實(shí)驗(yàn)四線性時(shí)不變離散時(shí)間系統(tǒng)的頻域分析

Name:Chenyifan20232121006Section:LaboratoryExercise4LINEAR,TIME-INVARIANTDISCRETE-TIMESYSTEMS:FREQUENCY-DOMAINREPRESENTATIONS4.1 TRANSFERFUNCTIONANDFREQUENCYRESPONSEProject4.1 TransferFunctionAnalysisAnswers:Q4.1 ThemodifiedProgramP3_1tocomputeandplotthemagnitudeandphasespectraofamovingaveragefilterofEq.(2.13)for02isshownbelow:<Insertprogramcodehere.Copyfromm-file(s)andpaste.>w=0:pi/511:2*pi;M=input('M=');num=ones(1,M)/M;h=freqz(num,1,w);subplot(2,1,1);plot(w/pi,abs(h));grid;title('H(e^{j\omega})幅度譜');xlabel('\omega/\pi');ylabel('振幅');subplot(2,1,2);plot(w/pi,angle(h));grid;title('相位譜H(e^{j\omega})');xlabel('\omega/\pi');ylabel('以弧度為單位的相位');ThisprogramwasrunforthefollowingthreedifferentvaluesofMandtheplotsofthecorrespondingfrequencyresponsesareshownbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>Thetypesofsymmetriesexhibitedbythemagnitudeandphasespectraaredueto-ThetypeoffilterrepresentedbythemovingaveragefilterisM=3-M=10;M=20;TheresultsofQuestionQ2.1cannowbeexplainedasfollows-Bythegraph,youcanseethatitrepresentsalow-passfilter.Q4.2 TheplotofthefrequencyresponseofthecausalLTIdiscrete-timesystemofQuestionQ4.2obtainedusingthemodifiedprogramisgivenbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>w=0:pi/511:pi;num=[0.150-0.15];den=[1-0.50.7];h=freqz(num,den,w);subplot(2,1,1);plot(w/pi,abs(h));grid;title('H(e^{j\omega})幅度譜');xlabel('\omega/\pi');ylabel('振幅');subplot(2,1,2);plot(w/pi,angle(h));grid;title('相位譜H(e^{j\omega})');xlabel('\omega/\pi');ylabel('以弧度為單位的相位');Thetypeoffilterrepresentedbythistransferfunctionis-ItsaysbandpassfilterisobtainedbydiagramscanbeBPFQ4.3 TheplotofthefrequencyresponseofthecausalLTIdiscrete-timesystemofQuestionQ4.3obtainedusingthemodifiedprogramisgivenbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>w=0:pi/511:pi;num=[0.150-0.15];den=[0.7-0.51]h=freqz(num,den,w);subplot(2,1,1);plot(w/pi,abs(h));grid;title('H(e^{j\omega})幅度譜');xlabel('\omega/\pi');ylabel('振幅');subplot(2,1,2);plot(w/pi,angle(h));grid;title('相位譜H(e^{j\omega})');xlabel('\omega/\pi');ylabel('以弧度為單位的相位');Thetypeoffilterrepresentedbythistransferfunctionis-Ocausallineartime-invariantdiscretetimesystemfrequencyresponse,determinethetypefilterThedifferencebetweenthetwofiltersofQuestions4.2and4.3is-Moreonthetopicoffigure,theamplitudespectrumisthesame,thephasespectrum,onthetopicthatiscontinuous,andkindsofphasejumpline,.IshallchoosethefilterofQuestionQ4.3forthefollowingreason-theamplitudespectrumisthesame,thephasespectrum,onthetopicthatiscontinuous,andkindsofphasejumpline.Q4.6 Thepole-zeroplotsofthetwofiltersofQuestions4.2and4.3developedusingzplaneareshownbelow:w=0:pi/511:pi;num=[0.150-0.15];den=[1-0.50.7];h=zplane(num,den);w=0:pi/511:pi;num=[0.150-0.15];den=[0.7-0.51];h=zplane(num,den);<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>Fromtheseplotswemakethefollowingobservations:Differentfunction,zerozeroandpolepolefigurewithdifferentrelativepositionofthecircle4.2 TYPESOFTRANSFERFUNCTIONSProject4.2 Filters AcopyofProgramP4_1isgivenbelow:<Insertprogramcodehere.Copyfromm-file(s)andpaste.