数字信号处理试题(英文版).docx

TestOfDigitalSignalProcessing1GivetheanswerofthefollowingproblemsaspossibleinEnglish.1 .Supposing一°'when”<°andn>N.peasegiveitsZTX(z),DTFTX")andN-pointDFT“【"Explaintherelationsamongthesethreetransfonns.1,O54n=>2 .Areal8-pointsequence-0,6'JIts8-pointDFTisXk.SUPPOSingy闵=MX内.SketchMm=5僧幻.(2)Determinethevalueofcos(),0«12764,3. A sequencethe 128-point DFTXkLGiVe the expression of .O,otherwise4. AcausalLTIdiscrete-limesystemhasIhetransferfunction/(Z)=!(z-0.5)(z2-z+0.5)(1) Determinethedifferenceequationofthesystem.(2) Give(hepole-zeroplotandtheROCofthesystem.(3) Isthissystemstable?(4) Developarealizationofthesysteminanycanonicform.5. DesignaFIRlinearphasecausalLPFwhichhascutofffrequency°25",transjtonbandwidth口0.1万andminimumstopbandattenuation40d8.DeveloptheimpulseresponseinclosedIbrmbyWindowedFourierSeriesmethod.6.IfyouusefollowingMATLABprogramtodesignaIowpassfilter,whichhastheSpecificationszpassbandedge800Hz,stopbandedgeIkHz,passbandripple0.5dB,minimunslopbandattenuation40dB,andsamplingrateis5kHz.Whatdatashouldyouinputtothecomputer?%ProgramForElIipticIIRLowpassFiIterDesign%Wp=input('Normalizedpassbandedge=')；Ws=input('Normalizedstopbandedge=,);Rp=input('PassbandrippleindB=');Rs=input('MinimumstopbandattenuationindB=,);N,Wn=e11ipord(Wp,Ws,Rp,Rs)b,a=ellip(N,Wn);h.omega=freqz(b,a,256):plot(omega/pi,20*1og10(abs(h):grid;×IabeI('omegapi,)；ylabel('Gain,dB,);title(,IIREllipticLowpassFiIter');7.InthesystemofFigure(a),"hasabandlimitedspectrumassketchedinFigure(b)andisbeingsampledatH(ej)theNyquistrate.ThedigitalidealLPFhasthefrequencyresponse'zasshowninFigure(c).SketchthespectrumX(")J(")and 工。)of M"M%nd 北 respectively.Figure(a)The DigitalProcessing of A Continuous Signal-2 oFigure (c)TestOfDigitalSignalProcessing2GivetheanswerofthefollowingproblemsinEnglishaspossibleasyoucan.f(n)=cos(-ri)+3sin(-ri)1. Determinethefundamentalperiodofsequence32X(Z)3TT1-2(IdZT)(Iz'H"z2)4482. Considerthez-transform(1) (1)Givethepole-zeroplotof'().(2) (2)DetennineallpossibleROCsof.Discussthetypeofinversez-transform(left-sided,right-sided,ortwo-sidedsequences)associatedwitheachoftheseROCs.(3) (3)Givetheexpressionofright-sidedsequence.÷l,0514n=0,6"73. Areal8-pointsequenceIJ.Its8-pointDFTisX伙.X0(1) Determinethevaluesof,"=°.Yk=WXkyn=IDFTYk(2) (2)Supposing.Sketchr4(3) (3)Determinethevalueofy = n*An4小瓦川="2W0"<3andM网=W0<<2.(1) (1)SketchthelinearconvolutionN> = 3 MY (N)式卜(2) (2)denotetheN-pointcircularconvolution.Determine%5thevaluesof,.Explainwhichvalueisequaltothe value of¼5Digital Signal ProcessingsNote: Give the answer of the following problems as possible in English.xn= <l,071 .(20 points J A length-8 sequence is given byQ. otherwiseX(Z).Please give its expressions of ZT2 . (20 points) Consider a length-8 real sequence, defined for ° << 7xn=3 -1 2 4 -3 0 1 -2Xk OVjtV7 with 8-point DFTL U 二八二/(1)CalculatethefollowingvaluesofX伙withoutcomputingtheDFTitself: X°7 X伙(b) =07°w闷(C) k=°yYk=W3kXkSketchthesequenceJwhose8-pointDFTisgivenbyLj4、3. (18points)AcausalLTIdiscrete-timesystemhasthedifferenceequationyri-yn-2=xn-4xn-24(4) (I)Detenninethetransferfunctionofthesystem.(5) (2)Givethepole-zeroplotofthesystem.(6) (3)Determinetheimpulseresponseofthesystem.4. (15points)ThefrequencyresponseofalinearphaseFIRfilterisgivenbyH(ej)=l+4cos(2)e-j2hn(1) (1)Detennineandsketchtheimpulseresponseofthefilter.(2) (2)WhichtypeisthislinearphaseFIRfilter?5. (15points)Usingbilinearmethod,designadigital2nd-orderButterworthIowpassfilterwhichhasthecut-offfrequencyCDc=c2(1) (1)Givethetransferfunctionofthedesignedfilter.