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27/01/2014, 03:20
... ,.The
coefficientsoftheexponentialFourierseriesrepresentedby(1.5b)canbeinterpretedasthespec-
tralrepresentationofs(t),becausethea
n
-thcoefficientrepresentsthecontributionofthe(nω
0
)-th
frequencytothetotalsignals(t).Becausethea
n
arecomplexvalued,theFourierdomainrepresen-
c
1999byCRCPressLLC
tationhasbothamagnitudeandaphasespectrum.Forexample,themagnitudeofthea
n
isplotted
inFig.1.4forthesawtoothwaveformofFig.1.3.Thefactthatthea
n
constituteadiscretesetis
consistentwiththefactthataperiodicsignalhasa“linespectrum,”i.e.,thespectrumcontainsonly
integermultiplesofthefundamentalfrequencyω
0
.Therefore,theequationpairgivenby(1.5a)
and( 1.5b)canbeinterpretedasatransformpairthatissimilartotheCTFouriertransformfor
periodicsignals.ThisleadstotheobservationthattheclassicalFourierseriescanbeinterpreted
asaspecialtransformthatprovidesaone-to-oneinvertiblemappingbetweenthediscrete-spectral
domainandtheCTdomain.Thenextsectionshowshowtheperiodicityconstraintcanberemoved
toproducethemoregeneralclassicalCTFouriertransform,whichappliesequallywelltoperiodic
andaperiodictimedomainwaveforms.
FIGURE1.3:PeriodicCTsignalusedinFourierseriesexample.
FIGURE1.4:MagnitudeoftheFouriercoefficientsforexampleofFigure1.3.
1.2.2 TheTrigonometricFourierSeries
AlthoughFourierseriesexpansionsexistforcomplexperiodicsignals,andFouriertheorycanbe
generalizedtothecaseofcomplexsignals,thetheoryandresultsaremoreeasilyexpressedforreal-
valuedsignals.Thefollowingdiscussionassumesthatthesignals(t)isreal-valuedforthesakeof
simplifyingthediscussion.However,allresultsarevalidforcomplexsignals,althoughthedetailsof
thetheorywillbecomesomewhatmorecomplicated.
Forreal-valuedsignalss(t),itispossibletomanipulatethecomplexexponentialformoftheFourier
seriesintoatrigonometricformthatcontainssin(ω
0
t)andcos(ω
0
t)termswithcorrespondingreal-
c
1999byCRCPressLLC
theCTFTisappliedtotheCTsamplingmodel,andthepropertieslistedaboveareusedtoproduce
thefollowingresult:
F{s
a
(t)}=F
s(t)
∞
n=−∞
δ(t−nT)
= ... TheDiscreteTimeFourierTransform
ThediscretetimeFouriertransform(DTFT)canbeobtainedbyusingtheDTsamplingmodeland
consideringtherelationshipobtainedin(1.12)tobethedefinitionoftheDTFT.LettingT=1so
thatthesamplingperiodisremovedfromtheequationsandthefrequencyvariableisreplacedwith
c
1999byCRCPressLLC
Jenkins, W.K. Fourier Series, Fourier Transforms, and the DFT”
Digital Signal Processing Handbook
Ed. Vijay K. Madisetti and Douglas B. Williams
Boca Raton: CRC Press ... ConvergenceoftheFourierSeries
TheFourierseriesrepresentationofaperiodicsignalisanapproximationthatexhibitsmeansquared
convergencetothetruesignal.Ifs(t)isaperiodicsignalofperiodT,ands
(t)denotestheFourier
seriesapproximationofs(t),thens(t)ands
(t)areequalinthemeansquaresenseif
MSE=
T/2
−T/2
|s(t)−s(t)
|
2
dt=0...