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SOLUTIONSMANUALtoaccompanyDigitalSignalProcessing:AComputer-BasedApproachThirdEditionSanjitK.MitraPreparedbyChowdaryAdsumilli,JohnBerger,MarcoCarli,Hsin-HanHo,RajeevGandhi,ChinKayeKoh,LucaLucchese,andMyleneQueirozdeFariasNotforsale.1Chapter22.1(a),81.4,1396.9,85.2212111===∞xxx(b).48.3,1944.7,68.1822212===∞xxx2.2Hence,Thus,⎩⎨⎧≥=µ.0,0,0,1][nnn⎩⎨⎧≥=−−µ.0,0,0,1]1[nnn].1[][][−−µ+µ=nnnx2.3(a)ConsiderthesequencedefinedbyIfn0,thenk=0isnotincludedinthesumandhence,x[n]=0forn0.Ontheotherhand,fork=0isincludedinthesum,andasaresult,x[n]=1forTherefore,.][][∑δ=−∞=nkknx,0≥n.0≥n].[,0,0,0,1][][nnnknxnkµ=⎩⎨⎧≥=∑δ=−∞=(b)SinceitfollowsthatHence,⎩⎨⎧≥=µ,0,0,0,1][nnn⎩⎨⎧≥=−µ.1,0,1,1]1[nnn].[,0,0,0,1]1[][nnnnnδ=⎩⎨⎧≠==−µ−µ2.4Recall].[]1[][nnnδ=−µ−µHence,]3[4]2[2]1[3][][−δ+−δ−−δ+δ=nnnnnx])4[]3[(4])3[]2[(2])2[]1[(3])1[][(−µ−−µ+−µ−−µ−−µ−−µ+−µ−µ=nnnnnnnn].4[4]3[6]2[5]1[2][−µ−−µ+−µ−−µ+µ=nnnnn2.5(a)},4512302{]2[][−−−=+−=↑nxnc(b)},006310872{]3[][↑−−−=−−=nynd(c)},003221025{][][↑−−=−=nwne(d)},27801332154{]2[][][−−−−=−+=↑nynxnu(e)},040342150{]4[][][−−=+⋅=↑nwnxnv(f)},221000543{]4[][][−−−=+−=↑nwnyns(g)}.75.248.205.35.1021{][5.3][−−−==↑nynr2.6(a)],3[2]1[3][2]1[]2[5]3[4][−δ+−δ−δ−+δ++δ++δ−=nnnnnnnx],5[2]4[7]3[8]1[][3]1[6][−δ−−δ+−δ+−δ−δ−+δ=nnnnnnny],8[5]7[2]5[]4[2]3[2]2[3][−δ+−δ−−δ−−δ+−δ+−δ=nnnnnnnw(b)Recall].1[][][−µ−µ=δnnnHence,])[]1[(])1[]2[(5])2[]3[(4][nnnnnnnxµ−+µ++µ−+µ++µ−+µ−=])4[]3[(2])2[]1[(3])1[][(2−µ−−µ+−µ−−µ−−µ−µ−nnnnnnNotforsale.2],4[2]3[2]2[3]1[][3]1[4]2[9]3[4−µ−−µ+−µ+−µ−µ−+µ−+µ++µ−=nnnnnnnn2.7(a)z1_z1_++h[0]h[1]h[2]x[n]y[n]x[n-1]x[n-2]Fromtheabovefigureitfollowsthat].2[]2[]1[]1[][]0[][−+−+=nxhnxhnxhny(b)h[0]z1_z1_++z1_z1_++11β12β22β21βx[n]y[n]x[n1]_x[n2]_w[n1]_w[n2]_w[n]Fromtheabovefigureweget])2[]1[][](0[][2111−β+−β+=nxnxnxhnwand].2[]1[][][2212−β+−β+=nwnwnwnyMakinguseofthefirstequationinthesecondwearriveat])2[]1[][](0[][2111−β+−β+=nxnxnxhny])3[]2[]1[](0[211112−β+−β+−β+nxnxnxh])4[]3[]2[](0[211122−β+−β+−β+nxnxnxh]2[)(]1[)(][]0[221112211211(−β+ββ+β+−β+β+=nxnxnxh.]4[]3[)()212211222112−ββ+−ββ+ββ+nxnx(c)FigureP2.1(c)isacascadeofafirst-ordersectionandasecond-ordersection.Theinput-outputrelationremainsunchangediftheorderingofthetwosectionsisinterchangedasshownbelow.z1_z1_++++0.60.30.2_0.80.5_y[n]w[n1]_w[n2]_w[n]z1_+0.4x[n]u[n]y[n+1]Notforsale.3Thesecond-ordersectioncanberedrawnasshownbelowwithoutchangingitsinput-outputrelation.z1_z1_++++0.60.30.2_0.80.5_w[n1]_w[n2]_w[n]x[n]y[n]z1_+0.4u[n]y[n+1]z1_z1_Thesecond-ordersectioncanbeseentobecascadeoftwosections.Interchangingtheirorderingwefinallyarriveatthestructureshownbelow:z1_z1_++++0.60.30.2_0.80.5_x[n]y[n]z1_+0.4u[n]y[n+1]z1_z1_x[n1]_x[n2]_u[n1]_u[n2]_s[n]Analyzingtheabovestructurewearriveat],2[2.0]1[3.0][6.0][−+−+=nxnxnxns],2[5.0]1[8.0][][−−−−=nununsnu].