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Chapter6:ApplicationsofFourierRepresentation
HoushouChen
Dept.ofElectricalEngineering,NationalChungHsingUniversityE-mail:houshou@ee.nchu.edu.tw
H.S.ChenChapter6:ApplicationsofFourierRepresentation1
ApplicationsofFS,DTFS,FT,andDTFT
•Inthepreviouschapters,wedevelopedtheFourierrepresentationsoffourdistinctclassesofsignals.
1.FSforperiodiccontinuous-timesignals.
2.DTFSforperiodicdiscrete-timesignals.
3.FTforaperiodiccontinuous-timesignals.
4.DTFTforaperiodicdiscrete-timesignals.
•Inthischapter,weconsidermixedsignalssuchas
1.periodicandaperiodicsignals
2.continuous-anddiscrete-timesignals
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H.S.ChenChapter6:ApplicationsofFourierRepresentation2
•IfweapplyaperiodicsignalstoastableLTIsystem,theconvolutionoperationinvolvesamixingofaperiodicimpulseresponseandperiodicinput.
•Asystemthatsamplescontinuous-timesignalsinvolvesbothcontinuous-anddiscrete-timesignals.
•InordertouseFouriermethodstoanalyzesuchinteractions,wemustbuildbridgesbetweentheFourierrepresentationofdifferentclassesofsignals.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation3
•WecandevelopFTandDTFTrepresentationsofcontinuous-anddiscrete-timeperiodicsignals,respectively.
•FTcanalsobeusedtoanalyzeproblemsinvolvingmixturesofcontinuous-anddiscrete-timesignals.
•FTandDFTTaremostcommonlyusedforanalysisapplications.
•TheDTFSistheprimaryrepresentationusedforcomputationalapplications.
•WewillconsiderthesamplingtheoremandFFTinthischapter.
•AthoroughunderstandingoftherelationshipbetweenthefourFourierrepresentationsisacriticalstepinusingFouriermethodstosolveproblemsinvolvingsignalsandsystems.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation4
•Strictlyspeaking,neithertheFTnortheDTFTconvergesforperiodicsignals.
•However,byincorporatingimpulseintotheFTandDTFT,wemaydevelopFTandDTFTrepresentationforperiodicsignals.
•WemayusethemandthepropertiesofFTandDTFTtoanalyzeproblemsinvolvingmixturesofperiodicandaperiodicsignals.
•Wewillconsidertheconvolutionalandmultiplicationofaperiodicandperiodicsignalsintimedomainandseewhathappensinfrequencydomain.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation5
RelatingFTandFS
•Givenacontinuous-timeperiodicsignalx(t)withFSrepresentation
x(t)=
∞∑
k=−∞
X[k]ejkw0t
.
•RecallthefollowingFTpair(withimpulseinfrequencydomain)
ejkw0tFT
←→2πδ(w−kw0)
•WethushavetheFTforx(t)asfollows.
x(t)=
∞∑
k=−∞
X[k]ejkw0tFT
←→X(jw)=2π
∞∑
k=−∞
X[k]δ(w−kw0)
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H.S.ChenChapter6:ApplicationsofFourierRepresentation6
•Thus,theFTofaperiodicsignalisaseriesofimpulsesspacedbythefundamentalfrequencyw0.
•Thekthimpulsehasstrength2πX[k],whereX[k]isthekthFScoefficient.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation7
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H.S.ChenChapter6:ApplicationsofFourierRepresentation8
FindtheFTrepresentationofx(t)=cosw0t
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H.S.ChenChapter6:ApplicationsofFourierRepresentation9
TheFTofaunitimpulsetrain
p(t)=∞∑
n=−∞
δ(t−nT)
Sincep(t)isperiodicwithfundamentalfrequencyw0=2π/TandtheFScoefficientsaregivenby
p[k]=1/T
∫T/2
−T/2
δ(t)e−jkw0t
dt=1/T.
ThereforetheFTofp(t)isgivenbyP(jw)
P(jw)=2π
T
∞∑
k=−∞
δ(w−kw0).
•Hence,theFTofp(t)isalsoanimpulsetrain;thatis,animpulsetrainisitsownFT.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation10
•Thespacingbetweentheimpulsesinthefrequencydomainisinverselyrelatedtothespacingbetweentheimpulsesinthetimedomain.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation11
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H.S.ChenChapter6:ApplicationsofFourierRepresentation12
RelatingDTFTandDTFS
•Givenadiscrete-timeperiodicsignalx[n]withDTFSrepresentation
x[n]=
N−1 ∑
k=0
X[k]ejkw0n
.
