专业英语电子版7 filters

Filters

Introduction

Linearfiltersformaclassofsystemwhichisofcrucialimportanceinsignalprocessing.Althoughinitsmostgeneralsensetheterm‘filter’impliesanyfrequencyselectivedeviceorprocessor,inpracticeitisgenerallyreservedforasystemwhichtransmitsacertainrangeoffrequencies,andrejectsother:suchfrequencyrangesarecalled‘passbands’and‘stopbands’respectively.Weshallseelaterthattheidealfilter,whichwouldintroducenoattenuationofinputsignalsfallingwithinthepassband,andinfiniteaattenuationofsignalsinthestopband,isnotaattainableinpractice.

Historically,boththetheoryandpracticalapplicationoffiltershavebeenverymuchtiedupwithelectroniccommunications.Forexample,aradioreceiverisrequiredtodiscriminateinfavourofjustoneofthemanyincomingsignalspickedupbyitsaerial;thisitdoesonthebasisoftheirdifferentfrequencybands,byuseofahighlyselectivefilter.Suchafilterprocessescontinuoussignalsandisthereforeanexampleofwhatwehavepreviouslycalleda‘continuous’linearsystem;itisalsowidelyreferredtoasan‘analogue’filter.Analoguefiltersareinvariablyconstructedfromlinearelectricalcircuitcomponents,andtheirdetaileddesignfallsoutsidefromlinearelectricalcircuitcomponents,andtheirdetaileddesignfallsoutsidethescopeofthisbook;ontheotherhandweareinapositiontodiscusstheoverallperformanceofcertainwell-knowntypesofanaloguefilter,andthisisdoneinsection7.3.

Althoughaknowledgeofelectricalnetworktheoryisneededforthedesignofanaloguefilters,thesameisfortunatelynottrueoffiltersforsampled-datasignals.Sampled-datafilters,generallyknownas‘digital’filters,mayberealizedbysuitableprogrammingofadigitalcomputerwhichisfedwithasampledversionoftheinputsignal.Theincreasinginterestindigitalfiltersislargelyareflectionoftheavailabilityofthedigitalcomputerasaresearchtoolinallbranchesofscienceandtechnology.Theworkwehavedoneonthez-transforminchapter4andonlinearsystemsinlesson5formsanadequatebackgroundforthedesignandimplementationofdigitalfilters.Howeverbeforegettinginvolvedindetail,wefirstinvestigatesomegeneralaspectsoffilterperformanceintimeandfrequencydomains.

Generalaspectsoffilterperformance

Filtercategories

Apartfromthedivisionoflinearfiltersintothetwobroadcategoriesofanalogueanddigitalfilters,theymaybefurtherclassifiedaccordingtothefrequencyrangeswhichtheytransmitorreject.A‘low-pass’filterhasapassbandinthelow-frequencyregion,whereasa‘high-pass’filtertransmitsonlyhigh-frequencyinputsignals;‘band-pass’and‘band-stop’filtersaredefinedbytheirabilitytodiscriminateinfavourof,oragainst,particularfrequencybands.Theactualfrequencyatwhichthetransitionfrompassbandtostopbandoccursvariesfromcasetocase,andisclearlyanimportantparameteroffilterdesign.

Sincethefrequencyresponseofalinearsampled-datasystemisalwaysaperiodicfunctionof,

itfollowsthatthetermslow-pass,high-pass,band-passandband-stophavetobeinterpretedslightlydifferentlyinthecaseofdigitalfilters.Wehavealreadynotedinsection7.2.1thatsamplingwithan

intervalofTsecondsallowsfaithfulrepresentationofacontinuoussignalhavingfrequencycomponentsuptoTradians/second.Adigitalfilteristhereforeclassifiedaccordingtoits

effectonfrequencycomponentsintherangeTT,whichisthemaximumrangeoccupiedbyanyadequately-sampledinputsignal.

