数字信号处理试题(英文版)

1、 TestOfDigitalSignalProcessing1GivetheanswerofthefollowingproblemsaspossibleinEnglish.Supposingxn=0,whennN-1.PleasegiveitsZTX(z),DTFTX(e妙)andN-pointDFTXk.Explaintherelationsamongthesethreetransforms.TOC o 1-5 h zf1,0n5xn=Areal8-pointsequence10,6n7丿.Its8-pointDFTisXk.SupposingYk-W4Xk.Sketchyn=IDFTYk.

2、DeterminethevalueofY4.5兀r、cos(n),0n127xn=the128-pointDFTXk.GivetheexpressionofXk. HYPERLINK l bookmark58 I0,otherwiseI.,transitionbandwidthw兀Asequencehas4.AcausalLTIdiscrete-timesystemhasthetransferfunctionH(z)z2-1(z-0.5)(z2z+0.5)(1)(2)(3)(4)Determinethedifferenceequationofthesystem.Givethepole-zero

3、plotandtheROCofthesystem.Isthissystemstable?Developarealizationofthesysteminanycanonicform.5.DesignaFIRlinearphasecausalLPFwhichhascutofffrequencyandminimumstopbandattenuation40dB.DeveloptheimpulseresponseinclosedformbyWindowedFourierSeriesmethod.6.IfyouusefollowingMATLABprogramtodesignalowpassfilte

4、r,whichhasthespecifications:passbandedge800Hz,stopbandedge1kHz,passbandripple0.5dB,minimunstopbandattenuation40dB,andsamplingrateis5kHz.Whatdatashouldyouinputtothecomputer?%ProgramForEllipticIIRLowpassFilterDesign%Wp=input(Normalizedpassbandedge=);Ws=input(Normalizedstopbandedge=);Rp=input(Passbandr

5、ippleindB=);Rs=input(MinimumstopbandattenuationindB=);N,Wn=ellipord(Wp,Ws,Rp,Rs)b,a=ellip(N,Wn);h,omega=freqz(b,a,256);plot(omega/pi,20*log10(abs(h);grid;xlabel(omega/pi);ylabel(Gain,dB);title(IIREllipticLowpassFilter);7.InthesystemofFigure(a),x(t)ahasabandlimitedspectrumassketchedinFigure(b)andisbe

6、ingsampledattheNyquistrate.ThedigitalidealLPFhasthefrequencyresponseasshowninFigure(c).Sketchy(t)axathespectrumX(e,Y(河and(丿。)ofX“,“and乙(t)respectively.Figure(a)TheDigitalProcessingofAContinuousSignalmmFigure(c)Figure(b)TestOfDigitalSignalProcessing2GivetheanswerofthefollowingproblemsinEnglishaspossi

7、bleasyoucan.21f(n)二cos(兀n)+3sin(兀n)321.Determinethefundamentalperiodofsequence2.Considerthez-transform31(1+z-1)(1-z-1+z-2)48(1)Givethepole-zeroplotofX(z).(2)DetermineallpossibleROCsofX(z).Discussthetypeofinversez-transform(left-sided,right-sided,ortwo-sidedsequences)associatedwitheachoftheseROCs.(3)

8、Givetheexpressionofright-sidedsequence.3.Areal8-pointsequencen+1,0n5xn二0,6n7.Its8-pointDFTisXk.X0(1)DeterminethevaluesofYk=WkXk(2)(2)Supposing4yn=IDFTYk.Sketch00 (3)(3)DeterminethevalueofY4hn0n3xn0n24.Let=1211,and=111,yn=xn*hn(1)(1)Sketchthelinearconvolution.(2)(2)denotetheN-pointcirculary5y556convo

9、lution.Determinethevaluesof,y5ExplainwhichvalueisequaltothevalueofDigitalSignalProcessing3Note:GivetheanswerofthefollowingproblemsaspossibleinEnglish.f1,0n7xn=0,otherwise1.(20points)Alength-8sequenceisgivenby.PleasegiveitsexpressionsofZTX(z)X(e)Xk,DTFTand8-pointDFTinclosed-form.0n72.(20points)Consid

10、eralength-8realsequence,definedfor,xn=3-124-301-2丿Xkwith8-pointDFT0k7(1)CalculatethefollowingvaluesofXkwithoutcomputingtheDFTitself:X0(a)fXkY|xk|2(b)k=0(c)k=0yn(2)Sketchthesequencewhose8-pointDFTisgivenbyYk=W3kXk43.(18points)AcausalLTIdiscrete-timesystemhasthedifferenceequation1yn一一yn一2=xn一4xn一24(1)

