esentation repr f pplications a 6: chapter

Chapter6:ApplicationsofFourierRepresentation

HoushouChen

Dept.ofElectricalEngineering,NationalChungHsingUniversityE-mail:[email protected]

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

DepartmentofElectricalEngineering,NationalChungHsingUniversity


•IfweapplyaperiodicsignalstoastableLTIsystem,theconvolutionoperationinvolvesamixingofaperiodicimpulseresponseandperiodicinput.

•Asystemthatsamplescontinuous-timesignalsinvolvesbothcontinuous-anddiscrete-timesignals.

•InordertouseFouriermethodstoanalyzesuchinteractions,wemustbuildbridgesbetweentheFourierrepresentationofdifferentclassesofsignals.



•WecandevelopFTandDTFTrepresentationsofcontinuous-anddiscrete-timeperiodicsignals,respectively.

•FTcanalsobeusedtoanalyzeproblemsinvolvingmixturesofcontinuous-anddiscrete-timesignals.

•FTandDFTTaremostcommonlyusedforanalysisapplications.

•TheDTFSistheprimaryrepresentationusedforcomputationalapplications.

•WewillconsiderthesamplingtheoremandFFTinthischapter.

•AthoroughunderstandingoftherelationshipbetweenthefourFourierrepresentationsisacriticalstepinusingFouriermethodstosolveproblemsinvolvingsignalsandsystems.



•Strictlyspeaking,neithertheFTnortheDTFTconvergesforperiodicsignals.

•However,byincorporatingimpulseintotheFTandDTFT,wemaydevelopFTandDTFTrepresentationforperiodicsignals.

•WemayusethemandthepropertiesofFTandDTFTtoanalyzeproblemsinvolvingmixturesofperiodicandaperiodicsignals.

•Wewillconsidertheconvolutionalandmultiplicationofaperiodicandperiodicsignalsintimedomainandseewhathappensinfrequencydomain.



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)



•Thus,theFTofaperiodicsignalisaseriesofimpulsesspacedbythefundamentalfrequencyw0.

•Thekthimpulsehasstrength2πX[k],whereX[k]isthekthFScoefficient.



FindtheFTrepresentationofx(t)=cosw0t



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.



•Thespacingbetweentheimpulsesinthefrequencydomainisinverselyrelatedtothespacingbetweentheimpulsesinthetimedomain.



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]



•OrasthefollowingDTFTpair(withimpulseinfrequencydomain)

ejkw0nDTFT

←→

∞∑

m=−∞

2πδ(w−kw0−m2π)



•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)



•Thus,theDTFTofaperiodicsignalisaseriesofimpulsesspacedbythefundamentalfrequencyw0.

•Thekthimpulsehasstrength2πX[k],whereX[k]isthekthDTFScoefficient.



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)



•TheformofY(jw)impliesthaty(t)correspondstoaperiodicsignalwiththesameperiodTasx(t)

•Indeed,ThestrengthofthekthimpulseinX(jw)isadjustedbythevalueofH(jw)evaluatedatthefrequencyatwhichitislocated,orH(jkw0),toyieldanimpulseinY(jw)atw=kw0.

•Theresultsshowthattheperiodicextensionintimedomaincorrespondstothediscreteoperationinfrequencydomain.



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)



•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)



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



originalfunciton,i.e.,


∞∑

k=−∞

X[k]G(j(w−kw0)).

•Multiplicationofg(t)withtheperiodicfunctionx(t)givesanFTconsistingofaweightedsumofshiftedversionofG(jw).

•Asexpected,theformofY(jw)correspondstotheFTofacontinuous-timeaperiodicsignal,sincetheproductofperiodicandaperiodicsignalsisaperiodic.



•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).



•Nowy(t)=g(t)x(t)isthesampledversionofg(t)andtheFTofy(t)

is


∞∑

k=−∞

2π

TG(j(w−kw0)).

•WeseethatY(jw)istheperiodicextensionofG(jw).Thecorrespondedresultiscalledthesamplingtheoremandwewilldiscussnow.

•Theresultsshowthatthediscreteoperationintimedomaincorrespondstotheperiodicextensioninfrequencydomain



RelatingtheFTtotheDTFT

•WecanderiveanFTrepresentationofdiscrete-timesignalsbyincorporatingimpulsesintothedescriptionofthesignalsinthetimedomain.

•Therefore,theFTisapowerfultoolforanalyzingproblemsinvolvingmixturesofdiscrete-andcontinuous-timesignals.

•CombinetheresultsoftherelationshipbetweenFTandFS,alsoFTandDTFT,theFTcanbeusedforthefourclassesofsignals.



•ConsidertheDTFTofanarbitrarydiscrete-timesignalsx[n]:

X(ejΩ

)=

∞∑

n=−∞

x[n]e−jΩn

.

•WeseekanFTpair

xδ(t)FT←→Xδ(jw)

thatcorrespondstotheDTFTpair

x[n]DTFT←→X(e

jΩ).



•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).



•Hence,wehave

x[n]DTFT←→X(e

jΩ)=

∞∑

n=−∞

x[n]e−jΩn

and

xδ(t)FT←→Xδ(jw)=

∞∑

n=−∞

x[n]e−jwTsn



SamplingTheorem:

•WeusetheFTrepresentationofdiscrete-timesignalstoanalyzetheeffectsofuniformlysamplingasignal.

•Thesamplingoperationgeneratesadiscrete-timesignalfromacontinuous-timesignal.

•WewillseetherelationshipbetweentheDTFTofthesampledsignalsandtheFTofthecontinuous-timesignal.

•Samplingofcontinuous-timesignalsisoftenperformedinordertomanipulatethesignalonacomputerormicroprocessor.



Asignalx(t)withX(jw)asfollows

w b w b w

) ( jw X

-

iscalledband-limitedsignalandcanbeexactlyreconstructedfromitssamplesx(nTS)

∞

n=−∞providedthatthesamplingfrequencyωs=

2πTS≥2ωB,

2ωB:Nyquistsamplingrate.



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



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



⇒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))



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



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)



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



FFT(FastFourierTransform)

•TheroleofDTFSasacomputationaltoolisgreatlyenhancedbytheavailabilityofefficientalgorithmsforevaluatingtheforwardandinverseDTFS.

•WecallthesealgorithmsfastFouriertransform(FFT)algorithm.

•FFTusethe”divideandconquer”principlebydividingtheDTFSintoaseriesoflowerorderDTFSandusingthesymmetryandperiodicitypropertiesofthecomplexsinusoide

jk2πn/N.

•ThetotalcomputationsofFFTissubstantiallylessthantheoriginalDTFS.



•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.



(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



∵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



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.



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



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



Forexample:8-pointFFT



OperationCountedinFFTalgorithm/DFTalgorithm

•TotaloperationsinFFT(∵2M

=N)

=(Msections)×(N/2butterfly/section)×(3operations/butterfly)=

32NM=

32N·log2N



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)



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



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



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



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 )



Example:



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


esentation repr f pplications a 6: chapter

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