Fourier Series, Fourier Transforms, and the DFT - ArchiMeDes

30 Pages·1999·0.42 MB·English

by  W. Kenneth Jenkins

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 LLC, 1999
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1
Fourier Series, Fourier Transforms,
and the DFT
1.1 Introduction
1.2 FourierSeriesRepresentationofContinuousTime
PeriodicSignals
ExponentialFourierSeries(cid:15)TheTrigonometricFourierSeries
(cid:15)ConvergenceoftheFourierSeries
1.3 TheClassicalFourierTransformforContinuousTime
Signals
Properties of the Continuous Time Fourier Transform (cid:15)
FourierSpectrumoftheContinuousTimeSamplingModel(cid:15)
FourierTransformofPeriodicContinuousTimeSignals(cid:15)The
GeneralizedComplexFourierTransform
1.4 TheDiscreteTimeFourierTransform
PropertiesoftheDiscreteTimeFourierTransform(cid:15)Relation-
shipbetweentheContinuousandDiscreteTimeSpectra
1.5 TheDiscreteFourierTransform
PropertiesoftheDiscreteFourierSeries(cid:15)FourierBlockPro-
cessing in Real-Time Filtering Applications (cid:15) Fast Fourier
TransformAlgorithms
1.6 FamilyTreeofFourierTransforms
1.7 SelectedApplicationsofFourierMethods
FastFourierTransforminSpectralAnalysis(cid:15)FiniteImpulse
ResponseDigitalFilterDesign(cid:15)FourierAnalysisofIdealand
W.KennethJenkins PracticalDigital-to-AnalogConversion
1.8 Summary
UniversityofIllinois,
Urbana-Champaign References
1.1 Introduction
Fouriermethodsarecommonlyusedforsignalanalysisandsystemdesigninmoderntelecommu-
nications,radar,andimageprocessingsystems. ClassicalFouriermethodssuchastheFourierseries
andtheFourierintegralareusedforcontinuoustime(CT)signalsandsystems,i.e.,systemsinwhich
acharacteristicsignal,s.t/,isdefinedatallvaluesoft onthecontinuum−1 < t < 1. Amore
recentlydevelopedsetofFouriermethods,includingthediscretetimeFouriertransform(DTFT)and
thediscreteFouriertransform(DFT),areextensionsofbasicFourierconceptsthatapplytodiscrete
time (DT) signals. A characteristic DT signal, sTnU, is defined only for values of n where n is an
integerintherange−1 < n < 1. Thefollowingdiscussionpresentsbasicconceptsandoutlines
importantpropertiesforboththeCTandDTclassesofFouriermethods,withaparticularemphasis
ontherelationshipsbetweenthesetwoclasses. TheclassofDTFouriermethodsisparticularlyuseful
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asabasisfordigitalsignalprocessing(DSP)becauseitextendsthetheoryofclassicalFourieranalysis
toDTsignalsandleadstomanyeffectivealgorithmsthatcanbedirectlyimplementedongeneral
computersorspecialpurposeDSPdevices.
TherelationshipbetweentheCTandtheDTdomainsischaracterizedbytheoperationsofsampling
andreconstruction. Ifsa.t/denotesasignals.t/thathasbeenuniformlysampledeveryT seconds,
thenthemathematicalrepresentationofsa.t/isgivenby
X1
sa.t/D s.t/(cid:14).t −nT/ (1.1)
nD−1
where(cid:14).t/isaCTimpulsefunctiondefinedtobezeroforallt 6D 0,undefinedatt D 0,andhas
unitareawhenintegratedfromt D−1tot DC1. Becausetheonlyplacesatwhichtheproduct
s.t/(cid:14).t−nT/isnotidenticallyequaltozeroareatthesamplinginstances,s.t/in(1.1)canbereplaced
withs.nT/withoutchangingtheoverallmeaningoftheexpression. Hence,analternateexpression
forsa.t/thatisoftenusefulinFourieranalysisisgivenby
X1
sa.t/D s.nT/(cid:14).t −nT/ (1.2)
nD−1
TheCTsamplingmodelsa.t/consistsofasequenceofCTimpulsefunctionsuniformlyspacedat
intervalsofT secondsandweightedbythevaluesofthesignals.t/atthesamplinginstants,asdepicted
inFig.1.1. Notethatsa.t/isnotdefinedatthesamplinginstantsbecausetheCTimpulsefunction
itselfisnotdefinedatt D 0. However,thevaluesofs.t/atthesamplinginstantsareimbeddedas
“areaunderthecurve”ofsa.t/,andassuchrepresentausefulmathematicalmodelofthesampling
process. IntheDTdomainthesamplingmodelissimplythesequencedefinedbytakingthevalues
ofs.t/atthesamplinginstants,i.e.,
sTnUDs.t/jtDnT (1.3)
Incontrasttosa.t/,whichisnotdefinedatthesamplinginstants,sTnUiswelldefinedatthesampling
instants,asillustratedinFig.1.2. Thus,itisnowclearthatsa.t/andsTnUaredifferentbutequivalent
modelsofthesamplingprocessintheCTandDTdomains,respectively. Theyarebothusefulfor
signalanalysisintheircorrespondingdomains. Theirequivalenceisestablishedbythefactthatthey
haveequalspectraintheFourierdomain,andthattheunderlyingCTsignalfromwhichsa.t/and
sTnUarederivedcanberecoveredfromeithersamplingrepresentation,providedasufficientlylarge
samplingrateisusedinthesamplingoperation(seebelow).