>clf;fc=0.25;n=[-6.5:1:6.5];y=2*fc*sinc(2*fc*n);k=n+6.5;stem(k,y);title('N=13');axis([013-0.20.6]);xlabel('時(shí)間序號n');ylabel('振幅');gridAnswers:Q4.7 TheplotoftheimpulseresponseoftheapproximationtotheideallowpassfilterobtainedusingProgramP4_1isshownbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>clf;fc=0.25;n=[-6.5:1:6.5];y=2*fc*sinc(2*fc*n);k=n+6.5;stem(k,y);title('N=13');axis([013-0.20.6]);xlabel('時(shí)間序號n');ylabel('振幅');gridThelengthoftheFIRlowpassfilteris-14ThestatementinProgramP4_1determiningthefilterlengthis-n=[-6.5:1:6.5];Theparametercontrollingthecutofffrequencyis-FcparameterscontrolthecutofffrequencyQ4.8 TherequiredmodificationstoProgramP4_1tocomputeandplottheimpulseresponseoftheFIRlowpassfilterofProject4.2withalengthof20andacutofffrequencyofc=0.45areasindicatedbelow:<Insertprogramcodehere.Copyfromm-file(s)andpaste.>clf;fc=0.45/(2*pi);n=[-9.5:1:9.5];y=2*fc*sinc(2*fc*n);k=n+9.5;stem(k,y);title('N=20');axis([020-0.20.6]);xlabel('時(shí)間序號n');ylabel('振幅');gridTheplotgeneratedbyrunningthemodifiedprogramisgivenbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>Q4.9 TherequiredmodificationstoProgramP4_1tocomputeandplottheimpulseresponseoftheFIRlowpassfilterofProject4.2withalengthof15andacutofffrequencyofc=0.65areasindicatedbelow:<Insertprogramcodehere.Copyfromm-file(s)andpaste.>clf;fc=0.65/(2*pi);n=[-7.5:1:6.5];y=2*fc*sinc(2*fc*n);k=n+7.5;stem(k,y);title('N=15');axis([014-0.20.6]);xlabel('時(shí)間序號n');ylabel('振幅');gridTheplotgeneratedbyrunningthemodifiedprogramisgivenbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>Q4.11 Aplotofthegainresponseofalength-2movingaveragefilterobtainedusingProgramP4_2isshownbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>unction[g,w]=gain(num,den)--gain函數(shù)w=0:pi/255:pi;h=freqz(num,den,w);g=20*log10(abs(h));M=2;--滑動平均低通濾波器的增益響應(yīng)程序num=ones(1,M)/M;[g,w]=gain(num,1);plot(w/pi,g);grid;axis([01-500.5])xlabel('\omega/\pi');ylabel('單位為db的增益');title(['M=',num2str(M)])Fromtheplotitcanbeseenthatthe3-dBcutofffrequencyisat-3dB。Q4.12 TherequiredmodificationstoProgramP4_2tocomputeandplotthegainresponseofacascadeofKlength-2movingaveragefiltersaregivenbelow:<Insertprogramcodehere.Copyfromm-file(s)andpaste.>function[g,w]=gainlink(num,den)\w=0:pi/255:pi;h=freqz(num,den,w);H=h.*h.*h;g=20*log10(abs(H));wc=2*acos(2^(-1/(2*3)))wc=0.9430wc=0.3*piTheplotofthegainresponseforacascadeof3sectionsobtainedusingthemodifiedprogramisshownbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>Fromtheplotitcanbeseenthatthe3-dBcutofffrequencyofthecascadeisat-\wc=0.3*pi\Q4.19 AcopyofProgramP4_3isgivenbelow:<Insertprogramcodehere.Copyfromm-file(s)andpaste.>clf;b=[1-8.530.5-63];num1=[b81fliplr(b)];num2=[b8181fliplr(b)];num3=[b0-fliplr(b)];num4=[b81-81-fliplr(b)];n1=0:length(num1)-1;n2=0:length(num2)-1;subplot(2,2,1);stem(n1,num1);xlabel('時(shí)間序號n');ylabel('振幅');grid;title('1型有限沖激響應(yīng)濾波器');subplot(2,2,2);stem(n2,num2);xlabel('時(shí)間序號n');ylabel('振幅');grid;title('2型有限沖激響應(yīng)濾波器');subplot(2,2,3);stem(n1,num3);xlabel('時(shí)間序號n');ylabel('振幅');grid;title('3型有限沖激響應(yīng)濾波器');subplot(2,2,4);stem(n2,num4);xlabel('時(shí)間序號n');ylabel('振幅');grid;title('4型有限沖激響應(yīng)濾波器');pausesubplot(2,2,1);zplane(num1,1);title('1型有限沖激響應(yīng)濾波器');subplot(2,2,2);zplane(num2,1);title('2型有限沖激響應(yīng)濾波器');subplot(2,2,3);zplane(num3,1);title('3型有限沖激