数字信号处理实验报告5.docx
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数字信号处理实验报告5
Name:
Yangbo20112121010
Section:
LaboratoryExercise4
LINEAR,TIME-INVARIANTDISCRETE-TIMESYSTEMS:
FREQUENCY-DOMAINREPRESENTATIONS
4.1TRANSFERFUNCTIONANDFREQUENCYRESPONSE
Project4.1TransferFunctionAnalysis
Answers:
Q4.1ThemodifiedProgramP2_1tocomputeandplotthemagnitudeandphasespectraofamovingaveragefilterofEq.（2.13）for0
ω
2isshownbelow:
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}）幅度谱'）;
xlabel（'\omega/\pi'）;ylabel（'振幅'）;
subplot（2,1,2）;
plot（w/pi,angle（h））;grid;
title（'相位谱H（e^{j\omega}）'）;
xlabel（'\omega/\pi'）;ylabel（'以弧度为单位的相位'）;
ThisprogramwasrunforthefollowingthreedifferentvaluesofMandtheplotsofthecorrespondingfrequencyresponsesareshownbelow:
M=5
M=20
Thetypesofsymmetriesexhibitedbythemagnitudeandphasespectraaredueto-
Thetypeoffilterrepresentedbythemovingaveragefilteris–LPF
Q4.2TheplotofthefrequencyresponseofthecausalLTIdiscrete-timesystemofQuestionQ4.2obtainedusingthemodifiedprogramisgivenbelow：
w=0:
pi/511:
pi;
num=[0.15 0 -0.15];
den=[1 -0.5 0.7];
h=freqz（num,den,w）;
subplot（2,1,1）;
plot（w/pi,abs（h））;grid;
title（'H（e^{j\omega}）幅度谱'）;
xlabel（'\omega/\pi'）;ylabel（'振幅'）;
subplot（2,1,2）;
plot（w/pi,angle（h））;grid;
title（'相位谱H（e^{j\omega}）'）;
xlabel（'\omega/\pi'）;
ylabel（'以弧度为单位的相位'）;
Thetypeoffilterrepresentedbythistransferfunctionis–BPF
Q4.3TheplotofthefrequencyresponseofthecausalLTIdiscrete-timesystemofQuestionQ4.3obtainedusingthemodifiedprogramisgivenbelow:
w=0:
pi/511:
pi;
num=[0.15 0 -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}）幅度谱'）;
xlabel（'\omega/\pi'）;ylabel（'振幅'）;
subplot（2,1,2）;
plot（w/pi,angle（h））;grid;
title（'相位谱H（e^{j\omega}）'）;
xlabel（'\omega/\pi'）;
ylabel（'以弧度为单位的相位'）;
Thetypeoffilterrepresentedbythistransferfunctionis-BPF
ThedifferencebetweenthetwofiltersofQuestions4.2and4.3is–
Ontheamplitudespectrumisthesame,thetopicofthephasespectrumiscontinuous,andkindsofphasejumpline,
IshallchoosethefilterofQuestionQ4.2
Q4.6Thepole-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）;
Fromtheseplotswemakethefollowingobservations:
Differentfunction,zerozeroandpolepolefigurewithdifferentrelativepositionofthecircle
4.2TYPESOFTRANSFERFUNCTIONS
Project4.2Filters
AcopyofProgramP4_1isgivenbelow:
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（'时间信号n'）;ylabel（'振幅'）;grid
Answers:
Q4.7TheplotoftheimpulseresponseoftheapproximationtotheideallowpassfilterobtainedusingProgramP4_1isshownbelow:
ThelengthoftheFIRlowpassfilteris14
ThestatementinProgramP4_1determiningthefilterlengthisn=[-6.5:
1:
6.5];
Theparametercontrollingthecutofffrequencyisfc
Q4.8TherequiredmodificationstoProgramP4_1tocomputeandplottheimpulseresponseoftheFIRlowpassfilterofProject4.2withalengthof20andacutofffrequencyofωc=0.45areasindicatedbelow:
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（[0 20 -0.2 0.6]）;
xlabel（'时间序号n'）;ylabel（'振幅'）;grid
Theplotgeneratedbyrunningthemodifiedprogramisgivenbelow:
Q4.9TherequiredmodificationstoProgramP4_1tocomputeandplottheimpulseresponseoftheFIRlowpassfilterofProject4.2withalengthof15andacutofffrequencyofωc=0.65areasindicatedbelow:
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（[0 14 -0.2 0.6]）;
