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–Part18.1Analysisyields€Y(z)=G(z)X(z)−C(z)Y(z)().Hence,€H(z)=Y(z)X(z)=G(z)1+G(z)C(z)=2(1+2K)+3z−1.TheoveralltransferfunctionH(z)isgivenby€H(z)=z−21+1.5z−1+(K+0.5)z−2.Thetransferfunctionisstableif€K+0.5<1and€1.5<1+K+0.5.Fromthefirstinequalitywehave€−1.5<K<0.5andfromthesecondinequalitywehaveK>0.HenceH(z)isstableif€0<K<0.5.8.2TheoveralltransferfunctionH(z)isgivenby€H(z)=z−11+(1.5+K)z−1+0.5z−2.ThusThetransferfunctionisstableif€1.5+K<1.5whichissatisfiedif€−3<K<0.8.3FromtheresultsofProblem8.3,wehave€H(z)=G(z)1+G(z)C(z)whichcanbesolvedyielding€C(z)=G(z)−H(z)G(z)H(z).Substitutingtheexpressionsfor€G(z)and€H(z)inthisexpressionweget€C(z)=1.2+0.4667z−1−1.8133z−2−4.2867z−3−3.735z−4−1.9275z−5−0.9z−61+2.3667z−1+3.65z−2+3.7617z−3+2.9217z−4+1.49z−5+0.56z−6.Pole-zeroplotsof€G(z),C(z),and€H(z)obtainedusingzplanecanbeeasilyobtained.8.4Thestructurewithinternalvariablesisshownbelow.Analysisofthisstructureyields€U(z)=KX(z)+d2z−1V(z),€V(z)=U(z)−d1z−1V(z),€Y(z)=d1z−1V(z)−z−2V(z)−d1V(z).Eliminatingtheinternalvariableswearriveat€H(z)=Y(z)X(z)=K(−d2+d1z−1+z−2)1++d1z−1−d2z−2.d12d_1++z1_z1_+z1_z1__1KX(z)Y(z)U(z)V(z)3(a)Sincethetransferfunctionissecond-order,thestructureisnon-canonic.(b)€H(ej0)=K(−d2+d1+1)1+d1−d2=K.Hencethestructurehasaunitygainatω=0ifK=1.(c)€H(ejπ)=K(−d2−d1+1)1−d1−d2=K.Hencethestructurehasaunitygainatω=πifK=1.(d)Ifwelet€D(z)=1+d1z−1−d2z−2,then€H(z)=Kz−2D(z−1)D(z).Now,€H(z)H(z−1)=Kz−2D(z−1)D(z)⋅Kz2D(z)D(z−1)=K2.Thisimplies€H(ejω)2=H(z)H(z−1)z=ejω=K2,orinotherwordsthetransferfunctionhasaconstantmagnitudeforallvaluesofω.8.5Thestructurewithinternalvariablesisshownbelow.Analysisofthisstructureyields€V(z)=γ2X(z)+U(z),€U(z)=k1Y(z)+γ1X(z),€Y(z)=z−1V(z)+k2z−1U(z).Eliminatingtheinternalvariableswearriveat€H(z)=Y(z)X(z)=[(1+k2)γ1+γ2]z−11+k1(1+k2)z−1.Forstabilitywerequire€k1(1+k2)<1.8.6AnequivalentrepresentationofthestructureofFigureP8.4withinternalvariablesisshownbelow.Analysisofthisstructureyields€U(z)=X(z)+α0(z−1−1)W(z),€W(z)=α1(z−1−1)U(z)−Y(z),€Y(z)=U(z)−α2(z−1−1)Y(z).z1_z1_++k1k21γ2γX(z)Y(z)+U(z)V(z)X(z)Y(z)1_U(z)W(z)0α(z1)__11α(z1)__12α(z1)__14Eliminatingtheinternalvariableswearriveat€H(z)=Y(z)X(z)=1D(z)where€D(z)=[1−(α0+α2)−α0α1+α0α1α2]+[(α0+α2)+2α0α1−3α0α1α2]z−1€+[−α0α1+3α0α1α2]z−2−α0α1α2z−3.8.7AnequivalentrepresentationofthestructureofFigureP8.4withinternalvariablesisshownbelow.Let€Ti(z)=βiz−11−αiz−1,i=1,2,3.Thenanalysisofthestructureyields€V(z)=X(z)+T2(z)U(z),€W(z)=T1(z)V(z),€U(z)=W(z)+T3(z)V(z),€Y(z)=α0X(z)+W(z).Eliminatingtheinternalvariableswearriveat€H(z)=Y(z)X(z)=1D(z)where8.8Thestructurewithinternalvariablesisshownbelow.Analysisofthisstructureyields(1):€W(z)=X(z)+k1Y(z),(2):€U(z)=11−z−1W(z)+k2Y(z),and(3):€Y(z)=−k11−z−1U(z).SubstitutingEq.(2)inEq.(3)weget(4):€Y(z)=−k11−z−111−z−1W(z)+k2Y(z)⎛⎝⎜⎞⎠⎟=−k1(1−z−1)2W(z)−k1k21−z−1Y(z).SubstitutingEq.(1)inEq.(4)wethenget€Y(z)=−k1(1−z−1)2[X(z)+k1Y(z)]−k1k21−z−1Y(z)€=−k1(1−z−1)2X(z)−k11−z−1k1+k2−k2z−11−z−1⎡⎣⎢⎢⎤⎦⎥⎥Y(z),or,X(z)Y(z)0α1β_1z_1z1α_12β_1z_1z2α_1V(z)W(z)3β_1z_1z3α_1U(z)5€1+k1(k1+k2−k2z−1)(1−z−1)2⎡⎣⎢⎢⎤⎦⎥⎥Y(z)=−k1(1−z−1)2X(z).Hence,€H(z)=Y(z)X(z)=−k1[1+k1(k1+k2)]−(2+k1k2)z−1+z−2.8.10FromFigureP8.7(a),theinput-outputrelationofthechannelisgivenby€Y1(z)Y2(z)⎡⎣⎢⎤⎦⎥=1H12(z)H21(z)1⎡⎣⎢⎤⎦⎥X1(z)X2(z)⎡⎣⎢⎤⎦⎥.Likewise,theinput-outputrelationofthechannelseparationcircuitofFigureP8.7(b)isgivenby€V1(z)V2(z)⎡⎣⎢⎤⎦⎥=1−G12(z)−G21(z)1⎡⎣⎢⎤⎦⎥Y1(z)Y2(z)⎡⎣⎢⎤⎦⎥.Hence,theoverallsystemischaracterizedby€V1(z)V2(z)⎡⎣⎢⎤⎦⎥=1−G12(z)−G21(z)1⎡⎣⎢⎤⎦⎥1H12(z)H21(z)1⎡⎣⎢⎤⎦⎥X1(z)X2(z)⎡⎣⎢⎤⎦⎥€=1−H21(z)G12(z)H12(z)−G12(z)H21(z)−G21(z)1−H12(z)G21(z)⎡⎣⎢⎤⎦⎥X1(z)X2(z)⎡⎣⎢⎤⎦⎥.Thecross-talkiseliminatedif€V1(z)isafunctionofeither€X1(z)or€X2(z),andsimilarly,if€V2(z)isafunctionofeither€X1(z)or€X2(z).Fromtheaboveequationitfollowsthatif€H12(z)=G12(z),and€H21(z)=G21(z),then€V1(z)=1−H21(z)G12(z)()X1(z),and€V2(z)=1−H12(z)G21(z)()X2(z).Alternately,if€G12(z)=H21−1(z),and€G21(z)=H12−1(z),then€V1(z)=H12(z)H21(z)−1H21(z)⎛⎝⎜⎞⎠⎟X2(z),and€V2(z)=H12(z)H21(z)−1H12(z)⎛⎝⎜⎞⎠⎟X1(z).8.11AnalyzingFigureP8.8,weget€w[n]=AX[n]+CDu[n]()and€y[n]=CABx[n]+u[n]().Adirectimplementationofthesetwoequationsleadstothestructureshownbelowwhichhasnodelay-freeloop.8.12(a)FigureP8.9(a)withinternaladderoutputvariablesisshownbelow:6Theoutputsofthe4addersaregivenby€Y(z)=z−1W3(z)+α1W1(z),W1(z)