[4.0][]1[nynuny+=+FromSubstitutingthisinthesecondequationwegetaftersomealgebra].[4.0]1[][nynynu−+=].2[8.0]1[18.0][4.0][]1[−+−−−=+nynynynsnyMakinguseofthefirstequationinthisequationwefinallyarriveatthedesiredinput-outputrelation].3[2.0]2[3.0]1[6.0]3[2.0]2[18.0]1[4.0][−+−+−=−−−+−+nxnxnxnynynyny(d)FigureP2.19(d)isaparallelconnectionofafirst-ordersectionandasecond-ordersection.Thesecond-ordersectioncanberedrawnasacascadeoftwosectionsasindicatedbelow:z1_z1_+++0.30.2_0.80.5_w[n1]_w[n2]_w[n]x[n]z1_z1_y[n]2Notforsale.4Interchangingtheorderofthetwosectionswearriveatanequivalentstructureshownbelow:q[n]z1_z1_++_0.80.5_x[n]+0.30.2z1_z1_y[n]2y[n1]_2_y[n2]2Analyzingtheabovestructureweget],2[2.0]1[3.0][−+−=nxnxnq].2[5.0]1[8.0][][222−−−−=nynynqnySubstitutingthefirstequationinthesecondwehave].2[2.0]1[3.0]2[5.0]1[8.0][222−+−=−+−+nxnxnynyny(2-1)Analyzingthefirst-ordersectionofFigureP2.1(d)givenbelow0.6x[n]z1_+0.4u[n]y[n]1u[n1]_weget],1[4.0][][−+=nunxnu].1[6.0][1−=nunySolvingtheabovetwoequationswehave].1[6.0]1[4.0][11−=−−nxnyny(2-2)TheoutputofthestructureofFigureP2.19(d)isgivenby][ny].[][][21nynyny+=(2-3)FromEq.(2-2)weget]2[48.0]2[32.0]1[8.011−=−−−nxnynyand].3[3.0]3[2.0]2[5.011−=−−−nxnynyAddingthelasttwoequationstoEq.(2-2)wearriveat]3[2.0]2[18.0]1[4.0][1111−−−+−+nynynyny].3[3.0]2[48.0]1[6.0−+−+−=nxnxnx(2-4)Similarly,fromEq.(2-1)weget].3[08.0]2[12.0]3[2.0]2[32.0]1[4.0222−−−−=−−−−−−nxnxnynynyAddingthisequationtoEq.(2-1)wearriveat]3[2.0]2[18.0]1[4.0][2222−−−+−+nynynyny].3[08.0]2[08.0]1[3.0−−−+−=nxnxnx(2-5)AddingEqs.(2-4)and(2-5),andmakinguseofEq.(2-3)wefinallyarriveattheinput-outputrelationofFigureP2.1(d)as:].3[22.0]2[56.0]1[9.0]3[2.0]2[18.0]1[4.0][−+−+−=−−−+−+nxnxnxnynynynyNotforsale.52.8(a)},137235241{][*1jjjjjnx−−−−+−−−=↑Therefore}.415223371{][*1jjjjjnx−−−+−−−−=−↑()},....{][][][*,51543545111211jjjjnxnxnxcs−−−+−=−+=↑()}.....{][][][*,521452245252111211jjjjjnxnxnxca+−+−−++=−−=↑(b)Hence,andthus,Therefore,.][3/2njenxπ=3/*2][njenxπ−=].[][23/*2nxenxnj==−π()],[][][][/*,nxenxnxnxnjcs23222212==−+=πand().][][][*,022212=−−=nxnxnxca(c)Hence,andthus,Therefore,.][5/3njejnxπ−=5/*3][njejnxπ−=].[][35/*3nxejnxnj−=−=−π−(),][][][*,033213=−+=nxnxnxcsand().][][][][/*,5333213njcaejnxnxnxnxπ−==−−=2.9(a)}.2032154{][−−−=↑nxHence,}.4512302{][−−−=−↑nxTherefore,}2524252{])[][(][2121−−−−−=