2023學年完整公開課版chapter7

Chapter7Objective-DeterminationofarealizabletransferfunctionG(z)approximatingagivenfrequencyresponsespecificationisanimportantstepinthedevelopmentofadigitalfilterIfanIIRfilterisdesired,G(z)shouldbeastablerealrationalfunctionDigitalfilterdesignistheprocessofderivingthetransferfunctionG(z)§7.1

DigitalFilterSpecificationsUsually,eitherthemagnitudeand/orthephase(delay)responseisspecifiedforthedesignofdigitalfilterformostapplicationsInsomesituations,theunitsampleresponseorthestepresponsemaybespecifiedInmostpracticalapplications,theproblemofinterestisthedevelopmentofarealizableapproximationtoagivenmagnituderesponsespecification§7.1

DigitalFilterSpecificationsWediscussinthiscourseonlythemagnitudeapproximationproblemTherearefourbasictypesofidealfilterswithmagnituderesponsesasshownbelow§7.1

DigitalFilterSpecificationsAstheimpulseresponsecorrespondingtoeachoftheseidealfiltersisnoncausalandofinfinitelength,thesefiltersarenotrealizableInpractice,themagnituderesponsespecificationsofadigitalfilterinthepassbandandinthestopbandaregivenwithsomeacceptabletolerancesInaddition,atransitionbandisspecifiedbetweenthepassbandandstopband§7.1

DigitalFilterSpecificationsForexample,themagnituderesponse|G(ej)|ofadigitallowpassfiltermaybegivenasindicatedbelow§7.1

DigitalFilterSpecificationsAsindicatedinthefigure,inthepassband,definedby0p,werequirethat|G(ej)|1withanerrorp,i.e.,1-p

|G(ej)|1+p,||pInthestopband,definedbys

,werequirethat|G(ej)|0withanerrors

i.e.,|G(ej)|p,s

||§7.1

DigitalFilterSpecificationsp-passbandedgefrequencys-stopbandedgefrequencyp-peakripplevalueinthepassbands-peakripplevalueinthestopbandSinceG(ej)isaperiodicfunctionofw,and|G(ej)|ofareal-coefficientdigitalfilterisanevenfunctionofwAsaresult,filterspecificationsaregivenonlyforthefrequencyrange0||§7.1

DigitalFilterSpecificationsSpecificationsareoftengivenintermsoflossfunctionG()=-20log10

|G(ej)|indBPeakpassbandripplep=-20log10

(1-p) dBMinimumstopbandattenuations=-20log10

(s)dB§7.1

DigitalFilterSpecificationsMagnitudespecificationsmayalternatelybegiveninanormalizedformasindicatedbelow§7.1

DigitalFilterSpecificationsHere,themaximumvalueofthemagnitudeinthepassbandisassumedtobeunity1/(1+2)-Maximumpassbanddeviation,givenbytheminimumvalueofthemagnitudeinthepassband1/A-Maximumstopbandmagnitude§7.1

DigitalFilterSpecificationsForthenormalizedspecification,maximumvalueofthegainfunctionortheminimumvalueofthelossfunctionis0dBMaximumpassbandattenuation

dBForp<<1,itcanbeshownthatdB§7.1

DigitalFilterSpecificationsInpractice,passbandedgefrequencyFpandstopbandedgefrequencyFsarespecifiedinHzFordigitalfilterdesign,normalizedbandedgefrequenciesneedtobecomputedfromspecificationsinHzusing§7.2SelectionofFilterTypeThetransferfunctionH(z)meetingthefrequencyresponsespecificationsshouldbeacausaltransferfunctionForIIRdigitalfilterdesign,theIIRtransferfunctionisarealrationalfunctionofz-1:H(z)mustbeastabletransferfunctionandmustbeoflowestorderNforreducedcomputationalcomplexity§7.2SelectionofFilterTypeForreducedcomputationalcomplexity,degreeNofH(z)mustbeassmallaspossibleIfalinearphaseisdesired,thefiltercoefficientsmustsatisfytheconstraint:h[n]=h[N-n]ForFIRdigitalfilterdesign,theFIRtransferfunctionisapolynomialinz-1withrealcoefficients:§7.2SelectionofFilterTypeAdvantagesinusinganFIRfilter- (1)Canbedesignedwithexactlinearphase, (2)FilterstructurealwaysstablewithquantizedcoefficientsDisadvantagesinusinganFIRfilter-OrderofanFIRfilter,inmostcases,isconsiderablyhigherthantheorderofanequivalentIIRfiltermeetingthesamespecifications,andFIRfilterhasthushighercomputationalcomplexity§7.3DigitalFilterDesign:

