Chapter7

LTIDiscrete-TimeSystemsintheTransformDomainTypesofTransferFunctionsThetime-domainclassificationofanLTIdigitaltransferfunctionsequenceisbasedonthelengthofitsimpulseresponse: -Finiteimpulseresponse(FIR)transferfunction -Infiniteimpulseresponse(IIR)transferfunction★

TypesofTransferFunctionsSeveralotherclassificationsarealsousedInthecaseofdigitaltransferfunctionswithfrequency-selectivefrequencyresponses,oneclassificationisbasedontheshapeofthemagnitudefunctionH(ei)ortheformofthephasefunction

q(w)Basedonthisfourtypesofidealfiltersareusuallydefined§7.1.1DigitalFilterswithIdealMagnitudeResponseOnecommonclassificationisbasedonanidealmagnituderesponseAdigitalfilterdesignedtopasssignalcomponentsofcertainfrequencieswithoutdistortionshouldhaveafrequencyresponseequalto1atthesefrequencies,andshouldhaveafrequencyresponseequalto0atallotherfrequencies§7.1.1DigitalFilterswithIdealMagnitudeResponseTherangeoffrequencieswherethefrequencyresponsetakesthevalueofoneiscalledthepassbandTherangeoffrequencieswherethefrequencyresponsetakesthevalueofzeroiscalledthestopband§7.1.1DigitalFilterswithIdealMagnitudeResponseFrequencyresponsesofthefourpopulartypesofidealdigitalfilterswithrealimpulseresponsecoefficientsareshownbelow:§7.1.1DigitalFilterswithIdealMagnitudeResponseThefrequenciesc,c1,andc2arecalledthecutofffrequenciesAnidealfilterhasamagnituderesponseequaltooneinthepassbandandzerointhestopband,andhasazerophaseeverywhere§7.1.1DigitalFilterswithIdealMagnitudeResponseLowpassfilter:Passband:0≤ω≤ωcStopband:ωc≤ω≤πHighpassfilter:Passband:ωc≤ω≤πStopband:0≤ω≤ωcBandpassfilter:Passband:ωc1≤ω≤ωc2

Stopband:0≤ω<ωc1and

ωc2<ω<π

Bandstopfilter:Stopband:ωc1<ω<ωc2

Passband:0≤ω≤ωc1and

ωc2≤ω≤π

§7.1.1DigitalFilterswithIdealMagnitudeResponseEarlierinthecoursewederivedtheinverseDTFTofthefrequencyresponseHLP(ej)oftheideallowpassfilter:(p132EMAPLE3.8)hLP[n]=sincn/n,-<n<Wehavealsoshownthattheaboveimpulseresponseisnotabsolutelysummable,andhence,thecorrespondingtransferfunctionisnotBIBOstable§7.1.1DigitalFilterswithIdealMagnitudeResponseAlso,hLP[n]isnotcausalandisofdoublyinfinitelengthTheremainingthreeidealfiltersarealsocharacterizedbydoublyinfinite,noncausalimpulseresponsesandarenotabsolutelysummableThus,theidealfilterswiththeideal“brickwall”frequencyresponsescannotberealizedwithfinitedimensionalLTIfilter§7.1.1DigitalFilterswithIdealMagnitudeResponseTodevelopstableandrealizabletransferfunctions,theidealfrequencyresponsespecificationsarerelaxedbyincludingatransitionbandbetweenthepassbandandthestopbandThispermitsthemagnituderesponsetodecayslowlyfromitsmaximumvalueinthepassbandtothezerovalueinthestopband§7.1.1DigitalFilterswithIdealMagnitudeResponseMoreover,themagnituderesponseisallowedtovarybyasmallamountbothinthepassbandandthestopbandTypicalmagnituderesponsespecificationsofalowpassfilterareshownas:§7.1.2

BoundedRealTransferFunctionsLetx[n]andy[n]denote,respectively,theinputandoutputofadigitalfiltercharacterizedbyaBRtransferfunctionH(z)withX(ejω)andY(ejω)denotingtheirDTFTsforallvaluesofwAcausalstablereal-coefficienttransferfunctionH(z)isdefinedasaboundedreal(BR)transferfunctionif§7.1.2

BoundedRealTransferFunctionsIntegratingtheabovefrom-πtoπ,andapplyingParseval’srelationwegetThentheconditionimpliesthat§7.1.2

