实验四线性时不变离散时间系统的频域分析

Name:Chenyifan20232121006

Section:

LaboratoryExercise4

LINEAR,TIME-INVARIANTDISCRETE-TIMESYSTEMS:FREQUENCY­

DOMAINREPRESENTATIONS

4.1TRANSFERFUNCTIONANDFREQUENCYRESPONSE

Project4.1TransferFunctionAnalysis

Answers:

Q4.1ThemodifiedProgram33_1tocomputeandplotthemagnitudeandphasespectraofamovingaverage

filterofEq.(2.13)forO<co<2nisshownbelow:

<Insertprogramcodehere.Copyfromm-file(s)andpaste.>

w=0:pi/51l:2*pi;

M=input('M=');

num=oncs(I,M)/M;

h=freqz(num,l,w);

subplot(2,l,l);

plot(w/pi,abs(h));grid;

出招(旧'人］j\omega})幅度谱)

xlabclC'omcgaApTXylabcl(振幅，)；

subplot(2,l,2);

plot(w/pi,angle(h));grid;

tilleC相位谱H(eA{j\omega}),);

xlabelCumegaApi，);ylabel(以弧度为单位的相位工

ThisprogramwasrunforthefallowingthreedifferentvaluesofMandtheplotsofthecorrespondingfrequency

responsesareshownbelow:

<Inserth/ATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>

Thetypesofsymmetriesexhibitedbythemagnitudeandphasespectraaredueto-

Thetypeoffilterrepresentedbythemovingaveragefilteris

M-3

M=10;

M=20;

TheresultsofQuestionQ2.1cannowbeexplainedasfollows-Bythegraph,youcanseethatitrepresentsalow-

passfilter.

Q4.2Theplotofthefrequenc/responseofthecausalLTIdiscrete-timesystemofQuestionQ4.2obtainedusing

themodifiedprogramisgivenbelow:

<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>

w=0:pi/51l:pi;

num=[0.150-0.15];

den-[l-0.50.7],

h=freqz(num,den,w);

subplot(2,l,l);

plot(w/pi,abs(h));grid;

(铲{j'omega})幅度谱)

xlabel(\omega八pi);ylabel(振幅)；

subplot(2,l,2);

plot(w/pi,angle(h));grid;

title，相位谱H(eA{j\omega}),);

xlabelCVomcgaApi1);

ylabelC以弧度为单位的相位，)：

Thetypeoffilterrepresentedbythistransferfunctionis-Itsaysbandpassfilterisobtainedbydiagrams

canbeBPF

Q4.3Theplotofthefrequenc/responseofthecausalLTIdiscrete-timesystemofQuestionQ4.3obtainedusing

themodifiedprogramisgivenbelow:

<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>

w=0:pi/511:pi;

num=[0.150-0.15];

dcn=[0.7-0.51]

h=frcqz(num,dcn,vv);

subplot(2,l,l);

plot(w/pi,abs(h));grid;

tiUe(H(e八{j'omega})幅度谱)；

xlabcl(*\omcga八pi);ylabclC振幅

subplot(2,l,2);

plot(w/pi,angle(h));grid;

tilleC相位谱H(eA{j\omega}),);

xlabel(,\omegaApi,);

ylabclC以弧度为单位的相位，)；

Thetypeoffilterrepresentedbythistransferfunctionis-Ocausallineartime-invariantdiscretetime

systemfrequencyresponse,determinethetypefilter

ThedifferencebetweenthetwofillersofQuestions4.2and4.3is-Moreonthetopicoffigure,the

amplitudespectrumisthesame,thephasespectrum,onthetopicthatiscontinuous,andkincsofphase

jumpline,.

IshallchoosethefilterofQuestionQ4.3forthefollowingreason-theamplitudespectrumisthesame,the

phasespectrum,onthetopicthatiscontinuous,andkindsofphasejumpline.

Q4.6Thepole-zeroplotsofthetwofiltersofQuestions4.2and4.3developedusingzplaneareshownbelow:

w=0:pi/511:pi;

num=[0.150-0.15];

den=[l-0.50.7];

h=zplane(num,den);

w=0:pi/511:pi;

num=[0.150-0.15];

dcn=[0.7-0.51];

h=zplanc(num,dcn);

<InsertIVATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>

Fromtheseplotswemakethefollowingobservations:Differentfunction,zerozeroandpolepolefigurewith

differentrelativepositionofthecircle

4.2TYPESOFTRANSFERFUNCTIONS

Project4.2Filters

AcopyofProgramP4_1isgivenbelow:

<Insertprogramcodehere.Copyfromm-file(s)andpaste.>

elf;

fc=0.25;

n=[-6.5:1:6.5];

y=2*fc*sinc(2*fc*n);k=n+6.5;

stcm(k,y);titlc('N=13');axis([013-0.20.6]);

xlabelC时间序号n«);ylab或振幅为grid

Answers:

Q4.7TheplotoftheimpulseresponseoftheapproximationtotheideallowpassfilterobtainedusingProgram

P4_1isshownbelow:

