matlab 滤波器 外文翻译 外文文献IIR数字滤波器的设计.docx

matlab 滤波器 外文翻译 外文文献IIR数字滤波器的设计.docx

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matlab滤波器外文翻译外文文献IIR数字滤波器的设计

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,themostwidelyusedtransformationisthebilineartransformationdescribedinSecti