7.2.2.Responseintimeandfrequencydomains

Likeanyotherlinearsystem,afrequency-selectivefiltermaybedescribedeitherbyitsfrequencyresponse,orbyitsimpulseresponse.Thefrequencyresponsedescriptionistheoneformallyused,becauseafilterisgenerallyspecifiedintermsofitsabilitytodiscriminateagainstcertainfrequencyrangesandinfavourofothers;butanyformoffrequencyresponseimpliesaparticularshapeofimpulseresponse,andthelattergivessomeimportantcluestofilterperformance.

Asarule,themorelimitedthebandoffrequenciestransmittedbyafilter,themoreextendedintimeisitsimpulseresponsewaveform:thisisjustareflectionofthegeneralantithesisbetweenfrequency-limitationandtime-limitation,discussedwithreferencetosignalwaveformsinsection

2.3.2.Itmeansthattheoutputfromahighlyselectivefiltermustalwaysbeexpectedtotakealongtimetosettletozeroaftertheinputhasbeenremoved,andthatitsresponsetoasinusoidalinputorsteadylevelwilltakealongtimetoreachits‘steadystate’aftertheinputhasbeenapplied.Thetransienteffectswhichaccompanythesuddenapplicationofaninputsignaltoaselectivefilteraresometimesreferredtoas‘ringing’.SincethefrequencyresponseandimpulseresponseofafilterarerelatedasaFouriertransformpair,theformoftheimpulseresponsemustbeexpectedtoreflectthosefrequencieswhicharestronglytransmittedbythefilter.

Thereasonwhynolinearfiltercandisplaytheidealcharacteristicsoflowpass,bandpass,bandstopandhighpassbecomesclearifitisrecalledthatanylinearsystemhasatransformfunctionexpressibleintermsofasetofpolesandzeros.Aswesawinsection6.2.2,givenasetofs-planepolesandzerositispossibletoinferthesystem’sresponsetoanysinusoidalfrequencybydrawingvectorsfromthevariouspolesandzerostoapointontheimaginaryaxisinthes-plane.Theresponseislargeiftheproductof‘zero-vector’magnitudesislarge,and/ortheproductof‘pole-vector’magnitudesissmall.Aninfinitelyfasttransitionformpassbandtostopbandthereforeimpliesaninfiniterateofchangeofoneorbothoftheseproductsasaparticularpointontheimaginaryaxisiscrossed.Itisintuitivelyclearthatsuchaneffectcannotbeachievedbyanyarrangementofafinitesetofpolesandzeros-althoughitmaybemorecloselyapproximatedwhenalargenumberofpolesandzerosisspecified.However,thenumberofpolesandzerosinatransferfunctionreflectsthecomplexityofthesystem,andcomplexityiscloselyrelatedtocost.

Thepole-zeroapproachmayalsobeusedtodemonstratetherelationshipbetweenthemagnitudeandphaseresponsesofalinearfilter.Letusstartbyconsideringananaloguefilterwhichhasallitspolesandzerosintheleft-handhalfofthes-plane,asinfigure7.1.Thisconfigurationgivesrisetoaparticularmagnitudeandphaseresponse,andthetwowillbeinterdependent.Itisinterestingtoconsiderwhetherthephaseresponsemaynowbeadjustedindependentlyofthemagnituderesponse.Actuallythismaybedoneintwoways,illustratedinpartsofthefigure.Inpartbthezerosofthetransferfunctionaremovedacrosstheimaginaryaxistomirrorimagepositionsintheright-halfs-plane.Thelengthofthezerovectorsdrawntoanypointontheimaginaryaxisisclearlyunalteredbythismove,buttheircontributiontothephaseresponseischanged.Inpartcofthefigure,theoriginalpole-zeropatternisaugmentedby

figure7.1.