xlabel（'时间序号n'）;ylabel（'振幅'）;grid
Theplotgeneratedbyrunningthemodifiedprogramisgivenbelow:
Q4.11Aplotofthegainresponseofalength-2movingaveragefilterobtainedusingProgramP4_2isshownbelow:
function [g,w]=gain（num,den） ----gain函数
w=0:
pi/255:
pi;
h=freqz（num,den,w）;
g=20*log10（abs（h））;
M=2;
num=ones（1,M）/M;
[g,w]=gain（num,1）;
plot（w/pi,g）;grid;
axis（[0 1 -50 0.5]）
xlabel（'\omega/\pi'）;ylabel（'单位为db的增益'）;
title（['M= ',num2str（M）]）
Fromtheplotitcanbeseenthatthe3-dBcutofffrequencyisatPI/2
Q4.12TherequiredmodificationstoProgramP4_2tocomputeandplotthegainresponseofacascadeofKlength-2movingaveragefiltersaregivenbelow:
function [g,w]=gainlink（num,den） –gainlink函数
w=0:
pi/255:
pi;
h=freqz（num,den,w）;
H=h.*h.*h;
g=20*log10（abs（H））;
M=2;
num=ones（1,M）/M;
[g,w]=gainlink（num,1）;
plot（w/pi,g）;grid;
axis（[0 1 -50 0.5]）
xlabel（'\omega/\pi'）;ylabel（'单位为db的增益'）;
title（['M= ',num2str（M）]）
Theplotofthegainresponseforacascadeof3sectionsobtainedusingthemodifiedprogramisshownbelow:
Fromtheplotitcanbeseenthatthe3-dBcutofffrequencyofthecascadeisat0.3*pi
Q4.19AcopyofProgramP4_3isgivenbelow:
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（'时间序号n'）;ylabel（'振幅'）;grid;
title（'1型有限冲激响应滤波器'）;
subplot（2,2,2）;stem（n2,num2）;
xlabel（'时间序号n'）;ylabel（'振幅'）;grid;
title（'2型有限冲激响应滤波器'）;
subplot（2,2,3）;stem（n1,num3）;
xlabel（'时间序号n'）;ylabel（'振幅'）;grid;
title（'3型有限冲激响应滤波器'）;
subplot（2,2,4）;stem（n2,num4）;
xlabel（'时间序号n'）;ylabel（'振幅'）;grid;
title（'4型有限冲激响应滤波器'）;
pause
subplot（2,2,1）;zplane（num1,1）;
title（'1型有限冲激响应滤波器'）;
subplot（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'）;
subplot（2,2,4）;zplane（num4,1）;
title（'4型有限冲激响应滤波器'）;
disp（'1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零点是'）;
disp（roots（num4））;
TheplotsoftheimpulseresponsesofthefourFIRfiltersgeneratedbyrunningProgramP4_3aregivenbelow:
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、0.3362
Filter#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、0.2289-0.0922i
Filter#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
Filter#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、0.2929
Q4.20TheplotsoftheimpulseresponsesofthefourFIRfiltersgeneratedbyrunningProgramP4_3aregivenbelow:
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,0.2457-0.2126i
Filter#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,0.2397-0.1934i
Filter#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,0.1655-0.7803i
Filter#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,0.2431-0.2399i
4.3STABILITYTEST
AcopyofProgramP4_4isgivenbelow:
clf;
den=input（‘分母系数=’）;
ki=poly2rc（den）;
disp（‘稳定性测试参数是’）;
disp（ki）;
Answers:
Q4.23Thepole-zeroplotsofH1（z）andH2（z）obtainedusingzplaneareshownbelow:
Fromtheabovepole-zeroplotsweobservethat-
Q4.24UsingProgramP4_4wetestedthestabilityofH1（z）andarriveatthefollowingstabilitytestparameters{ki}:
FromtheseparametersweconcludethatH1（z）is_____stable________.
UsingProgramP4_4wetestedthestabilityofH2（z）andarriveatthefollowingstabilitytestparameters{ki}:
FromtheseparametersweconcludethatH2（z）is_____stable________.
Q4.25UsingProgramP4_4wetestedtherootlocationsofD（z）andarriveatthefollowingstabilityt
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