數字信號處理 -上機報告題目I

1、上機報告題目I姓名學號第一題Questions:Q3,lWhalislheexpressionofIlieDTFTbeingevaluatedinProgramP3_l?WhatisthefunclionoflheMATLABcommandpause?Q3.2RunProgramP3_landcomputelherealandimaginarypartsoftheDTFT,andlhemagnitudeandphasespectra.IstheDTFTaperiodicfunctionoflj?Ifitis,whatistheperiodExplainlhetype01symmclricsex

2、hibitedbylhelourplots.Q3.3MixlilyProgramP3loevalualeinlherange0ttlhefollowingDTFT:y_0,7-0.血”+碁+E-仙1+0.3e_Jw0.5ej2+0.7e_J3wandrepciUQuestion(J3.2.CommenIonyourresults.Cunyouexplainlhejumpinlhephasespectru111?ThejumpcanberemovedusingtheMATLABcommandunwrap.Evaluatelhephusespeclruniwithlhejumpremoved.Q3

3、.4ModiIyProgramP3toevaluulclheDTFTofthefollowinglinile-lengtlisequence:gn=1357911131517,andrepcal(ucslionQ?.2.Commentonyourresults.Canyouexplainlhejumpsinlhephase斗卩eclrum?Q3,5HowwouldyoumodifyProgramP3_ltoplotthephaseindegrees?Answers.Q3.1TheexpressionoftheDTFTbeingevaluatedinProgramP3_1is-Hej32+ejs

4、106qjsThefunctionofthepausecommandispause命令用于暫時中止程序的運行。當程序運行到此命令時，程序暫時中止，然后等待用戶按任意鍵繼續(xù)進行。在本程序中，pause用來顯示兩組不同的圖（實虛部圖和幅頻相頻圖），并在中間暫停以供用戶查看。Q3.2TheplotsgeneratedbyrunningProgramP3_1areshownbelow:44Realpartof-2-1012343山ImaginarypartofHi：)TheDTFTisa_周期MagnitudeSpectrumH(e)|-2-1012343/tTPhaseSpectrumargH(e)

5、|-2-1012wAt_functionofItsperiodis-2piThetypesofsymmetriesexhibitedbythefourplotsareasfollows:共軛對稱Q3.3TherequiredmodificationstoProgramP3_1toevaluatethegivenDTFTofQ3.3aregivenbelow：num=--0.5TheplotsgeneratedbyrunningthemodifiedProgramP3_1areshownbelow:15051a心-Dpn七-dEHi-10123心/ttIm白ginarypart

6、ofH(d)CDpn七-dE15051a4;/ttMagnitudeSpectrumH(eM)|PhaseSpectrumargH(ejLL，)Itsperiodis-2piDTFT經過虛部等于0,實Thejumpinthephasespectrumiscausedby-部小于0的點o5-20-253023Thephasespectrumevaluatedwiththejumpremovedbythecommandunwrapisasgivenbelow：3ITTPhaseSpectrumargH(eJQ,)Q3.4TherequiredmodificationstoProgramP3_1to

7、evaluatethegivenDTFTofQ3.4aegivenbelow：h=freqz(1357911131517,1,w);TheplotsgeneratedbyrunningthemodifiedProgramP3_1areshownbelow:RealpartofH(ej)oo5QP2一-duj-3-2u；/Z伽ImaginarypartofTheDTFTisa_周期如p-dEfunctionofItsperiodis-2pi實areThejumpinthephasespectrumiscausedby-DTFT經過虛部等于0,部小于0的點Q3.5Therequiredmodifi

8、cationstoProgramP3_1toplotthephaseindegreesindicatedbelow:%PlottheDTFTsubplot(2,1,1)plot(w/pi*180,real(h);gridtitle(RealpartofH(eAjomega)xlabel(omega/degree);ylabel(Amplitude);subplot(2,1,2)plot(w/pi*180,imag(h);gridtitle(ImaginarypartofH(eAjomega)xlabel(omega/degree);ylabel(Amplitude);pausesubplot(

