understanding cascaded integrator-comb filters

2/22/2016 Understandingcascadedintegratorcombfilters

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UnderstandingcascadedintegratorcombfiltersByRichardLyons,CourtesyofEmbeddedSystemsProgrammingMar312005(14:49PM)URL:http://www.embedded.com/showArticle.jhtml?articleID=160400592

ThepreviouslyobscureCICfilterisnowvitaltomanyhighvolumewirelesscommunicationstasksandequipment.UsingCICfilterscancutcosts,improvereliability,andhelpperformance.Here'saprimertogetyoustarted.

Cascadedintegratorcomb(CIC)digitalfiltersarecomputationallyefficientimplementationsofnarrowbandlowpassfiltersandareoftenembeddedinhardwareimplementationsofdecimationandinterpolationinmoderncommunicationssystems.CICfilterswereintroducedtothesignalprocessingcommunity,byEugeneHogenauer,morethantwodecadesago,

buttheirapplicationpossibilitieshavegrowninrecentyears.1

Improvementsinchiptechnology,theincreaseduseofpolyphasefilteringtechniques,advancesindeltasigmaconverterimplementations,andthesignificantgrowthinwirelesscommunicationshaveallspurredmuchinterestinCICfilters.

Whilethebehaviorandimplementationofthesefiltersisn'tcomplicated,theircoveragehasbeenscarceintheliteratureofembeddedsystems.Thisarticleattemptstoaugmentthebodyofliteratureforembeddedsystemsengineers.AfterdescribingafewapplicationsforCICfilters,I'llintroducetheirstructureandbehavior,presentthefrequencydomainperformanceofCICfilters,anddiscussseveralimportantpracticalissuesinbuildingthesefilters.

CICfilterapplicationsCICfiltersarewellsuitedforantialiasingfilteringpriortodecimation(sampleratereduction),asshowninFigure1aandforantiimagingfilteringforinterpolatedsignals(samplerateincrease)asinFigure1b.Bothapplicationsareassociatedwithveryhighdataratefiltering,suchashardwarequadraturemodulationanddemodulationinmodernwirelesssystemsanddeltasigmaA/DandD/Aconverters.

Figure1:CICfilterapplicationsViewfullsizedimage

Becausetheirfrequencymagnituderesponseenvelopesaresin(x)/xlike,CICfiltersaretypicallyeitherfollowedorprecededbyhigherperformancelinearphaselowpasstappeddelaylineFIRfilterswhosetasksaretocompensatefortheCICfilter'snonflatpassband.Thatcascadedfilterarchitecturehasvaluablebenefits.Forexample,withdecimation,youcangreatlyreducecomputationalcomplexityofnarrowbandlowpassfilteringcomparedwithifyou'dusedasinglelowpassfiniteimpulseresponse(FIR)filter.Inaddition,thefollowonFIRfilteroperatesatreducedclockratesminimizingpowerconsumptioninhighspeedhardwareapplications.

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Understanding cascaded integrator-comb filters

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AcrucialbonusinusingCICfilters,andacharacteristicthatmakesthempopularinhardwaredevices,isthattheyrequirenomultiplication.Thearithmeticneededtoimplementthesedigitalfiltersisstrictlyadditionsandsubtractionsonly.Withthatsaid,let'sseehowCICfiltersoperate.

Figure2:DpointaveragingfiltersViewfullsizedimage

RecursiverunningsumfilterCICfiltersoriginatefromthenotionofarecursiverunningsumfilter,whichisitselfanefficientformofanonrecursivemovingaverager.RecallthestandardDpointmovingaverageprocessinFigure2a.ThereweseethatD1summations(plusonemultiplyby1/D)arenecessarytocomputetheaverageroutputy(n).

