数字信号处理-基于计算机的方法(第四版)答案--8-11章

1SOLUTIONSMANUALtoaccompanyDigitalSignalProcessing:AComputer-BasedApproachFourthEditionSanjitK.MitraPreparedbyChowdaryAdsumilli,JohnBerger,MarcoCarli,Hsin-HanHo,RajeevGandhi,MartinGawecki,ChinKayeKoh,LucaLucchese,MyleneQueirozdeFarias,andTravisSmithCopyright©2011bySanjitK.Mitra.Nopartofthispublicationmaybereproducedordistributedinanyformorbyanymeans,orstoredinadatabaseorretrievalsystem,withoutthepriorwrittenconsentofSanjitK.Mitra,including,butnotlimitedto,inanynetworkorotherelectronicStorageortransmission,orbroadcastfordistancelearning.2Chapter8-Part28.40When€HN(z)isanallpasstransferfunctionoftheform€HN(z)=AN(z)=dN+dN−1z−1++d1z−N+1+z−N1+d1z−1++dN−1z−N+1+dNz−N,thenfromEq.(8.151a),thenumeratorcoefficientsof€HN−1(z)aregivenby€pk'=p0dk+1−pk+1p0dN−pN=dNdk+1−dN−k−1dN2−1,andfromEq.(8.151b)thedenominatorcoefficientsof€HN−1(z)aregivenby€dN−k−1'=pN−k−1dN−dN−k−1p0dN−pN=dk+1dN−dN−k−1dN2−1=pk',implying€HN−1(z)isanallpasstransferfunctionoforder€N−1.Sincehere€pN=1and€p0=dN,thelatticestructureofProblem8.38thenreducestothelatticestructureemployedintheGray-Markelrealizationprocedure.8.41(a)ConsidertherealizationofType1Ballpassstructure.FromitstransferparametersgiveninEq.(8.50b)wearriveat€Y1=z−1X1+(1+z−1)X2=z−1(X1+X2)+X2,and€Y2=(1−z−1)X1−z−1X2=X1−z−1(X1+X2).Arealizationofthetwo-pairbasedonthesetwoequationsisasshownbelowwhichleadstothestructureofFigure8.24(b).(b)FromthetransferparametersofType1AtallpassgiveninEq.(8.50c)weobtain€Y1=z−1X1+X2,and€Y2=(1−z−2)X1−z−1X2=X1−z−1(z−1X1+X2)=X1−z−1Y1.Arealizationofthetwo-pairbasedonthesetwoequationsisasshownonnextpagewhichleadstothestructureofFigure8.24(c).(c)FromthetransferparametersofType1AtallpassgiveninEq.(8.50d)weobtain€Y1=z−1X1+(1−z−1)X2=z−1(X1−X2)+X2,and€Y2=(1+z−1)X1−z−1X2€=X1+z−1(X1−X2).Arealizationofthetwo-pairbasedonthesetwoequationsisasshownbelowwhichleadstothestructureofFigure8.24(d).38.42(a)Acascadeconnectionof3Type1Afirst-orderallpassnetworksisshownbelow:Simpleblockdiagrammanipulationoftheabovestructureleadsto:Finally,bydelaysharingbetweenadjacentallpasssectionswearriveatthefollowingequivalentrealizationrequiringnow4delayscomparedto6delaysinthedirectrealizationshownonthepreviouspage.(b)Acascadeconnectionof3Type1Atfirst-orderallpassnetworksisshownbelowwhichrequires8delays.4Bydelaysharingbetweenadjacentallpasssectionswearriveatthefollowingequivalentrealizationrequiringnow4delays.8.43ThestructureofFigureP8.15withinternalvariableslabeledisshownbelow:Itsanalysisyields(1):€W(z)=X(z)−z−1U(z),(2):€U(z)=aW(z)+z−1U(z),and(3):€Y(z)=−W(z)+U(z).FromEq.(2)weobtain(4):€U(z)=a1−z−1W(z).SubstitutingEq.(4)inEq.(3)weget(5):€Y(z)=−1−a−z−11−z−1⎛⎝⎜⎜⎞⎠⎟⎟W(z).SubstitutingEq.(4)inEq.(1)weget€X(z)=1−a−z−1+az−11−z−1⎛⎝⎜⎜⎞⎠⎟⎟W(z).Finally,fromEqs.(5)and(6)wearriveat€H(z)=Y(z)X(z)=−(1−a)+z−11−(1−a)z−1.8.44Werealize€A2(z)=d1d2+d1z−1+z−21++d1z−1+d1d2z−2intheformofaconstrainedthree-pairasindicatedbelow:Y(z)Y25Fromtheabovefigure,wehave€Y1Y2Y3⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥=t11t12t13t21t22t23t31t32t33⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥X1X2X3⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥,and€X2=−d1Y2,€X3=d2Y3.Fromtheseequations,wegetaftersomealgebra€A2(z)=Y1X1=N(z)D(z),where€D(z)=1+d1t22−d2t33+d1d2(t23t32−t22t33),and€N(z)=t11+d1(t11t22−t12t21)−d2(t11t33−t13t31)€+d1d2t21(t12t33−t13t32)+t31(t22t13−t12t23)+t11(t23t32−t22t33){}.Comparingthedenominatorofthedesiredallpasstransferfunctionwith€D(z)weget€t22=z−1,t33=0,t23t32=z−2.Next,comparingthenumeratorofthedesiredallpasstransferfunctionwith€N(z)weget€t11=z−2,t12t21=z−1(z−2−1),t13t31=0,and€t32(t11t23−t21t13)+t31(t22t13−t12t23)=1.Substitutingtheappropriatetransferparametersfromthepreviousequationsintothelastequationwesimplifyitto€t13t21t32+t31t12t23=z−4−1.Since€t13t31=0,either€t13=0,or€t31=0.(Bothcannotbesimultaneouslyequaltozero,asthiswillviolatethecondition€t13t21t32+t31t12t23€=z−4−1.Considerthecase€t13=0.Thentheaboveequationreducesto€t31t12t23=z−4−1.Fromthisequationand€t23t32=z−2,itfollowsthat€t32=z−2,t23=1,t31t12=z−4−1€=(z−1−1)(z−1+1)(z−2+1).Therearefourpossiblerealizablesetsofvaluesof€t21and€t31satisfyingthelastequationand€t12t21=z−1(z−2−1).Theseleadto4differentrealizabletransfermatricesforthethree-pair:Type2A:€z−2z−2−10z−1z−11z−2+1z−20⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥,Type2B:€z−2z−1+10z−1(z−1−1)z−11(z−2+1)(z−1−1)z−20⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥,Type2C:€z−2z−1−10z−1(z−1+1)z−11(z−2+1)(z−1+1)z−20⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥,Type2D:€z−210z−1(z−2−1)z−11z−4−1z−20⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥.ArealizationofeachoftheaboveType2allpassstructuresisobtainedbyimplementingitsrespectivetransfermatrix,andthenconstrainingthe€Y2and€X2variablesthroughthemultiplier€d1andconstrainingthe€Y3and€X3variablesthroughthemultiplier€−d2resultinginthefourstructuresshowninFigure8.25ofthetext.Itcanbeeasilyshownthattheallpassstructuresobtainedforthecase€t31=0arepreciselythetransposeofthestructuresofFigure8.25.68.45AcascadeoftwoType2Dsecond-orderallpassnetworksisshownbelowwhichrequires8delays.Bydelaysharingbetweenadjacentallpasssectionswearriveatthefollowingequivalentrealizationsrequiringnow6delays.TheminimumnumberofdelaysneededtoimplementacascadeofType2Dsecond-orderal