9、2,1,1)plot(w/pi*180,abs(h);gridtitle(MagnitudeSpectrum|H(eAjomega)|)xlabel(omega/degree);ylabel(Amplitude);subplot(2,1,2)plot(w/pi*180,angle(h)/pi*180);gridtitle(PhaseSpectrumargH(eAjomega)xlabel(omega/degree);ylabel(Phaseindegrees);MagnitudeSpectrumupm一-dlulc-zcroplotolarulionalz-lRinsibrinG(z)canb

10、ereadilyobtainedusinglheliinclionzplane.Therearetwoversionsolthisfunclion.IIlhe-IrLinsIbrinisiveninlhelbrmofamlionallundionasinEq.(3.32).lhecommmidlouseiszplane(num,den)wheienumanddenarcixnvvectorscontaininglheeocflieienlsoflhenumcRiloranddcnoniiiiLilorpolynoniialsofG(z)inascending|xweil()nlheoilier

11、hand,iflhezerosandpleisindictUcdbvlhesvmbolxandlhelocalionofazeroisindiciUedbvlhesymbolG(z).Theprogramslalcmenllouseisz,p,k=tf2zp(num,den)wheicnumanddenarcixwvcelorseonlaininglhecocflicienlsoflhenunicraloranddcnoniiiiLilorpolynoniialsofinusccnding|xwersofs_1andlheoulpullileeonlainslhegainconstantkan

12、dthecomputedzerosandpolesgivenascolumnvectorszandp,respectively.Thefactoredformofthez-transformcanbeobtainedfromthezero-poledescriptionusinglhelunclionsos=zp2sos(z,p.Thefunclioncomputesthecoefficientsofeachsecond-orderfactorgivenasanLx6matrixsoswhere臥ii切1剛1盹1H212切2剛2125sos=boLbLb-2LfOLflL2Lwherethet

13、hrowcontainsthecoefficientsofthenumeratorandthedenominatorofthe&hsecond-orderlacloroflhe2-lraiislbrniG(z).血+b-2(z2Q12_1+口2?_2Questions:Q3,47WriteaMATLABprogramlocoinpuleunddisplaythepolesandzeros,lowmpiileanddisplaythefactoredtbrni,andtogeneratethepole-zeroplotofa-transformthatisaratiooftwopolynomia

14、lsin:_1.Usingthisprogram,analyzethez-lrunsformGofEq.(3.32).Q3.48Fromthepole-zeroplotgeneratedinQuestionQ3.47,determinethenumberofregionsofconvergence(R()C)訂G.SIkwexplicitlyallpossibleR()Cs.Canyoutelllionithepole-zeroplotwhetherornottheDTFTexists?Thereverseprocessofconvertinga2-transformgiveninthefor

15、mofzeros,poles,andthegninconstrmIloaralionalIbrmcanbeimpleinciilcdusinglhelimclioiizp2tf.TheprogramsliUemenllouseisnumjden=zp2tf(z,k).Question:Q3,49WriteaMATLABprogranilocoinpuleanddisplaylheralional2-Irunsformfromitszeros,polesandgainconstant.Usingthisprogram,determinetherationalformofaz-transformw

16、hosezerosareat4i=0.3,fa=2.5,3=0.2+j0.4,and4=0.2j0.4;lhepolesarcalAi=0.5.=(175.A：CL6+jCl7,andA4=0,6j0,7:andIliegainconslanlkis3.9.Answers:Q3.46Thefrequencyresponseofthez-transformobtainedusingProgramP3_1isplottedbelow：Q3.47TheMATLABprogramtocomputeanddisplaythepolesandzeros,tocomputeanddisplaythefact

17、oredform,andtogeneratethepole-zeroplotofarationalz-transformis21012七Ed當匸一Emlu-0X0-II0Q-9-8-7-654-3-2-101RealPartQ3.48Fromthepole-zeroplotgeneratedinQuestionQ3.47,thenumberofregionsofconvergence(ROC)ofG(z)are-4AllpossibleROCsofthisz-transformaresketchedbelow:|z|0.27180.2718|z|0.28660.2866|z|8.9576存在8