TheDpointmovingaveragefilter'soutputintimeisexpressedas:

Equation1

wherenisourtimedomainindex.Thezdomainexpressionforthismovingaverageris:

Equation2

whileitszdomainH(z)transferfunctionis:

Equation3

Iprovidetheseequationsnottomakethingscomplicated,butbecausethey'reuseful.Equation1tellsushowtobuildamovingaverager,andEquation3isintheformusedbycommercialsignalprocessingsoftwaretomodelthefrequencydomainbehaviorofthemovingaverager.

ThenextstepinourjourneytowardunderstandingCICfiltersistoconsideranequivalentformofthemovingaverager,therecursiverunningsumfilterdepictedinFigure2b.Thereweseethatthecurrentinputsamplex(n)isadded,andtheoldestinputsamplex(nD)issubtractedfromthepreviousoutputaveragey(n1).It'scalled"recursive"becauseithasfeedback.Eachfilteroutputsampleisretainedandusedtocomputethenextoutputvalue.Therecursiverunningsumfilter'sdifferenceequationis:

Equation4

havingazdomainH(z)transferfunctionof:

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Equation5

WeusethesameH(z)variableforthetransferfunctionsofthemovingaveragefilterandtherecursiverunningsumfilterbecausetheirtransferfunctionsareequaltoeachother!It'strue.Equation3isthenonrecursiveexpressionandEquation5istherecursiveexpressionforaDpointaverager.Themathematicalproofofthiscanbefoundinmybookondigitalsignalprocessing,butshortlyI'lldemonstratethatequivalencywith

anexample.2

Here'swhywecareaboutrecursiverunningsumfilters:thestandardmovingaveragerinFigure2amustperformD1additionsperoutputsample.Therecursiverunningsumfilterhasthesweetadvantagethatonlyoneadditionandonesubtractionarerequiredperoutputsample,regardlessofthedelaylengthD.Thiscomputationalefficiencymakestherecursiverunningsumfilterattractiveinmanyapplicationsseekingnoisereductionthroughaveraging.Nextwe'llseehowaCICfilteris,itself,arecursiverunningsumfilter.

CICfilterstructuresIfwecondensethedelaylinerepresentationandignorethe1/DscalinginFigure2bweobtaintheclassicformofa1storderCICfilter,whosecascadestructureisshowninFigure2c.ThefeedforwardportionoftheCICfilteriscalledthecombsection,whosedifferentialdelayisD,whilethefeedbacksectionistypicallycalledanintegrator.Thecombstagesubtractsadelayedinputsamplefromthecurrentinputsample,andtheintegratorissimplyanaccumulator.TheCICfilter'sdifferenceequationis:

Equation6

anditszdomaintransferfunctionis:

Equation7

Figure3:SinglestageCICfiltertimedomainresponseswhenD=5Viewfullsizedimage

ToseewhytheCICfilterisofinterest,firstweexamineitstimedomainbehavior,forD=5,showninFigure3.Ifaunitimpulsesequence,aunityvaluedsamplefollowedbymanyzerovaluedsamples,wasappliedtothecombstage,thatstage'soutputisasshowninFigure3a.Nowthink,whatwouldbetheoutputoftheintegratorifitsinputwasthecombstage'simpulseresponse?Theinitialpositiveimpulsefromthecombfilterstartstheintegrator'sallonesoutput,asinFigure3b.Then,Dsampleslater,thenegativeimpulsefromthecombstagearrivesattheintegratortozeroallfurtherCICfilteroutputsamples.

ThekeyissueisthatthecombinedunitimpulseresponseoftheCICfilter,beingarectangularsequence,isidenticaltotheunitimpulseresponsesofamovingaveragefilterandtherecursiverunningsumfilter.(Movingaveragers,recursiverunningsumfilters,andCICfiltersareclosekin.Theyhavethesamezdomainpole/zerolocations,theirfrequencymagnituderesponseshaveidenticalshapes,theirphaseresponsesareidentical,andtheirtransferfunctionsdifferonlybyaconstantscalefactor.)Ifyouunderstandthetimedomainbehaviorofamovingaverager,thenyounowunderstandthetimedomainbehavioroftheCICfilterinFigure2c.



