线性滤波器.docx
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线性滤波器.docx
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线性滤波器
Linearfilter
FromWikipedia,thefreeencyclopedia
Thisarticleincludesalistofreferences,relatedreadingorexternallinks,butitssourcesremainunclearbecauseitlacksinlinecitations.Pleaseimprovethisarticlebyintroducingmoreprecisecitationswhereappropriate.(March2011)
Linearfiltersinthetimedomainprocesstime-varyinginputsignalstoproduceoutputsignals,subjecttotheconstraintoflinearity.Thisresultsfromsystemscomposedsolelyofcomponents(ordigitalalgorithms)classifiedashavingalinearresponse.
Mostfiltersimplementedinanalogelectronics,indigitalsignalprocessing,orinmechanicalsystemsareclassifiedascausal,timeinvariant,andlinear.Howeverthegeneralconceptoflinearfilteringisbroader,alsousedinstatistics,dataanalysis,andmechanicalengineeringamongotherfieldsandtechnologies.Thisincludesnoncausalfiltersandfiltersinmorethanonedimensionsuchaswouldbeusedinimageprocessing;thosefiltersaresubjecttodifferentconstraintsleadingtodifferentdesignmethods,whicharediscussedelsewhere.
Alineartime-invariant(LTI)filtercanbeuniquelyspecifiedbyitsimpulseresponseh,andtheoutputofanyfilterismathematicallyexpressedastheconvolutionoftheinputwiththatimpulseresponse.Thefrequencyresponse,givenbythefilter'stransferfunctionH(ω),isanalternativecharacterizationofthefilter.Thefrequencyresponsemaybetailoredto,forinstance,eliminateunwantedfrequencycomponentsfromaninputsignal,ortolimitanamplifiertosignalswithinaparticularbandoffrequencies.Thereareanumberofparticularlydesirableorusefulfiltertransferfunctions,ofwhichthisarticlewillpresentanoverview.
Amongthetime-domainfilterswehereconsider,therearetwogeneralclassesoffiltertransferfunctionsthatcanapproximateadesiredfrequencyresponse.Verydifferentmathematicaltreatmentsapplytothedesignoffilterstermedinfiniteimpulseresponse(IIR)filters,characteristicofmechanicalandanalogelectronicssystems,andfiniteimpulseresponse(FIR)filters,whichcanbeimplementedbydiscretetimesystemssuchascomputers(thentermeddigitalsignalprocessing).
Impulseresponseandtransferfunction
Theimpulseresponsehofalineartime-invariantcausalfilterspecifiestheoutputthatthefilterwouldproduceifitweretoreceiveaninputconsistingofasingleimpulseattime0.An"impulse"inacontinuoustimefiltermeansaDiracdeltafunction;inadiscretetimefiltertheKroneckerdeltafunctionwouldapply.Theimpulseresponsecompletelycharacterizestheresponseofanysuchfilter,inasmuchasanypossibleinputsignalcanbeexpressedasa(possiblyinfinite)combinationofweighteddeltafunctions.Multiplyingtheimpulseresponseshiftedintimeaccordingtothearrivalofeachofthesedeltafunctionsbytheamplitudeofeachdeltafunction,andsummingtheseresponsestogether(accordingtothesuperpositionprinciple,applicabletoalllinearsystems)yieldstheoutputwaveform.
Mathematicallythisisdescribedastheconvolutionofatime-varyinginputsignalx(t)withthefilter'simpulseresponseh,definedas:
Thefirstformisthecontinuous-timeformwhichdescribesmechanicalandanalogelectronicsystems,forinstance.Thesecondequationisadiscrete-timeversionused,forexample,bydigitalfiltersimplementedinsoftware,so-calleddigitalsignalprocessing.Theimpluseresponsehcompletelycharacterizesanylineartime-invariant(orshift-invariantinthediscrete-timecase)filter.Theinputxissaidtobe"convolved"withtheimpulseresponsehhavinga(possiblyinfinite)durationoftimeT(orofNsamplingperiods).
ThefilterresponsecanalsobecompletelycharacterizedinthefrequencydomainbyitstransferfunctionH(ω),whichistheFouriertransformoftheimpulseresponseh.Typicalfilterdesigngoalsaretorealizeaparticularfrequencyresponse,thatis,themagnitudeofthetransferfunction|H(ω)|;theimportanceofthephaseofthetransferfunctionvariesaccordingtotheapplication,inasmuchastheshapeofawaveformcanbedistortedtoagreaterorlesserextentintheprocessofachievingadesired(amplitude)responseinthefrequencydomain.
Filterdesignconsistsoffindingapossibletransferfunctionthatcanbeimplementedwithincertainpracticalconstraintsdictatedbythetechnologyordesiredcomplexityofthesystem,followedbyapracticaldesignthatrealizesthattransferfunctionusingthechosentechnology.Thecomplexityofafiltermaybespecifiedaccordingtotheorderofthefilter,whichisspecifieddifferentlydependingonwhetherwearedealingwithanIIRorFIRfilter.Wewillnowlookatthesetwocases.
