英文版数字信号处理去年考试题.doc

英文版数字信号处理去年考试题.doc

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1.ConsideranLTIsystemwithinputx[n]andoutputy[n]thatsatisfiesthedifferenceequation

a.FindthesystemfunctionH（z）.HowmanyROCsareassociatedwithH（z）?

Foreachcase,determinewhattypeofthecorrespondingimpulseresponseh[n]?

b.Ifthissystemiscausal,thenisitstable?

Justifyyouranswer.Andwhetherexistsornot?

Ifitexists,determine.

2.Considerthefinite-lengthsequencex[n]={1,0,2,1},0≤n≤3,withan4-pointDFTgivenbyx[k].

a.Ifthe4-pointDFTy[k]oflength-4sequencey[n]isgivenbyy（k）=,determiney[n].

b.Ifthe4-pointDFTw[k]oflength-4sequencew[n]isdenotedbyw（k）=x[k],determinew[n].

c.IfN-pointDFTsareusedinthetwo-stepprocedure,howshouldwechooseNsothatw[n]=x[n]*x[n]for0≦n≦N-1?

Determinew[n]inthecasealso.

Note:

usingtheDFTpropertieswithoutcomputingx[k].

3.Givenx[n]={0,1,2,3,4,5,6}bealength-7sequencedefinedfor0≦n≦6,withXdenotingitsDTFT

a）Evaluatethefollowingfunctionwithoutcomputingthetransformitself:

X（）;;

b）DefineY[k]=X（e）,0≦n≦4,withy[n]denotingits5-pointIDFT.Determiney[n]withoutcomputingY[k]anditsIDFT.CanyourecoverX[n]fromy[n].

4.AnIIRfilterisdescribedbythefollowingsystemfunction:

H（z）=

Determineanddrawthefollowingstructures:

（a）DirectfromII

（b）Cascadeform.

5.Verifytheidentityequation

6.Acontinuous-timesignalX=cos（2x300t）+（2x500t）+（2x1200t）issampledata2kHzrate,andthesampledsequenceispassedthroughanideallowpassfilterwithacutofffrequencyof900Hz,generatingacontinuous-timesignaly.

a.Determinethediscrete-timesignalx[n]generatedbyperiodicallysamplingXatF=2khz

b.Whatarethefrequencycompontspresentinthereconstructedsignaly.

c.Ifyisequaltotheoriginalcontinuoussignal,determinethesamplingfrequencyinthiscase

d.Ifwewanttopassthefrequencycomponentsat300Hz,whattypefilterweshouldchoose?

Ifthetransition-widthisassumedtoas100Hz,theminimumstopandattenuation,whichwindowfunctionscanwechoosefromTable1?

Determinethelengthofthefilterforthewindowyouselectd.

Table1

WindowName

TransitionWidth

Min.stopbandattenuation

Rectangular

20.9dB

Hanning

43.9dB

Hamming

54.5dB

Blackman

75.3dB

一、（20points）1.Theinput-outputpairofastableLTIsystemisshowninFig.1（a）.

（a）Determinetheresponsetotheinputx1[n]inFig.1（b）.

（b）Determinetheimpulseresponseandfrequencyresponseofthesystem.

（c）Sketchthemagnitude-frequencyresponseofthesystem.

（a）

（b）

Fig.1

二、（20points）Leth[n]beaType-4real-coefficientlinear-phaseFIRfilter.

（1）Ifthisfilterhasthefollowingzeros:

pleasedeterminethelocationsoftheremainingzeros.

（2）PleasedeterminetheFIRtransferfunctionandrealizeitincascadeformanddirectformI.

三、（20points）ThedifferenceequationofaLTIdiscrete-timesystemis:

wherex[n]andy[n]aretheinputandoutputsequencesrespectively.

（a）PleasegivethetransformfunctionH（z）aswellasitspolesandzeros;

（b）Ifthesystemiscasualandstable,pleasegivetheROCofthez-transform;

（c）Givetheimpulseresponseh[n]ofthiscasualstablesystem.

四、（20points）DesignaDIGITALlowpassfiltertomeetthefollowingrequirements:

Selectawindowtodesignalinear-phaseFIRfilter,usingthelowestorderfiltertomeetthespecifications.

五、（20points）Fig.2（a）showsa6-pointdiscretetimesequencex[n].Assumethatx[n]=0outsidetheintervalshown.Thevalueofx[4]isnotknownandisrepresentedasb.LetX（ejω）betheDTFTofx[n],andX1[k]besamplesofX（ejω）withsamplingintervalπ/2,i.e.,

The4-pointsequencex1[n]isthe4-pointInverseDFTofX1[k],andisshowninFig.2（b）.

（a）（b）

Fig.2

（a）Pleasedeterminethevalueofb.

（b）LetX2[k]besamplesofX（ejω）withsamplingintervalπ/3,i.e.,

Pleasedetermineandsketchthefinite-lengthsequencey[n]whose6-pointDFTisY[k]=W64kX2[k].

Determineandsketchthefinite-lengthseque

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