1.2 Fourier Series Representation of Continuous Time Periodic
Signals
ItisconvenienttobeginthisdiscussionwiththeclassicalFourierseriesrepresentationofaperiodic
timedomainsignal,andthenderivetheFourierintegralfromthisrepresentationbyfindingthelimit
oftheFouriercoefficientrepresentationastheperiodgoestoinfinity. Theconditionsunderwhicha
periodicsignals.t/canbeexpandedinaFourierseriesareknownastheDirichetconditions. They
requirethatineachperiods.t/hasafinitenumberofdiscontinuities,afinitenumberofmaxima
andminima,andthats.t/satisfiesthefollowingabsoluteconvergencecriterion[1]:
Z
T=2
js.t/jdt <1 (1.4)
−T=2
Itisassumedinthefollowingdiscussionthatthesebasicconditionsaresatisfiedbyallfunctionsthat
willberepresentedbyaFourierseries.
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FIGURE1.1: CTmodelofasampledCTsignal.
FIGURE1.2: DTmodelofasampledCTsignal.
1.2.1 Exponential Fourier Series
IfaCTsignals.t/isperiodicwithaperiodT,thentheclassicalcomplexFourierseriesrepresentation
ofs.t/isgivenby
X1
s.t/D anejn!0t (1.5a)
nD−1
where!0 D2(cid:25)=T,andwheretheanarethecomplexFouriercoefficientsgivenby
Z
T=2
an D.1=T/ s.t/e−jn!0tdt (1.5b)
−T=2
It is well known that for every value of t where s.t/ is continuous, the right-hand side of (1.5a)
converges to s.t/. At values of t where s.t/ has a finite jump discontinuity, the right-hand side
of(1.5a)convergestotheaverageofs.t −/ands.tC/,wheres.t−/(cid:17)lim(cid:15)!0s.t −(cid:15)/ands.tC/(cid:17)
lim(cid:15)!0s.t C(cid:15)/.
Forexample,theFourierseriesexpansionofthesawtoothwaveformillustratedinFig.1.3ischar-
acterizedbyT D 2(cid:25),!0 D 1,a0 D 0,andan D a−n D Acos.n(cid:25)/=.jn(cid:25)/forn D 1;2;:::,. The
coefficientsoftheexponentialFourierseriesrepresentedby(1.5b)canbeinterpretedasthespec-
tralrepresentationofs.t/,becausethean-thcoefficientrepresentsthecontributionofthe(n!0)-th
frequencytothetotalsignals.t/. Becausetheanarecomplexvalued,theFourierdomainrepresen-
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tationhasbothamagnitudeandaphasespectrum. Forexample,themagnitudeoftheanisplotted
in Fig. 1.4 for the sawtooth waveform of Fig. 1.3. The fact that the an constitute a discrete set is
consistentwiththefactthataperiodicsignalhasa“linespectrum,”i.e.,thespectrumcontainsonly
integermultiplesofthefundamentalfrequency! . Therefore, theequationpairgivenby(1.5a)
0
and(1.5b)canbeinterpretedasatransformpairthatissimilartotheCTFouriertransformfor
periodic signals. This leads to the observation that the classical Fourier series can be interpreted
asaspecialtransformthatprovidesaone-to-oneinvertiblemappingbetweenthediscrete-spectral
domainandtheCTdomain. Thenextsectionshowshowtheperiodicityconstraintcanberemoved
toproducethemoregeneralclassicalCTFouriertransform,whichappliesequallywelltoperiodic
andaperiodictimedomainwaveforms.
