武汉大学数字信号实验.docx
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武汉大学数字信号实验
姓名:
全权学号：
*************班级：
信安四班日期：
5月8日
实验2离散时间系统-时域分析
实验内容：
2.1SIMULATIONOFDISCRETE-TIMESYSTEMS
Project2.1TheMovingAverageSystem（移动平均值系统）
AcopyofProgramP2_1isgivenbelow:
%ProgramP2_1
%SimulationofanM-pointMovingAverageFilter
%Generatetheinputsignal
n=0:
100;
s1=cos（2*pi*0.05*n）;%Alow-frequencysinusoid
s2=cos（2*pi*0.47*n）;%Ahighfrequencysinusoid
x=s1+s2;
%Implementationofthemovingaveragefilter
M=input（'InputDesiredlengthofthefilter='）;
num=ones（1,M）;
y=filter（num,1,x）/M;
%注：
上式用滤波函数实现移动平均
%当M=2时，y=（x（n）+x（n-1））/M
%当M=3时，y=（x（n）+x（n-1）+x（n-2））/M
%Displaytheinputandoutputsignals
clf;
subplot（2,2,1）;
plot（n,s1）;
axis（[0,100,-2,2]）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'Signal#1'）;
subplot（2,2,2）;
plot（n,s2）;
axis（[0,100,-2,2]）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'Signal#2'）;
subplot（2,2,3）;
plot（n,x）;
axis（[0,100,-2,2]）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'InputSignal'）;
subplot（2,2,4）;
plot（n,y）;
axis（[0,100,-2,2]）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'OutputSignal'）;
axis;
Answers:
Q2.1TheoutputsequencegeneratedbyrunningtheaboveprogramforM=2withx[n]=s1[n]+s2[n]astheinputisshownbelow.
s2=cos（2*pi*0.47*n）;%Ahighfrequencysinusoid
Thecomponentoftheinputx[n]suppressed（抑制）bythediscrete-timesystemsimulatedbythisprogramis-
s2=cos（2*pi*0.47*n）;%Ahighfrequencysinusoid
Q2.2ProgramP2_1ismodifiedtosimulatetheLTIsystemy[n]=0.5（x[n]–x[n–1]）andprocesstheinputx[n]=s1[n]+s2[n]resultingintheoutputsequenceshownbelow:
%ProgramP2_1
%SimulationofanM-pointMovingAverageFilter
%Generatetheinputsignal
n=0:
100;
s1=cos（2*pi*0.05*n）;%Alow-frequencysinusoid
s2=cos（2*pi*0.47*n）;%Ahighfrequencysinusoid
x=s1+s2;
%Implementationofthemovingaveragefilter
M=input（'InputDesiredlengthofthefilter='）;
%num=ones（1,M）;
num=（-1）.^[0:
M-1];
y=filter（num,1,x）/M;
%注：
上式用滤波函数实现移动平均
%当M=2时，y=（x（n）+x（n-1））/M
%当M=3时，y=（x（n）+x（n-1）+x（n-2））/M
%Displaytheinputandoutputsignals
clf;
subplot（2,2,1）;
plot（n,s1）;
axis（[0,100,-2,2]）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'Signal#1'）;
subplot（2,2,2）;
plot（n,s2）;
axis（[0,100,-2,2]）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'Signal#2'）;
subplot（2,2,3）;
plot（n,x）;
axis（[0,100,-2,2]）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'InputSignal'）;
subplot（2,2,4）;
plot（n,y）;
axis（[0,100,-2,2]）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'OutputSignal'）;
axis;
TheeffectofchangingtheLTIsystemontheinputis-过滤掉高频信号留下低频信号
Q2.3ProgramP2_1isrunforthefollowingvaluesoffilter（滤波器）lengthMandfollowingvaluesofthefrequenciesofthesinusoidalsignalss1[n]ands2[n].TheoutputgeneratedforthesedifferentvaluesofMandthefrequenciesareshownbelow.Fromtheseplotswemakethefollowingobservations-f1=0.05;f2=0.47;M=15
Project2.2（Optional）ASimpleNonlinearDiscrete-TimeSystem
AcopyofProgramP2_2isgivenbelow:
%ProgramP2_2
%Generateasinusoidalinputsignal
clf;
n=0:
200;
x=cos（2*pi*0.05*n）;
%Computetheoutputsignal
x1=[x00];%x1[n]=x[n+1]
x2=[0x0];%x2[n]=x[n]
x3=[00x];%x3[n]=x[n-1]
y=x2.*x2-x1.*x3;
y=y（2:
202）;
%Plottheinputandoutputsignals
subplot（2,1,1）
plot（n,x）
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'InputSignal'）
subplot（2,1,2）
plot（n,y）
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'Outputsignal'）;
Answers:
Q2.4Thesinusoidalsignalswiththefollowingfrequenciesastheinputsignalswereusedtogeneratetheoutputsignals:
长度101，最小频率0.最大0.5
n=0:
100;
a=pi/2/100;
b=0;
arg=a*n.*n+b*n;
x=cos（arg）;
M=input（'Desiredlengthofthefilter='）;
num=ones（1,M）;
y=filter（num,1,x）/M;
clf;
subplot（2,1,1）;
plot（n,x）;
axis（[0,100,-1.5,1.5]）;
xlabel（'Timeindexn'）;
ylabel（'Amplitude'）;
title（'inputsignal'）;
subplot（2,1,2）;
plot（n,y）;
axis（[0,100,-1.5,-1.5]）;
xlabel（'Timeindexn'）;
ylabel（'Amplitude'）;
title（'inputsignal'）;
axis;