•AsintheFScase,thekeyobservationisthattheinverseDTFTofafrequency-shiftedimpulseisadiscrete-timecomplexsinusoid.
•TheDTFTisa2π-periodicfunctionoffrequency,sowemayexpress
ejkw0nDTFT
←→δ(w−kw0)
forw∈[−π,π]andkw0∈[−π,π]
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H.S.ChenChapter6:ApplicationsofFourierRepresentation13
•OrasthefollowingDTFTpair(withimpulseinfrequencydomain)
ejkw0nDTFT
←→
∞∑
m=−∞
2πδ(w−kw0−m2π)
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H.S.ChenChapter6:ApplicationsofFourierRepresentation14
•WethushavetheDTFTforx[n]asfollows.
x[n]=
N−1 ∑
k=0
X[k]ejkw0n
DTFT←→X(e
jw)=2π
N−1 ∑
k=0
X[k]
∞∑
m=−∞
δ(w−kw0−m2π)
•Orequivalentlyasfollows.
x[n]=
N−1 ∑
k=0
X[k]ejkw0nDTFT
←→X(ejw
)=2π
∞∑
m=−∞
X[k]δ(w−kw0)
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H.S.ChenChapter6:ApplicationsofFourierRepresentation15
•Thus,theDTFTofaperiodicsignalisaseriesofimpulsesspacedbythefundamentalfrequencyw0.
•Thekthimpulsehasstrength2πX[k],whereX[k]isthekthDTFScoefficient.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation16
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H.S.ChenChapter6:ApplicationsofFourierRepresentation17
Convolutionofperiodicandaperiodicsignals
•Usethefactthatconvolutioninthetimedomaincorrespondstomultiplicationinthefrequencydomain.Thatis,
y(t)=x(t)⊗h(t)FT←→Y(jw)=X(jw)H(jw).
•Iftheinputx(t)isperiodicwithperiodT,then
x(t)FT←→X(jw)=2π
∞∑
k=−∞
X[k]δ(w−kw0),
whereX[k]aretheFScoefficientsofx(t).
•Wesubstitutethisrepresentationintotheconvolutionpropertytoobtain
y(t)=x(t)⊗h(t)FT←→Y(jw)=2π
∞∑
k=−∞
H(jkw0)X[k]δ(w−kw0)
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H.S.ChenChapter6:ApplicationsofFourierRepresentation18
•TheformofY(jw)impliesthaty(t)correspondstoaperiodicsignalwiththesameperiodTasx(t)
•Indeed,ThestrengthofthekthimpulseinX(jw)isadjustedbythevalueofH(jw)evaluatedatthefrequencyatwhichitislocated,orH(jkw0),toyieldanimpulseinY(jw)atw=kw0.
•Theresultsshowthattheperiodicextensionintimedomaincorrespondstothediscreteoperationinfrequencydomain.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation19
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H.S.ChenChapter6:ApplicationsofFourierRepresentation20
Example:
LettheinputsignalappliedtoanLTIsystemwithimpulseresponseh(t)=1/(πt)sin(πt)betheperiodicsquarewave.Findtheoutputofthesystem.
•Thefrequencyresponseofh(t)canbeshowntobelowpassfilter
h(t)FT←→H(jw)=
1,|w|≤π
0,|w|>π
•TheFTofsquarewavecanbefoundbytheFScoefficientsasfollows.
x(t)FT←→X(jw)=
∞∑
k=−∞
2sin(kπ/2)
kδ(w−kπ/2)
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H.S.ChenChapter6:ApplicationsofFourierRepresentation21
•TherearefivetermsofX(jw)inside[−π,π],i.e.,k=−2,−1,0,1,,2,
2sin(−2π/2)
−2δ(w−(−2)π/2)+
2sin(−1π/2)
(−1)δ(w−(−1)π/2)
+2sin(−0π/0)
0δ(w−0π/2)+
2sin(1π/2)
1δ(w−1π/2)+
2sin(2π/2)
2δ(w−2π/2)
=2δ(w+π/2)+πδ(w)+2δ(w−π/2)
•Finally,Y(jw)isobtainedbythefactthatH(jw)actsasalow-passfilter,passingtheharmonicsat−π/2,0,andπ/2,whilesuppressingallothers.