=X(z)+α2W3(z),W2(z)=W1(z)+k2W2(z),and€W3(z)=k1W2(z)+z−1W2(z).Fromthe3rdequationgivenaboveweobtain€W2(z)=W1(z)1−k2andfromthe4thequationgivenaboveweobtain€W3(z)=(z−1+k1)W2(z)=(z−1+k1)1−k2W1(z).Substitutingtheexpressionfor€W3(z)inthefirstequationgivenabovewearriveat€Y(z)=z−1(z−1+k1)1−k2W1(z)+α1W1(z)=z−1(z−1+k1)+(1−k2)α11−k2W1(z).Substitutingtheexpressionfor€W3(z)givenaboveinthesecondequationweget€W1(z)=X(z)+α2(z−1+k1)1−k2W1(z)whichleadsto€W1(z)=(1−k2)X(z)1−k2−α2z−1−α2k1.Substitutingthisexpressionfor€W1(z)inthelastexpressionforY(z)wefinallyarriveattheexpressionforinput-outputrelationofthestructureofFigureP8.9(a)€Y(z)=z−1(z−1+k1)+(1−k2)α11−k2−α2z−1−α2k1X(z).Itcanbeseenthattherearetwodelay-freeloopsinthestructure:onegoingthroughthemultipliers€k1and€α2,andtheotheronegoingthroughthemultiplier€k2.Todeveloptheequivalentrealizationwithoutanydelay-freeloopwefirstremovethetwounitdelaysasshownbelow:X(z)Y(z)k1k2z1_α2α1z1_W(z)1W(z)2W(z)3X(z)Y(z)k1k2α2α11X(z)1Y(z)2X(z)2Y(z)7WethenremovetheoutputvariableY(z)andthetwoinputstoitsadderasshownbelow:Ascanbeseenfromtheabovestructure,therearetwodelay-freeloopswithloopgains€α2k1and€k2.Thecorrespondingexpressionforthedeterminantis€Δ=1−α2k1−k2.Theabovestructureisnextredrawnasindicatedbelow:Thetransfermatrixoftheabovethree-pairisgivenby€Y(z)Y1(z)Y2(z)⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥=t11t12t13t21t22t23t31t32t33⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥=1Δα1(1−k2)Δα1α2(1−k2)k10(1−k2)10α2⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥X(z)X1(z)X2(z)⎡⎣⎢⎢⎢⎤⎦⎥⎥⎥.Arealizationoftheabovestructurewithoutanydelay-freeloopisthusasshownbelow:(b)FigureP8.9(b)isredrawnbelowwiththedashedblockcontainingthetwodelays.X(z)k1k2α21Y(z)2X(z)2Y(z)X(z)k2α21Y(z)2X(z)2Y(z)1k/k2Y(z)X(z)Y(z)1X(z)1X(z)2Y(z)2t1121t23t33t31t13t8FigureP8.9(b)withdelaysremovedisasshownbelowTheonlydelay-freeloopgoesthroughthemultipliers€γand€δ.Thecorrespondingloop-gainis€γδandthedeterminantis€Δ=1−loopgain=1−γδ.Thetransfermatrixoftheabovetwo-pairisgivenby€Y(z)Y1(z)⎡⎣⎢⎤⎦⎥=t11t12t21t22⎡⎣⎢⎤⎦⎥X(z)X1(z)⎡⎣⎢⎤⎦⎥=1Δγ11δ⎡⎣⎢⎤⎦⎥X(z)X1(z)⎡⎣⎢⎤⎦⎥.ArealizationofthestructureofFigureP8.9(b)withoutanydelay-freeloopisthusasshownbelow:8.13(a)€H(z)=(1+0.4z−1)4(1−0.2z−1)2€=1+1.2z−1+0.36z−2−0.64z−3−0.0384z−4+0.001z−6.AdirectformrealizationofH(z)isshownbelow:Y(z)βγδz–1z–1X(z)Y(z)1+X(z)1Y(z)γδX(z)Y(z)1+X(z)1Y(z)βγδz–1z–1X(z)Y(z)1+X(z)1∆1/∆1/9Thetransposedformoftheabovestructureyieldsanotherdirectformrealizationasindicatedbelow:(b)Arealizationintheformofcascadeofsixfirst-ordersectionsisshownbelow:(c)Arealizationintheformofcascadeofthreesecond-ordersectionsisshownbelow:(d)Arealizationintheformofcascadeoftwothird-ordersectionsisshownbelow:(e)Arealizationintheformofcascadeoftwofirst-ordersectionsandtwosecond-ordersectionsisshownbelow:z1_z1_z1_+++z1_+x[n]y[n]z1_+z1_1.20.360.0010.0384_0.064_x[n]y[n]z–1z–1z–1z–1z–1z–11.20.360.0010.0384_0.064_x[n]y[n]z1_+z1_+z1_+z1_+z1_+z1_+0.40.40.40.4_0.2_0.2x[n]y[n]z1_+z1_+z1_+z1_+0.80.80.160.16z1_+z1_+0.12_0.12_z1_z1_++z1_+x[n]y[n]z1_z1_+z1_+1.20.480.0640.0160.12_x[n]y[n]z1_+z1_+z1_+z1_+0.80.80.160.16z1_+z1_+0.12_0.12_108.14€H(z)=E0(z4)+z−1E1(z4)+z−2E2(z4)+z−3E3(z4)where,€E0(z)=a+ez−1,E1(z)=b+fz−1,E2(z)=c+gz−1,E3(z)=d+hz−1.Adirect4-bandpolyphaserealizationofH(z)isshownbelowwhichrequires18unitdelaysandhenceisanon-canonicstructure.Aminimum-delayrealizationoftheabovestructureisshownbelowwhichrequires7unitdelaysandhenceitisacanonicstructure.8.15€H(z)=E0(z3)+z−1E1(z3)+z−2E2(z3)where,€E0(z)=a+dz−1+gz−2,E1(z)=b+ez−1+hz−2,E2(z)=c+fz−1.Adirect3-bandpolyphaserealizationofH(z)isshownbelowwhichrequires17unitdelaysandhenceisanon-canonicstructure.x[n]y[n]z1_z1_4_a+ez4_c+gz4_b+fz4_d+hz++z1_+x[n]y[n]z4_z1_++z1_+z1_++++abcdefgh11Aminimum-delayrealizationoftheabovestructureisshownbelowwhichrequires8unitdelaysandhenceitisanon-canonicstructure.8.16€H(z)=E0(z2)+z−1E1(z2)where,€E0(z)=a+cz−1+ez−2+gz−3,E1(z)=b+dz−1+fz−2+hz−3.Adirect2-bandpolyphaserealizationofH(z)isshownbelowwhichrequires20unitdelaysandhenceisanon-canonicstructure.x[n]y[n]z1_+z1_+3_a+dz+gz6_3_b+ez+hz6_3_c+fzx[n]y[n]z3_z1_++z1_+++abcdefghz3_x[n]y[n]z1_+2_a+cz+ez+gz4_6_2_b+dz+fz+hz4_6_12Aminimum-delayrealizationoftheabovestructureisshownbelowwhichrequires7unitdelaysandhenceitisacanonicstructure.8.178.188.19€G(z)=z−N/2−H(z).Acanonicrealizationofboth€G(z)and€H(z)isshownbelowfor€N=6.x[n]y[n]z2_+z1_+abcdefghz2_z2_+++z1_+h[0]h[1]z1_z1_z1_++x[n]h[3]z1_z1_++y[n]h[2]z1_z1_++_1_1_1_1z1_h[0]h[1]h[2]z1_z1_z1_z1_++h[3]+++z1_z1_+z1_z1_++h[4]+_1_1_1_1_1x[n]y[n]138.20Withoutanylossofgenerality,assume€M=5whichmeans€N=11.Inthiscase,thetransferfunctionisgivenby€=z−5h[5]+h[4](z+z−1)+h[3](z2+z−2)+h[2](z3+z−3)+h[1](z4+z−4)+h[0](z5+z−5)[].Now,therecursionrelationfortheChebyshevpolynomialisgivenby€Tr(x)=2xTr−1(x)−Tr−2(x),r≥2with€T0(x)=1and€T1(x)=x.Hence,€T2(x)=2xT1(x)−T0(x)=2x2−1,€T3(x)=2xT2(x)−T1(x)=2x(2x2−1)−x=4x3−3x,€T4(x)=2xT3(x)−T2(x)=2x(4x3−3x)−(2x2−1)=8x4−8x2+1,€T5(x)=2xT4(x)−T3(x)=2x(8x4−8x2+1)−(4x3−3x)=16x5−20x3+5x.Wecanthusrewritetheexpressioninsidethesquarebracketsgivenaboveas€h[5]+2h[4]T1z+z−12⎛⎝⎜⎞⎠⎟+2h[3]T2z+z−12⎛⎝⎜⎞⎠⎟+2h[2]T3z+z−12⎛⎝⎜⎞⎠⎟+2h[1]T4z+z−12⎛⎝⎜⎞⎠⎟+2h[0]T5z+z−12⎛⎝⎜⎞⎠⎟€=h[5]+2h[4]z+z−12⎛⎝⎜⎞⎠⎟+2h[3]2z+z−12⎛⎝⎜⎞⎠⎟2−1⎡⎣⎢⎢⎤⎦⎥⎥+2h[2]4z+z−12⎛⎝⎜⎞⎠⎟3−3z+z−12⎛⎝⎜⎞⎠⎟⎡⎣⎢⎢⎤⎦⎥⎥€+2h[1]8z+z−12⎛⎝⎜⎞⎠⎟4−8z+z−12⎛⎝⎜⎞⎠⎟2+1⎡⎣⎢⎢⎤⎦⎥⎥+2h[0]16z+z−12⎛⎝⎜⎞⎠⎟5−20z+z−12⎛⎝⎜⎞⎠⎟3+5z+z−12⎛⎝⎜⎞⎠⎟⎡⎣⎢⎢⎤⎦⎥⎥€=a[n]z+z−12⎛⎝⎜⎞⎠⎟nn=05∑,where€a[0]=h[5]−2h[3]+2h[1],a[1]=2h[4]−6h[2]+10h[0],€a[2]=4h[3]−16h[1],a[3]=8h[2]−40h[1],a[4]=16h[1],and€a[5]=32h[0].Arealizationof€H(z)=z−5a[n]z+z−12⎛⎝⎜⎞⎠⎟nn=05∑=a[0]z−5+a[1]1+z−22⎛⎝⎜⎞⎠⎟z−4€+a[2]1+z−22⎛⎝⎜⎞⎠⎟2z−3€+a[3]1+z−22⎛⎝⎜⎞⎠⎟3z−2+a[4]1+z−22⎛⎝⎜⎞⎠⎟4z−1+a[5]1+z−22⎛⎝⎜⎞⎠⎟5isshownbelow:148.21Consider€H(z)=P(z)D(z)=P1(z)D1(z)⋅P2(z)D2(z)⋅P3(z)D3(z).Assumeallzerosof€P(z)and€D(z)arecomplex.Notethatthenumeratorofthefirststagecanbeoneofthe3factors,€P1(z),€P2(z),and€P3(z).Likewise,thenumeratorofthesecondstagecanbeoneoftheremaining2factors,andthenumeratorofthethirdstageistheremainingfactor.Similarly,thatthedenominatorofthefirststagecanbeoneofthe3factors,€D1(z),€D2(z),and€D3(z).Likewise,thedenominatorofthesecondstagecanbeoneoftheremaining2factors,andthedenominatorofthethirdstageistheremainingfactor.Hence,thereare€(3!)2=36differenttypesofcascaderealizations.Ifthezerosof€P(z)and€D(z)areallreal,then€P(z)has6realzerosand€D(z)has6realzeros.Inthiscase,thenthereare€(6!)2=518400differenttypesofcascaderealizations.8.22€H(z)=Pi(z)Di(z)i=1K∏.Here,thenumeratorofthefirststagecanbechosenin€K1⎛⎝⎜⎞⎠⎟ways,thenumeratorofthesecondstagecanbechosenin€K−11⎛⎝⎜⎞⎠⎟ways,andsoon,untilthereisonlyonepossiblechoiceforthenumeratorofthe€K–thstage.Likewise,thedenominatorofthefirststagecanbechosenin€K1⎛⎝⎜⎞⎠⎟ways,thedenominatorofthesecondstagecanbechosenin€K−11⎛⎝⎜⎞⎠⎟ways,andsoon,untilthereisonlyonepossiblechoiceforthedenominatorofthe€K–thstage.Hence,thetotalnumberofpossiblecascaderealizationsareequalto€K1⎛⎝⎜⎞⎠⎟2K−11⎛⎝⎜⎞⎠⎟2K−21⎛⎝⎜⎞⎠⎟2€21⎛⎝⎜⎞⎠⎟211⎛⎝⎜⎞⎠⎟2=(K!)2.8.23Arealizationoftheringingdelayisshownbelow:Thesecond-orderdirectformIIstructurewitheachdelayreplacedbytheaboveringingdelayisshownbelow:+zD_α158.24(a)AdirectformIIrealizationof€H1(z)isshownbelow:(b)AdirectformIIrealizationof€H2(z)isshownbelow:8.25AdirectformIIrealizationof€H(z)isshownbelow:+zD_α+zD_α++++1p1_d2p0p2d_z1_z1_+++x[n]20.60.18_0.9_y[n]z1_z1_+++x[n]30.15_y[n]0.6_0.8++z1_z1_z1_+++x[n]y[n]34.5_2.9_2.20.81_5.116AdirectformII€trealizationof€H(z)isshownbelow:8.26(a)AdirectformIIrealizationof€H1(z)isshownbelow:AdirectformII€trealizationof€H1(z)isshownbelow:(b)AdirectformIIrealizationof€H2(z)isshownbelow:AdirectformII€trealizationof€H2(z)isshownbelow:+++z1_+z1_z1_x[n]y[n]34.5_2.9_2.2_5.10.81z–1z–1z–130.6_0.5_41.50.3_1x[n]y[n]+z1_+z1_0.3_1_41.5++z1_3_0.51.8z–1z–1z–1z–131.5_4.9_0.84.20.617(c)AdirectformIIrealizationof€H3(z)isshownbelow:AdirectformII€trealizationof€H3(z)isshownbelow:8.27ThetransferfunctionofthestructureofFigureP8.11is€H(z)=(1−a0z−M)⋅p0+p1z−1+p2z−21+d1z−1+d2z−2.(a)M-pointmovingaveragefilter:€HMA(z)=1M1−z−M1−z−1⎛⎝⎜⎜⎞⎠⎟⎟.Choose€a0=1,p0=1/M,p1=0,p2=0,d1=−1,d2=0.(b)Firstdifferencedifferentiator:€HFD(z)=1+z−1.Choose€a0=0,p0=1,p1=1,p2=0,d1=0,d2=0.(c)Centraldifferencedifferentiator:€HCD(z)=0.5(1−z−2).+z1_z1_+++z1_z1_+31.5_4.94.20.6_0.8z–1z–1z–140.4750.754/3_0.251/122.917_x[n]y[n]+z1_+z1_++z1_4_1.90.75_2.917_0.25+4/31/1218Choose€a0=0,p0=0.5,p1=−0.5,p2=0,d1=0,d2=0.(d)Runningsumintegrator:€HRS(z)=11−z−1.Choose€a0=0,p0=1,p1=0,p2=0,d1=−1,d2=0.(e)Leakyintegrator:€HLI(z)=α1−(1−α)z−1.Choose€a0=0,p0=α,p1=0,p2=0,d1=−(1−α),d2=0.(f)DCblocker:€HDC(z)=1−z−11−αz−1.Choose€a0=0,p0=1,p1=−1,p2=0,d1=−α,d2=0.