響應(yīng)濾波器');subplot(2,2,4);zplane(num4,1);title('4型有限沖激響應(yīng)濾波器');disp('1型有限沖激響應(yīng)濾波器的零點(diǎn)是');disp(roots(num1));disp('2型有限沖激響應(yīng)濾波器的零點(diǎn)是');disp(roots(num2));disp('3型有限沖激響應(yīng)濾波器的零點(diǎn)是');disp(roots(num3));disp('4型有限沖激響應(yīng)濾波器的零點(diǎn)是');disp(roots(num4));TheplotsoftheimpulseresponsesofthefourFIRfiltersgeneratedbyrunningProgramP4_3aregivenbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>Fromtheplotswemakethefollowingobservations:Filter#1isoflength_____8_____witha___80_______impulseresponseandisthereforeaType__linear-phaseFIRfilter.Filter#2isoflength____10______witha___80_______impulseresponseandisthereforeaType__linear-phaseFIRfilter.Filter#3isoflength______8____witha___60_______impulseresponseandisthereforeaType__linear-phaseFIRfilter.Filter#4isoflength______10____witha___80_______impulseresponseandisthereforeaType__linear-phaseFIRfilter.FromthezerosofthesefiltersgeneratedbyProgramP4_3weobservethat:Filter#1haszerosatz=2.9744,2.0888,0.9790+1.4110i,0.9790-1.4110i,0.3319+0.4784i,0.3319-0.4784i,0.4787,0.3362Filter#2haszerosatz=3.7585+1.5147i,3.7585-1.5147i,0.6733+2.6623i,0.6733-2.6623i,-1.0000,0.0893+0.3530i,0.0893-0.3530i,0.2289+0.0922i,0.2289-0.0922iFilter#3haszerosatz=4.7627,1.6279+3.0565i,1.6279-3.0565i,-1.0000,1.00000.1357+0.2549i,0.1357-0.2549i,0.2100Filter#4haszerosatz=3.4139,1.6541+1.5813i,1.6541-1.5813i,-0.0733+0.9973i,-0.0733-0.9973i,1.0000,0.3159+0.3020i,0.3159-0.3020i,0.2929PlotsofthephaseresponseofeachofthesefiltersobtainedusingMATLABareshownbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>Fromtheseplotsweconcludethateachofthesefiltershave______linear____phase.Q4.20 TheplotsoftheimpulseresponsesofthefourFIRfiltersgeneratedbyrunningProgramP4_3aregivenbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>Fromtheplotswemakethefollowingobservations:Filter#1isoflength_____8_____witha___80_______impulseresponseandisthereforeaType__linear-phaseFIRfilter.Filter#2isoflength____10______witha___80_______impulseresponseandisthereforeaType__linear-phaseFIRfilter.Filter#3isoflength_____8_____witha_____5_____impulseresponseandisthereforeaType__linear-phaseFIRfilter.Filter#4isoflength____10______witha____80______impulseresponseandisthereforeaType__linear-phaseFIRfilter.FromthezerosofthesefiltersgeneratedbyProgramP4_3weobservethat:Filter#1haszerosatz=2.3273+2.0140i,2.3273-2.0140i,-1.2659+2.0135i,-1.2659-2.0135i,-0.2238+0.3559i,-0.2238-0.3559i,0.2457+0.2126i,0.2457-0.2126iFilter#2haszerosatz=2.5270+2.0392i,2.5270-2.0392i,-1.0101+2.1930i.-1.0101-2.1930i,-1.0000,-0.1733+0.3762i,-0.1733-0.3762i,0.2397+0.1934i,0.2397-0.1934iFilter#3haszerosatz=-1.0000,0.2602+1.2263i,0.2602-1.2263i,1.0000,0.6576+0.7534i,0.6576-0.7534i,0.1655+0.7803i,0.1655-0.7803iFilter#4haszerosatz=2.0841+2.0565i,2.0841-2.0565i,-1.5032+1.9960i,-1.5032-1.9960i,1.0000,-0.2408+0.3197i,-0.2408-0.3197i,0.2431+0.2399i,0.2431-0.2399iPlotsofthephaseresponseofeachofthesefiltersobtainedusingMATLABareshownbelow:<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>Fromtheseplotsweconcludethateachofthesefiltershave___linear_______phase.Answers:4.3 STABILITYTEST AcopyofProgramP4_4isgivenbelow:<Insertprogramcodehere.Copyfromm-file(s)andpaste.>clf;den=input(‘分