Y(jw)=πδ(w)+2δ(w−π/2)+2δ(w+pi/2)
•TakingtheinverseFTofY(jw)givestheoutput.Thusy(t)=1/2+2/πcos(π/2t)
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H.S.ChenChapter6:ApplicationsofFourierRepresentation22
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H.S.ChenChapter6:ApplicationsofFourierRepresentation23
Multiplicationofperiodicandaperiodicsignals
•RecallthemultiplicationpropertyoftheFT,representedas
y(t)=g(t)x(t)FT←→Y(jw)=
1
2πG(jw)⊗X(jw).
•Ifthesignalx(t)isperiodicwithperiodT,then
x(t)FT←→X(jw)=2π
∞∑
k=−∞
X[k]δ(w−kw0).
•Therefore,theFTofy(t)is
y(t)=g(t)x(t)FT←→Y(jw)=G(jw)⊗
∞∑
k=−∞
X[k]δ(w−kw0).
•Finally,bythesiftingpropertyoftheimpulsefunction,theconvolutionofanyfunctionwithashiftedimpulseresultsinashiftedversionofthe
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H.S.ChenChapter6:ApplicationsofFourierRepresentation24
originalfunciton,i.e.,
y(t)=g(t)x(t)FT←→Y(jw)=
∞∑
k=−∞
X[k]G(j(w−kw0)).
•Multiplicationofg(t)withtheperiodicfunctionx(t)givesanFTconsistingofaweightedsumofshiftedversionofG(jw).
•Asexpected,theformofY(jw)correspondstotheFTofacontinuous-timeaperiodicsignal,sincetheproductofperiodicandaperiodicsignalsisaperiodic.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation25
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H.S.ChenChapter6:ApplicationsofFourierRepresentation26
•Letx(t)bethecontinuous-timeimpulsetrain.
x(t)=
∞∑
n=−∞
δ(t−nT)
•Remark:Byintroducingthedeltafunctionintimedomain,wecanrepresentadiscrete-timefunctionasacontinuous-timefunction.Forexample,thediscrete-timeperiodicsignalx[n]=1,foralln
correspondstocontinuous-timeperiodicx(t)asabove.
•TheFTofx(t)isalsoaperiodicimpulsetraininfrequencydomain
X(jw)=2π
T
∞∑
k=−∞
δ(w−kw0).
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H.S.ChenChapter6:ApplicationsofFourierRepresentation27
•Nowy(t)=g(t)x(t)isthesampledversionofg(t)andtheFTofy(t)
is
y(t)=g(t)x(t)FT←→Y(jw)=
∞∑
k=−∞
2π
TG(j(w−kw0)).
•WeseethatY(jw)istheperiodicextensionofG(jw).Thecorrespondedresultiscalledthesamplingtheoremandwewilldiscussnow.
•Theresultsshowthatthediscreteoperationintimedomaincorrespondstotheperiodicextensioninfrequencydomain
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H.S.ChenChapter6:ApplicationsofFourierRepresentation28
RelatingtheFTtotheDTFT
•WecanderiveanFTrepresentationofdiscrete-timesignalsbyincorporatingimpulsesintothedescriptionofthesignalsinthetimedomain.
•Therefore,theFTisapowerfultoolforanalyzingproblemsinvolvingmixturesofdiscrete-andcontinuous-timesignals.
•CombinetheresultsoftherelationshipbetweenFTandFS,alsoFTandDTFT,theFTcanbeusedforthefourclassesofsignals.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation29
•ConsidertheDTFTofanarbitrarydiscrete-timesignalsx[n]:
X(ejΩ
)=
∞∑
n=−∞
x[n]e−jΩn
.
•WeseekanFTpair
xδ(t)FT←→Xδ(jw)
thatcorrespondstotheDTFTpair
x[n]DTFT←→X(e
jΩ).
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H.S.ChenChapter6:ApplicationsofFourierRepresentation30
•Now,letΩ=wTs,wherex[n]=xδ(nTs).I.e.,x[n]isequaltothesamplesofx(t)takenatintervalsofTs.
•Bythissubstitution,Ω=wTs,wetransformX(ejΩ
)ofΩintoXδ(jw)
ofw
Xδ(jw)=X(ejΩ
)|Ω=wTs=
∞∑
n=−∞
x[n]e−jwTsn
.