additionalpairsofpolesandzerosarrangedsymmetricallywithrespecttotheimaginaryaxis.Theseadditionalpairscausenoalterationtothemagnituderesponseofthefilter,sincevariationsinlengthofoneofthenewzerovectorsareexactlycounterbalancedbythoseofthecorrespondingolevector;ontheotherhand,theirphasecontributionsdonotcancel.Weshouldnotethatphasevariationcannotbeobtainedbyplacingpolesintheright–halfs-plane,becausethiswouldgiverisetoanunstablefilter.Formthisbriefdiscussionwemaydrawseveralconclusions.Firstly,ifzerosaretoberestrictedtotheleft-halfs-plane,magnitudeandphaseresponseareuniquelyrelated,andmaynotbeadjustedindependently:afilterofthistypeisknownasa‘minimum-phase’system.Ifafiltertransferfunctionhaszeros,thesemaybeplacedineithertheleftorright-halfs-plane,givingacorrespondingflexibilityinthephaseresponse,butwithoutalteringthemagnituderesponse.Andfinally,thephaseresponsemaybeadjustedbyaddingmirror-imagepole-zeropairs,althoughthisaddstothefiler’scomplexityandshouldthereforebeavoidedifpossible.Inpractice,adjustmentofthephaseresponseofafilterisoftenaccomplishedbyaseparatesystemknownasan‘all-passnetwork’,whichprovides,

ineffect,justtherequiredmirror-imagepole-zeropairs.Suchaprocessissometimesreferredtoasphase‘equalisation’.However,itisfortunatelytruethatthephaseresponseofafilterisrelativelyunimportantinmayapplications;filterdesignthereforetendstoconcentrateonmagnituderesponse,andthephaseresponseislefttolookafteritself.

Analoguefilters

General

Inprinciple,itispossibletorealizeanydesireds-planepole-zeroconfigurationusinganelectricalnetwork.Howeverthenumberofpolesandzerosusedbearsadirectrelationshiptofiltercomplexity,andhencecost,sothatitisdesirabletoachieveanacceptablefilterperformanceusingasfewpolesandzerosaspossible.Inthissectiononanaloguefilters,wediscussbrieflythepole-zeroconfigurationandfrequencyresponsesofsomecommontypesoffilter,eachofwhichrepresentsausefulcompromisebetweenidealperformanceanddesigneconomy.Ourdiscussionwillbesomewhatbiasedtowardsfiltershavinglow-passcharacteristics;however,itisgenerallypossibletoconvertalow-passfilterintoahigh-pass,bandpass,orbandstoponewithsimilarpassbandandstopbandperformance,bymodificationstotheelectricalcircuitcomponents.Themathematicaltechniquesinvolvedinsuchconversionsfallundertheheadingof‘frequencytransformations’,andarepartofthestock-in-tradeoftheanaloguefilterdesigner.

Whatmightbetermed‘conventional’analoguefiltersarecomposedofpassivelinearelectricalelements-resistors,inductors,andcapacitors.Inthedesignofsuchfiltersaccountmustgenerallybetakenoftheelectricalimpedanceofthedevicestowhichthefilteristobeconnected;inotherwordstheperformanceofthefilterisgenerallyaffectedbytheelectricalcharacteristicsofthesignalsourceconnectedtoitsinputside,andofthe‘load’connectedtoitsoutput.Dueattentionmustbepaidtothisquestion,becauseincorrectterminationofananaloguefiltermayleadtoseriousdiscrepanciesbetweenitsadvertisedandactualfrequencyresponses.In

morerecentyears,‘active’filtershavebecomeincreasinglypopular.Anactivefilterincorporatesoneormoreactivedevicessuchasamplifiers—whichneedapowersupply,anditsmainadvantagesoverthemoreconventionalpassivefilteraretwofold:theuseofactivedevicesallowsinductorstobeeliminatedfromthenetwork,andinductorstendtobebulky,expensive,anddifficulttorealizesinidealform,particularlywhenthefilteristoworkatlowfrequencies;furthermore,itisarelativelysimplemattertodesignanactivefilterwhichisnotsensitivetothepreciseelectricalcharacteristicsofthesignalsourceandloadtowhichitisconnected.

Somecommonfiltertypes

figure7.2.

