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matlabdtftdft等代码实验报告
LaboratoryReportofDigitalSignalProcessing
LabII.FourieranalysisinMATLAB
Name:
廖艺灵
No.:
5100309716
Date:
2013/4/6
SHANGHAIJIAOTONGUNIVERSITY
DepartmentofInstrumentScience&Engineering
Content
1Introduction1
2Exercises1
Ex.1CreateaTriangleSignal1
Ex.2CTFTAndCTFS1
Ex.3DTFT1
Ex.4DFT1
3ResultsandDiscussion2
Ex.1CreateaTriangleSignal2
Ex.2CTFTAndCTFS2
Ex.3DTFT4
Ex.4DFT5
4Summary7
5Reference7
6Appendix8
Ex.1CreateaTriangleSignal8
Ex.3DTFT9
Ex.4DFT10
1Introduction
ThislabconcentratesmainlyontheFourieranalysisinMATLAB.Foreachkindofsignals,e.g.discreteorcontinuoustimesignalsandperiodicornon-periodicsignals,thecorrespondingFourieranalysismaybeContinuous-TimeFourierSeries（CTFS）,Discrete-TimeFourierSeries（DTFS）,Continuous-TimeFourierTransform（CTFT）,Discrete-TimeFourierTransform（DTFT）andDiscreteFourierTransform（DFT）.Refertotheexamplewhichusessquaresignalastheinputsignal,wewilluseatrianglewavetoachievesamegoals,thatisfinditsCTFS,DTFS,CTFT,DTFTandDFTtocomparethedifferencesbetweenthem.TheMatlabcodesfortheseFourieranalysiscalculationsareprovidedinscriptfilesasfollows.
2Exercises
Ex.1CreateaTriangleSignal
1.1RefertomySquare.m,developasub-functionmyTriangle.mforaperiodicisoscelestriangle（seeFig.1asdemonstration）withproperinputsandoutputs.Providethewholecodesinthereportappendix.
1.2RunmyTriangle.mtocreateacontinuousperiodictrianglewiththeheightH=1.5,lengthL=2andperiodT=5.Showtheresultwithatleast3cycles.
Fig.1TriangleWave
Ex.2CTFTAndCTFS
2.1RunCTFS.mforthisperiodictriangleandCTFT.mforonlyonetrianglesignal.CalculateandshowtheFouriercoefficients（includingatleast99%averagepower）andspectrum（includingatleast99%ofenergy）,respectively.
2.2ExplaintherelationshipbetweentheCTFTandCTFS.
Ex.3DTFT
3.1RefertoCTFT.m,developDTFT.mfordiscrete-timeFouriertransform.Adiscrete-timetrianglesignalobtainedfrommyTriangleischosenasanexamplesignalinDTFT.Providethewholecodesinthereportappendix.
3.2RunDTFT.mofthesampledonetriangleofEx.1withasamplingfrequencyoffs=10.ShowandcomparetheDTFTresultswiththosefromCTFTofthecorrespondingCTtriangle.
3.3ExplaintherelationshipbetweentheDTFTandCTFT.
Ex.4DFT
4.1RefertoCTFT.m,developasub-functionofmyDFT.mfordiscreteFouriertransformwithproperinputsandoutputs.Providethewholecodesinthereportappendix.
4.2RunDFT.mofthesampledonetriangleofEx.1withasamplingfrequencyoffs=10.ShowandcomparetheDFTresultswiththosefromDTFT.
4.3ExplaintherelationshipbetweentheDFTandDTFT.
3ResultsandDiscussion
Ex.1CreateaTriangleSignal
Thefunctioncodesareputintheappendix.Followingistheoutputfigureofthefunction.
Fig.2TheTriangleSignal
Ex.2CTFTAndCTFS
CTFT
CTFS
TimeDomain
Continuous
Aperiodic
Continuous
Periodic
FrequencyDomain
Continuous
Aperiodic
Discrete
Aperiodic
BetweentheCTFSandCTFT,weprefertohavetheCTFTbecauseithasallthe
informationcontainedintheCTFSontheassumptionthattheCTFSconsistsof
thesamplesofCTFT.Besides,theCTFSisnotsopracticalbecauseitishardtofindtheperiod.
