数字信号处理实验报告.docx
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数字信号处理实验报告
Name:
Section:
LaboratoryExercise2
DISCRETE-TIMESYSTEMS:
TIME-DOMAINREPRESENTATION
2.1SIMULATIONOFDISCRETE-TIMESYSTEMS
Project2.1TheMovingAverageSystem
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（'Desiredlengthofthefilter='）;
num=ones（1,M）;
y=filter（num,1,x）/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.
Thecomponentoftheinputx[n]suppressedbythediscrete-timesystemsimulatedbythisprogramis–s2
Q2.2ProgramP2_1ismodifiedtosimulatetheLTIsystemy[n]=0.5（x[n]–x[n–1]）andprocesstheinputx[n]=s1[n]+s2[n]resultingintheoutputsequenceshownbelow:
s3=cos（2*pi*0.05*（n-1））;
s4=cos（2*pi*0.47*（n-1））;
z=s3+s4;
y=0.5*（x-z）;
TheeffectofchangingtheLTIsystemontheinputis-
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:
Q2.5Thesinusoidalsignalswiththefollowingfrequenciesastheinputsignalswereusedtogeneratetheoutputsignals:
Theoutputsignalsgeneratedforeachoftheaboveinputsignalsaredisplayedbelow:
Theoutputsignalsdependonthefrequenciesoftheinputsignalaccordingtothefollowingrules:
Thisobservationcanbeexplainedmathematicallyasfollows:
Project2.3LinearandNonlinearSystems
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:
Q2.7Theoutputsy[n],obtainedwithweightedinput,andyt[n],obtainedbycombiningthetwooutputsy1[n]andy2[n]withthesameweights,areshownbelowalongwiththedifferencebetweenthetwosignals:
Thetwosequencesare–same;wecanregard10（-15）as0
Thesystemis–alinersystem
Q2.9Program2_3wasrunwiththefollowingnon-zeroinitialconditions-ic=[22];
Theplotsgeneratedareshownbelow-
Basedontheseplotswecanconcludethatthesystemwithnonzeroinitialconditionsis–assameasthezeroinitialconditionwiththetimegone
Project2.4Time-invariantandTime-varyingSystems
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:
Q2.12Theoutputsequencesy[n]andyd[n-10]generatedbyrunningProgramP2_4areshownbelow-
Thesetwosequencesarerelatedasfollows–same,theoutputdon’tchangewiththetime
Thesystemis-Timeinvariantsystem
Q2.15Theoutputsequencesy[n]andyd[n-10]generatedbyrunningProgramP2_4fornon-zeroinitialconditionsareshownbelow-ic=[52];
Thesetwosequencesarerelatedasfollows–justasthesequencesabove
Thesystemis–notrelatedtotheinitialconditions
2.2LINEARTIME-INVARIANTDISCRETE-TIMESYSTEMS
Project2.5ComputationofImpulseResponsesofLTISystems
AcopyofProgramP2_5isshownbelow:
%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;
Answers:
Q2.19Thefirst41samplesoftheimpulseresponseofthediscrete-timesystemofProject2.3generatedbyrunningProgramP2_5isgivenbelow:
Project2.6CascadeofLTISystems
AcopyofProgramP2_6isgivenbelow:
%ProgramP2_6
%CascadeRealization
clf;
x=[1zeros（1,40）];%Generatetheinput
n=0:
40;
%Coefficientsof4thordersystem
den=[11.62.281.3250.68];
num=[0.06-0.190.27-0.260.12];
%Computetheoutputof4thordersystem
y=filter（num,den,x）;
%Coefficientsofthetwo2ndordersystems
num1=[0.3-0.20.4];den1=[10.90.8];
num2=[0.2-0.50.3];den2=[10.70.85];
%Outputy1[n]ofthefirststageinthecascade
y1=filter（num1,den1,x）;
%Outputy2[n]ofthesecondstageinthecascade
y2=filter（num2,den2,y1）;
%Differencebetweeny[n]andy2[n]
d=y-y2;
%Plotoutputanddifferencesignals
subplot（3,1,1）;
stem（n,y）;
ylabel（'Amplitude'）;
title（'Outputof4thorderRealization'）;grid;
subplot（3,1,2）;
stem（n,y2）
ylabel（'Amplitude'）;
title（'OutputofCascadeRealization'）;grid;
subplot（3,1,3）;
stem（n,d）
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'DifferenceSignal'）;grid;
Answers:
Q2.23Theoutputsequencesy[n],y2[n],andthedifferencesignald[n]generatedbyrunningProgramP2_6areindicatedbelow:
Therelationbetweeny[n]andy2[n]is–y[n]istheConvolutionofy2[n]andy1[n]
The4thordersystemcandothesamejobasthecascadesystem
Q2.24ThesequencesgeneratedbyrunningProgramP2_6withtheinputchangedtoasinusoidalsequenceareasfollows:
x=sin（2*pi*0.05*n）;
Therelationbetweeny[n]andy2[n]inthiscaseis–sameastherelationabove
Project2.7Convolution
AcopyofProgramP2_7isreproducedbelow:
%ProgramP2_7
clf;
h=[321-210-403];%impulseresponse
x=[1-23-4321];%inputsequence
y=conv（h,x）;
n=0:
14;
subplot（2,1,1）;
stem（n,y）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'OutputObtainedbyConvolution'）;grid;
x1=[xzeros（1,8）];
y1=filter（h,1,x1）;
subplot（2,1,2）;
stem（n,y1）;
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
title（'OutputGeneratedbyFiltering'）;grid;
Answers:
Q2.28Thesequencesy[n]andy1[n]generatedbyrunningProgramP2_7areshownbelow:
Thedifferencebetweeny[n]andy1[n]is-same
Thereasonforusingx1[n]astheinput,obtainedbyzero-paddingx[n],forgeneratingy1[n]is–thelengthofxis7,butthelengthoftheConvolutionis14,andn=14,weneedthelengthoffiltertobe14
Project2.8StabilityofLTISystems
AcopyofProgramP2_8isgivenbelow:
%ProgramP2_8
%Stabilitytestbasedonthesumoftheabsolute
%valuesoftheimpulseresponsesamples
clf;
num=[1-0.8];den=[11.50.9];
N=200;
h=impz（num,den,N+1）;
parsum=0;
fork=1:
N+1;
parsum=parsum+abs（h（k））;
ifabs（h（k））<10^（-6）,break,end
end
%Plottheimpulseresponse
n=0:
N;
stem（n,h）
xlabel（'Timeindexn'）;ylabel（'Amplitude'）;
%Printthevalueofabs（h（k））
disp（'Value='）;disp（abs（h（k）））;
Answers:
Q2.32Thediscrete-timesystemofProgramP2_8is-
[h,t]=impz（hd）computestheinstantaneousimpulseresponseofthediscrete-timefilterhdchoosingthenumberofsamplesforyou,andreturnstheresponseincolumnvectorhandavectoroftimesorsampleintervalsintwhere（t=[012...]'）.impzreturnsamatrixhifhdisavector.Eachcolumnofthematrixcorrespondstoonefilterinthevector.Whenhdisavectorofdiscrete-timefilters,impzreturnsthematrixh.Eachcolumnofhcorrespondstoonefilterinthevectorhd.
impz（hd）usesFVTooltoplottheimpulseresponseofthediscrete-timefilterhd.Ifhdisavectoroffilters,impzplotstheresponseandforeachfilterinthevector.
TheimpulseresponsegeneratedbyrunningProgramP2_8isshownbelow:
Thevalueof|h（K）|hereis-1.6761e-005
Fromthisvalueand
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