Filterscomposedentirelyofinductorsandcapacitors,sometimesreferredtoas‘reactive’filter,havetransferfunctionpolesandzeroswhicharerestrictedtotheimaginaryaxisinthes-plane.Thetheoryofsuchfiltershasbeenextensivelyinvestigatedeversincetheearlydaysofradio,andtheyhavefoundwidespreadpracticalapplication.Thepole-zeroconfigurationandfrequencyresponsecharacteristicofatypicalbandpassfilterofthistypeareshowninfigure7.2.Thesearetheoreticaldiagramsbaseduponidealfiltercomponentsandidealterminations,neitherofwhichmayberealizedinpractice.Theeffectofpracticalcomponentsandterminationsistomovethepolesandzerosslightlyawayfromtheimaginaryaxis:thismeansthatinfinitepeaksandnullssuchasthoseshownintheresponsemagnitudecharacteristicoffigure7.2willnotbeobservedinpractice;andthesuddenjumpsshowninthephasecharacteristicwillbesomewhatsmoothedout.Apartfromthedifficultyofdesignatingreactivefilterstoaccountfornonidealelectricalcomponentsandtermination,theirmaindisadvantageisthattheirperformancegenerallydepartsconsiderablyfromtheidealcharacteristics.

Figure7.2Characteristicsofareactivebandpassfilter

Filterscomposedofresistorsandcapacitorsonly(orresistorsandinductorsonly)havetheirpolesconfinedtothenegativerealaxisinthes-plane.Thisrestrictionmeansthatitisdifficulttoachievehighlyselectivebandpassorbandstopcharacteristics,althoughresistor-capacitorfiltersarewildlyusedforrelativelysimplefilteringtasks.Althoughzerosmaybeplacedanywhereinthes-plane,acommonsubclassofsuchfilters(knownasresistor-capacitor‘addernetworks’)haszeros,likepoles,confinedtotherealaxis.Itisworthinvestigatingthelow-passandhigh-passcharacteristicsalittlemorecarefully,inordertoexplainthecommonuseoftheterms‘integrator’and‘differentiator’todescribethem.Anideal‘integrator’wouldproduceanoutputsignalproportionaltothetime-integraloftheinput,andcontinuallyupdateitastimeproceeded:theoutputofanidealdifferentiatorwouldbeproportionalateveryinstanttotherateofchangeoftheinputsignal.Wehavealreadyshowninsection3.4.4thatdifferentiationofatimefunctionisequivalenttomultiplyingitsLaplacetransformbys.Similarly,integrationofatimefunctionisequivalenttodividingtheLaplacetransformbys.Hencewemaywritethetransferfunctionofanidealintegratoranddifferentiatoras

H1s1s,foranintegratorand

H2ss,forandifferentiator

Theuseofallthreetypesoflinearelectriccircuitelement-resistors,capacitorsandinductors-enableszerostobeplacedanywhereinthes-plane,andpolestobeplacedanywheretotheleftoftheimaginaryaxis:thisallowsamuchmoreflexibleapproachtowardsfilterdesign.Asalreadynoted,theuseofactivedevicesinafilterobviatestheneedforinductors,sothatfiltersofthismore

generaltypeareincreasinglyrealizedusingresistors,capacitorsandactiveelements.Themodernapproachtowardsanaloguefilterdesigninvolvesspecifyingthes-planelocationsofagivennumbersofpolesandzerossoastoapproximateadesiredfrequencyresponsemagnitude(orphase)characteristicascloselyaspossible.Insuchdesignitisnormaltotakeaccountoftheactualelectricalpropertiesofthedevicestowhichthefilterisconnected.

