1、1. ConsideranLTIsystemwithinputxnandoutputynthatsatisfiesthedifferenceequation311yn+-M11-l+-yn-2=xn-xn-14o2a. FindthesystemfunctionH（三）.HowmanyROCsareassociatedwithH（三）?Foreachcase,determinewhattypeofthecorrespondingimpulseresponsehn?b. Ifthissystemiscausal,thenisitstable?Justifyyouranswer.Andwhethe
2、rH(ejw)existsornot?Ifitexists,determineH(*).!.Considerthefinite-lengthsequencexn=l,0,2,l,0113,withan4-pointDFTgivenbyxk.a.Ifthe4-pointDFTykoflength-4sequenceynisgivenbyy(k)=,determineyn.b.Ifthe4-pointDFTwkoflength-4sequencewnisdenotedbyw(k)=xk,determinewn.c. IfN-pointDFTsareusedinthetwo-stepprocedur
3、e,howshouldwechooseNsothatwn=xn*xnforOn=N-l?Determinewninthecasealso.Note:usingtheDFTpropertieswithoutcomputingxk.3.Givenxn=0,l,2,3,4,5,6bealength-7sequencedefinedfor0116,withX(ejw)denotingitsDTFTa) Evaluatethefollowingfunctionwithoutcomputingthetransformitself:x(/)；x()2Jvv；J-11.2Jkb) DefineYk=X(e),
4、0=n=4,withyndenotingits5-pointIDFTDetermineynwithoutcomputingYkanditsIDFT.CanyourecoverXnfromyn.4 .AnHRfilterisdescribedbythefollowingsystemfunction:/、QMz2+0.36z+0.020.44z2+0.362z+0.02z3+OAz2+0.18z-0.2(z2+0.8z+0.5)(z-0.4)Determineanddrawthefollowingstructures:(a) DirectfromII(b) Cascadeform.NI5 .Ver
5、ifytheidentityequation/7八(atfork-l=rNris即integerw-(Z:-l)n_NJVN-I。otherwise=06 .Acontinuous-timesignalX=cos(211x300t)+(211x500t)+(211xl200t)issampledata2kHzrate,andthesampledsequenceispassedthroughanidealIowpassfilterwithacutofffrequencyof900Hz,generatingacontinuous-timesignalyQQ).a.Determinethediscr
6、ete-timesignalxngeneratedbyperiodicallysamplingXQatFr=2khzb.Whatarethefrequencycompontspresentinthereconstructedsignaly.c.Ifya(t)isequaltotheoriginalcontinuoussignal,determinethesamplingfrequencyinthiscased.Ifwewanttopassthefrequencycomponentsat300Hz,whattypefilterweshouldchoose?Ifthetransition-widt
7、hisassumedtoas100Hz,theminimumStopandattenuationas52dB,whichwindowfunctionscanwechoosefromTablel?Determinethelengthofthefilterforthewindowyouselectd.TablelWindowNameTransitionWidthwMin.stopbandattenuationRectangular0.92%M20.9dBHanning3.11M43.9dBHamming03.32M54.5dBBlackman5.5611M75.3dB一、(20points)1.T
8、heinput-outputpairofastableLTIsystemisshowninFig.l（八）.(a) DeterminetheresponsetotheinputxninFig.l(b).(b) Determinetheimpulseresponseandfrequencyresponseofthesystem.(c) Sketchthemagnitude-frequencyresponseofthesystem.什2yQnT1-2-14-012n-2o(a)(b)Fig.l二、(20points)LethnbeaType-4real-coefficientlinear-phas
9、eFIRfilter.(I)Ifthisfilterhasthefollowingzeros:Zl=2,z?=0.6J,z3=0.6+2),pleasedeterminethelocationsoftheremainingzeros.(2)PleasedeterminetheFIRtransferfunctionandrealizeitincascadeformanddirectformI.三、(20points)ThedifferenceequationofaLTIdiscrete-timesystemis:311yn-yn-1+-yn-2=xn+-xn.14o3wherexnandynar
10、etheinputandoutputsequencesrespectively.(a) PleasegivethetransformfunctionH（三）aswellasitspolesandzeros;(b) Ifthesystemiscasualandstable,pleasegivetheROCofthez-transform;(c) Givetheimpulseresponsehnofthiscasualstablesystem.四、(20points)DesignaDIGITALlowpassfiltertomeetthefollowingrequirements:3p=0.2CO
11、S=057ap=3dBocs=60dBSelectawindowtodesignalinear-phaseFIRfilter,usingthelowestorderfiltertomeetthespecifications.五、(20points)Fig.2（八）showsa6-pointdiscretetimesequencexn.Assumethatxn=0outsidetheintervalshown.Thevalueofx4isnotknownandisrepresentedasb.LetX(ej)betheDTFTofxn,andXkbesamplesofX(ej)withsampl
12、inginterval1125i.e.5X1k=X(ej)11k(O,k=0,1,2,3The4-pointsequencexnisthe4-pointInverseDFTofXk,andisshowninFig2(b).-101234567,4巧卬J212I”-1012345（八）(b)Fig.2(a) Pleasedeterminethevalueofb.(b) LetX2kbesamplesofX(ej)withsamplinginterval113,i.e.5X2k=X(ej)11k=yk=0,1,5Pleasedetermineandsketchthefinite-lengthsequenceynwhose6-pointDFTisYk=W64kX2k.Determineandsketchthefinite-lengthseque
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