FIGURE1.3: PeriodicCTsignalusedinFourierseriesexample.
FIGURE1.4: MagnitudeoftheFouriercoefficientsforexampleofFigure1.3.
1.2.2 The Trigonometric Fourier Series
Although Fourier series expansions exist for complex periodic signals, and Fourier theory can be
generalizedtothecaseofcomplexsignals,thetheoryandresultsaremoreeasilyexpressedforreal-
valuedsignals. Thefollowingdiscussionassumesthatthesignals.t/isreal-valuedforthesakeof
simplifyingthediscussion. However,allresultsarevalidforcomplexsignals,althoughthedetailsof
thetheorywillbecomesomewhatmorecomplicated.
Forreal-valuedsignalss.t/,itispossibletomanipulatethecomplexexponentialformoftheFourier
seriesintoatrigonometricformthatcontainssin.! t/andcos.! t/termswithcorrespondingreal-
0 0
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valuedcoefficients[1]. ThetrigonometricformoftheFourierseriesforareal-valuedsignals.t/is
givenby
X1 X1
s.t/D bncos.n!0t/C cnsin.n!0t/ (1.6a)
nD0 nD1
where!0 D2(cid:25)=T. Thebnandcnarereal-valuedFouriercoefficientsdeterminedby
FIGURE1.5: PeriodicCTsignalusedinFourierseriesexample2.
FIGURE1.6: FouriercoefficientsforexampleofFigure1.5.
Z
T=2
b D .1=T/ s.t/dt
0
−T=2
Z
T=2
bn D .2=T/ s.t/cos.n!0t/dt; nD1;2;:::; (1.6b)
−T=2
Z
T=2
cn D .2=T/ s.t/sin.n!0t/dt; nD1;2;:::;
−T=2
Anarbitraryreal-valuedsignals.t/canbeexpressedasasumofevenandoddcomponents,s.t/D
s .t/ C s .t/, where s .t/ D s .−t/ and s .t/ D −s .−t/, and where s .t/ D
even odd even even odd odd even
Ts.t/Cs.−t/U=2ands .t/ D Ts.t/−s.−t/U=2. ForthetrigonometricFourierseries, itcanbe
odd
shownthats .t/isrepresentedbythe(even)cosinetermsintheinfiniteseries,s .t/isrepresented
even odd
bythe(odd)sineterms,andb istheDClevelofthesignal. Therefore,ifitcanbedeterminedby
0
inspectionthatasignalhasDClevel,orifitisevenorodd,thenthecorrectformofthetrigonometric
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seriescanbechosentosimplifytheanalysis. Forexample,itiseasilyseenthatthesignalshownin
Fig.1.5isanevensignalwithazeroDClevel. Thereforeitcanbeaccuratelyrepresentedbythecosine
serieswithbn D2Asin.(cid:25)n=2/=.(cid:25)n=2/;nD1;2;:::;asillustratedinFig.1.6. Incontrast,notethat
thesawtoothwaveformusedinthepreviousexampleisanoddsignalwithzeroDClevel;thus,itcan
becompletelyspecifiedbythesinetermsofthetrigonometricseries. Thisresultcanbedemonstrated
bypairingeachpositivefrequencycomponentfromtheexponentialserieswithitsconjugatepartner,
i.e.,cn Dsin.n!0t/Danejn!0t Ca−ne−jn!0t,wherebyitisfoundthatcn D2Acos.n(cid:25)/=.n(cid:25)/for
thisexample. Ingeneralitisfoundthatan D.bn−jcn/=2fornD1;2;:::;a0 Db0,anda−n Dan(cid:3).
ThetrigonometricFourierseriesiscommoninthesignalprocessingliteraturebecauseitreplaces
complexcoefficientswithrealonesandoftenresultsinasimplerandmoreintuitiveinterpretation
oftheresults.