BoundedRealTransferFunctionsThus,forallfinite-energyinputs,theoutputenergyislessthanorequaltotheinputenergyimplyingthatadigitalfiltercharacterizedbyaBRtransferfunctioncanbeviewedasapassivestructureIf,thentheoutputenergyisequaltotheinputenergy,andsuchadigitalfilteristhereforealosslesssystem§7.1.2

BoundedRealTransferFunctionsTheBRandLBRtransferfunctionsarethekeystotherealizationofdigitalfilterswithlowcoefficientsensitivityAcausalstablereal-coefficienttransferfunctionH(z)withisthuscalledalosslessboundedreal(LBR)transferfunction§7.1.2

BoundedRealTransferFunctionsExample7.1:ConsiderthecausalstableIIRtransferfunctionwhereKisarealconstantItssquare-magnitudefunctionisgivenby§7.1.2

BoundedRealTransferFunctionsThus,forα>0,themaximumvalueof|H(ejω)|2isequaltoK2/(1-α)2atω=0andtheminimumvalueisequaltoK2/(1-α)2atω=πOntheotherhand,forα<0,themaximumvalueof2αcosωisequalto-2α

atω=πandtheminimumvalueisequalto2α

atω=0§7.1.2

BoundedRealTransferFunctionsHere,themaximumvalueof|H(ejω)|2isequaltoK2/(1-α)2atω=πandtheminimumvalueisequaltoK2/(1-α)2atω=0Hence,themaximumvaluecanbemadeequalto1bychoosingK=±(1-α),inwhichcasetheminimumvaluebecomes(1-α)2/(1+α)2

§7.1.2

BoundedRealTransferFunctionsHence,isaBRfunctionforK=±(1-α),

Plotsofthemagnitudefunctionforα=±0.5withvaluesofKchosentomakeH(z)aBRfunctionareshownonthenextslide§7.1.2

BoundedRealTransferFunctionsHighpassfilterLowpassfilter§7.1.3AllpassTransferFunction

iscalledanallpasstransferfunctionAnM-thordercausalreal-coefficientallpasstransferfunctionisoftheformDefinition:AnIIRtransferfunctionA(z)withunitymagnituderesponseforallfrequencies,i.e.,§7.1.3AllpassTransferFunctionIfwedenotethedenominatorpolynomialsofAM(z)asDM(z):

Notefromtheabovethatifz=rejisapoleofarealcoefficientallpasstransferfunction,thenithasazeroatz=(1/r)e-jthenitfollowsthatAM(z)canbewrittenas:§7.1.3AllpassTransferFunctionThenumeratorofareal-coefficientallpasstransferfunctionissaidtobethemirror-imagepolynomialofthedenominator,andviceversa~~

Weshallusethenotationtodenotethemirror-imagepolynomialofadegree-MpolynomialDM(z),i.e.,§7.1.3AllpassTransferFunction

Theexpressionimpliesthatthepolesandzerosofareal-coefficientallpassfunctionexhibitmirror-imagesymmetryinthez-plane§7.1.3AllpassTransferFunctionTherefore

Hence

Toshowthatweobservethat§7.1.3AllpassTransferFunctionNow,thepolesofacausalstabletransferfunctionmustlieinsidetheunitcircleinthez-planeHence,allzerosofacausalstableallpasstransferfunctionmustlieoutsidetheunitcircleinamirror-imagesymmetrywithitspolessituatedinsidetheunitcircle§7.1.3AllpassTransferFunctionFigurebelowshowstheprincipalvalueofthephaseofthe3rd-orderallpassfunctionNotethediscontinuitybytheamountof2πinthephaseθ(ω)§7.1.3AllpassTransferFunctionIfweunwrapthephasebyremovingthediscontinuity,wearriveattheunwrappedphasefunctionθc(ω)indicatedbelowNote:Theunwrappedphasefunctionisacontinuousfunctionofω§7.1.3AllpassTransferFunctionTheunwrappedphasefunctionofanyarbitrarycausalstableallpassfunctionisacontinuousfunctionofω