(g)Trapezoidalintegrator:€HTI(z)=0.5+0.5z−11−z−1.Choose€a0=0,p0=0.5,p1=0.5,p2=0,d1=−1,d2=0.(h)First-orderallpassdelaynetwork:€A1(z)=α+z−11+αz−1.Choose€a0=0,p0=α,p1=0.5,p2=0,d1=−α,d2=0.(i)Second-orderallpassdelaynetwork:€A2(z)=α+βz−1+z−21+βz−1+αz−2.Choose€a0=0,p0=α,p1=β,p2=1,d1=−β,d2=α.(j)Simpsonintegrator:€HSI(z)=13+43z−1+13z−21−z−2.Choose€a0=0,p0=1/3,p1=4/3,p2=1/3,d1=0,d2=−1.8.28(a)€H(z)=1−0.6z−11+0.25z−1⋅0.2+z−11+0.3z−1⋅21+0.25z−1=0.4+1.76z−1−1.2z−21+0.8z−1+0.2125z−2+0.0187z−3.(b)€y[n]=0.4x[n]+1.76x[n−1]−1.2x[n−2]−0.8y[n−1]−0.2125y[n−2]−0.0187y[n−3].(c)Acascaderealizationofisshownbelow:19(d)Apartial-fractionexpansionofin€z−1obtainedusingresiduezisgivenby€H(z)=5481+0.25z−1+129.2(1+0.25z−1)2+−676.81+0.3z−1.TheParallelFormIrealizationbasedonthisexpansionisshownonthenextpage.(e)Theinverse€z–transformofthepartial-fractionof€H(z)giveninPart(d)yields€h[n]=548(−0.25)nµ[n]+129.2(n+1)(−0.25)nµ[n]−676.8(−0.3)nµ[n].8.29Acascaderealizationofbasedonthefactoredformgivenby€H(z)=z−11+0.8z−1+0.5z−2⎛⎝⎜⎜⎞⎠⎟⎟0.44+0.362z−1+0.02z−21−0.4z−1⎛⎝⎜⎜⎞⎠⎟⎟isshownbelow:z–1z–1z–1_0.6_0.25_0.25_0.30.22++z1_z1_z1_+++z1__676.8_0.3_0.25_0.5_0.0625548129.220Theabovestructureuses4unitdelays,6multipliers,and5two-inputadders,whereasthestructureofFigure8.19uses3unitdelays,6multipliers,and5two-inputadders.8.30(a)€H1(z)=0.9z−1+3.545z−2+1.8z−31+4.5z−1+0.5z−2−0.75z−3.Apartialfractionexpansionof€H1(z)usingresiduezyields€H1(z)=−2.4+−0.04351+4.3452z−1+0.15691+0.5z−1+2.28661−0.3452z−1whichleadstotheParallelFormIrealizationshownbelow:Apartialfractionexpansionof€H1(z)usingresidueyields€H1(z)=0.9+−0.8217z−11+4.3452z−1+0.0392z−11+0.5z−1+0.2725z−11−0.3452z−1whichleadstotheParallelFormIIrealizationshownbelow:++z1_+z1_+z1_+_2.4_0.0435_4.34520.1569_0.52.28660.3452+z1_+z1_+z1_++0.9_0.8217_4.34520.0392_0.50.27250.345221(b)€H2(z)=1.5z−1+7.05z−2+4.05z−3+0.45z−41+5.7z−2+3.92z−3.Apartialfractionexpansionof€H2(z)usingresiduezyields€H2(z)=0.1148+−1.6971+0.8049z−11+4.9z−2+1.5823+0.6951z−11+0.8z−2whichleadstotheParallelFormIrealizationshownbelow:Apartialfractionexpansionof€H2(z)usingresidueyields€H2(z)=0.8049z−1+8.3159z−21+4.9z−2+0.6951z−1−1.2659z−21+0.8z−2whichleadstothePar