequency:fC≤(fS/R)/2.Whenthecutofffrequencyisexactly(fS/R)/2,asthecaseshowninFigure4,thecompensationfilterhasaninversesincresponseacrossitsentirebandwidth,thusitisalsocalleda“widebandcompensationfilter”(Figure5).Inamultiratecompensationfilter,adecimationcompensationfilterwitharatechangefactorof2hasinputsamplingratefS/Randoutputsamplingrate(fS/R)/2.Toavoidaliasing,thecompensationfiltermusthaveacutofffrequencythatisnomorethanhalfof(fS/R)/2,thatis,(fS/R)/4.8AlteraCorporationPreliminaryUnderstandingCICCompensationFiltersFigure5.WidebandCompensationFilterResponseOverfSforR=4,N=4,andM=1Note(1)NotetoFigure5(1)Theblackdiamondmarkerindicatesstopbandamplificationduetothecompensationfilterfrequencyresponsecorrection.ChoosingPassBandEdgeNarrowtransitionbandwidthandgoodstopbandattenuationaredesirablepropertiesofanarrowbandrateconversionsystem.Unfortunately,aCICfilteralonedoesnotoffertheseproperties.AsdemonstratedinFigure4,inadditiontolackingaflatpassband,theCICfilterbyitselfalsosuffersfromthelackofawell-definedtransitionband.Thisproblemcanbealleviatedbycontrollingthecompensationfilterresponse.Insteadofhavingthewidebandcompensationresponse,asdemonstratedinFigure4andFigure5,youcanimposeconstraintsonthelowpasscompensationfilter,suchaspassbandedgeandstopbandattenuation.Forexample,youcanmodifytherequirementofthecompensationfilterexample(referto“CompensationFilterExample”onpage6).Insteadofhavingwidebandcompensation,apassbandedgeisimposedataquarterofthenormalizedlowfrequencyfS/R:fC=(fS/R)/4.Theidealresponseofthecompensationfilterisinversesincwithinthepassband,andzerooutsidethepassband,asshowninFigure6.AlteraCorporation9PreliminaryCICCompensationFilterDesignFigure6.IdealCompensationFilterResponsewithaCutoffFrequencyof(fS/R)/4Basedontheidealfrequencyresponse,thenewcompensationfiltercoefficientscanbegeneratedusingfir2.TheresponsesareplottedinFigure7.10AlteraCorporationPreliminaryUnderstandingCICCompensationFiltersFigure7.DesignedCompensationFilterResponsewithaNormalizedCutoffFrequencyof(fS/R)/4Note(1)NotetoFigure7(1)Theblackdiamondmarkerindicatesstopbandamplificationduetothecompensationfilterfrequencyresponsecorrection.Figures5and7alsodemonstratetheimpactofchoosingpassbandwidthonnoiseamplification.Intheseexamples,theCICfilterhasitsfirstnullat(1/M)/RonthehighfrequencyscalefS(thatis,thefirstnulloccursat1/MatlowfrequencyfS/R).Asthepassbandedgegetsclosertothenull,theCICfilterattenuatesmoreandthecompensationfilterneedstoprovidemorecorrection.Thiscorrectioncausesnoiseamplificationinthestopband.TheblackdiamondmarkersinFigures5and7illustratethispoint.Thetotalresponseinthestopbandhasaspikeatimagesofthecutofffrequency.Thecloserthepassbandedgeistothefirstnullandthemorestagesthereare,thehigherthespikeis.Intheextremecase—thewidebandcompensationcaseshowninFigure5—thenoiseamplificationcanbesignificant.Therefore,acarefulchoiceofpassbandwidthisimportantwhendesigningtheCICfilterandcompensationfiltercascade.1AgoodpracticeistochoosethepassbandedgetobelessthanaquarterofthefirstnullonthelowfrequencyscalefS/R.AlteraCorporation11PreliminaryDataRateDownConversionExampleDataRateDownConversionExampleThissectioncontainsinstructionsfordesigningapracticaldecimationfilterchainbasedontheIEEE802.16d(WiMAX)digitaldownconversion(DDC)requirement.TheWiMAXDDCfunctionhasatightsystemrequirementforfrequencyresponsesandisagoodexampleofhowtodesignamulti-ratesystemandchoosefilterparameters.fFormoreinformationabouttheIEEEstandard,refertoIEEEStandardforLocalandMetropolitanAreaNetworks,Part16:AirInterfaceforFixedBroadbandWirelessAccessSystems,IEEEP802.16-REVd/D5-2004,May2004.Thedesignusesadecimation-by-8filterchain.Table