IIR数字滤波器外文翻译.docx
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IIR数字滤波器外文翻译
中英文资料翻译
IIRDigitalFilterDesign
AnimportantstepinthedevelopmentofadigitalfilteristhedeterminationofarealizabletransferfunctionG(z)approximatingthegivenfrequencyresponsespecifications.IfanIIRfilterisdesired,itisalsonecessarytoensurethatG(z)isstable.TheprocessofderivingthetransferfunctionG(z)iscalleddigitalfilterdesign.AfterG(z)hasbeenobtained,thenextstepistorealizeitintheformofasuitablefilterstructure.Inchapter8,weoutlinedavarietyofbasicstructuresfortherealizationofFIRandIIRtransferfunctions.Inthischapter,weconsidertheIIRdigitalfilterdesignproblem.ThedesignofFIRdigitalfiltersistreatedinchapter10.
Firstwereviewsomeoftheissuesassociatedwiththefilterdesignproblem.AwidelyusedapproachtoIIRfilterdesignbasedontheconversionofaprototypeanalogtransferfunctiontoadigitaltransferfunctionisdiscussednext.Typicaldesignexamplesareincludedtoillustratethisapproach.WethenconsiderthetransformationofonetypeofIIRfiltertransferfunctionintoanothertype,whichisachievedbyreplacingthecomplexvariablezbyafunctionofz.Fourcommonlyusedtransformationsaresummarized.Finallyweconsiderthecomputer-aideddesignofIIRdigitalfilter.Tothisend,werestrictourdiscussiontotheuseofmatlabindeterminingthetransferfunctions.
9.1preliminaryconsiderations
TherearetwomajorissuesthatneedtobeansweredbeforeonecandevelopthedigitaltransferfunctionG(z).Thefirstandforemostissueisthedevelopmentofareasonablefilterfrequencyresponsespecificationfromtherequirementsoftheoverallsysteminwhichthedigitalfilteristobeemployed.ThesecondissueistodeterminewhetheranFIRorIIRdigitalfilteristobedesigned.Inthesection,weexaminethesetwoissuesfirst.NextwereviewthebasicanalyticalapproachtothedesignofIIRdigitalfiltersandthenconsiderthedeterminationofthefilterorderthatmeetstheprescribedspecifications.Wealsodiscussappropriatescalingofthetransferfunction.
9.1.1DigitalFilterSpecifications
Asinthecaseoftheanalogfilter,eitherthemagnitudeand/orthephase(delay)responseisspecifiedforthedesignofadigitalfilterformostapplications.Insomesituations,theunitsampleresponseorstepresponsemaybespecified.Inmostpracticalapplications,theproblemofinterestisthedevelopmentofarealizableapproximationtoagivenmagnituderesponsespecification.Asindicatedinsection4.6.3,thephaseresponseofthedesignedfiltercanbecorrectedbycascadingitwithanallpasssection.Thedesignofallpassphaseequalizershasreceivedafairamountofattentioninthelastfewyears.
Werestrictourattentioninthischaptertothemagnitudeapproximationproblemonly.Wepointedoutinsection4.4.1thattherearefourbasictypesoffilters,whosemagnituderesponsesareshowninFigure4.10.Sincetheimpulseresponsecorrespondingtoeachoftheseisnoncausalandofinfinitelength,theseidealfiltersarenotrealizable.OnewayofdevelopingarealizableapproximationtothesefilterwouldbetotruncatetheimpulseresponseasindicatedinEq.(4.72)foralowpassfilter.ThemagnituderesponseoftheFIRlowpassfilterobtainedbytruncatingtheimpulseresponseoftheideallowpassfilterdoesnothaveasharptransitionfrompassbandtostopbandbut,rather,exhibitsagradual"roll-off."
Thus,asinthecaseoftheanalogfilterdesignproblemoutlinedinsection5.4.1,themagnituderesponsespecificationsofadigitalfilterinthepassbandandinthestopbandaregivenwithsomeacceptabletolerances.Inaddition,atransitionbandisspecifiedbetweenthepassbandandthestopbandtopermitthemagnitudetodropoffsmoothly.Forexample,themagnitude
ofalowpassfiltermaybegivenasshowninFigure7.1.Asindicatedinthefigure,inthepassbanddefinedby0
werequirethatthemagnitudeapproximatesunitywithanerrorof
i.e.,
.
Inthestopband,definedby
werequirethatthemagnitudeapproximateszerowithanerrorof
.e.,
for
.
Thefrequencies
and
are,respectively,calledthepassbandedgefrequencyandthestopbandedgefrequency.Thelimitsofthetolerancesinthepassbandandstopband,
and
areusuallycalledthepeakripplevalues.Notethatthefrequencyresponse
ofadigitalfilterisaperiodicfunctionof
andthemagnituderesponseofareal-coefficientdigitalfilterisanevenfunctionof
.Asaresult,thedigitalfilterspecificationsaregivenonlyfortherange
.
Digitalfilterspecificationsareoftengivenintermsofthelossfunction,
indB.Herethepeakpassbandripple
andtheminimumstopbandattenuation
aregivenindB,i.e.,thelossspecificationsofadigitalfilteraregivenby
.