•TakingtheinverseFTofXδ(jw)andusethefollowingfact
δ(t−nTs)DTFT←→e
−jwTsn,
weobtainthecontinuous-timexδ(t)
xδ(t)=∞∑
n=−∞
x[n]δ(t−nTs).
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H.S.ChenChapter6:ApplicationsofFourierRepresentation31
•Hence,wehave
x[n]DTFT←→X(e
jΩ)=
∞∑
n=−∞
x[n]e−jΩn
and
xδ(t)FT←→Xδ(jw)=
∞∑
n=−∞
x[n]e−jwTsn
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H.S.ChenChapter6:ApplicationsofFourierRepresentation32
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H.S.ChenChapter6:ApplicationsofFourierRepresentation33
SamplingTheorem:
•WeusetheFTrepresentationofdiscrete-timesignalstoanalyzetheeffectsofuniformlysamplingasignal.
•Thesamplingoperationgeneratesadiscrete-timesignalfromacontinuous-timesignal.
•WewillseetherelationshipbetweentheDTFTofthesampledsignalsandtheFTofthecontinuous-timesignal.
•Samplingofcontinuous-timesignalsisoftenperformedinordertomanipulatethesignalonacomputerormicroprocessor.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation34
Asignalx(t)withX(jw)asfollows
w b w b w
) ( jw X
-
iscalledband-limitedsignalandcanbeexactlyreconstructedfromitssamplesx(nTS)
∞
n=−∞providedthatthesamplingfrequencyωs=
2πTS≥2ωB,
2ωB:Nyquistsamplingrate.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation35
IdealSampling:
H(jw) Xr(t) = X(t) ? X(t) Xs(t)
P(t)
First,wemultiplyx(t)bytheimpulsetrain
P(t)=
∞∑
n=−∞
δ(t−nTs)(periodTs)
⇒xs(t)=p(t)x(t)
X(t) Xs(t)
-2Ts -Ts 0 Ts 2Ts
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H.S.ChenChapter6:ApplicationsofFourierRepresentation36
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H.S.ChenChapter6:ApplicationsofFourierRepresentation37
Now,
xs(t)=x(t)p(t)=x(t)
∞∑
n=−∞
δ(t−nTs)
=∑
n
x(t)δ(t−nTs)
=
∞∑
n=−∞
x(nTs)δ(t−nTs)
Next,wewanttofindXs(jw)
p(t)=
∞∑
n=−∞
δ(t−nTs)=
∞∑
n=−∞
Ckejkwst
whereCk=1
Ts
∫
<Ts>
P(t)e−jkwst
dt
=1
Ts
∫Ts
0
δ(t)e−jkwst
dt=1
Ts
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H.S.ChenChapter6:ApplicationsofFourierRepresentation38
⇒p(t)=
∞∑
n=−∞
1
Tsejkwst
⇒P(jw)=
∞∑
n=−∞
2π
Tsδ(w−kws)
Sincexs(t)=x(t)p(t)
⇒Xs(jw)=1
2πX(jw)⊗p(jw)
=1
2πX(jw)⊗
∞∑
k=−∞
2π
Tsδ(w−kws)
=1
Ts
∞∑
k=−∞
X(jw)⊗δ(w−kws)
=1
Ts
∞∑
k=−∞
X(j(w−kws))
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H.S.ChenChapter6:ApplicationsofFourierRepresentation39
Thesamplingtheoremsaysthatthediscreteoperationintimedomain
xs(t)=x(t)p(t)
correspondstotheperiodicextensioninfrequencydomain
Xs(jw)=1
Ts
∞∑
k=−∞
X(j(w−kws))
1
W B -W B
X(jw)
1/Ts
Xs(jw)
W B Ws
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H.S.ChenChapter6:ApplicationsofFourierRepresentation40
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H.S.ChenChapter6:ApplicationsofFourierRepresentation41
WecanrecoverX(jw)fromXs(jw)ifandonlyif
ws−wB≥wB⇒ws≥2wB
Torecoverx(t),wemultiplyXs(jw)by
W B -W B
H(jw)
Ts
w
⇒Xr(jw)=Xs(jw)·H(jw)
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H.S.ChenChapter6:ApplicationsofFourierRepresentation42
SinceH(jw)=Tsrect(w
2wB)
⇒h(t)=Ts1
2π2wBsinc(
2wB
2t)
=TswB
πsinc(wBt)
=2wB
wssinc(wBt)
⇒xr(t)=xs(t)⊗h(t)
=[
∞∑
n=−∞
x(nTs)δ(t−nTs)]⊗[2wB
wssinc(wBt)]
=
∞∑
n=−∞
x(nTs)·2wB
ws·sinc(wB(t−nTS))
=x(t)iffws≥2wB
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H.S.ChenChapter6:ApplicationsofFourierRepresentation43
FFT(FastFourierTransform)
•TheroleofDTFSasacomputationaltoolisgreatlyenhancedbytheavailabilityofefficientalgorithmsforevaluatingtheforwardandinverseDTFS.
