Digital-filters-(FIR)-數(shù)字濾波器課件

DigitalFilterSpecificationsThesefiltersareunealisablebecause(oneofthefollowingissufficient)theirimpulseresponsesinfinitelylongnon-causalTheiramplituderesponsescannotbeequaltoaconstantoverabandoffrequenciesAnotherperspectivethatprovidessomeunderstandingcanbeobtainedbylookingattheidealamplitudesquared.1DigitalFilterSpecificationsConsidertheidealLPresponsesquared(sameasactualLPresponse)2DigitalFilterSpecificationsTherealisablesquaredamplituderesponsetransferfunction(anditsdifferential)iscontinuousinSuchfunctionsifIIRcanbeinfiniteatpointbutaroundthatpointcannotbezero.ifFIRcannotbeinfiniteanywhere.Hencepreviousdefferentialofidealresponseisunrealisable3DigitalFilterSpecificationsArealisableresponsewouldeffectivelyneedtohaveanapproximationofthedeltafunctionsinthedifferentialThisisanecessarycondition4DigitalFilterSpecificationsForexamplethemagnituderesponseofadigitallowpassfiltermaybegivenasindicatedbelow5DigitalFilterSpecificationsInthepassbandwerequirethat withadeviationInthestopbandwerequirethat withadeviation

6DigitalFilterSpecificationsFilterspecificationparameters-passbandedgefrequency-stopbandedgefrequency-peakripplevalueinthepassband-peakripplevalueinthestopband7DigitalFilterSpecificationsPracticalspecificationsareoftengivenintermsoflossfunction(indB)

Peakpassbandripple

dBMinimumstopbandattenuation

dB8DigitalFilterSpecificationsInpractice,passbandedgefrequencyandstopbandedgefrequencyarespecifiedinHzFordigitalfilterdesign,normalizedbandedgefrequenciesneedtobecomputedfromspecificationsinHzusing9DigitalFilterSpecificationsExample-LetkHz,kHz,and kHzThen10Thetransferfunction

H(z)

meetingthespecificationsmustbeacausaltransferfunctionForIIRrealdigitalfilterthetransferfunctionisarealrationalfunctionofH(z)mustbestableandoflowestorder

N

orM

forreducedcomputationalcomplexitySelectionofFilterType11SelectionofFilterTypeFIRrealdigitalfiltertransferfunctionisapolynomialin(orderN)withrealcoefficientsForreducedcomputationalcomplexity,degreeNofH(z)mustbeassmallaspossibleIfalinearphaseisdesiredthenwemusthave:(Moreonthislater)12SelectionofFilterTypeAdvantagesinusinganFIRfilter-

(1)Canbedesignedwithexactlinearphase

(2)FilterstructurealwaysstablewithquantisedcoefficientsDisadvantagesinusinganFIRfilter-OrderofanFIRfilterisconsiderablyhigherthanthatofanequivalentIIRfiltermeetingthesamespecifications;thisleadstohighercomputationalcomplexityforFIR13FIRDesign

FIRDigitalFilterDesign ThreecommonlyusedapproachestoFIRfilterdesign-

(1)

WindowedFourierseriesapproach

(2)Frequencysamplingapproach

(3)Computer-basedoptimizationmethods14FiniteImpulseResponseFiltersThetransferfunctionisgivenbyThelengthofImpulseResponseisNAllpolesareat.Zeroscanbeplacedanywhereonthez-plane15FIR:LinearphaseForphaselinearitytheFIRtransferfunctionmusthavezerosoutsidetheunitcircle16FIR:LinearphaseTodevelopexpressionforphaseresponsesettransferfunction(ordern)Infactoredform

Where,isreal&zerosoccurinconjugates17FIR:LinearphaseLet whereThus

18FIR:LinearphaseExpandinaLaurentSeriesconvergentwithintheunitcircleTodosomodifythesecondsumas19FIR:LinearphaseSothatThus

where20FIR:Linearphasearetherootmomentsoftheminimumphasecomponent

aretheinverserootmomentsofthemaximumphasecomponentNowontheunitcirclewehave and21FundamentalRelationshipshence(noteFourierform)22FIR:LinearphaseThusforlinearphasethesecondterminthefundamentalphaserelationshipmustbeidenticallyzeroforallindexvalues.Hence1)themaximumphasefactorhaszeroswhicharetheinversesofthethoseoftheminimumphasefactor2)thephaseresponseislinearwithgroupdelay(normalised)equaltothenumberofzerosoutsidetheunitcircle23FIR:LinearphaseItfollowsthatzerosoflinearphaseFIRtrasferfunctionsnotonthecircumferenceoftheunitcircleoccurintheform24FIR:LinearphaseForLinearPhase

t.f.(orderN-1)

sothatforNeven:25FIR:LinearphaseforNodd:I)OnwehaveforNeven,and+vesign26FIR:LinearphaseII)Whilefor–vesign[Note:antisymmetriccaseaddsradstophase,withdiscontinuityat]III)ForNoddwith+vesign27FIR:LinearphaseIV)Whilewitha–vesign[Noticethatfortheantisymmetriccasetohavelinearphasewerequire ThephasediscontinuityisasforNeven]28FIR:LinearphaseThecasesmostcommonlyusedinfilterdesignare(I)and(III),forwhichtheamplitudecharacteristiccanbewrittenasapolynomialin29DesignofFIRfilters:Windows(i)Startwithidealinfiniteduration(ii)Truncatetofinitelength.(Thisproducesunwantedripplesincreasinginheightneardiscontinuity.)(iii)ModifytoWeightw(n)isthewindow30WindowsCommonlyusedwindowsRectangularBartlett Hann Hamming

Blackman

Kaiser 31KaiserwindowKaiserwindowβTransitionwidth(Hz)

Min.stopattndB

2.121.5/N304.542.9/N506.764.3/N708.965.7/N9032ExampleLowpassfilteroflength51and33FrequencySamplingMethodInthisapproachwearegivenandneedtofindThisisaninterpolationproblemandthesolutionisgivenintheDFTpartofthecourseIthassimilarproblemstothewindowingapproach34Linear-PhaseFIRFilterDesignbyOptimisationAmplituderesponseforall4typesoflinear-phaseFIRfilterscanbeexpressedas where35Linear-PhaseFIRFilterDesignbyOptimisationModifiedformofweightederrorfunction where36Linear-PhaseFIRFilterDesignbyOptimisationOptimisationProblem-Determinewhichminimisethepeakabsolutevalueof overthespecifiedfrequencybandsAfterhasbeendetermined,constructtheoriginalandhenceh[n]37Linear-PhaseFIRFilterDesignbyOptimisationSolutionisobtainedviatheAlternationTheoremTheoptimalsolutionhasequiripplebehaviourconsistentwiththetotalnumberofavailableparameters.ParksandMcClellanusedtheRemezalgorithmtodevelopaprocedurefordesigninglinearFIRdigitalfilters.38FIRD