9.1PreliminaryConsiderations
Asinthecaseofananaloglowpassfilter,thespecificationsforadigitallowpassfiltermayalternativelybegivenintermsofitsmagnituderesponse,asinFigure7.2.Herethemaximumvalueofthemagnitudeinthepassbandisassumedtobeunity,andthemaximumpassbanddeviation,denotedas1/
isgivenbytheminimumvalueofthemagnitudeinthepassband.Themaximumstopbandmagnitudeisdenotedby1/A.
Forthenormalizedspecification,themaximumvalueofthegainfunctionortheminimumvalueofthelossfunctionistherefore0dB.Thequantity
givenby
Iscalledthemaximumpassbandattenuation.For
1,asistypicallythecase,itcanbeshownthat
Thepassbandandstopbandedgefrequencies,inmostapplications,arespecifiedinHz,alongwiththesamplingrateofthedigitalfilter.Sinceallfilterdesigntechniquesaredevelopedintermsofnormalizedangularfrequencies
and
thesepcifiedcriticalfrequenciesneedtobenormalizedbeforeaspecificfilterdesignalgorithmcanbeapplied.Let
denotethesamplingfrequencyinHz,andFPandFsdenote,respectively,thepassbandandstopbandedgefrequenciesinHz.Thenthenormalizedangularedgefrequenciesinradiansaregivenby
9.1.2SelectionoftheFilterType
Thesecondissueofinterestistheselectionofthedigitalfiltertype,i.e.,whetheranIIRoranFIRdigitalfilteristobeemployed.TheobjectiveofdigitalfilterdesignistodevelopacausaltransferfunctionH(z)meetingthefrequencyresponsespecifications.ForIIRdigitalfilterdesign,theIIRtransferfunctionisarealrationalfunctionof
.
H(z)=
Moreover,H(z)mustbeastabletransferfunction,andforreducedcomputationalcomplexity,itmustbeoflowestorderN.Ontheotherhand,forFIRfilterdesign,theFIRtransferfunctionisapolynomialin
:
Forreducedcomputationalcomplexity,thedegreeNofH(z)mustbeassmallaspossible.Inaddition,ifalinearphaseisdesired,thentheFIRfiltercoefficientsmustsatisfytheconstraint:
ThereareseveraladvantagesinusinganFIRfilter,sinceitcanbedesignedwithexactlinearphaseandthefilterstructureisalwaysstablewithquantizedfiltercoefficients.However,inmostcases,theorderNFIRofanFIRfilterisconsiderablyhigherthantheorderNIIRofanequivalentIIRfiltermeetingthesamemagnitudespecifications.Ingeneral,theimplementationoftheFIRfilterrequiresapproximatelyNFIRmultiplicationsperoutputsample,whereastheIIRfilterrequires2NIIR+1multiplicationsperoutputsample.Intheformercase,iftheFIRfilterisdesignedwithalinearphase,thenthenumberofmultiplicationsperoutputsamplereducestoapproximately(NFIR+1)/2.Likewise,mostIIRfilterdesignsresultintransferfunctionswithzerosontheunitcircle,andthecascaderealizationofanIIRfilteroforder
withallofthezerosontheunitcirclerequires[(3
+3)/2]multiplicationsperoutputsample.Ithasbeenshownthatformostpracticalfilterspecifications,theratioNFIR/NIIRistypicallyoftheorderoftensormoreand,asaresult,theIIRfilterusuallyiscomputationallymoreefficient[Rab75].However,ifthegroupdelayoftheIIRfilterisequalizedbycascadingitwithanallpassequalizer,thenthesavingsincomputationmaynolongerbethatsignificant[Rab75].Inmanyapplications,thelinearityofthephaseresponseofthedigitalfilterisnotanissue,makingtheIIRfilterpreferablebecauseofthelowercomputationalrequirements.
9.1.3BasicApproachestoDigitalFilterDesign
InthecaseofIIRfilterdesign,themostcommonpracticeistoconvertthedigitalfilterspecificationsintoanaloglowpassprototypefilterspecifications,andthentotransformitintothedesireddigitalfiltertransferfunctionG(z).Thisapproachhasbeenwidelyusedformanyreasons:
(a)Analogapproximationtechniquesarehighlyadvanced.
(b)Theyusuallyyieldclosed-formsolutions.
(c)Extensivetablesareavailableforanalogfilterdesign.
(d)Manyapplicationsrequirethedigitalsimulationofanalogfilters.
Inthesequel,wedenoteananalogtransferfunctionas
Wherethesubscript"a"specificallyindicatestheanalogdomain.ThedigitaltransferfunctionderivedformHa(s)isdenotedby
ThebasicideabehindtheconversionofananalogprototypetransferfunctionHa(s)intoadigitalIIRtransferfunctionG(z)istoapplyamappingfromthes-domaintothez-domainsothattheessentialpropertiesoftheanalogfrequencyresponsearepreserved.Theimpliesthatthemappingfunctionshouldbesuchthat
(a)Theimaginary(j
)axisinthes-planebemappedontothecircleofthez-plane.
(b)Astableanalogtransferfunctionbetransformedintoastabledigitaltransferfunction.
Tothisend,themostwidelyusedtransformationisthebilineartransformationdescribedinSection9.2.