[edit]Infiniteimpulseresponsefilters
Consideraphysicalsystemthatactsasalinearfilter,suchasasystemofspringsandmasses,orananalogelectroniccircuitthatincludescapacitorsand/orinductors(alongwithotherlinearcomponentssuchasresistorsandamplifiers).Whensuchasystemissubjecttoanimpulse(oranysignaloffiniteduration)itwillrespondwithanoutputwaveformwhichlastspastthedurationoftheinput,eventuallydecayingexponentiallyinoneoranothermanner,butnevercompletelysettlingtozero(mathematicallyspeaking).Suchasystemissaidtohaveaninfiniteimpulseresponse(IIR).Theconvolutionintegral(orsummation)aboveextendsoveralltime:
T(orN)mustbesettoinfinity.
Forinstance,consideradampedharmonicoscillatorsuchasapendulum,oraresonantL-Ctankcircuit.Ifthependulumhasbeenatrestandweweretostrikeitwithahammer(the"impulse"),settingitinmotion,itwouldswingbackandforth("resonate"),say,withanamplitudeof10cm.Butafter10minutes,say,itwouldstillbeswingingbuttheamplitudewouldhavedecreasedto5cm,halfofitsoriginalamplitude.Afteranother10minutesitsamplitudewouldbeonly2.5cm,then1.25cm,etc.Howeveritwouldnevercometoacompleterest,andwethereforecallthatresponsetotheimpulse(strikingitwithahammer)"infinite"induration.
ThecomplexityofsuchasystemisspecifiedbyitsorderN.Nisoftenaconstraintonthedesignofatransferfunctionsinceitspecifiesthenumberofreactivecomponentsinananalogcircuit;inadigitalIIRfilterthenumberofcomputationsrequiredisproportionaltoN.
[edit]Finiteimpulseresponsefilters
Afilterimplementedinacomputerprogram(oraso-calleddigitalsignalprocessor)isadiscrete-timesystem;adifferent(butparallel)setofmathematicalconceptsdefinesthebehaviorofsuchsystems.AlthoughadigitalfiltercanbeanIIRfilterifthealgorithmimplementingitincludesfeedback,itisalsopossibletoeasilyimplementafilterwhoseimpulsetrulygoestozeroafterNtimesteps;thisiscalledafiniteimpulseresponse(FIR)filter.
Forinstance,supposewehaveafilterwhich,whenpresentedwithanimpulseinatimeseries:
0,0,0,1,0,0,0,0,0,0,0,0,0,0,0.....
willoutputaserieswhichrespondstothatimpulseattime0untiltime4,andhasnofurtherresponse,suchas:
0,0,0,1,1,1,1,1,0,0,0,0,0,0,0.....
Althoughtheimpulseresponsehaslasted4timestepsaftertheinput,startingattime5ithastrulygonetozero.Theextentoftheimpulseresponseisfinite,andthiswouldbeclassifiedasa4thorderFIRfilter.Theconvolutionintegral(orsummation)aboveneedonlyextendtothefulldurationoftheimpulseresponseT,ortheorderNinadiscretetimefilter.
[edit]Implementationissues
ClassicalanalogfiltersareIIRfilters,andclassicalfiltertheorycentersonthedeterminationoftransferfunctionsgivenbyloworderrationalfunctions,whichcanbesynthesizedusingthesamesmallnumberofreactivecomponents.[1]Usingdigitalcomputers,ontheotherhand,bothFIRandIIRfiltersarestraightforwardtoimplementinsoftware.
AdigitalIIRfiltercangenerallyapproximateadesiredfilterresponseusinglesscomputingpowerthanaFIRfilter,howeverthisadvantageismoreoftenunneededgiventheincreasingpowerofdigitalprocessors.TheeaseofdesigningandcharacterizingFIRfiltersmakesthempreferabletothefilterdesigner(programmer)whenamplecomputingpowerisavailable.AnotheradvantageofFIRfiltersisthattheirimpulseresponsecanbemadesymmetric,whichimpliesaresponseinthefrequencydomainwhichhaszerophaseatallfrequencies(notconsideringafinitedelay),whichisabsolutelyimpossiblewithanyIIRfilter.[2]
[edit]Frequencyresponse
Thefrequencyresponseortransferfunction|H(ω)|ofafiltercanbeobtainediftheimpulseresponseisknown,ordirectlythroughanalysisusingLaplacetransforms,orindiscrete-timesystemstheZ-transform.Thefrequencyresponsealsoincludesthephaseasafunctionoffrequency,howeverinmanycasesthephaseresponseisoflittleornointerest.FIRfilterscanbemadetohavezerophase,butwithIIRfiltersthatisgenerallyimpossibleWithmostIIRtransferfunctionstherearerelatedtransferfunctionshavingafrequencyresponsewiththesamemagnitudebutadifferentphase;inmostcasetheso-calledminimumphasetransferfunctionispreferred.
Filtersinthetimedomainaremostoftenrequestedtofollowaspecifiedfrequencyresponse.Thenamathematicalprocedureisusedtofindafiltertransferfunctionwhichcanberealized(withinsomeconstraints)andwhichapproximatesthedesiredresponsetowithinsomecriterion.Commonfilterresponsespecificationsaredescribedasfollows:
∙Alow-passfilterpasseslowfrequencieswhileblockinghigherfrequencies.
∙Ahigh-passfilterpasseshighfrequencies.
∙Aband-passfilterpassesaband(range)offrequencies.
∙Aband-stopfilter
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