1.2.3 Convergence of the Fourier Series
TheFourierseriesrepresentationofaperiodicsignalisanapproximationthatexhibitsmeansquared
convergencetothetruesignal. Ifs.t/isaperiodicsignalofperiodT,ands0.t/denotestheFourier
seriesapproximationofs.t/,thens.t/ands0.t/areequalinthemeansquaresenseif
Z
T=2
MSE D js.t/−s.t/0j2dt D0 (1.7)
−T=2
Even with (1.7) satisfied, mean square error (MSE) convergence does not mean that s.t/ D s0.t/
at every value of t. In particular, it is known that at values of t, where s.t/ is discontinuous, the
Fourierseriesconvergestotheaverageofthelimitingvaluestotheleftandrightofthediscontinuity.
For example, if t is a point of discontinuity, then s0.t / D Ts.t−/Cs.tC/U=2, where s.t−/ and
0 0 0 0 0
s.tC/weredefinedpreviously. (Notethatatpointsofcontinuity,thisconditionisalsosatisfiedby
0
thedefinitionofcontinuity.) BecausetheDirichetconditionsrequirethats.t/haveatmostafinite
number of points of discontinuity in one period, the set St, defined as all values of t within one
periodwheres.t/ 6D s0.t/,containsafinitenumberofpoints,andSt isasetofmeasurezerointhe
formalmathematicalsense. Therefore, s.t/anditsFourierseriesexpansions0.t/areequalalmost
everywhere,ands.t/canbeconsideredidenticaltos0.t/fortheanalysisofmostpracticalengineering
problems.
Convergencealmosteverywhereissatisfiedonlyinthelimitasaninfinitenumberoftermsare
included in the Fourier series expansion. If the infinite series expansion of the Fourier series is
truncatedtoafinitenumberofterms,asitmustbeinpracticalapplications,thentheapproximation
willexhibitanoscillatorybehavioraroundthediscontinuity,knownastheGibbsphenomenon[1].
Lets0 .t/denoteatruncatedFourierseriesapproximationofs.t/,whereonlythetermsin(1.5a)
N
fromnD−N tonDN areincludedifthecomplexFourierseriesrepresentationisused,orwhere
onlythetermsin(1.6a)fromnD0tonDN areincludedifthetrigonometricformoftheFourier
seriesisused. Itiswellknownthatinthevicinityofadiscontinuityatt theGibbsphenomenon
0
causess0 .t/tobeapoorapproximationtos.t/. ThepeakmagnitudeoftheGibbsoscillationis13%
N
ofthesizeofthejumpdiscontinuitys.t−/−s.tC/regardlessofthenumberoftermsusedinthe
0 0
approximation. AsN increases,theregionthatcontainstheoscillationbecomesmoreconcentrated
in the neighborhood of the discontinuity, until, in the limit as N approaches infinity, the Gibbs
oscillationissqueezedintoasinglepointofmismatchatt .
0
Ifs0.t/isreplacedbysN0 .t/in(1.7),itisimportanttounderstandthebehavioroftheerrorMSEN
asafunctionofN,where
Z
T=2
MSEN D js.t/−sN0 .t/j2dt (1.8)
−T=2
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AnimportantpropertyoftheFourierseriesisthattheexponentialbasisfunctionsejn!0t(orsin.n!0t/
and cos.n! t/ for the trigonometric form) for n D 0;(cid:6)1;(cid:6)2;::: (or n D 0;1;2;::: for the
0
trigonometricform)constituteanorthonormalset,i.e.,tnk D 1forn D k,andtnk D 0forn 6D k,
where Z
T=2
tnk D.1=T/ .e−jn!0t/.ejk!0t/dt (1.9)
−T=2
AstermsareaddedtotheFourierseriesexpansion,theorthogonalityofthebasisfunctionsguarantees
thattheerrordecreasesinthemeansquaresense,i.e.,thatMSEN monotonicallydecreasesasN is
increased. Therefore,apractitionercanproceedwiththeconfidencethatwhenapplyingFourierseries
analysismoretermsarealwaysbetterthanfewerintermsoftheaccuracyofthesignalrepresentations.