1liststhekeyparametersofthesystem.Table1.WiMAXDDCExampleTotalSpectrumRequirementParameterValueInputSamplingFrequency91.392MHzOutputSamplingFrequency11.424MHzPassBandEdge4.75MHzPassBandRipple<0.05dBStopBandAttenuation>90dB12AlteraCorporationPreliminaryUnderstandingCICCompensationFiltersTheoverallfilterresponsefollowstheWiMAXdownlinkspectrummask,asshowninFigure8.Figure8.WiMAXTransmitSpectrumMaskSolution1:SingleStageDecimationInthissolution,thedecimation-by-8CICfilterimplementstheratechangewithdifferentialdelayM=1andnumberofstagesN=9(toachievethestopbandattenuation).ThecompensationfilterisasinglerateFIRfilteroperatingat11.424MHz.Itspassbandedgeischosentobe4.85MHz,slightlylargerthantherequiredpassbandedge4.75MHztoguaranteegoodperformancefordatasubcarriersat4.75MHz.Thefilterorder,L,ischosentobe110.Thelargefilterorderisnecessarytomeetthesmallpassbandripplerequirementandthenarrowtransitionbandwidthrequirement.Usingfir2and18-bitwidthtorepresentfiltercoefficients,thedesignedCICfilterandcompensationfiltercascadehastheresponseshowninFigure9.TheperformanceissummarizedinTable2onpage16.AlteraCorporation13PreliminaryDataRateDownConversionExampleFigure9.CICandCompensationFilterResponseswithWiMAXSpectrumMaskforSingleStageDecimationDesignNote(1)NotetoFigure9(1)TheblackdiamondmarkersindicatestopbandamplificationduetothecompensationfilterfrequencyresponsecorrectionThenoiseamplificationduetocompensationfilterfrequencycorrectionisevidentintheabovedesign,asillustratedbytheblackdiamondmarkers.TheoverallfrequencyresponseappearstocomplywiththeWiMAXspectrummask.However,acloserlookatthepassbandrevealsthatthepassbandrippleexceedstheallowedvariationof0.05dB,asshowninFigure10.14AlteraCorporationPreliminaryUnderstandingCICCompensationFiltersFigure10.PassBandofDesignedSignalStageDecimationFilterChainIncreasingthecompensationfilterorderLimprovesthepassbandvariationslightly,butnotenoughtomeetsystemrequirements.Ontheotherhand,ahighcompensationfilterorderisnotonlyimpractical,butalsocauseslargenoiseamplificationinthetransitionband.Inthissolutionsetup,whenLexceeds160,themagnituderesponsearound6.6MHz(demonstratedbythefirstblackdiamondmarkerinFigure9)canexceedtheWiMAXspectrummask.AsthelastcolumnofTable2onpage16demonstrates,inthissingledecimationstagesolution,thenormalizedpassbandedgeis0.42onthelowfrequencyscale(fS/R),whichistooclosetothefirstnull.Therecommendednormalizeddigitalpassbandwidthis0.25.Solution2:Multi-StageDecimationDigitalfilterdesigntheorystatesthatwithallotherparametersfixed,thepassbandvariationisproportionaltothenormalizedtransitionbandwidth.ThenormalizedtransitionbandwidthinthisexampleisdeterminedbytheWiMAXspectrummaskandtheoperatingfrequencyofthecompensationfilter.Thesinglestagedecimationarchitecturewasnotabletomeetthefilterdesignrequirementfortworeasons:1)theoperatingfrequencyofthecompensationfilteristoolow,causingawidenormalizedtransitionbandwidth,and2)thenormalizedcutofffrequencyistooclosetothefirstnull.AlteraCorporation15PreliminaryDataRateDownConversionExampleThenaturalsolutiontotheseproblemsistodividethedecimationintomultiplestages.Inthesecondsolution,adecimation-by-4CICfilterisfollowedbyacompensationFIRfilterthatimplementsadditionaldecimation-by-2.TheCICfilterhasN=8andthedifferentialdelay,M,is1.Thecompensationfilterinputhasasamplefrequencyof22.848MHzandoutputsignalsat11.424MHz.Thecompensationfilter,L,hasanorderof110,andthepassbandedgeisstill4.85MHz.ComparedwithSolution1,thenormalizedtransitionbandwidthishalved,andthenormalizedpassbandedgeisalsohalved.TheperformanceparametersaresummarizedinTable2onpage16.TheresponseofthefiltercascadeisshowninFigures11and12.Figure11.CICandCompensationFilterResponseswithWiMAXSpectrumMaskforMulti-StageDecimationDesignNote(1)NotetoFigure11(1)TheblackdiamondmarkerindicatesstopbandamplificationduetothecompensationfilterfrequencyresponsecorrectionThenoiseamplificationinthemulti-stagedecimationsolutionismuchmilderthaninthesinglestagedecimation,duetoreducednormalizedcutofffrequency.TheblackdiamondmarkerinFigure11marksthefrequencyresponsespikecausedbytheimageofthecompensationfilterpassbandedge.Acloserlookofthepassbandconfirmsthatthepassbandvariationrequirementinthemulti-stagedecimationsetuphasbeenmet.Figure12showsthepassbandresponsesofSolution2.16AlteraCorporationPreliminaryUnderstandingCICCompensationFiltersFigure12.PassBandResponsesoftheMulti-StageDesignTable2showstheWiMAXDDCfilterdesignperformances.Table2.WiMAXDDCFilterDesignPerformancesNote(1)CasesFilterOrderPassBandRipple(dB)MagnitudeatImageofPassBandEdge(dB)StopBandAttenuation(dB)PassBandEdge@fS/RSolution11100.14(≤0.05)27.3(≥27)92(≥90)0.42(≤0.25)Solution11600.072(≤0.05)26.3(≥27)91.7(≥90)0.42(≤0.25)Solution21100.048(≤0.05)91.5(≥90)91.5(≥90)0.2(≤0.25)NotetoTable2:(1)Valuesinparenthesesarerequiredvaluesfortheparameter.AlteraCorporation17Preliminary101InnovationDriveSanJose,CA95134www.altera.comTechnicalSupport:www.altera.com/support/LiteratureServices:literature@altera.comCopyright©2007AlteraCorporation.Allrightsreserved.Altera,TheProgrammableSolutionsCompany,thestylizedAlteralogo,specificdevicedesignations,andallotherwordsandlogosthatareidentifiedastrademarksand/orservicemarksare,unlessnotedotherwise,thetrademarksandservicemarksofAlteraCorporationintheU.S.andothercountries.Allotherproductorservicenamesarethepropertyoftheirre-spectiveholders.AlteraproductsareprotectedundernumerousU.S.andforeignpatentsandpendingapplications,maskworkrights,andcopyrights.AlterawarrantsperformanceofitssemiconductorproductstocurrentspecificationsinaccordancewithAltera'sstandardwarranty,butreservestherighttomakechang-estoanyproductsandservicesatanytimewithoutnotice.Alteraassumesnoresponsibilityorliabilityarisingoutoftheapplicationoruseofanyinformation,product,orservicedescribedhereinexceptasexpresslyagreedtoinwritingbyAlteraCorporation.Alteracustomersareadvisedtoobtainthelatestversionofdevicespecificationsbeforerelyingonanypub-lishedinformationandbeforeplacingordersforproductsorservices.ConclusionConclusionThisapplicationnotepresentsdesignconsiderationsforCICcompensationfiltersinasamplerateconversionsystem.AlthoughCICfilterscanimplementdecimationandinterpolationefficientlyinhardwareforawiderangeofratechangefactors,aCICfilterresponselacksaflatpassbandresponseandgoodtransitionbandwidth.Toalleviatetheseproblems,acompensationFIRfiltercanbedesignedtoprovidefrequencycorrectionandspectrumshaping,evenwithanadditionalratechangeoffactor2.ThisapplicationnotedemonstrateshowtousetheMATLABSignalProcessingToolboxtodesignacompensationfilterusingthefrequencysamplingmethodfir2.Asamplescriptisgivenforreference.DetailedexamplesusingWiMAXdigitaldownconversionparametersareshowntoillustratehowtochooseparametersforthecompensationfilterandCICandFIRfiltercascade.DocumentRevisionHistoryTable3showstherevisionhistoryforthisapplicationnote.Table3.DocumentRevisionHistoryDateandDocumentVersionChangesMadeSummaryofChangesApril2007v1.0Initialrelease.— UnderstandingCICCompensationFilters Introduction Prerequisites CICFilterStructure CICCompensationFilterDesign GeneratingCompensationFilterCoefficientsUsingMATLAB CompensationFilterExample ChoosingPassBandEdge DataRateDownConversionExample Solution1:SingleStageDecimation Solution2:Multi-StageDecimation Conclusion DocumentRevisionHistory