Figure4:CharacteristicsofasinglestageCICfilterwhenD=5Viewfullsizedimage

ThefrequencymagnitudeandlinearphaseresponseofaD=5CICfilterisshowninFigure4awherethefrequencysistheinputsignalsamplerateinHz.

WecanobtainanexpressionfortheCICfilter'sfrequencyresponsebyevaluatingEquation7'sHcic(z)transferfunctiononthezplane'sunitcircle,

bysettingz=ej2,yielding:

Equation8

UsingEuler'sidentity2jsin()=ejej,wecanwrite:

Equation9

IfweignorethephasefactorinEquation9,thatratioofsin()termscanbeapproximatedbyasin(x)/xfunction.ThismeanstheCICfilter'sfrequencymagnituderesponseisapproximatelyequaltoasin(x)/xfunctioncenteredat0HzasweseeinFigure4a.(ThisiswhyCICfiltersaresometimescalledsincfilters.)

Digitalfilterdesignersliketoseezplanepole/zeroplots,soweprovidethezplanecharacteristicsofaD=5CICfilterinFigure4c,wherethecombfilterproducesDzeros,equallyspacedaroundtheunitcircle,andtheintegratorproducesasinglepolecancelingthezeroatz=1.Eachofthecomb'szeros,beingaDthrootof1,arelocatedatz(m)=ej2m/D,wherem=0,1,2,...,D1,correspondingtoamagnitudenullinFigure4a.

ThenormallyriskysituationofhavingafilterpoledirectlyontheunitcircleneednottroubleusherebecausethereisnocoefficientquantizationerrorinourHcic(z)transferfunction.CICfiltercoefficientsareonesandcanberepresentedwithperfectprecisionwithfixedpointnumberformats.Althoughrecursive,happilyCICfiltersareguaranteedstable,linearphaseshowninFigure4b,andhavefinitelengthimpulseresponses.At0Hz(DC)thegainofaCICfilterisequaltothecombfilterdelayD.Thisfact,whosederivationisavailable,willbeimportanttouswhenweactuallyimplement

aCICfilterinhardware.2

Figure5:SinglestageCICfiltersusedindecimationandinterpolationViewfullsizedimage

Again,CICfiltersareprimarilyusedforantialiasingfilteringpriortodecimationandforantiimagingfilteringforinterpolatedsignals.WiththosenotionsinmindweswaptheorderofFigure2c'scombandintegratorwe'repermittedtodosobecausethoseoperationsarelinearandincludedecimationbyasampleratechangefactorRinFigure5a.(Youmaywishtoprovethattheunitimpulseresponseoftheintegrator/comb



combination,priortothesampleratechange,inFigure5aisequaltothatinFigure3c.)InmostCICfilterapplicationstheratechangeRisequaltothecomb'sdifferentialdelayD,butwe'llkeepthemasseparatedesignparametersfornow.

Figure6:Magnituderesponseofa1storder,D=8,decimatingCICfilter:beforedecimationaliasiingafterR=8decimationViewfullsizedimage

ThedecimationoperationRmeansdiscardallbuteveryRthsample,resultinginanoutputsamplerateofs,out=s,in/R.ToinvestigateaCICfilter'sfrequencydomainbehaviorinmoredetail,Figure6ashowsthefrequencymagnituderesponseofaD=8CICfilterpriortodecimation.Thespectralband,ofwidthB,centeredat0Hzisthedesiredpassbandofthefilter.AkeyaspectofCICfiltersisthespectralfoldingthattakesplaceduetodecimation.

ThoseBwidthshadedspectralbandscenteredaboutmultiplesofs,in/RinFigure6awillaliasdirectlyintoourdesiredpassbandafterdecimationbyR=8asshowninFigure6b.Noticehowthelargestaliasedspectralcomponent,inthisexample,isroughly16dBbelowthepeakofthebandofinterest.OfcoursethealiasedpowerlevelsdependonthebandwidthBthesmallerBis,thelowerthealiasedenergyafterdecimation.

