chapter digital filters四章数字滤波器

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,