Theoutputsignalsgeneratedforeachoftheaboveinputsignalsaredisplayedbelow:
Theoutputsignalsdependonthefrequenciesoftheinputsignalaccordingtothefollowingrules:
Thisobservationcanbeexplainedmathematicallyasfollows:
Project2.3LinearandNonlinearSystems（线性和非线性系统）
AcopyofProgramP2_3isgivenbelow:
%ProgramP2_3
%Generatetheinputsequences
clf;
n=0:
40;
a=2;b=-3;
x1=cos（2*pi*0.1*n）;
x2=cos（2*pi*0.4*n）;
x=a*x1+b*x2;
num=[2.24032.49082.2403];
den=[1-0.40.75];
ic=[00];%Setzeroinitialconditions
y1=filter（num,den,x1,ic）;%Computetheoutputy1[n]
y2=filter（num,den,x2,ic）;%Computetheoutputy2[n]
y=filter（num,den,x,ic）;%Computetheoutputy[n]
yt=a*y1+b*y2;
d=y-yt;%Computethedifferenceoutputd[n]
%Plottheoutputsandthedifferencesignal
subplot（3,1,1）
stem（n,y）;
ylabel（'Amplitude'）;
title（'OutputDuetoWeightedInput:
a\cdotx_{1}[n]+b\cdotx_{2}[n]'）;
subplot（3,1,2）
stem（n,yt）;
ylabel（'Amplitude'）;
title（'WeightedOutput:
a\cdoty_{1}[n]+b\cdoty_{2}[n]'）;
subplot（3,1,3）
stem（n,d）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'DifferenceSignal'）;
Answers:
Q2.5Theoutputsy[n],obtainedwithweighted（加权的）input,andyt[n],obtainedbycombiningthetwooutputsy1[n]andy2[n]withthesameweights,areshownbelowalongwiththedifferencebetweenthetwosignals:
Thetwosequencesare-数字取舍后相同
Thesystemis-线性
Q2.6ProgramP2_3wasrunforthefollowingthreedifferentsetsofvaluesoftheweightingconstants（常数）,aandb,andthefollowingthreedifferentsetsofinputfrequencies:
Theplotsgeneratedforeachoftheabovethreecasesareshownbelow:
Basedontheseplotswecanconcludethatthesystemwithdifferentweightsis-_线性__
Q2.7ProgramP2_3wasmodifiedtosimulatethesystem:
y[n]=x[n]x[n–1]
Theoutputsequencesy1[n],y2[n],andy[n]oftheabovesystemgeneratedbyrunningthemodifiedprogramareshownbelow:
Comparingy[n]withyt[n]weconcludethatthetwosequencesare-_不同____
Thissystemis-_非线性
Project2.4Time-invariant（时不变）andTime-varying（时变）Systems
AcopyofProgramP2_4isgivenbelow:
%ProgramP2_4
%Generatetheinputsequences
clf;
n=0:
40;D=10;a=3.0;b=-2;
x=a*cos（2*pi*0.1*n）+b*cos（2*pi*0.4*n）;
xd=[zeros（1,D）x];
num=[2.24032.49082.2403];
den=[1-0.40.75];
ic=[00];%Setinitialconditions
%Computetheoutputy[n]
y=filter（num,den,x,ic）;
%Computetheoutputyd[n]
yd=filter（num,den,xd,ic）;
%Computethedifferenceoutputd[n]
d=y-yd（1+D:
41+D）;
%Plottheoutputs
subplot（3,1,1）
stem（n,y）;
ylabel（'Amplitude'）;
title（'Outputy[n]'）;grid;
subplot（3,1,2）
stem（n,yd（1:
41））;
ylabel（'Amplitude'）;
title（['OutputduetoDelayedInputx[n-',num2str（D）,']']）;grid;
subplot（3,1,3）
stem（n,d）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'DifferenceSignal'）;grid;
Answers:
Q2.8Theoutputsequencesy[n]andyd[n-10]generatedbyrunningProgramP2_4areshownbelow-
Thesetwosequencesarerelatedasfollows-
Thesystemis-___时不变系统_
Q2.9Theoutputsequencesy[n]andyd[n-D]generatedbyrunningProgramP2_4forthefollowingvaluesofthedelayvariableD-
areshownbelow-____________
Ineachcase,thesetwosequencesarerelatedasfollows-
Thesystemis-__时不变系统
2.2LINEARTIME-INVARIANTDISCRETE-TIMESYSTEMS
Project2.5ComputationofImpulseResponses（冲击响应）ofLTISystems
AcopyofProgramP2_5isshownbelow:
Answers:
Q2.10Thefirst41samplesoftheimpulseresponseofthediscrete-timesystemofProject2.3generatedbyrunningProgramP2_5isgivenbelow:
%ProgramP2_5
%Computetheimpulseresponsey
clf;
N=40;
num=[2.24032.49082.2403];
den=[1-0.40.75];
y=impz（num,den,N）;
%Plottheimpulseresponse
stem（y）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'ImpulseResponse'）;grid;
Q2.11TherequiredmodificationstoProgramP2_5togeneratetheimpulseresponseofthefollowingcausalLTIsystem:
y[n]+0.71y[n-1]–0.46y[n-2]–0.62y[n-3]
=0.9x[n]–0.45x[n-1]+0.35x[n-2]+0.002x[n-3]
aregivenbelow:
clf;
N=45;
num=[0.9-0.450.350.002];
den=[1.00.71-0.46-0.62];
y=impz（num,den,N）;
stem（y）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'ImpulseResponse'）;grid;
Thefirst45samplesoftheimpulseresponseofthisdiscrete-timesystemgeneratedbyrunningthem
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