18、.9576|z|Fromthepole-zeroplotitcanbeseenthattheDTFTQ3.49TheMATLABprogramtocomputeanddisplaytherationalz-transformfromitszgros,polesandgainconstantisgivenbelow:z=0.32.5-0.2+0.4j-0.2-0.4j.;p=0.5-0.750.6+0.7j0.6-0.7j.;k=3.9;num,den=zp2tf(z,p,k);numdennum=3.9000-9.3600-0.6630-1.01400.5850den=1.0000-0.950

19、00.17500.6625-0.3187Therationalformofaz-transformwiththegivenpoles,zeros,andgainisfoundtobe3.99.36zi0.663z21.0140z-3+0.5850z-4G(z)=10.95z-i+0.1750z-2+0.6625z-30.3187z-4第三題Questions:Q44ModifyProgramP3_ltocomputeundplotthemagnitudeandphasespectraofamovingaveragefilterofEq.(2.13)forthreedifferentvalues

20、oflengthMandfor0wJuslilyIlietypeofsymmclricsexhibiledbytheinagiiiludemidphasespectra.Wh:iltypeoffilterdoesitrepresent?CanyounowexplaintheresultsofQuestionQ2.1?ThemodifiedProgramP3_lcanalsobeusedtocomputelhefrequencyresponseofanLTIJiscrclc-liinesyslcnifromitsiRinsfcrliincliondescriptionasinEq.(4.12).

21、Questions:Q4,2UsinglhemodiliedProgramP3_lcomputeandplotlhefrequencyresponseofacausalLTIdiscrclc-limcsystemwithaIrunsIcrluiicliongivenby(4.36)lor0a；esilrcprcsenl?Project2.1TheMovingAverageSystemExjLiriLnLLionofEq.f1.21reveals;thatthethree-pointjinioothingtillerconfiiJrcdix1TlFIRMoreover,川“l(fā)eprriiiKti

22、nniLhEihrtrinpLifsample.?|n+Inthenonciual.AcausalversinnofThethree-pcintmn(xhingtillerisnbiainedbysimplydekiyinLheouipuibyonesiiinpl.epcriinj.resultininhi?FIRliberdesciihcdbj/n(工審+工代f1+日“2)r(2-12)CiLhiii?raizinglheabnvtwetibuin1站仏=和e*-虬(2+13)it0whichtlchnesaCiuiHulA/-pointsiinoolTiinFIRfilterThesyst

23、emofEq.(2.Ealsokiion;恬lirHavingtiverticfiifcr.WeilltLslndci低ttseitilillcrin*;hii.h-frcquencjcoitifxincnlsironiacomposctjoE7asutnofwveralsinusoidulsignah.Answers:Q4.1ThemodifiedProgramP3_1tocomputeandplotthemagnitudeandphasespectraofamovingaveragefilterofEq.(2.13)for02兀isshownbelow：w=-4*pi:8*pi/511:4

24、*pi;M=5;h=freqz(ones(1,M)/M,1,w);subplot(2,1,1)plot(w/pi,abs(h);gridtitle(MagnitudeSpectrum|H(eAjomega)|)xlabel(omega/pi);ylabel(Amplitude);subplot(2,1,2)plot(w/pi,angle(h)/pi);gridtitle(PhaseSpectrumargH(eAjomega)xlabel(omega/pi);ylabel(Phaseinrad);Thisprogramwasrunforthefollowingthreedifferentvalu

25、esofMandtheplotsofthecorrespondingfrequencyresponsesareshownbelow：M=5：11A-3-2-1012345a也pn三dujid/7TPhaseSpectrum古gH（R）M=10：PE=邊EL-dM=20：o-Opm-l匚一們ineqdMagnitudeSpectrum-2-101234PhaseSpectrum白rgH（J）-0.5V-3-2-101iU/T234-1012MagnitudeSpectrumH佗5aaipn七-duJ0pe匚一DZJeud低通濾波器Thetypesofsymmetriesexhibitedbyth