DigitalFilter

General

Adigitalfiltermayberealizedineither‘hardware’or‘software’form.Inthefirst,asuitablesetofdigital(logic)electroniccircuitsisinterconnectedtoprovidetheessentialbuildingblocksofadigitalfilteringoperation–storage,delay,addition/subtraction,andmultiplicationbyconstants.Recentdevelopmentsinelectronicsallowacompletedigitalfiltertobeconstructedinintegratedcircuitform.Amajoradvantageofusingsuchspecial-purposehardware,dedicatedtoaparticularsignalprocessingtask,isspeed-particularifsomeofthenecessaryoperationsareperformedinparallel.Ontheotherhand,large-scaleintegratedcircuitsareverycostlyunlessproducedinhighvolume.Thealternative,‘software’,approachistoprogramageneral-purposedigitalcomputerasadigitalfilter.Sincesuchcomputersaregeneralserialmachines,whichcanonlycarryoutasetofprogrammedinstructionsoneatatime,operatingspeedsaremuchslower.However,powerfulmicrocomputersarenowsocheapandwidelyavailablethatwemustexpect‘software’digitalfilterstobecomeincreasinglycommonforreal-timeapplicationsrequiringsamplingratesupto(say)10kHz;andthisapproximateupperlimitmaybeexpectedtoincreaseascomputerhardwareisfurtherdeveloped.Itmustalsoberememberedthatsignals,recordsanddatamaybestoredandsubsequentlyprocessed‘off-line’.Insuchcases,adigitalfilter’soperatingspeedisoftennotamajorconsideration.

Beforediscussingdigitalfiltersinanydetail,itshouldbeemphasizedthattherearetworatherdistinctwaysinwhichasampled-datasignalmaybefiltered.ThefirstinvolvestakingtheDiscreteFourierTransform(DFT)ofthesignal,probablybyuseoftheFastFourierTransform(FFT)algorithm.BoththeDFTandtheFFThavebeencoveredinourdiscussionofsampled-datasignalsinlesson5.Havingfoundthesignalspectrum,themagnitudesandphasesofitsvariousfrequencycomponentsmaythenbeadjustedinaccordancewiththedesiredfiltercharacteristics,andthefilteredtime-domainsignalevaluatedbyinversetransformation–againusingtheFFT.Inthisfirstmethod,thefilteringmaybeconsideredtotakeplaceinthefrequencydomain.Itisanimportantandwidelyusedapproach,whichallowsgreatflexibilityinthechoiceoffiltercharacteristics.Sincethesignalspectrumissimplymultipliedbythedesiredfiltercharacteristic,itoftenprovesfasterthantheequivalenttime-domainconvolution.Ontheotherhand,itsflexibilityisnotneededformanypracticalfilteringtasks,andtime-domainfiltering,asdiscussedbelow,isoftenpreferable.Although

theprincipleoftheFFThasbeencoveredinsection3.3.2.,itsdetailedimplementationisratherspecializedmatter,includedinvariousothertexts.Weshallthereforenotdiscussitfurtherhere.However,itisworthnotingthatmanygeneralpurposecomputersincludeFFTprogrammersaspartoftheirstandardsoftware,andthatspecialpurposeFFThardwareisnowcommerciallyavailable.

Thesecondmethodofimplementingadigitalfilter,whichweshallbeconsideringinsomedetailinthissection,istoworkentirelyinthetime-domain.Ineffect,thisisdonebyconvolutionoftheinputsignalwiththeimpulseresponseoftheappropriatefilter.Whetheratime-domainorafrequency-domainfilterismoreappropriatedependsuponanumberoffactorssuchasthestorageavailableinthecomputer,thedurationofsignal,theoperatingspeed,andwhetherornota‘real-time’operationisrequired.Insuchanoperation,anewoutputsamplevalueiscalculatedeverytimeaninputsampleisgenerated.Thisimpliesdigitalfilteringinthetime-domain,becauseefficientuseoftheFFTrequiresthestorageandprocessingofdatainsubstantialblocks.

ApowerfulwayofchoosingasuitabletransferfunctionHzforaparticularfilteringtaskisto

specifyasetofz-planepolesandzeros.Wesawinsection3.4.2howasampled-datasignalcouldberepresentedbysuchpolesandzeros,andhowitsfrequencyspectrumcouldbeinferredbyconsideringthechangesinlengthsandphaseofvectorsdrawnformthevariouspolesandzerostopointsontheunitcircle.Whenweturnourattentiontodigitalfiltersthesamegeneralconceptsapply,exceptthatwearenowconcernedwithfrequencyresponsesratherthanfrequencyspectra.