Properties(1)Acausalstablereal-coefficientallpasstransferfunctionisalosslessboundedreal(LBR)functionor,equivalently,acausalstableallpassfilterisalosslessstructure§7.1.3AllpassTransferFunction(2)ThemagnitudefunctionofastableallpassfunctionA(z)satisfies:(3)Letτ(ω)denotethegroupdelayfunctionofanallpassfilterA(z),i.e.,§7.1.3AllpassTransferFunctionTheunwrappedphasefunctionθc(ω)ofastableallpassfunctionisamonotonicallydecreasingfunctionofwsothatt(w)iseverywherepositiveintherange0<w<pThegroupdelayofanM-thorderstablereal-coefficientallpasstransferfunctionsatisfies:§7.1.3AllpassTransferFunctionASimpleApplicationAsimplebutoftenusedapplicationofanallpassfilterisasadelayequalizerLetG(z)bethetransferfunctionofadigitalfilterdesignedtomeetaprescribedmagnituderesponseThenonlinearphaseresponseofG(z)canbecorrectedbycascadingitwithanallpassfilterA(z)sothattheoverallcascadehasaconstantgroupdelayinthebandofinterest§7.1.3AllpassTransferFunctionG(z)A(z)

OverallgroupdelayisthegivenbythesumofthegroupdelaysofG(z)andA(z)

Since,wehave§7.1.3AllpassTransferFunctionExample:Figurebelowshowsthegroupdelayofa4thorderellipticfilterwiththefollowingspecifications:ωp=0.3π,δp=1dB,δs=35dB§7.1.3AllpassTransferFunctionFigurebelowshowsthegroupdelayoftheoriginalellipticfiltercascadedwithan8thorderallpasssectiondesignedtoequalizethegroupdelayinthepassband§7.2ClassificationBasedonPhaseCharacteristicsAsecondclassificationofatransferfunctioniswithrespecttoitsphasecharacteristicsInmanyapplications,itisnecessarythatthedigitalfilterdesigneddoesnotdistortthephaseoftheinputsignalcomponentswithfrequenciesinthepassband§7.2.1Zero-PhaseTransferFunctionsOnewaytoavoidanyphasedistortionistomakethefrequencyresponseofthefilterrealandnonnegative,i.e.,todesignthefilterwithazerophasecharacteristicHowever,itisimpossibletodesignacausaldigitalfilterwithazerophase§7.2.1Zero-PhaseTransferFunctionsFornon-real-timeprocessingofreal-valuedinputsignalsoffinitelength,zero-phasefilteringcanbeverysimplyimplementedbyrelaxingthecausalityrequirementOnezero-phasefilteringschemeissketchedbelowx[n]v[n]u[n]w[n]H(z)H(z)u[n]=v[-n],y[n]=w[-n]§7.2.1Zero-PhaseTransferFunctionsItiseasytoverifytheaboveschemeinthefrequencydomainLetX(ej),V(ej),U(ej),W(ej),andY(ej) denotetheDTFTsofx[n],v[n],u[n],w[n],andy[n],respectivelyFromthefigureshownearlierandmakinguseofthesymmetryrelationswearriveattherelationsbetweenvariousDTFTsasgivenonthenextslide§7.2.1Zero-PhaseTransferFunctionsV(ej)=H(ej)X(ej),W(ej)=H(ej)U(ej)

U(ej)=V*(ej),Y(ej)=W*(ej)CombiningtheaboveequationswegetY(ej)=W*(ej)=H*(ej)U*(ej)

=H*(ej)V(ej)=H*(ej)H(ej)X(ej)=|H(ej)|2X(ej)Thisisazero-phasefilterwithafrequencyresponse|H(ej)|2x[n]v[n]u[n]w[n]H(z)H(z)u[n]=v[-n],y[n]=w[-n]§7.2.1Zero-PhaseTransferFunctionsThefunctionfiltfiltimplementstheabovezero-phasefilteringschemeInthecaseofacausaltransferfunctionwithanonzerophaseresponse,thephasedistortioncanbeavoidedbyensuringthatthetransferfunctionhasaunitymagnitudeandalinear-phasecharacteristicinthefrequencybandofinterest§7.2.2Linear-PhaseTransferFunctionsThemostgeneraltypeofafilterwithalinearphasehasafrequencyresponsegivenbyH(ej)=e-jD whichhasalinearphasefromw=0tow=2pNotealso|H(ej)|=1

()=D

§7.2.2Linear-PhaseTransferFunctionsTheoutputy[n]ofthisfiltertoaninput x[n]=Aejnisthengivenbyy[n]=

Aej(n-D)