Figure7:1storder,D=R=8,interpolatingCICfilterspectra:inputspectrumoutputspectralimagesViewfullsizedimage

Figure5bshowsaCICfilterusedforinterpolationwheretheRsymbolmeansinsertR1zerosbetweeneachx(n)sample,yieldingay(n)outputsamplerateofs,out=Rs,in.(InthisCICfilterdiscussion,interpolationisdefinedaszerosinsertionfollowedbyfiltering.)Figure7ashowsanarbitrarybasebandspectrum,withitsspectralreplications,ofasignalappliedtotheD=R=8interpolatingCICfilterofFigure5b.Thefilter'soutputspectruminFigure7bshowshowimperfectfilteringgivesrisetotheundesiredspectralimages.

Afterinterpolation,unwantedimagesoftheBwidthbasebandspectrumresideatthenullcenters,locatedatintegermultiplesofs,out/R.IfwefollowtheCICfilterwithatraditionallowpasstappeddelaylineFIRfilter,whosestopbandincludesthefirstimageband,fairlyhighimagerejectioncanbeachieved.



Figure8:3rdorderCICdecimationfilterstructure,andmagnituderesponsebeforedecimationwhenD=R=8Viewfullsizedimage

ImprovingCICattenuationThemostcommonmethodtoimproveCICfilterantialiasingandimagerejectattenuationisbyincreasingtheorderMoftheCICfilterusingmultiplestages.Figure8showsthestructureandfrequencymagnituderesponseofa3rdorder(M=3)CICdecimatingfilter.

Noticetheincreasedattenuationats,out/RinFigure8bcomparedwiththe1storderCICfilterinFigure6a.BecausetheM=3CICstagesareincascade,theoverallfrequencymagnituderesponsewillbetheproductoftheirindividualresponsesor:

Equation10

ThepricewepayforimprovedantialiasattenuationisadditionalhardwareaddersandincreasedCICfilterpassbanddroop.Anadditionalpenaltyofincreasedfilterordercomesfromthegainofthefilter,whichisexponentialwiththeorder.BecauseCICfiltersgenerallymustworkwithfullprecisiontoremainstable,thenumberofbitsintheaddersisMlog2(D),whichmeansalargedatawordwidthpenaltyforhigherorderfilters.Evenso,thismultistageimplementationiscommonincommercial

integratedcircuits,whereanMthorderCICfilterisoftencalledasincM

filter.

Figure9:SinglestageCICfilterimplementations:fordecimationforinterpolationViewfullsizedimage

BuildingaCICfilterInCICfilters,thecombsectioncanprecede,orfollow,theintegratorsection.It'ssensible,however,toputthecombsectiononthesideofthefilteroperatingatthelowersampleratetoreducethestoragerequirementsinthedelay.SwappingthecombfiltersfromFigure5withtheratechangeoperationsresultsinthemostcommonimplementationofCICfilters,asshowninFigure9.Noticethedecimationfilter'scombsectionnowhasadelaylength(differentialdelay)ofN=D/R.That'sbecauseanNsampledelayafterdecimationbyRisequivalenttoaDsampledelaybeforedecimationbyR.LikewisefortheinterpolationfilteranNsampledelaybeforeinterpolationbyRisequivalenttoaDsampledelayafterinterpolationbyR.

ThoseFigure9configurationsyieldtwomajorbenefits:first,thecombsection'snewdifferentialdelayisdecreasedtoN=D/Rreducingdatastoragerequirementssecond,thecombsectionnowoperatesatareducedclockrate.Bothoftheseeffectsreducehardwarepowerconsumption.



Figure10:CICdecimationfilterresponses:forvariousvaluesofdifferentialdelayN,whenR=8fortwodecimationfactorswhenN=2Viewfullsizedimage

Thecombsection'sdifferentialdelaydesignparameterNistypically1or2forhighsamplerateratiosasisoftenusedinup/downconverters.Neffectivelysetsthenumberofnullsinthefrequencyresponseofadecimationfilter,asshowninFigure10a.