BasicApproachesMostcommonapproachtoIIRfilterdesign–(1)Convertthedigitalfilterspecificationsintoananalogprototypelowpassfilterspecifications(2)DeterminetheanaloglowpassfiltertransferfunctionHa(s)(3)TransformHa(s)intothedesireddigitaltransferfunctionG(z)§7.3DigitalFilterDesign:

BasicApproachesThisapproachhasbeenwidelyusedforthefollowingreasons: (1)Analogapproximationtechniquesarehighlyadvanced (2)Theyusuallyyieldclosed-formsolutions (3)Extensivetablesareavailableforanalogfilterdesign (4)Manyapplicationsrequiredigitalsimulationofanalogsystems§7.3DigitalFilterDesign:

BasicApproachesAnanalogtransferfunctiontobedenotedasHa(s)=Pa(s)/Da(s) wherethesubscript“a”specificallyindicatestheanalogdomainAdigitaltransferfunctionderivedfromHa(s)shallbedenotedasG(z)=P(z)/D(z)§7.3DigitalFilterDesign:

BasicApproachesBasicideabehindtheconversionofHa(s)intoG(z)istoapplyamappingfromthes-domaintothez-domainsothatessentialpropertiesoftheanalogfrequencyresponsearepreservedThusmappingfunctionshouldbesuchthatImaginary(j)axisinthes-planebemappedontotheunitcircleofthez-planeAstableanalogtransferfunctionbemappedintoastabledigitaltransferfunction§7.3DigitalFilterDesign:

BasicApproachesFIRfilterdesignisbasedonadirectapproximationofthespecifiedmagnituderesponse,withtheoftenaddedrequirementthatthephasebelinearThedesignofanFIRfilteroforderNmaybeaccomplishedbyfindingeitherthelength-(N+1)impulseresponsesamples{h[n]}orthe(N+1)samplesofitsfrequencyresponseH(ej)§7.3DigitalFilterDesign:

BasicApproachesThreecommonlyusedapproachestoFIRfilterdesign- (1)WindowedFourierseriesapproach (2)Frequencysamplingapproach (3)Computer-basedoptimizationmethods§7.4IIRDigitalFilterDesign:BilinearTransformationMethodAbovetransformationmapsasinglepointinthes-planetoauniquepointinthez-planeandvice-versaRelationbetweenG(z)andHa(s)isthengivenbyBilineartransformation§7.4IIRDigitalFilterDesign:BilinearTransformationMethodDigitalfilterdesignconsistsof3steps: (1)DevelopthespecificationsofHa(s)byapplyingtheinversebilineartransformationtospecificationsofG(z) (2)DesignHa(s) (3)DetermineG(z)byapplyingbilineartransformationtoHa(s)Asaresult,theparameterThasnoeffectonG(z)andT=2ischosenforconvenience§7.4IIRDigitalFilterDesign:BilinearTransformationMethodMappingofs-planeintothez-plane§7.4IIRDigitalFilterDesign:BilinearTransformationMethodForz=ejwithT=2wehaveor=tan(/2)§7.4IIRDigitalFilterDesign:BilinearTransformationMethodMappingishighlynonlinearCompletenegativeimaginaryaxisinthes-planefrom=-to=0ismappedintothelowerhalfoftheunitcircleinthez-planefromz=-1toz=1Completepositiveimaginaryaxisinthes-planefrom=0to=ismappedintotheupperhalfoftheunitcircleinthez-planefrom

z=1toz=-1

§7.4IIRDigitalFilterDesign:BilinearTransformationMethodNonlinearmappingintroducesadistortioninthefrequencyaxiscalledfrequencywarpingEffectofwarpingshownright§7.4IIRDigitalFilterDesign:BilinearTransformationMethodStepsinthedesignofadigitalfilter- (1)Prewarp(p,s)tofindtheiranalogequivalents(p,s) (2)DesigntheanalogfilterHa(s) (3)DesignthedigitalfilterG(z)byapplyingbilineartransformationtoHa(s)TransformationcanbeusedonlytodesigndigitalfilterswithprescribedmagnituderesponsewithpiecewiseconstantvaluesTransformationdoesnotpreservephaseresponseofanalogfilter§7.4IIRDigitalFilterDesign:BilinearTransformationMethodApplyingbilineartransformationtotheabovewegetthetransferfunctionofafirst-orderdigitallowpassButterworthfilter