Ifxa(t)andya(t)representthecontinuous-timesignalswhosesampledversions,sampledatt=nT,arex[n]andy[n]givenabove,thenthedelaybetweenxa(t)andya(t)ispreciselythegroupdelayofamountD§7.2.2Linear-PhaseTransferFunctionsIfDisaninteger,theny[n]isidenticaltox[n],butdelayedbyDsamplesIfDisnotaninteger,y[n],beingdelayedbyafractionalpart,isnotidenticaltox[n]Inthelattercase,thewaveformoftheunderlyingcontinuous-timeoutputisidenticaltothewaveformoftheunderlyingcontinuous-timeinputanddelayedDunitsoftime

§7.2.2Linear-PhaseTransferFunctionsFigurerightshowsthefrequencyresponseifalowpassfilterwithalinear-phasecharacteristicinthepassband§7.2.2Linear-PhaseTransferFunctionsSincethesignalcomponentsinthestopbandareblocked,thephaseresponseinthestopbandcanbeofanyshapeExample-Determinetheimpulseresponseofanideallowpassfilterwithalinearphaseresponse:§7.2.2Linear-PhaseTransferFunctionsApplyingthefrequency-shiftingpropertyoftheDTFTtotheimpulseresponseofanidealzero-phaselowpassfilterwearriveat

Asbefore,theabovefilterisnoncausalandofdoublyinfinitelength,andhence,unrealizable§7.2.2Linear-PhaseTransferFunctionsBytruncatingtheimpulseresponsetoafinitenumberofterms,arealizableFIRapproximationtotheideallowpassfiltercanbedevelopedThetruncatedapproximationmayormaynotexhibitlinearphase,dependingonthevalueofn0chosen§7.2.2Linear-PhaseTransferFunctionsIfwechoosen0=N/2withNapositiveinteger,thetruncatedandshiftedapproximation^willbealengthN+1causallinear-phaseFIRfilter§7.2.2Linear-PhaseTransferFunctionsFigurebelowshowsthefiltercoefficientsobtainedusingthefunctionsincfortwodifferentvaluesofNN=12N=13§7.2.2Linear-PhaseTransferFunctionsBecauseofthesymmetryoftheimpulseresponsecoefficientsasindicatedinthetwofigures,thefrequencyresponseofthetruncatedapproximationcanbeexpressedas:^^~where,calledthezero-phaseresponseoramplituderesponse,isarealfunctionofw~★§7.2.3Minimum-PhaseandMaximum-PhaseTransferFunctions

Bothtransferfunctionshaveapoleinsidetheunitcircleatthesamelocationz=-aandarestableButthezeroofH1(z)isinsidetheunitcircleatz=-b,whereas,thezeroofH2(z)isatz=1/b situatedinamirror-imagesymmetry

Considerthetwo1st-ordertransferfunctions:§7.2.3Minimum-PhaseandMaximum-PhaseTransferFunctionsFigurebelowshowsthepole-zeroplotsofthetwotransferfunctionsH1(z)H2(z)§7.2.3Minimum-PhaseandMaximum-PhaseTransferFunctionsHowever,bothtransferfunctionshaveanidenticalmagnitudefunctionas

Thecorrespondingphasefunctionsare§7.2.3Minimum-PhaseandMaximum-PhaseTransferFunctionsAcausalstabletransferfunctionwithallzerosinsidetheunitcircleiscalledaminimum-phasetransferfunctionAcausalstabletransferfunctionwithallzerosoutsidetheunitcircleiscalledamaximum-phasetransferfunctionAnynonminimum-phasetransferfunctioncanbeexpressedastheproductofaminimum-phasetransferfunctionandastableallpasstransferfunction§7.2.3Minimum-PhaseandMaximum-PhaseTransferFunctionsGeneralizingtheaboveresult,let

Hm(z)beacausalstabletransferfunctionwithallzerosinsidetheunitcircleandletH(z)beanothercausalstabletransferfunctionsatisfying|H(ejω)|=|Hm(ejω)|ThesetwotransferfunctionsarethenrelatedthroughH(z)=Hm(z)A(z)whereA(z)isacausalstableallpassfunction§7.2.3Minimum-PhaseandMaximum-PhaseTransferFunctionsExample7.4(p367):considerthemixed-phasetransferfunction