Filterswithfiniteimpulseresponses(FIRs)

Introduction

Thefilterhasmdelaystages,(m+1)positiveornegativemultipliers,andanadderorsummingjunction,isoftenreferredtoasadigitaltransversalfilter.Itisnon-recursive,withatime-domainrecurrenceformulagivenby:

m

yna0xna1xn1 amxnmaixni

i0

(7-1)

Ifweconsiderasingle,unit-valued,inputsampletothisfilter,itwillclearlygenerateasequenceofsamplevaluesa0,a1,…,amattheoutput.Thereforethefilter’simpulseresponseisjustmadeupofthemultipliercoefficientsequencea0toam,andisfiniteinduration.Theartofdesigning

suchafilteristospecifytheminimumnumberofdelayandmultiplierelementstoachieveanacceptableperformance.Thefirstofthelow-passfiltersdiscussedinpriorsectionisofthistype,butwithonlytowunitmultipliers;theseprovideaveryelementarylow-passfunction.MostusefulFIR

filtersrequirebetween(say)15and150multipliers,andhaveanequivalentnumberofzeros(butnopoles)intheirtransferfunctions.Theirdesigncanhardlythereforebebaseduponasuitablechoiceofz-planezeroconfiguration,andothermethodsmustbeused.Althoughnon-recursiveFIRfiltersusuallyrequiremanymoredelayandmultiplierelementsthanrecursivefiltersofcomparableperformance,theyhavetwomajorcompensatingadvantages:

Animpulseresponsewhichisfiniteindurationcanalsobesymmetricalinform.Thisproducesa

purelinear-phasecharacteristic.Thereis,inotherwords,nophasedistortion.FIRfiltersdonothavetobelinear-phase,butmostpracticaldesignstakeadvantageofthispossibility,whichisnotavailableinanaloguefiltersbaseduponlumpedcircuitelementssuchasresistors,capacitors,andinductors.

Non-recursivedesignsareinherentlystable,sincetheydonotinvolvefeedbackfromoutputtoinput.Thereisnoriskthatinaccuratespecificationofoneormoreofthemultipliersmightleadtoinstability.

Themoving-averagefilter

Westartbyconsideringasimpleformofdigitaltransversalfilterinwhichallthemultipliersareequal.Thisisoftencalledamoving-averagefilter.Suppose,forexample,weuse19delayelementsand20multipliersallequalto1/20or0.05.Convolutionofthisresponsewithaninputsignalyieldsoutputsamplevalues,eachofwhichistheaverageof20consecutiveinputs.Atypicalcaseisthatthe

inputsequenceisassumedtohaveitsfirstnon-zerovalueatt0,anddisplaysasteadydownward

trendtogetherwithsuperimposedrandomhigh-frequencyfluctuations.Thesefluctuationsmightrepresentobservationormeasurementerrors.Thefilter’soutputhasastart-uptransientbetweent0andt19T,whichcorrespondstothedurationofitsimpulseresponse;thenafterit

transmitstheslowtrendoftheinputbutgreatlyreducestherapidfluctuations.Thissmoothing,or

averaging,actionisequivalenttolow-passfiltering.

Thepuredelayimposedbyaliner-phasefiltermaybeshowntoequalhalfthedurationofitssymmetricalimpulseresponse;inthiscase,thedelayis9.5T.Iftheoutputwaveformisadvancedintimebythisamount,itisseentofollowtheslowtrendoftheinputveryclosely,withagainofaboutunity.Thefilter’slow-frequencygainofunityisconfirmedbyitsfrequency-responsemagnitudecharacteristic.Asexpected,thisisbroadlyofthelow-passtype,althoughthepresenceofsubstantialsidelobesmakesitrathernon-ideal.Nevertheless,moving-averagefiltersarewidelyusedforundemandinglow-passfilteringtasks.Notethat,sinceallmultipliercoefficientsareequal,asinglemultiplierattheoutputoftheadderwouldhavethesameeffectasthemanyindividualonesontheinputside.Alternatively,forevengreatersimplicity,themultipliersmaybeomittedaltogether.Thismerelyalterstheoutputbyascalefactor,butdoesnototherwiseaffectfilterperformance.Inthisexample,wehaveillustratedamoving-averagedesignwith20multipliers,ortappingpoints,givinga