Fig.3FourierCoefficients
Fig.4TheReconstructedTriangleWavewith10-orderHarmonics.
Fig.5FourierTransformofaContinuousTriangleSignal.
Ex.3DTFT
CTFT
DTFT
TimeDomain
Continuous
Aperiodic
Discrete
Aperiodic
FrequencyDomain
Continuous
Aperiodic
Continuous
Periodic
theDTFTofx[n]=x（nT）isqualitativelytheperiodicextensionoftheCTFTofx（t）（withperiod2π/Tinanalogfrequencyωor2πindigitalfrequencyΩ）,thesumofinfinitelymanyshiftedversionofCTFT.ThisexplainshowtheDTFTstraysfromtheCTFTbecauseoffrequency-aliasingDiscrete-TimeFourierAnalysis
Fig.6DTFTofaTriangleSignal
Ex.4DFT
DTFT
DFT
TimeDomain
Discrete
Aperiodic
Discrete
Aperiodic
FrequencyDomain
Continuous
Periodic
Discrete
Periodic
DTFTaftertransformationfrequencyofagraphicisgenerallysequential,andintervalsampling,suchasDFTisDTFTisdiscretepoints.Fromthesaid,youcanseethatitsfunctionisexpressedasX（k）,andDTFTfunctionisexpressedastheX（exp（jw））.（heregivesprominencetothesamplinginterval,suchasDFTisDTFTDTFTfrequencyresponseisgenerallystraightafterthechange,afterDFTtransformoffrequencyresponseisdiscrete）.DTFTis2PIforthecycle.AndsequenceofDFTX（k）islimited.
Fig.7ComparisonofDFTandDTFTofaTriangleSignal.
4Summary
WecouldmakeatabletoshowthemajorfeatureoftheseFouriertransforms.
CTFT
CTFS
DTFT
DTFS
DFT
TimeDomain
Continuous
Aperiodic
Continuous
Periodic
Discrete
Aperiodic
Discrete
Periodic
Discrete
Aperiodic
FrequencyDomain
Continuous
Aperiodic
Discrete
Aperiodic
Continuous
Periodic
Discrete
Periodic
Discrete
Periodic
Whenanalyzeanysignal,weshoulduseproperFouriertransformtohelpusfindtheexactanswer.
5Reference
[1]MATLABGettingStartedGuideCOPYRIGHT1984-2010byTheMathWorks,Inc
[2]A.V.Oppenheim,A.S.WillskyandS.H.Nawab,SignalsandSystems（SecondEdition）,PublishingHouseofElectricsIndustry
6Appendix
Ex.1CreateaTriangleSignal
%%==================================================================
%MYTRIANGLEusedtocreateatrianglewavewithncylces.
%byTramps,30/03/2013
%%==================================================================
%%
function[x,t]=myTriangle（para）
%%inputparameters
paraDef=struct（'H',1.5,'T',5,'L',2,'n',3,'fs',100）;
bDef=0;%plotthesignal?
0:
no,1:
yes
%noinputparameter
ifnargin<1,
para=paraDef;
bDef=1;
end
%gettheparametersforthesquaresignal
ifisstruct（para）
H=para.H;
T=para.T;
L=para.L;
n=para.n;
fs=para.fs;
else
para=paraDef;
bDef=1;
end
%%createoneperiodofthesignal
dt=1/fs;
tp=（-T/2+dt）:
dt:
T/2;
xp=zeros（1,length（tp））
lp=-L/2:
dt:
L/2;
forii=1:
（length（lp）-1）
xp（fs*（T-L）/2+ii）=H*（fs*L-ii）/fs;
xp（fs*（T+L）/2-ii）=H*（fs*L-ii）/fs;
end
%plot（tp,xp）;
%return
%%Createall-timesignalbyreplicatingonecyclethroughconvolution
ta=（-n*T/2+dt）:
dt:
n*T/2;
xa=zeros（1,length（ta））;
forii=0:
n-1
tmp=ii*fs*T;
xa（（tmp+1）:
（tmp+T*fs））=xp;
end
%plot（ta,xa）;
%return
%%outputresults
t=ta;
x=xa;
if（bDef）
plot（t,x）
set（gca,'YLim',[-H*0.1,H*1.1]）
title（'DefaultTriangleSignal'）
end
end
Ex.3DTFT
%========================================================================
%1）CalculatetheDiscrete-TimeFourierTransform（DTFT）ofatrianglesignal.