26、emagnitudeandphasespectraaredueto-滑動窗口平均濾波器的單位脈沖響應序列是實偶對稱序列。ThetypeoffilterrepresentedbythemovingaveragefilterisTheresultsofQuestionQ2.1cannowbeexplainedasfollows-波器實際上是一種低通濾波器，所以能夠去掉信號中的高頻正弦波而保留低頻正弦波。Q4.2TheplotofthefrequencyresponseofthecausalLTIdiscrete-timesystemofQuestionQ4.2obtainedusingthemod

27、ifiedprogramisgivenbelow:Thetypeoffilterrepresentedbythistransferfunctionis帶通濾波器gpa-dlLJVdMagnitudeSpectrum|H(e)|(J/TPhaseSpectrumargH(e)(J/T第四題Questions:Q4.23UsingMATLABgcnemlethepole-zeroplotsoflhefollowingtwocausalIransIcrfunctions.Hi11-18z-1+0,S5-2並=11-L851Z-1+0.S52-3Canyouinfertheirstabilitybye

28、xaminingthegeneratedpole-zeroplots?Q4,24TesttheslabililyolIheIrunsIerfunclionsin(JucslionQ4.23usingProgmmP4_4.Whichoneofthetwotransferfunctionsisstable?Q4.25UsingProgramP4_4determinewhetherornotalltherootsofthefollowingpolynomialareinsidetheunitcircle:D(z)=1+2,52_1+2,5.7-3+0.3125+0,03125Answers:Q4.2

29、3Thepole-zeroplotsofH1(z)andH2(z)obtainedusingzplaneareshownbelow:zplane(1,1-1.8480.85);pausezplane(1,1-1.8510.85);-O.B-O.S-13B42d.240.亡rodAJEBBLU一-0.6-0.B-1-0.50.5186420240.0.000.0.-ii一md已raUWPE-異-050051RealPart0RealPartFromtheabovepole-zeroplotsweobservethatH1(z)的所有極點都在單位圓內，穩(wěn)定；H2(z)有一個極點在單位圓外，不穩(wěn)定。

30、Q4.24UsingProgramP4_4wetestedthestabilityofH1(z)andarriveatthefollowingstabilitytestparametersk,：Denominatorcoefficients=1-1.8480.85Stabilitytestparametersare-0.99890.8500FromtheseparametersweconcludethatH1(z)is_.UsingProgramP4_4wetestedthestabilityofH2(z)andarriveatthefollowingstabilitytestparamete

31、rs$：Denominatorcoefficients=1-1.8510.85Stabilitytestparametersare-1.00050.8500FromtheseparametersweconcludethatH2(z)is_.Q4.25UsingProgramP4_4wetestedtherootlocationsofD(z)andarriveatthefollowingstabilitytestparameterski:Denominatorcoefficients=50.31250.03125Stabilitytestparametersare0.96300.87260.62

32、250.23460.0313_theunitFromtheseparametersweconcludethatallrootsofD(z)are_circle.第五題Questions:Q5,lRunProgramP5_ltogenerateboththecontinuous-limesignalanditssampledvcrsioiLanddisplaythem.Q5,2Wh；uisiliefrequencyinHzofthesinusoidalsignal?Whalisthesamplingperiodinseconds?Q5.3Explaintheeffectsofthetwoaxis

33、commands.Q5,4RunProgramP5_lforfourothervaluesofthesamplingperiodwithtwolowerandtwohigherthanthatlistedinProgramP5_l.Commentonyourresults.Q5,5RepeatProgramP5_lbychangingthefrequencyofthesinusoidalsignalto3Hzanil1Hz.rcspcclivcly.Is(hereanydifferencebetweenthecorrcsixnidingequivalentdiscrete-timesignal

34、sandtheonegeneratedinQuestionQ5.1?Ifnot,whynol?Answers:A,copyofProgramP51isgivenbelow:%ProgramP5_1%IllustrationoftheSamplingProcess%intheTime-Domainclf;t=0:0.0005:1;f=13;xa=cos(2*pi*f*t);subplot(2,1,1)plot(t,xa);gridxlabel(Time,msec);ylabel(Amplitude);title(Continuous-timesignalx_a(t);axis(01-1.21.2