11、Determinethetransferfunctionofthesystem.(2)Givethepole-zeroplotofthesystem.(6)(3)Determinetheimpulseresponseofthesystem.4.(15points)ThefrequencyresponseofalinearphaseFIRfilterisgivenbyH(ej)=1+4cos(2)e-j2hn(1)Determineandsketchtheimpulseresponseofthefilter.(2)WhichtypeisthislinearphaseFIRfilter?(15po

12、ints)Usingbilinearmethod,designadigital2nd-orderButterworthlowpassfilterwhichhasthecut-off兀二一c2frequency.(1)Givethetransferfunctionofthedesignedfilter.(2)Developarealizationstructureofthedesignedfilterincannonicform.QNote:Thetransferfunctionofanalog2nd-orderButterworthlowpassfilterwith3-dBcut-offfre

13、quencycis1H(s)=TOC o 1-5 h zaSS()2+J2()+1QQcc. HYPERLINK l bookmark71 hnxn(12points)Letbeanorder-63FIRfilter,andbealength-64sequence.Theinput-outputrelationshipofthefilterisshownasfollowingfigure.ynPleasewriteaMATLABprogramtocomputeynbyFFT,plotthemagnitudeandphasespectrumofhnH(ej)5.5.AsequencehasDTF

14、Tshownasfollowingfigure.SketchDTFTY(ej)ofyn=(-1)nh(n)Hk=H(ejQYk=Y(ejk)2n=kkNand.Ifand,where0kN-1YkHk.Givetheexpressionbetweenand.2兀一兀一co0兀2兀TestOfDigitalSignalProcessing411(GivetheanswerofthefollowingproblemsinEnglishaspossibleasyoucan)xnAsequenceisknownasbelow:xn=A(1+苗+j)n,where1A=()n422xnSketchthe

15、realpartof.IfA=1,isitperiodic?Ifitis,giveitsperiod.Foreachofthefollowingdiscrete-timesystems,whereynandxnare,respectively,theoutputandtheinputsequences,determinewhetherornotthesystemis(1)linear,(2)causal,(3)stable,(4)shift-invariant.(a)y(n)=sin2(n+1)x(n)(b)yn=a2x1一n+ba,bisanonzeroconstantWhere33Calc

16、ulatethecorrespondingtransformationoffollowingfunctions:x(n)=cos(1(1)Known5兀n),0n102364,its1024-pointDFTiswanted.(2)Known(3)Known44x(n)20,otherwise,itsDTFTiswanted.1X(ej)=(一+4coso+333osin2o)cos()e2,itsIDTFTiswanted.Xk,0k7,isthe8-pointDFTofsequencexn=-1,2,-3,2,0,-4,6,2,0k7.Pleasecalculatethefollowing

17、valueswithoutcomputingDFT:7工e-j(k/4)XkX4(a);(b)k=0;(c)k=0(d)k=055Giventwosequences:gn=1234(0n3),hn=503(0n2).yn=gnhn(1)DeterminethelinearconvolutionofthetwosequenceL* yn=gn*hn(2)DeterminethecircularconvolutionofthetwosequenceCeNeforN=4andN=6,WhilethelengthsofsequenceslessthanN,dothezero-padding.(3)Fr

18、omtheresultsof(1)and(2),maketheconclusionofhowtousecircularconvolutiontohelpcalculatinglinearconvolution.6.Considerthez-transformasbelow:3(1+z-i)(l-5z-1+6z-2)4X(z)(7)(1)Givethepole-zeroplotof.X(z)(8)(2)DetermineallpossibleROCsof.Discussthetypeofinversez-transform(left-sided,right-sided,ortwo-sidedse

19、quences)associatedwitheachoftheseROCs.X(ej)(9)(3)Canyoufindthe?Ifitispossible,whatisit?TestOfDigitalSignalProcessing(GivetheanswerofthefollowingproblemsinEnglishaspossibleasyoucan)61xnAsequenceisknownasbelow:xn=A(1+八/3)(苗+j)n,where1A=()n472xn(1)Sketchtherealpartof.(2)IfA=1,isitperiodic?Ifitis,giveit

20、speriod.Foreachofthefollowingdiscrete-timesystems,whereynandxnare,respectively,theoutputandtheinputsequences,determinewhetherornotthesystemis(1)linear,(2)causal,(3)stable,(4)shift-invariant.(a)y(n)=sin2(n+1)x(n)(b)yn=a2x1-n+ba,bisanonzeroconstantWhere83Calculatethecorrespondingtransformationoffollowingfunctions:5兀x(n)=cos(n),0n1023164(1)Known,its1024-pointDFTiswanted.1,0n15x(n)=220,otherwise(2)Known,itsDTFTiswanted.13迴X(ej)=(+4cos+sin2)cos()e2332(3)Known94,itsIDTFTiswanted.Xk,0k7,isthe8-pointDFTofsequencexn=-1,2,-3,2,0,-4,6,2,0k7.Plea