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H62SPC–SignalProcessingandControlEngineering–SignalPartH62SPC2016-2017Chapter9:DigitalFilterChengboWangConvolution:DefinitionandEquations
Page2Example1
Page3ConvolutionTheorem
Page4LTIsystem
Page5Z-TransformFourierTransformistoevaluatez-transformonaunitcircle.StableSystemAstablesystemrequiresthatitsFouriertransformisuniformlyconvergent.Fact:Fouriertransformistoevaluatez-transformonaunitcircle.AstablesystemrequirestheROCofz-transformtoincludetheunitcircle.Page7ShiftPage8Example:x(n)=anu(n)
Page9KeyPointsFlowChartforfiltersUnitCircleZerosandPolesFrequencyResponseofafilterStabilityofafilterIIRandFIRdesign.Page10BasicOperationsAddition:
Multiplicationbyaconstant:DelaysignalbyatimeT(timebetweensamples):Page11Exampley[n]=y[n-1]+3x[n]Page12ReviewQuestionDF1Drawadiagramshowinghowthefollowingdigitalfiltersareimplemented:
(i) y[n]=0.4y[n-1]+0.6x[n]
(ii) y[n]=x[n]+0.5x[n-1]
Page13ReviewQuestionDerivetherelationshipbetweenx[n]andy[n]Page14LTISystem:TransformFunctionh(n)H(z)x(n)y(n)=x(n)*h(n)X(z)Y(z)=X(z)H(z)Page15FrequencyResponseofadigitalFilter
UnitcirclePage16FrequencyResponse
Page17ReviewQuestionDF51. Thetransferfunctionofaparticulardigitalfilterisgivenby:Writedownthefrequencyresponseforthisfilter.2.Samplesforaparticulardigitalsignalaregivenby
x[0]=1.3,x[1]=-2.3,x[2]=4,x[3]=3
(i) Determinethez-transformforthissignal
(ii) Usingyouranswerto(i),determinetheFouriertransformforthissignal.
Page18UnitCircle
Page19GeneralFormofH(z)
whereP(z)andQ(z)arepolynomialsinz.Zeros:
Thevaluesofz’ssuchthatH(z)=0markedonthez-planebycircles:'O‘Poles:
Thevaluesofz’ssuchthatH(z)=
markedonthez-planebycrosses:'X'Page20ExampleSoithasonezero:(z=0)andonepole:(z=0.7)Page21Example2
Page22Example3
Page23REVIEWQUESTIONSDF6Drawtheplacementofpolesandzerosinthez-planeforthefollowingdigitalfilters:(i) H(z)=z2–1(ii)(iii)
Page24StabilityofAFilterConditions:Allthepolesmustbeinsidetheunitcircle.Exceptwhenthereisaco-incidentpoleandzero(pole-zeropair)on,oroutside,theunitcircleIsolatedpolesonoroutsidetheunitcirclearenotallowed.Page25REVIEWQUESTIONSDF7Whichsystemisstable?(i) H(z)=z2–1(ii)(iii)
Page26SymmetryPropertyThepolesandzerosforadigitalfilteraredistributedsymmetricallyabouttherealaxisinthez-plane:Ifthereisazero(orpole)atinthez-plane,thereisalsoanotherzero(orpole)at.Solitarypolesandzerosonlyexistalongtherealaxisinthez-plane.
Page27ReviewQuestionDF8Thefollowingisanpletediagramshowingtheplacementofpolesandzerosintheupperhalfofthez-planeandtherealaxisonlyforaparticulardigitalfilter.Completethisdiagram.Page28DeriveAmplitudeResponsefromzerosandpolesRule1: Ifazeroisontheunitcircle,thentheamplituderesponsegoestozeroatthefrequencycorrespondingtothatpointontheunitcircle.Rule2: Ifazeroiscloseto(ratherthanon)theunitcircle,thentheamplituderesponsewilldecreasetoaminimumatthefrequencycorrespondingtothatpointontheunitcircleclosesttothezero.
Rule3: Ifapoleisclosetotheunitcircle(mustbeinsideforstability),thentheamplituderesponseesverylargeatthepointontheunitcircleclosesttothatpole.
Rule4: Ifapoleandazeroareco-incident,thentheydonotcontributetothetransferfunction.Page29Example1Page30Example2Page31Example3Page32VectorMethodCheckthebelowlink!!!PurdueEngineeringPolesandZerosMethodDemonsPage33DeterminationofFrequencyResponsefrompole-zeropatternALTIsystemiscompletelycharacterizedbyitspole-zeropattern.Example:ReImz1p1p2DeterminationofFrequencyResponsefrompole-zeropatternALTIsystemiscompletelycharacterizedbyitspole-zeropattern.Example:ReImz1p1p2|H(ej
)|=?
H(ej
)=?DeterminationofFrequencyResponsefrompole-zeropatternALTIsystemiscompletelycharacterizedbyitspole-zeropattern.Example:ReImz1p1p2|H(ej
)|=?
H(ej
)=?|H(ej
)|=||||||
1
2
3
H(ej
)=
1(2+3)ExampleReImadBReviewQuestionDF9Sketchtheamplituderesponseofadigitalfilterwiththefollowingplacementofpolesandzerosinthez-plane–assumeasamplingfrequencyof1Hz:Page38LinearPhaseFilterItcanbeshownthatadigitalfiltercanbemadetobelinearphaseifany,oracombination,ofthefollowingconditionshold:
(1)Zerosonunitcircle(2) Zerosorpolesatoriginofz-planeAnypolesnotattheoriginofthez-planeshouldbecancelledbyazero.UsingVectormethod,youcanproveit!Page39ReviewQuestionDF10Isthefilterwithpole-zeroplacementshowninthebelowfigurelinearphase–givereasonsforyouranswer?Isthefilterwithpole-zeroplacementshowninQuestionDF9linearphase–givereasonsforyouranswer?Page40FIR
Page41IIR
Page42DesignofDigitalFiltersUsing
Pole-ZeroMethod
Page43Step2
Page44Step3
Page45DrawitsFrequencyResponseadecideshowsharpthespectrumcanbe!Page46Exercise
Page47FIRdesignBandpassfilter,maximumat125Hz,samplingfrequencyfs=1000Hz.Nosinglepolesallowed.Step1:Distributezerosaroundunitcircle,
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