(2) (2)Developarealizationstructureofthedesignedfilterincannonicform.r°QCNote:Thetransferfunctionofanalog2nd-orderButterworthIowpassfilterwith3-dBcut-offfrequency乩G)=6. (12 points) Lethnbe an order-63 FIR filter, andxnbe a length-64 sequence. The input-output relationship of thexn> hnPlease write a MATLAB program to computeynbyFFT, plot the magnitude and phase spectrum ofy5.5,zA sequencehas DTFTH(ej)shown as following figure. SketchDTFTfilterisshownasfollowingfigure.ynY(ej)fyV= (-1),7z(h) f Hk= H(ejk)Yk = Y(ejk),whereSk=生 kk N4kWN-and *_* *>.Give the expression betweenYk HkTestOfDigitalSignalProcessing4(GivetheanswerofthefollowingproblemsinEnglishaspossibleasyoucan)x1. 1.Asequenceisknownasbelow:Mh=A(1+j3)(3+jhA=(%,wherex(1) Sketchtherealpartof(2) IfA=I,isitperiodic?Ifitis,givei(speriod.2. 2.Foreachofthefollowingdiscrete-timesystems,whereynandxnare,respectively,theoutputandtheinputsequences,determinewhetherornotIhesystemis(I)Iinear,(2)causal,(3)Slable.(4)Shifl-invariant.Q()y(")=sm(n+I)x(n)S)yn=a2xl-n+ba.bisanonzeroconstantWhere3.3.Calculate the corresponding transformation of following functions:(1) Known5 TTx1 ()= cos(- n),Onl 02364, its 1024-point DFT is wanted. Knownr-,015毛()=2O, otherwise,its DTFT is wanted.19 3 J-X3(ej) = (- + 4cos69 + sin co)cos(-)e 2 Known,its IDTFT is wanted.4. 4.Xk, 0 k 7, is the 8-point DFT of sequence (xn)= -1, 2, -3, 2, 0, -4, 6. 2), 0 k <7. Please calculate thefollowing values without computing DFT:7ye-jxkX46(a) L；(b)一ZX幻 Xk2(c) k=Q；(d) k=05. 5.Given two sequences: gn= 1 2 34(0723)"hn=5 0 3 (°<"<2)ylri=griM11(1) Determinethelinearconvolutionofthetwosequence*yn=g/m*¼n(2) DeterminethecircularconvolutionofthetwosequenceNforN=4andN=6,WhilethelengthsofsequenceslessthanN,dothezero-padding.(3) Fromtheresultsof(1)and(2),maketheconclusionofhowtousecircularconvolutiontohelpcalculatinglinearconvolution.6.ConsidertheZ-transformasbelow:X(Z) =l+z-2(l+-z1)(l-5z1+6z-2)4(7) (1)Give(hepole-zeroplotof,X。)，(8) (2)DetermineallpossibleROCsof.DiscussthetypeofinverseZ-transfbn(left-sided,right-sided,ortwo-sidedsequences)associatedwitheachoftheseROCs.X(ej')(9) (3)Canyoufindthe?Ifitispossible,whatisit?TestOfDigitalSignalProcessing(GivetheanswerofthefollowingproblemsinEnglishaspossibleasyoucan)xz6. 1.Asequenceisknownasbelow:=A(l+73)(3+jrA=5,wherex(1) Sketchtherealpartof(2) IfA=I,isitperiodic?Ifitis,giveitsperiod.7. 2.Foreachofthefollowingdiscrete-timesystems,whereynandxnare,respectively,theoutputandtheinputsequences,determinewhetherornotthesystemis(I)Iinear.(2)causal.(3)SIable.(4)Shifi-invariant.O(a)y(n)=sin(n+l)x(n)byn=a2xl-n+ba.bisanonzeroconstantWhere8. 3.Calculatethecorrespondingtransformationoffollowingfunctions:5TTxx()=cos(),O1023(1) Known,its1024-pointDFTiswanted.r-90n15芍()=2O,otherwiseKnown,itsDTFTiswanted.193J-X3(ej)=(-+4cos69+sin*)cos()e2(3) Known,itsIDTFTiswanted.9. 4.Xk,0k<7,isthe8-pointDFTofsequencexn=-1,2,-3,2.O,-4,6,2),0k7.PleasecalculatethefollowingvalueswithoutcomputingDFT:7ZXM YXk2；(d) k=。ye-jiu4yxkX4£(a)LJ；(b)k-v1().5.Giventwosequences:gnj=1234(0九3)"hn=503)(0<<2)(1)Determinethelinearconvolutionofthetwosequenceyln=gnhnyc11=gen*¼n(2) DeterminethecircularconvolutionofthetwosequenceNforN=4andN=6,WhilethelengthsofsequenceslessthanN,dothezero-padding.(3) Fromtheresultsof(1)and(2),maketheconclusionofhowtousecircularconvolutiontohelpcalculatinglinearconvolution.H.6.ConsidertheZ-transformasbelow:X(Z)=l+z23(1+-z1)(1-5z1+6z2)4X(Z)(10)(1)Givethepole-zeroplotofX(Z)(11)(2)DeiennineallpossibleROCsof.Discussthetypeofinversez-transfon(left-sided,right-sided,ortwo-sidedsequences)associatedwitheachoftheseROCs.X(ej)(12)(3)Canyoufindthe?Ifitispossible,whatisit?