Example-Consider§7.4IIRDigitalFilterDesign:BilinearTransformationMethodRearrangingtermsweget where§7.4IIRDigitalFilterDesign:BilinearTransformationMethod forwhich|Ha(j0)|=0|Ha(j0)|=|Ha(j)|=00iscalledthenotchfrequencyIf|Ha(j2)|=|Ha(j1)|=1/2thenB=2-1isthe3-dBnotchbandwidth

Example-Considerthesecond-orderanalognotchtransferfunction§7.4IIRDigitalFilterDesign:BilinearTransformationMethodThenwhere§7.4IIRDigitalFilterDesign:BilinearTransformationMethodExample-Designa2nd-orderdigitalnotchfilteroperatingatasamplingrateof400Hzwithanotchfrequencyat60Hz,3-dBnotchbandwidthof6HzThus0=2(60/400)=0.3Bw=2(6/400)=0.03Fromtheabovevaluesweget

=0.90993=0.587785§7.4IIRDigitalFilterDesign:BilinearTransformationMethodThegainandphaseresponsesareshownbelowThus§7.4IIRDigitalFilterDesign:BilinearTransformationMethodExample-DesignalowpassButterworthdigitalfilterwithp=0.25,s=0.55,p0.5dB,ands15dBThus2=0.1220185,A2=31.622777If|G(ej0)|=0thisimplies20log10|G(ej0.25)|-0.520log10|G(ej0.55)|-15§7.4IIRDigitalFilterDesign:BilinearTransformationMethodPrewarpingwegetp=tan(p/2)=tan(0.25/2)=0.4142136s=tan(s/2)=tan(0.55/2)=1.1708496Theinversetransitionratiois1/k=s/p=2.8266809Theinversediscriminationratiois1/k1=(A2-1)/=15.841979§7.4IIRDigitalFilterDesign:BilinearTransformationMethodThusN=log10(1/k1)/log10(1/k)=2.6586997ChooseN=3Todeterminecweuse|Ha(jp)|2=1/[1+(p/c)2N]=1/(1+2)§7.4IIRDigitalFilterDesign:BilinearTransformationMethodWethengetc=1.419915(p)=0.5881483rd-orderlowpassButterworthtransferfunctionforc=1isHan(s)=1/[(s+1)(s2+s+1)]Denormalizingtogetc=0.588148wearriveatHa(s)=Han(s/0.588148)§7.4IIRDigitalFilterDesign:BilinearTransformationMethodApplyingbilineartransformationtoHa(s)wegetthedesireddigitaltransferfunctionMagnitudeandgainresponsesofG(z)shownbelow:§7.5IIRHighpass,Bandpass,andBandstop

DigitalFilterDesignFirstApproach- (1)PrewarpdigitalfrequencyspecificationsofdesireddigitalfilterGD(z)toarriveatfrequencyspecificationsofanalogfilterHD(s)ofsametype (2)ConvertfrequencyspecificationsofHD(s)intothatofprototypeanaloglowpassfilterHLP(s) (3)Designanaloglowpassfilter

HLP(s)

§7.5IIRHighpass,Bandpass,andBandstop

DigitalFilterDesign(4)ConvertHLP(s)intoHD(s)usinginversefrequencytransformationusedinStep2 (5)DesigndesireddigitalfilterGD(z)byapplyingbilineartransformationtoHLP(s)§7.5IIRHighpass,Bandpass,andBandstop

DigitalFilterDesignSecondApproach- (1)PrewarpdigitalfrequencyspecificationsofdesireddigitalfilterGD(z)toarriveatfrequencyspecificationsofanalogfilterHD(s)ofsametype (2)ConvertfrequencyspecificationsofHD(s)intothatofprototypeanaloglowpassfilterHLP(s)§7.5IIRHighpass,Bandpass,andBandstop

DigitalFilterDesign

(3)DesignanaloglowpassfilterHLP(s) (4)ConvertHLP(s)intoanIIRdigitaltransferfunctionGLP(z)usingbilineartransformation (5)TransformGLP(z)intothedesireddigitaltransferfunctionGD(z)Weillustratethefirstapproach§7.5IIRHighpass,Bandpass,andBandstop

DigitalFilterDesignDesignofaType1ChebyshevIIRdigitalhighpassfilterSpecifications:Fp=700Hz,Fs=500Hz,

p=1dB,s=32dB,FT=2kHzNormalizedangularbandedgefrequenciesp

=2Fp/FT=2700/2000=0.7s

=2Fs/FT=2500/2000=0.5§7.5.1IIRHighpassDigitalFilterDesignAnaloglowpassfilterspecifications:p=1,s=1.926105,p=1dB,s=32dBPrewarpingthesefrequencieswegetFortheprototypeanaloglowpassfilterchoose

p=1Usingwegets=1.962105§7.5.1IIRHighpassDigitalFilterDesignMATLABcodefragmentsusedforthedesign