35、)subplot(2,1,2);T=0.1;n=0:T:1;xs=cos(2*pi*f*n);k=0:length(n)-1;stem(k,xs);grid;xlabel(Timeindexn);ylabel(Amplitude);title(Discrete-timesignalxn);axis(0(length(n)-1)-1.21.2)Q5.1Theplotsofthecontinuous-timesignalanditssampledversiongeneratedbyrunningProgramP5_1areshownbelow:ao.p2-dEV012345678910Timein

36、dexn-0.1SQ5.2ThefrequencyofthesinusoidalsignalinHzis13HzThesamplingperiodinsecondsisQ5.3Theeffectsofthetwoaxiscommandsare使圖像邊緣留有一定空隙不至于變成這樣：uprl七一dEVOPFI4一-dulvDiscrete-timesignalxnQ5.4Theplotsofthecontinuous-timesignalanditssampledversiongeneratedbyrunningProgramP5_1forthefollowingfourvaluesofthesa

37、mplingperiodareshownbelow:Ts=0.01s：ContinuOLis-timesignalxit)3.5051o.o-=0705一-dEOOJ0.201Time,msecDiscrete-timesignalxnTs=0.03s：Continuous-timesignalx(I)呂1505*uprq_duj0510250.102030.4050.60.7OBD.91Tim已msecDiscrete-timesignalxnDpn_=dul1520TimeindexnTs=0.2s：Gontinuous-timesignetlx(t)tlup2-duj述flL7c3Tim

38、e,msecDiscrete-timesignalxn15050a00.511.522.533.544.55TimeindexnTs=0.25s：ContinuouE-timgsignalx(t)a-O.pnlKEQ0OJ00.91Time,msecDiscrete-timesignalxnoep5-dlu0051L522533.54TimeindexnBasedontheseresultswemakethefollowingobservations-只要采樣頻率低于于信號最高頻率的兩倍，就不能根據采樣信號還原原信號了。Q5.5Theplotsofthecontinuous-timesinus

39、oidalsignaloffrequency3HzanditssampledversiongeneratedbyrunningamodifiedProgramP5_1areshownbelow:Continuous-timesignalx(t)日15051a-O.-mpB-dluv0010.20.3OX0.80.91Time,rnsecDiscrete-timesignalxno-Oopr!4=dE012345678910TimeindexnTheplotsofthecontinuous-timesinusoidalsignaloffrequency7Hzanditssampledversio

40、ngeneratedbyrunningamodifiedProgramP5_1areshownbelow:Continuous-timesignalx(t)日15051ad-aip2KUJ0Od0.80.91Tim已msecDiscrete-limesignalxno-OapE一-dulndinequivalentdisercle-limesignalsandlhenegenemlcdin(Jues-lionQ5.0?Ifnot.whynot?Answers:AcopyofProgramP52isgivenbelow:%ProgramP5_2%IllustrationofAliasingEff

41、ectintheTime-Domain%ProgramadaptedfromKra94withpermissionfrom%TheMathworks,Inc.,Natick,MA.clf;T=0.1;f=13;n=(0:T:1);xs=cos(2*pi*f*n);t=linspace(-0.5,1.5,500);ya=sinc(1/T)*t(:,ones(size(n)-(1/T)*n(:,ones(size(t)*xs;plot(n,xs,o,t,ya);grid;xlabel(Time,msec);ylabel(Amplitude);title(Reconstructedcontinuou

42、s-timesignaly_a(t);axis(01-1.21.2);Q5.6Theplotsofthediscrete-timesignalanditscontinuous-timeequivalentobtainedbyrunningProgramP5_2areshownbelow:epn莖dERecanstnjctpdcontinuQLi&tiiTiesign白Iy(t)clIE5oII00.050.507O.B0.91Tima,msecQ5.7TherangeoftintheProgramis-0.5,1.5u)=0.004050010,1Thevalueofthetimeincrem