AnimportantcharacteristicofaCICdecimatoristhattheshapeofthefilterresponsechangesverylittle,asshowninFigure10b,asafunctionofthedecimationratio.ForvaluesofRlargerthanroughly16,thechangeinthefiltershapeisnegligible.ThisallowsthesamecompensationFIRfiltertobeusedforvariabledecimationratiosystems.

TheCICfiltersuffersfromregisteroverflowbecauseoftheunityfeedbackateachintegratorstage.Theoverflowisofnoconsequenceaslongasthefollowingtwoconditionsaremet:

therangeofthenumbersystemisgreaterthanorequaltothemaximumvalueexpectedattheoutput,andthefilterisimplementedwithtwo'scomplement(nonsaturating)arithmetic.

Becausea1storderCICfilterhasagainofD=NRat0Hz(DC),McascadedCICdecimationfiltershaveanetgainof(NR)M.EachadditionalintegratormustaddanotherNRbitswidthforstages.InterpolatingCICfiltershavezerosinsertedbetweeninputsamplesreducingitsgainbyafactorof1/Rtoaccountforthezerovaluedsamples,sothenetgainofaninterpolatingCICfilteris(NR)M/R.Becausethefiltermustuseintegerarithmetic,thewordwidthsateachstageinthefiltermustbewideenoughtoaccommodatethemaximumsignal(fullscaleinputtimesthegain)atthatstage.

AlthoughthegainofanMthorderCICdecimationfilteris(NR)Mindividualintegratorscanexperienceoverflow.(TheirgainisinfiniteatDC.)Assuch,theuseoftwo'scomplementarithmeticresolvesthisoverflowsituationjustsolongastheintegratorwordwidthaccommodatesthemaximumdifferencebetweenanytwosuccessivesamples(inotherwords,thedifferencecausesnomorethanasingleoverflow).Usingthetwo'scomplementbinaryformat,withitsmodularwraparoundproperty,thefollowoncombfilterwillproperlycomputethecorrectdifferencebetweentwosuccessiveintegratoroutputsamples.

Forinterpolation,thegrowthinwordsizeisonebitpercombfilterstageandoverflowmustbeavoidedfortheintegratorstoaccumulateproperly.So,wemustaccommodateanextrabitofdatawordgrowthineachcombstageforinterpolation.Thereissomesmallflexibilityindiscardingsomeoftheleastsignificantbits(LSBs)withinthestagesofaCICfilter,attheexpenseofaddednoiseatthefilter'soutput.ThespecificeffectsofthisLSBremovalare,however,acomplicatedissueyoucanlearnmoreabout

theissuebyreadingHogenauer'spaper.1

WhiletheprecedingdiscussionfocusedonhardwiredCICfilters,thesefilterscanalsobeimplementedwithprogrammablefixedpointDSPchips.Althoughthosechipshaveinflexibledatapathsandwordwidths,CICfilteringcanbeadvantageousforhighsampleratechanges.Largewordwidthscanbeaccommodatedwithmultiwordadditionsattheexpenseofextrainstructions.Evenso,forlargesampleratechangefactorsthecomputationalworkloadperoutputsample,infixedpointDSPchips,maybesmall.

CompensationfiltersIntypicaldecimation/interpolationfilteringapplicationswewantreasonablyflatpassbandandnarrowtransitionregionfilterperformance.ThesedesirablepropertiesarenotprovidedbyCICfiltersalone,withtheirdroopingpassbandgainsandwidetransitionregions.Wealleviatethisproblem,indecimationforexample,byfollowingtheCICfilterwithacompensationnonrecursiveFIRfilter,asinFigure1a,tonarrowtheoutputbandwidthandflattenthepassbandgain.