%2）ChecktheParseval'srelationforFouriertransform.
%byTramps,30/03/2013
%========================================================================
%%
clc,clear,closeall
%%Createatrianglewavewithjustonecycle
%
（1）setparameters
H=1;%height
T=5;%cycletime
L=2;%length
n=1;%numberofcycles（1.2.3...）
fs=10;%samplingfrequency
%
（2）createthetriangle
para=struct（'H',H,'T',T,'L',L,'n',n,'fs',fs）;
[xp,tp]=myTriangle（para）;
%stem（tp,xp）;
%%FT
N=fs*T;
k=-N:
N;
FT_w=-3*pi:
0.1:
3*pi;
DTFT_FT=0;
forii=1:
N
DTFT_FTtmp=xp（ii）*exp（-1j*FT_w*k（ii））;
DTFT_FT=DTFT_FT+DTFT_FTtmp;
end
%%showresults
figure;
%
subplot（211）;%timedomain
stem（tp,xp,'r'）;
xlabel（'t（s）'）;
ylabel（'Amplitude'）;
title（'Discrete-TimeTriangleSignal'）;
gridon;
%
subplot（212）;%frequencydomain
plot（FT_w/（2*pi）,abs（DTFT_FT）,'r'）;
xlabel（'f'）;
ylabel（'|X（j\omega）|'）;
title（'Spectrum（DTFT）'）;
gridon;
%%end
Ex.4DFT
%%=====================================================================
%DTFTvsDFTbytrianglesignal
%byTramps,30/03/2013
%%=====================================================================
%%
clc,clear,closeall
%%createatrianglesignal
%
（1）setparameters
H=1;%height
T=5;%cycletime
L=2;%length
n=1;%numberofcycles（1.2.3...）
fs=10;%samplingfrequency
%
（2）createthesignals
para=struct（'H',H,'T',T,'L',L,'n',1,'fs',fs）;
[x_D,t_D]=myTriangle（para）;%DTFT
%stem（t_D,x_D）;
%%DTFT
N=fs*T;
k=-N:
N;
DTFT_w=-3*pi:
0.1:
3*pi;
DTFT_FT=0;
forii=1:
N
DTFT_FTtmp=x_D（ii）*exp（-1j*DTFT_w*k（ii））;
DTFT_FT=DTFT_FT+DTFT_FTtmp;
end
%plot（DTFT_w/（2*pi）,abs（DTFT_FT）,'r'）;
%%DFT
N_DFT=64;
[X_DFT,w,f]=myDFT（x_D,N_DFT,100）;
%stem（w/（2*pi）,abs（X_DFT）,'r'）;
%%showresults
figure;
%
subplot（311）;
stem（t_D,x_D,'r'）;%DTFTtimedomain
xlabel（'t（s）'）;
ylabel（'Amplitude'）;
title（'Discrete-TimeTriangleSignal'）;
gridon;
%
subplot（312）;
plot（DTFT_w/（2*pi）,abs（DTFT_FT）,'r'）;%DTFTfrequencydomain
xlabel（'f'）;
ylabel（'|X（j\omega）|'）;
title（'Spectrum（DTFT）'）;
gridon;
%
subplot（313）;
stem（w/（2*pi）,abs（X_DFT）,'r'）;%DFTfrequencydomain
xlabel（'f'）;
ylabel（'|X（j\omega）|'）;
title（'DFT'）;
gridon;
%%end
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