WecanrewriteH(z)as§7.2.3Minimum-PhaseandMaximum-PhaseTransferFunctionsAmin-phasecausalstabletransferfunctionHm(z)alsohasadelaythatissmallerthanthegroupdelayofanonmin-phasesystemwhichhavethesamemagnituderesponse.Forsamemagnituderesponse,and§7.3TypesofLinear-PhaseFIRTransferFunctionsItisnearlyimpossibletodesignalinear-phaseIIRtransferfunctionItisalwayspossibletodesignanFIRtransferfunctionwithanexactlinear-phaseresponseWenowdeveloptheformsofthelinear-phaseFIRtransferfunctionH(z)withrealimpulseresponseh[n]ConsideracausalFIRtransferfunctionH(z)oflengthN+1,i.e.,oforderN:§7.3TypesofLinear-PhaseFIRTransferFunctionsIfH(z)istohavealinear-phase,itsfrequencyresponsemustbeoftheformWherecandβareconstants,and,calledtheamplituderesponse,alsocalledthezero-phaseresponse,isarealfunctionofω

★§7.3TypesofLinear-PhaseFIRTransferFunctionsForarealimpulseresponse,themagnituderesponse|H(ejω)|isanevenfunctionof,i.e.,|H(ejω)|=|H(e-jω)|Since,theamplituderesponseistheneitheranevenfunctionoranoddfunctionofω,i.e.§7.3TypesofLinear-PhaseFIRTransferFunctionsThefrequencyresponsesatisfiestherelationH(ejω)=H*(e-jω),orequivalently,therelationIfisanevenfunction,thentheaboverelationleadstoejβ=e-jβimplyingthateitherβ=0orβ=π§7.3TypesofLinear-PhaseFIRTransferFunctionsFromWehaveSubstitutingthevalueofβintheaboveweget§7.3TypesofLinear-PhaseFIRTransferFunctionsReplacingωwith–ω

inthepreviousequationwegetMakingachangeofvariablel=N-n,werewritetheaboveequationas§7.3TypesofLinear-PhaseFIRTransferFunctionsAs,wehaveh[n]e-jω(c+n)=h[N-n]ejω(c+N-n)Theaboveleadstotheconditionh[n]=h[N-n],0≤n≤NWithc=-N/2Thus,theFIRfilterwithanevenamplituderesponsewillhavealinearphaseifithasasymmetricimpulseresponse§7.3TypesofLinear-PhaseFIRTransferFunctionsIfisanoddfunctionof

ω,thenfromWegetejβ=-e-jβas

Theaboveissatisfiedifβ=2π

orβ=-2πThenReducesto§7.3TypesofLinear-PhaseFIRTransferFunctionsThelastequationcanberewrittenasAs,fromtheaboveweget§7.3TypesofLinear-PhaseFIRTransferFunctionsMakingachangeofvariablel=N-n,werewritethelastequationasEquatingtheabovewithWearriveattheconditionforlinearphaseas§7.3TypesofLinear-PhaseFIRTransferFunctionsh[n]=-h[N-n],0nN withc=-N/2ThereforeaFIRfilterwithanoddamplituderesponsewillhavelinear-phaseresponseifithasanantisymmetricimpulseresponse§7.3TypesofLinear-PhaseFIRTransferFunctionsSincethelengthoftheimpulseresponsecanbeeitherevenorodd,wecandefinefourtypesoflinear-phaseFIRtransferfunctionsForanantisymmetricFIRfilterofoddlength,i.e.,Neven h[N/2]=0Weexaminenexttheeachofthe4cases§7.3TypesofLinear-PhaseFIRTransferFunctionsType1:N=8Type2:N=7Type3:N=8Type4:N=7§7.3TypesofLinear-PhaseFIRTransferFunctionsType1:SymmetricImpulseResponsewithOddLengthInthiscase,thedegreeNisevenAssumeN=8forsimplicityThetransferfunctionH(z)isgivenby§7.3TypesofLinear-PhaseFIRTransferFunctionsBecauseofsymmetry,wehaveh[0]=h[8],h[1]=h[7],h[2]=h[6],andh[3]=h[5]Thus,wecanwrite§7.3TypesofLinear-PhaseFIRTransferFunctionsThecorrespondingfrequencyresponseisthengivenby

Thequantityinsidethebracesisarealfunctionofw,andcanassumepositiveornegativevaluesin