<Inserth/ATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>

elf;

fc=0.25;

n=[-6.5:l:6.5];

y=2*fc*sinc(2*fc*n);k=n+6.5;

stem(k,y);title(,N=13,);axis([013-0.20.6]);

xlabelf舟间序号n'ylabelC振幅)grid

ThelengthoftheFIRlowpassfilteris-14

ThestatementinProgramP4_1determiningthefilterlengthis-n=[-6,5:1:6,5];

Theparametercontrollingthecutcfffrequencyis-Fcparameterscontrolthecutofffrequency

Q4.8TherequiredmodificationstoProgramP4_1tocomputeandplottheimpulseresponseoftheFIRlowpass

filterofProject4.2withalengthof20andacutofffrequencyof⑴「=0.45areasindicatedbelow:

<Insertprogramcodehere.Copyfromm-file(s)andpaste.>

elf;

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]);

xlabelf0\t间序号rT);ylabelC振幅上grid

Theplotgeneratedbyrunningthemodifiedprogramisgivenbelov/:

<Inserth/ATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>

Q4.9TherequiredmodificationstoProgramP4_1tocomputeandplottheimpulseresponseoftheFIRlowpass

filterofProject4.2withalengthof15andacutofffrequencyok。-=0.65areasindicatedbelow:

<Insertprogramcodehere.Copyfromm-file(s)andpaste.>

elf;

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]);

xlabelC舟间序号rT);ylabel，振幅'grid

Theplotgeneratedbyrunningthemodifiedprogramisgivenbelov/:

<InsertK/ATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>

Q4.11Aplotofthegainresponseofalength-2movingaveragefilterobtainedusingProgramP4_2isshown

below:

<InsertI7ATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>

unction[g,w]=gain(num,den)—gain函数

w=0:pi/255:pi;

h=freqz(nuin,den,w);

g=20*logl0(abs(h));

M=2;一虐动平均低通滤波器的增益响应程序

num=ones(1

[g,w]=gain(num,l);

plot(w/pi,g);grid;

axis([01-500.5])

xlabelC\omcgaApiXylabclC单位为db的增益，)；

titlc(['M=\num2str(M)])

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title('2型有限冲激响应滤波器')；

scbplot(2,2,3);stem(n1,num3);

xlabe(时间序号n);ylabel。振幅)grid;

title('3型有限冲激响应滤波器')；

subplot®2,4);stem(n2,num4);

xlabel('时间序号n');ylabel(f振幅)grid;

title('4型有限冲激响应滤波器')；

pause

SLbplot(2,2,1);zplane(num1,1);

titlef'1型有限冲激响应滤波器')；

sibplot(2,2,2);zplane(num2,1);

title('2型有限冲激响应滤波器')；

subplot(2,2,3);zplane(num3,1);

title('3型有限冲激响应滤波器')；

scbplot(2,2,4);zplane(num4,1);

title('4型有限冲激响应滤波器');

dispfl型有限冲激响应滤波器的零点是');

disp(roots(num1));

disp('2型有限冲激响应滤波器的零点是');

disp(roots(num2));

disp('3型有限冲激响应滤波器的零点是');

disp(roots(num3));

disp('4型有限冲激响应滤波器的考点是');

disp(roots(num4));

TheplotsoftheimpulseresponsesofthefourFIRfiltersgeneratedbyrunningProgramP4_3aregivenbelow:

<InsertMATLABfigure(s)here.Copyfromfigurewindow(s)andpaste.>

Fromtheplotswemakethefollowingobservations:

Fiter#1isoflength8witha___80impulseresponseandisthereforeaType_linear-phase

FIRfilter.

Fiter#2isoflength10witha___80impulseresponseandisthereforeaType_linear-

phaseFIRfilter.

Filer#3isoflength______8____witha____60________impulseresponseandisthereforeaType_linear-phase

FIRfilter."

Fiter#4isoflength10witha___80impulseresponseandisthereforeaType_linear-

phaseFIRfilter.

FromthezerosofthesefiltersgeneratedbyProgramP43weobservethat:

Filter#1haszerosatz=2.9744,2.0888,0.9790+1.4110i,0.9790-1.41lOi,0.3319+0.4784i,0.3319-

0.4784i,0.4787,0.3362

Filter#2haszerosatz=3.7585+1.5147i,3.7585-1.5147i,0.6733+2.6623i,0.6733-2.6623i,-

1.0000,0.0893+O.353Oi,0.0893-0.3530i,0.2289+0.0922i,0.2289-0.0922i

Fiter#3haszerosatz=4.7627,1.6279+3.0565i,1.6279-3.0565i,-1.0000,1.0000

0.1357+0.2549i,0.1357-0.2549i,0.2100

Fiter#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,0.3159-0.30201,0.2929

PlotsofthephaseresponseofeachofthesefiltersobtainedusingMATLABareshownbelow:

<InsertMATLABfigurfi(s)here.Copyfromfigurewindow(s)andpaste.>

Fromtheseplotsweconcludethateachofthesefiltershave