43、entis-Therangeoftintheplotis-TheplotgeneratedbyrunningProgramP5_2againwiththerangeoftchangedsoastodisplaythefullrangeofya(t)isshownbelow:.51Reconstruct自dcontinuoij$tiirisiqn刮y(t)EL.5-5051o.o.-aprn-dr00511.5Time,msecBasedontheseresultswemakethefollowingobservations號可以復原，區(qū)間外的信號會發(fā)生時域混疊。采樣區(qū)間內的信Q5.8Thepl

44、otsofthediscrete-timesignalanditscontinuous-timeequivalentobtainedbyrunningProgramP5_2withtheoriginaldisplayrangerestoredandwiththefrequencyofthesinusoidalsignalchangedto3Hzareshownbelow:Reconstructedcontinuous-time合ign自Iy(t)呂01Time,msecTheplotsofthediscrete-timesignalanditscontinuous-timeequivalent

45、obtainedbyrunningProgramP5_2withtheoriginaldisplayrangerestoredandwiththefrequencyofthesinusoidalsignalchangedto7Hzareshownbelow:Reconstructedcontinuous-timesignaly(t)1505D-o-fl3PB=dlu0OJ0D.1Time,msecBasedontheseresultswemakethefollowingobservations即頻率之和等于采樣頻率的兩個正弦信號恢復后的連續(xù)信號相同。Theseresultscanbeexpla

46、inedasfollows-即頻率之和等于米樣頻率的兩個正弦信號采樣值是相同的，恢復后的連續(xù)信號當然也相同。而且由于混疊，其中頻率高于采樣頻率一半的那個信號是沒有正?；謴偷?。第七題Questions:Q5,9Wh；uislliecontinuous-limeIunctionxa(t)inProgramP5_3?HowistheCTFTolbeingcomputed?Q5,10RunProgramP5_3togenerateanddisplaybothIliediscrete-timesignalanditsconlinuous-liinecquivalcnl,andIheirrcspccliv

47、eFourierIrunsforms.Is(hereanyvisibleeffectofaliasing?Q5,l1RepeatProgramP5_3byincreasingthesamplingperiodto1.5.Isthereanyvisibleeffectofaliasing?Q5,12ModifyProgramP5_3lurlhecase山(t)=andrepealQuestionsQ.1()andQ5.11.”Answers:AcopyofProgramP53isgivenbelow:%ProgramP5_3%IllustrationoftheAliasingEffect%int

48、heFrequency-Domainclf;t=0:0.005:10;xa=2*t.*exp(-t);subplot(2,2,1)plot(t,xa);gridxlabel(Time,msec);ylabel(Amplitude);title(Continuous-timesignalx_a(t);subplot(2,2,2)wa=0:10/511:10;ha=freqs(2,121,wa);plot(wa/(2*pi),abs(ha);grid;xlabel(Frequency,kHz);ylabel(Amplitude);title(|X_a(jOmega)|);axis(05/pi02)

49、;subplot(2,2,3)T=1;n=0:T:10;xs=2*n.*exp(-n);k=0:length(n)-1;stem(k,xs);grid;xlabel(Timeindexn);ylabel(Amplitude);title(Discrete-timesignalxn);subplot(2,2,4)wd=0:pi/255:pi;hd=freqz(xs,1,wd);plot(wd/(T*pi),T*abs(hd);grid;xlabel(Frequency,kHz);ylabel(Amplitude);title(|X(eAjomega)|);axis(01/T02)Q5.9Thec

50、ontinuous-timefunctionxa(t)inProgramP5_3isx(t)=tetaTheCTFTofxa(t)isbeingcomputedbywa=0:10/511:10;ha=freqs(2,121,wa);Q5.10TheplotsgeneratedbyrunningProgramP5_3areshownbelow:|XaQQ)|XaQQ)|Continuous-time合ign自Ix(t)呂o6420D.o.0.豆-duly510Time,msec冊prqi-duuvDiscrete-timesignalxnTimeindexn86420.00D.CDpn七-aEV10observations-兩者頻譜有定Basedont