1.3 The Classical Fourier Transform for Continuous
Time Signals
TheperiodicityconstraintimposedontheFourierseriesrepresentationcanberemovedbytakingthe
limitsof(1.5a)and(1.5b)astheperiodT isincreasedtoinfinity. Somemathematicalpreliminaries
arerequiredsothattheresultswillbewelldefinedafterthelimitistaken. Itisconvenienttoremove
the.1=T/factorinfrontoftheintegralbymultiplying(1.5b)throughbyT, andthenreplacing
Tanbyan0 inboth(1.5a)and(1.5b). Because! 0 D2(cid:25)=T,asT increasestoinfinity,!0becomes
infinitesimallysmall,aconditionthatisdenotedbyreplacing! with1!. Thefactor.1=T/in(1.5a)
0
becomes.1!=2(cid:25)/. Withthesealgebraicmanipulationsandchangesinnotation(1.5a)and(1.5b)
takeonthefollowingformpriortotakingthelimit:
X1
s.t/ D .1=2(cid:25)/ an0ejn1!t1! (1.10a)
nD−1
Z
T=2
an0 D s.t/e−jn1!tdt (1.10b)
−T=2
ThefinalstepinobtainingtheCTFouriertransformistotakethelimitofboth(1.10a)and(1.10b)
asT ! 1. Inthelimittheinfinitesummationin(1.10a)becomesanintegral,1!becomesd!,
n1!becomes!,andan0 becomestheCTFouriertransformofs.t/,denotedbyS.j!/. Theresult
issummarizedbythefollowingtransformpair,whichisknownthroughoutmostoftheengineering
literatureastheclassicalCTFouriertransform(CTFT):
Z
1
s.t/ D .1=2(cid:25)/ S.j!/ej!td! (1.11a)
Z −1
1
S.j!/ D s.t/e−j!tdt (1.11b)
−1
Often(1.11a\)iscalledtheFourierintegraland(1.11b)issimplycalledtheFouriertransform. The
relationshipS.j!/DFfs.t/gdenotestheFouriertransformationofs.t/,whereFf(cid:1)gisasymbolic
notationfortheFouriertransformoperator,andwhere!becomesthecontinuousfrequencyvariable
after the periodicity constraint is removed. A transform pair s.t/ $ S.j!/ represents a one-to-
oneinvertiblemappingaslongass.t/satisfiesconditionswhichguaranteethattheFourierintegral
converges.
From(1.11a)itiseasilyseenthatFf(cid:14).t −t 0/gDe−j!t0,andfrom(1.11b)thatF −1f2(cid:25)(cid:14).!−
!0/g D ej!0t,sothat(cid:14).t −t0/ $ e−j!t0 andej!0t $ 2(cid:25)(cid:14).!−!0/arevalidFouriertransform
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pairs. UsingtheserelationshipsitiseasytoestablishtheFouriertransformsofcos.! t/andsin.! t/,
0 0
aswellasmanyotherusefulwaveformsthatareencounteredincommonsignalanalysisproblems.
AnumberofsuchtransformsareshowninTable1.1.
TheCTFTisusefulintheanalysisanddesignofCTsystems,i.e.,systemsthatprocessCTsignals.
FourieranalysisisparticularlyapplicabletothedesignofCTfilterswhicharecharacterizedbyFourier
magnitudeandphasespectra, i.e., byjH.j!/jandargH.j!/, whereH.j!/iscommonlycalled
thefrequencyresponseofthefilter. Forexample,anidealtransmissionchannelisonewhichpasses
asignalwithoutdistortingit. ThesignalmaybescaledbyarealconstantAanddelayedbyafixed
time increment t , implying that the impulse response of an ideal channel is A(cid:14).t −t /, and its
0 0
correspondingfrequencyresponseisAe−j!t0. Hence,thefrequencyresponseofanidealchannelis
specifiedbyconstantamplitudeforallfrequencies,andaphasecharacteristicwhichislinearfunction
givenby!t .
0
1.3.1 Properties of the Continuous Time Fourier Transform
TheCTFThasmanypropertiesthatmakeitusefulfortheanalysisanddesignoflinearCTsystems.