frequencyresponsewithitsfirsttransmissionnullat1/20THz.Anyothernumberofmultipliersmay,

ofcourse,bespecified:thegreaterthenumber,themorepronouncedthelow-passfilteringaction,andthemorecloselythefrequency-responsemagnitudecharacteristicapproximatesaform,inrange

TT.

Filterswithinfiniteimpulseresponse(IIRs)

Anydigitalfilterspecifiedintermsofoneormorez-planepoleshasaninfiniteimpulseresponse(IIR).Actually,thisstatementshouldbeslightlyqualifiedbysayinguncancelledpoles,because,aswehaveseenintheprevioussection,certaintypesofFIRfilterhavez-planepoleswhicharecanceledbycoincidentzeros.TheIIRofanyrealizablefiltercannotbesymmetricalinform,becauseitcannotstartbeforet=0.Unfortunately,therefore,noIIRfiltercandisplaypurelinear-phasecharacteristics.ThemajoradvantageofIIRdesignsisthatthepositioningofjustoneorafewpolesinside,butcloseto,theunitcircleallowsustoachieveveryselectivefilterpassbandcharacteristics.Asimpleexamplehasalreadybeengiveninsection7.4.2:alow-passfilterbaseduponasinglez-planepole.Iftheparameterismadeclosetounity,thepassbandbecomesverynarrow.Comparableselectivitycouldonlybeachievedusingalargenumberofz-planezeros.

SinceIIRfiltersemployuncancelledz-planepoles,theyarerecursive;indeed,notinfiniteimpulseresponsecouldbeimplementedinapurelynon-recursiveoperation.Since,inmostpracticalcases,onlyafewpolesareneededtoachieveacceptableperformance,therecurrenceformulaeofIIRfiltersnormallyinvolverelativelyfewterms;thiscontrastswithFIRtransversalfilterswhereupto100or150termsmaybeneeded.However,therecursivemultipliersofanIIRdesignaregenerallydecimalnumbers,needingspecificationwithconsiderableaccuracyiftherequiredfrequencyresponseistobeachieved,andinstabilityavoided.TherearethereforebothadvantagesanddisadvantagesinIIR,asopposedtoFIR,designs.

ManypracticalIIRfiltersarebaseduponanalogueequivalents.Thereasonforthisislargelyhistorical:anextensivetheoryofanaloguefiltershasgrownupoverthelasthalf-century,anditwasnaturalthat,inthemuchmorerecentdevelopmentofdigitalfilters,equivalentdesignsshouldbesought.Sinceanaloguefiltersbaseduponlumpedcircuitelementshaveinfiniteimpulseresponses,thisapproachleadsnaturallytodigitalfiltersoftheIIRtype.Essentially,theproblemistofindsuitabletransformationsformappingthes-planepolesandzerosofananaloguefilterintothez-plane.Valuablethoughthisapproachisforderivingdigitalfilterswith,forexample,ButterworthorChebychevcharacteristics,itmustberememberedthatsomeoftheconstraintsofanaloguefilterdesigndonotapplytothedigitalcase.Forexample,thelinear-phaseFIRdesignsdiscussedintheprevioussectiondonothavedirectlumped-elementanalogueequivalents.Inaddition,itisquitepossibletochooseasuitablez-planepole-zeroconfigurationforafilter,withoutdirectreferencetoanaloguedesigns.

Furtheraspectsoffilterimplement

Thepracticalimplementationofadigitalfilterisoftenaffectedbytheproblemoffinite‘wordlength’.Becauseanumericalvaluemustberepresentedintermsofafinitenumberofbinarydigits,orbits,itisalwayssubjecttosomeinaccuracy.T