•WecallthesealgorithmsfastFouriertransform(FFT)algorithm.
•FFTusethe”divideandconquer”principlebydividingtheDTFSintoaseriesoflowerorderDTFSandusingthesymmetryandperiodicitypropertiesofthecomplexsinusoide
jk2πn/N.
•ThetotalcomputationsofFFTissubstantiallylessthantheoriginalDTFS.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation44
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H.S.ChenChapter6:ApplicationsofFourierRepresentation45
•Thecomputationofx[n]fromX[k]orthecomputationofX[k]fromx[n]requiresN
2complexmultiplicationsandN
−Ncomplex
additions.
•AssumeNisapowerof2,wecanthussplitx[n],0≤n≤N−1,intoevenandoddindexedsignals,i.e.,x[2n]andx[2n+1],0≤n≤N/2−1.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation46
(1)Decimationintime
Fk=
N−1 ∑
n=0
fne−j
2πnkNDFT
ifNisapowerof2e.q.fn=f0,f1,f2,f3,f4,f5,f6,f7N=8
Define
gn=f2n(even-numbersamples)
hn=f2n+1(odd-numbersamples)n=0,1,···,N2−1
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H.S.ChenChapter6:ApplicationsofFourierRepresentation47
∵NFk=
N−1 ∑
n=0
fne−j
2πnkN=DFTNfn
=
N2−1
∑
n=0
f2ne−j
2π(2n)kN+
N2−1
∑
n=0
f2n+1e−j
2π(2n+1)kN
=
N2−1
∑
n=0
gne−j
2πnkN/2+
N2−1
∑
n=0
hne−j
2πnkN/2e−j
2πkN
=DFTN/2gn+e−j
2πkN·DFTN/2hn
DefineWN=e−j
2πN∴W
kN=e
−j2πkN
NFk=[N2Gk]+W
kN[N
2Hk]0≤k≤
N
2−1
NFk=[N2Gk−
N2]+W
kN[N
2Hk−
N2]
N
2≤k≤N−1
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H.S.ChenChapter6:ApplicationsofFourierRepresentation48
Insummary,wehavethefollowing
=⇒
NFk=[N2Gk]+W
kN[N
2Hk]0≤k≤
N2−1
NFk=[N2Gk−
N2]+W
kN[N
2Hk−
N2]
N2≤k≤N−1
ThisindicatesthatF[k]andF[k+N/2],0≤k≤N/2−1,areaweightedcombinationofG[k]andH[k]
Thisstructureiscalledabutterflystructure.
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H.S.ChenChapter6:ApplicationsofFourierRepresentation49
Forexample:N=8
k=18F1=4G1+W18(4H1)
k=58F5=4G1+W58(4H1)
G 0
G 3
G 2
G 1
H 0
H 3
H 2
H 1
F 0
F 7
F 6
F 5
F 4
F 3
F 2
F 1 1
W 8 1
1
W 8 1
=W 8 4 W 8
1
= - W 8 1
This structure is call¡°Butterfly¡– =2 complex adds
+1 complex multiplication
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H.S.ChenChapter6:ApplicationsofFourierRepresentation50
Wecanfurthersimplifytheresultsbyexploitingthefollowingfact.
