matlab-滤波器-外文翻译-外文文献-英文文献-FIR数字滤波器的设计

FIRDigitalFilterDesign作者：SanjitK.Mitra国籍：USA出处：DigitalSignalProcessing-AComputer-BasedApproach3eInchapter9weconsideredthedesignofIIRdigitalfilters.Forsuchfilters,itisalsonecessarytoensurethatthederivedtransferfunctionG(z)isstable.Ontheotherhand,inthecaseofFIRdigitalfilterdesign,thestabilityisnotadesignissueasthetransferfunctionisapolynomialinz-1andisthusalwaysguaranteedstable.Inthischapter,weconsidertheFIRdigitalfilterdesignproblem.UnliketheIIRdigitalfilterdesignproblem,itisalwayspossibletodesignFIRdigitalfilterswithexactlinear-phase.First,wedescribeapopularapproachtothedesignofFIRdigitalfilterswithlinear-phase.Wethenconsiderthecomputer-aideddesignoflinear-phaseFIRdigitalfilters.Tothisend,werestrictourdiscussiontotheuseofmatlabindeterminingthetransferfunctions.SincetheorderoftheFIRtransferfunctionisusuallymuchhigherthanthatofanIIRtransferfunctionmeetingthesamefrequencyresponsespecifications,weoutlinetwomethodsforthedesignofcomputationallyefficientFIRdigitalfiltersrequiringfewermultipliersthanadirectformrealization.Finally,wepresentamethodofdesigningaminimum-phaseFIRdigitalfilterthatleadstoatransferfunctionwithsmallergroupdelaythanthatofalinear-phaseequivalent.Theminimum-phaseFIRdigitalfilteristhusattractiveinapplicationswherethelinear-phaserequirementisnotanissue.10.1preliminaryconsiderationsInthissection,wefirstreviewsomebasicapproachestothedesignofFIRdigitalfiltersandthedeterminationofthefilterordertomeettheprescribedspecifications.10.1.1BasicApproachestoFIRDigitalFilterDesignUnlikeIIRdigitalfilterdesign,FIRfilterdesigndoesnothaveanyconnectionwiththedesignofanalogfilters.ThedesignofFIRfiltersisthereforebasedonadirectapproximationofthespecifiedmagnituderesponse,withtheoftenaddedrequirementthatthephaseresponsebelinear.RecallacausalFIRtransferfunctionH(z)oflengthN+1isapolynomialinz-1ofdegreeN:NnnznhzH0][)((10.1)ThecorrespondingfrequencyresponseisgivenbyNnnjjenheH0][)((10.2)Ithasbeenshowninsection5.3.1thatanyfinitedurationsequencex[n]oflengthN+1iscompletelycharacterizedbyN+1samplesofitsdiscrete-timeFouriertransformXje.Asaresult,thedesignofanFIRfilteroflengthN+1canbeaccomplishedbyfindingeithertheimpulseresponsesequence{h[n]}orN+1samplesofitsfrequencyresponseHje.Also,toensurealinear-phasedesign,thecondition][][nNhnh,mustbesatisfied.TwodirectapproachestothedesignofFIRfiltersarethewindowedFourierseriesapproachandthefrequencysamplingapproach.WedescribetheformerapproachinSection10.2.ThesecondapproachistreatedinProblems10.31and10.32.Insection10.3,weoutlinecomputer-baseddigitalfilterdesignmethods.10.1.2EstimationoftheFilterOrderAfterthetypeofthedigitalfilterhasselected,thenextstepinthefilterdesignprocessistoestimatethefilterordershouldbethesmallestintegergreaterthanorequaltotheestimatedvalue.FIRDigitalFilterOrderEstimationForthedesignoflowpassFIRdigitalfilters,severalauthorshaveadvancedformulasforestimatingtheminimumvalueofthefilterorderNdirectlyfromthedigitalfilterspecifications:normalizedpassbandedgeangularfrequencyp,normalizefstopbandedgeangularfrequencys,peakpassbandripplep,andpeakstopbandripples.Wereviewthreesuchformulas.Kaiser'sFormula.ArathersimpleformuladevelopedbyKaiser[Kai74]isgivenby2/)(6.1413)(log2010psspN.WeillustratetheapplicationoftheaboveformulainExample10.1.Bellanger'sFormula.AnothersimpleformulaadvancedbyBellangerisgivenby[Bel81]10.1PreliminaryConsiderations12/)(3)10(log210psspN.ItsapplicationisconsideredinExample10.2.Hermann'sFormula.TheformuladuetoHermannetal.[Her73]givesaslightlymoreaccuratevaluefortheorderandisgivenby2/)(]2/))[(,(,2ppspsspsFDN）（,Where]6)(log5)(log4[log]3)(log2)(log1[),(102101010210aaaaaaDppsppsp,And]log[log21),(1010spspbbF,Witha1=0.005309,a2=0.07114,a3=-0.4761,a4=0.00266,a5=0.5941,a6=0.4278,b1=11.01217,b2=0.51244.TheformulagiveninEq.(10.5)isvalidforsp.Ifsp,thenthefilterorderformulatobeusedisobtainedbyinterchangingpandsinEq.(10.6a)and(10.6b).Forsmallvaluesofpands,alloftheaboveformulasprovidereasonablycloseandaccurateresults.Ontheotherhand,whenthevaluesofpandsarelarge,Eq.(10.5)yieldsamoreaccuratevaluefortheorder.AComparisonofFIRFilterOrderFormulasNotethatthefilterordercomputedinExamples10.1,10.2and10.3,usingEqs.(10.3),(10.3),and(10.5),Respectively,arealldifferent.Eachofthesethreeformulasprovideonlyanestimateoftherequiredfilterorder.ThefrequencyresponseoftheFIRfilterdesignedusingthisestimatedordermayormaynotmeetthegivenspecifications.Ifthespecificationsarenotmet,itisrecommendedthatthefilterorderbegraduallyincreaseduntilthespecificationsaremet.EstimationoftheFIRfilterorderusingMATLABisdiscussedinSection10.5.1.AnimportantpropertyofeachoftheabovethreeformulasisthattheestimatedfilterorderNoftheFIRfilterisinverselyproportionaltothetransitionbandwidth(ps)anddoesnotdependontheactuallocationofthetransitionband.ThisimpliesthatasharpcutoffFIRfilterwithanarrowtransitionbandwouldbeofveryhighorder,whereasanFIRfilterwithawidetransitionbandwillhaveaveryloworder.AnotherinterestingpropertyofKaiser'sandBellan