Figure11:CompensationFIRfilterresponseswitha1storderdecimationCICfilterwitha3rdorderdecimationViewfullsizedimage

ThecompensationFIRfilter'sfrequencymagnituderesponseisideallyaninvertedversionoftheCICfilterpassbandresponsesimilartothatshownbythedashedcurveinFigure11aforasimplethreetapFIRfilterwhosecoefficientsare[1/16,9/8,1/16].Withthedottedcurverepresentingtheuncompensatedpassbanddroopofa1storderR=8CICfilter,thesolidcurverepresentsthecompensatedresponseofthecascadedfilters.IfeitherthepassbandbandwidthorCICfilterorderincreasesthecorrectionbecomesgreater,requiringmorecompensationFIRfiltertaps.AnexampleofthissituationisshowninFigure11bwherethedottedcurverepresentsthepassbanddroopofa3rdorderR=8CICfilterandthedashedcurve,takingtheformof[x/sin(x)]3,istheresponseofa15tapcompensationFIRfilterhavingthecoefficients[1,4,16,32,64,136,352,1312,352,136,64,32,16,4,1].

Awidebandcorrectionalsomeanssignalsnears,out/2areattenuatedwiththeCICfilterandthenmustbeamplifiedinthecorrectionfilter,addingnoise.Assuch,practitionersoftenlimitthepassbandwidthofthecompensationFIRfiltertoroughly1/4thefrequencyofthefirstnullintheCICfilterresponse.

ThosedashedcurvesinFigure11representthefrequencymagnituderesponsesofcompensatingFIRfilterswithinwhichnosampleratechangetakesplace.(TheFIRfilters'inputandoutputsampleratesareequaltothes,outoutputrateofthedecimatingCICfilter.)IfacompensatingFIRfilterweredesignedtoprovideanadditionaldecimationbytwo,itsfrequencymagnituderesponsewouldlooksimilartothatinFigure12,where>s,inisthecompensationfilter'sinputsamplerate.

Figure12:Frequencymagnituderesponseofadecimateby2compensationFIRfilterViewfullsizedimage

AdvancedtechniquesHere'sthebottomlineofourCICfilterdiscussion:adecimatingCICfilterismerelyaveryefficientrecursiveimplementationofamovingaveragefilter,withNRtaps,whoseoutputisdecimatedbyR.Likewise,theinterpolatingCICfilterisinsertionofR1zerosamplesbetweeneachinputsamplefollowedbyanNRtapmovingaveragefilterrunningattheoutputsamplerates,out.ThecascadeimplementationsinFigure1resultintotalcomputationalworkloadsfarlessthanusingasingleFIRfilteraloneforhighsampleratechangedecimationandinterpolation.CICfilterstructuresaredesignedtomaximizetheamountoflowsamplerateprocessingtominimizepowerconsumptioninhighspeedhardwareapplications.Again,CICfiltersrequirenomultiplicationtheirarithmeticisstrictlyadditionandsubtraction.Theirperformanceallowsustostatethat,technicallyspeaking,CICfiltersarelean,meanfilteringmachines.

Inclosing,therearewaystobuildnonrecursiveCICfiltersthateasethewordwidthgrowthproblemofthetraditionalrecursiveCICfilters.ThoseadvancedCICfilterarchitecturesarediscussedinmybookUnderstanding

DigitalSignalProcessing,2E.2

RichardLyonsisaconsultingsystemsengineerandlecturerwithBesserAssociatesinMountainView,Ca.HeistheauthorofUnderstandingDigitalSignalProcessing2/EandanassociateeditorfortheIEEESignalProcessingMagazinewherehecreatedandeditsthe"DSPTips&Tricks"



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Endnotes

1.Hogenauer,Eugene."AnEconomicalClassofDigitalFiltersForDecimationandInterpolation,"IEEETransactionsonAcoustics,SpeechandSignalProcessing,Vol.ASSP29,pp.155162,April1981.

2.Lyons,Richard,UnderstandingDigitalSignalProcessing,2ndEd.,PrenticeHall,UpperSaddleRiver,NewJersey,2004,pp.556561.

understanding cascaded integrator-comb filters

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