Fork≥N/2,wehave
WkN=W
N2
N·Wk−
N2
N=−Wk−
N2
N∵W
N2
N=e−j
2πN
·N2=e
−jπ=−1
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H.S.ChenChapter6:ApplicationsofFourierRepresentation51
Forexample:8-pointFFT
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H.S.ChenChapter6:ApplicationsofFourierRepresentation52
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H.S.ChenChapter6:ApplicationsofFourierRepresentation53
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H.S.ChenChapter6:ApplicationsofFourierRepresentation54
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H.S.ChenChapter6:ApplicationsofFourierRepresentation55
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H.S.ChenChapter6:ApplicationsofFourierRepresentation56
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H.S.ChenChapter6:ApplicationsofFourierRepresentation57
OperationCountedinFFTalgorithm/DFTalgorithm
•TotaloperationsinFFT(∵2M
=N)
=(Msections)×(N/2butterfly/section)×(3operations/butterfly)=
32NM=
32N·log2N
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H.S.ChenChapter6:ApplicationsofFourierRepresentation58
DecimationinFrequency
Fk=
N−1 ∑
n=0
fne−j
2πnkN
ifNisapowerof2,eg.N=8
fn=f0,f1,f2,f3,f4,f5,f6,f7
Define
gn=fn(firsthalfsamples)n=0,1,···,N2−1
hn=fn+N2
(secondhalfsample)
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H.S.ChenChapter6:ApplicationsofFourierRepresentation59
Fk=
N2−1
∑
n=0
fne−j
2πnkN+
N2−1
∑
n=0
fn+N2·e
−j2π(n+N
2)k
N
=
N2−1
∑
n=0
gne−j
2πnkN+
N2−1
∑
n=0
hne−j
2πnkN·e
−jπk
=
N2−1
∑
n=0
(gn+hne−jπk
)·e−j
2πnkN
N=8Fk=F0,F1,F2,F3,F4,F5,F6,F7
Define2sequences
Rk=F0,F2,F4,F6evensamples
Sk=F1,F3,F5,F7oddsamples
DepartmentofElectricalEngineering,NationalChungHsingUniversity
H.S.ChenChapter6:ApplicationsofFourierRepresentation60
evennumbergroup
k=2k′
F2k′=
N2−1
∑
n=0
(gn+hne−jπ·2k
′
︸︷︷︸
=1
)e−j
2πn(2k′)
N
=
N2−1
∑
n=0
(gn+hn)e−j
2πnk′
N/2∼N
2−pointDFT
∴NF2k=DFTN2gn+hnk=0,1,···,
N
2−1
DepartmentofElectricalEngineering,NationalChungHsingUniversity
H.S.ChenChapter6:ApplicationsofFourierRepresentation61
oddnumbergroup
k=2k′+1F2k
′+1=
N2−1
∑
n=0
(gn+hne−jπ(2k
′+1)
︸︷︷︸
=−1
)e−j
2πn(2k′+1)
N
=
N2−1
∑
n=0
[(gn−hn)e−j
2πnN]e
−j2πnk
′
N/2
∴NF2k+1=DFTN2(gn−hn)e
−j2πnN
=DFTN2(gn−hn)W
nNk=0,1,··,
N
2−1
=⇒
NF2k=DFTN2[gn+hn︸︷︷︸
g′
n
]k=0,1,··,N2−1
NF2k+1=DFTN2[(gn−hn)W
nN
︸︷︷︸
h′
n
]k=0,1,··,N2−1
DepartmentofElectricalEngineering,NationalChungHsingUniversity
H.S.ChenChapter6:ApplicationsofFourierRepresentation62
WealreadyreduceN-pointDFTtoN2-pointDFT.wecanrepeatthisprocess
untilwegetone-pointDFTifNisapowerof2.N=2M
g 0
h 3
h 2
h 1
h 0
g 3
g 2
g 1
f 0
f 7
f 6
f 5
f 4
f 3
f 2
f 1
g 0 ’
h 0 ’
g 3 ’
g 2 ’
g 1 ’
h 3 ’
h 2 ’
h 1 ’
1
1
1
-1 W’
g n + h n = g n ¡fl
(g n - h n )W N n = h n ¡fl
Butterfly = 2 complex adds +1 complex multiplication (W N
n )
DepartmentofElectricalEngineering,NationalChungHsingUniversity
H.S.ChenChapter6:ApplicationsofFourierRepresentation63
Example:
DepartmentofElectricalEngineering,NationalChungHsingUniversity
H.S.ChenChapter6:ApplicationsofFourierRepresentation64
TotaloperationsinFFT(2M
=N)=(Msections)×(
N2butterfly/section)×(3operations/butterfly)
=32NM=
32N·log2N
OperationscountedinFFTalgorithm:
totaloperationsinDFT∝N2
(∵Xk=∑N−1
n=0xne−j
2πnkN,k=0,1,···,N−1)
totaloperationsinFFT∝N·log2N
ImprovementRatio=DFTFFT=
Nlog2N
DepartmentofElectricalEngineering,NationalChungHsingUniversity
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