[N,Wn]=cheb1ord(1,1.9626105,1,32,’s’) [B,A]=cheby1(N,1,Wn,’s’); [BT,AT]=lp2hp(B,A,1.9626105); [num,den]=bilinear(BT,AT,0.5);§7.5.2IIRBandpassDigitalFilterDesignDesignofaButterworthIIRdigitalbandpassfilterSpecifications:p1=0.45,p1=0.65,s1=0.3,s2=0.75,p=1dB,s=40dBPrewarpingweget§7.5.2IIRBandpassDigitalFilterDesignFortheprototypeanaloglowpassfilterwechoosep=1WidthofpassbandWesetWethereforemodifysothatandexhibitgeometricsymmetrywithrespectto§7.5.2IIRBandpassDigitalFilterDesignSpecificationsofprototypeanalogButterworthlowpassfilter:p=1,s=2.3617627,p=1dB,s=40dBUsingweget§7.5.2IIRBandpassDigitalFilterDesignMATLABcodefragmentsusedforthedesign

[N,Wn]=buttord(1,2.3617627,1,40,’s’) [B,A]=butter(N,Wn,’s’); [BT,AT]=lp2bp(B,A,1.1805647,0.777771); [num,den]=bilinear(BT,AT,0.5);§7.5.3IIRBandstopDigitalFilterDesignDesignofanellipticIIRdigitalbandstopfilterSpecifications:s1=0.45,s2=0.65,p1=0.3,p2=0.75,p=1dB,s=40dBPrewarpingweget

Widthofstopband§7.5.3IIRBandstopDigitalFilterDesignFortheprototypeanaloglowpassfilterwechooses=1WethereforemodifysothatandexhibitgeometricsymmetrywithrespecttoWesetUsingweget§7.5.3IIRBandstopDigitalFilterDesignMATLABcodefragmentsusedforthedesign

[N,Wn]=

ellipord(0.4234126,1,1,40,’s’); [B,A]=ellip(N,1,40,Wn,’s’); [BT,AT]=

lp2bs(B,A,1.1805647,0.777771); [num,den]=bilinear(BT,AT,0.5);§7.6

FixedWindowFunctionsUsingataperedwindowcausestheheightofthesidelobestodiminish,withacorrespondingincreaseinthemainlobewidthresultinginawidertransitionatthediscontinuityHann:W[n[=0.5+0.5cos[2n/(2M+1)],-MnMHamming:W[n[=0.54+0.46cos[2n/(2M+1)],-MnM

Blackman:W[n[=0.42+0.5cos[2n/(2M+1)]+0.08cos[4n/(2M+1)]§7.6

FixedWindowFunctionsPlotsofmagnitudesoftheDTFTsofthesewindowsforM=25areshownbelow00.20.40.60.81-100-80-60-40-200w/pGain,dBRectangularwindow00.20.40.60.81-100-80-60-40-200w/pGain,dBHanningwindow00.20.40.60.81-100-80-60-40-200w/pGain,dBHammingwindow00.20.40.60.81-100-80-60-40-200w/pGain,dBBlackmanwindow§7.6

FixedWindowFunctionsMagnitudespectrumofeachwindowcharacterizedbyamainlobecenteredatw=0followedbyaseriesofsidelobeswithdecreasingamplitudesParameterspredictingtheperformanceofawindowinfilterdesignare:MainlobewidthRelativesidelobelevel§7.6

FixedWindowFunctionsMainlobewidthML-givenbythedistancebetweenzerocrossingsonbothsidesofmainlobeRelativesidelobelevelAsl-givenbythedifferenceindBbetweenamplitudesoflargestsidelobeandmainlobe§7.6

FixedWindowFunctionsObserveThus,Passbandandstopbandripplesarethesame§7.6

FixedWindowFunctionsDistancebetweenthelocationsofthemaximumpassbanddeviationandminimumstopbandvalueMLWidthoftransitionbandw=s-p<ML§7.6

FixedWindowFunctionsToensureafasttransitionfrompassbandtostopband,windowshouldhaveaverysmallmainlobewidthToreducethepassbandandstopbandrippled,theareaunderthesidelobesshouldbeverysmallUnfortunately,thesetworequirementsarecontradictory§7.6

FixedWindowFunctionsInthecaseofrectangular,Hann,Hamming,andBlackmanwindows,thevalueofrippled