Someofthemoreusefulpropertiesarestatedbelow. AmorecompletelistoftheCTFTpropertiesis
giveninTable1.2. Proofsofthesepropertiescanbefoundin[2]and[3]. Inthefollowingdiscus-
sionFf(cid:1)gdenotestheFouriertransformoperation, F−1f(cid:1)gdenotestheinverseFouriertransform
operation,and(cid:3)denotestheconvolutionoperationdefinedas
Z
1
f .t/(cid:3)f .t/D f .t −(cid:28)/f .(cid:28)/d(cid:28)
1 2 1 2
−1
1. Linearity(superposition): Ffaf .t/Cbf .t/gDaFff .t/gCbFff .t/g
1 2 1 2
(aandb,complexconstants)
2. Timeshifting: Fff.t −t0/gDe−j!t0Fff.t/g
3. Frequencyshifting: ej!0tf.t/DF−1fF.j.!−!0//g
4. Timedomainconvolution: Fff .t/(cid:3)f .t/gDFff .t/gFff .t/g
1 2 1 2
5. Frequencydomainconvolution: Fff .t/f .t/gD.1=2(cid:25)/Fff .t/g(cid:3)Fff .t/g
1 2 1 2
6. Timedifferentiation: −j!F.j!/DFfd.f.t//=dtg
R
7. Timeintegration: Ff t f.(cid:28)/d(cid:28)gD.1=j!/F.j!/C(cid:25)F.0/(cid:14).!/
−1
TheabovepropertiesareparticularlyusefulinCTsystemanalysisanddesign,especiallywhenthe
systemcharacteristicsareeasilyspecifiedinthefrequencydomain,asinlinearfiltering. Notethat
properties1,6,and7areusefulforsolvingdifferentialorintegralequations. Property4providesthe
basisformanysignalprocessingalgorithmsbecausemanysystemscanbespecifieddirectlybytheir
impulseorfrequencyresponse. Property3isparticularlyusefulinanalyzingcommunicationsystems
inwhichdifferentmodulationformatsarecommonlyusedtoshiftspectralenergytofrequencybands
thatareappropriatefortheapplication.
1.3.2 Fourier Spectrum of the Continuous Time Sampling Model
BecausetheCTsamplingmodelsa.t/,givenin(1.1),isinitsownrightaCTsignal,itisappropriate
toapplytheCTFTtoobtainanexpressionforthespectrumofthesampledsignal:
( )
X1 X1
Ffsa.t/gDF s.t/(cid:14).t −nT/ D s.nT/e−j!Tn (1.12)
nD−1 nD−1
Becausetheexpressionontheright-handsideof(1.12)isafunctionofej!T itiscustomarytodenote
thetransformasF.ej!T/ D Ffsa.t/g. Laterinthechapterthisresultiscomparedtotheresultof
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TABLE1.1 SomeBasicCTFTPairs
FourierSeriesCoefficients
Signal FourierTransform (ifperiodic)
CX1 CX1
akejk!0t 2(cid:25) ak(cid:14).!k!0/ ak
kD−1 kD−1
ej!0t 2(cid:25)(cid:14).!C!0/ a1D1
akD0; otherwise
cos!0t (cid:25)T(cid:14).!−!0/C(cid:14).!C!0/U a1Da−1D 21
akD0; otherwise
sin!0t (cid:25)jT(cid:14).!−!0/−(cid:14).!C!0/U a1D−a−1D 21j
akD0; otherwise
a0(cid:18)D1; akD0; k6D0 (cid:19)
x.t/D1 2(cid:25)(cid:14).!/ hasthisFourierseriesrepresentationforany
choiceofT0>0
Periodicsquarewave
8
x.t/D<: 1; jtj<T1 T CX1 2sinkk!0T1(cid:14).!k!0/ !0(cid:25)T1sinc(cid:18)k!(cid:25)0T1(cid:19)D sinkk!(cid:25)0T1
0; T1<jtj(cid:20) 20 kD−1
and
x.tCT /Dx.t/
0
CX1 2(cid:25)CX1 (cid:18) 2(cid:25)k(cid:19) 1
(cid:14).t−nT/ T kD−1(cid:14) !− T akD T forallk
nD−1
(cid:26) (cid:18) (cid:19)
x.t/D 10;; jjttjj<>TT11 2T1sinc !(cid:25)T1 D 2sin!!T1 —
(cid:18) (cid:19) (
W Wt sinWt 1; j!j<W
sinc D X.!/D —
(cid:25) (cid:25) (cid:25)t 0; j!j>W
(cid:14).t/ 1 —
1
u.t/ C(cid:25)(cid:14).!/ —
j!
(cid:14).t−t0/ ej!t0 —
e−atu.t/;Refag>0 1 —
aCj!
te−atu.t/;Refag>0 1 —
.aCj!/2
.ntn−−11/We−atu.t/; .aC1j!/n —
Refag>0
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