Fourier Series

26 Pages·2012·0.37 MB·English

Chapter 14
Fourier Series
14.1 GeneralProperties
Periodicphenomenainvolvingwaves[∼sin(2πx/λ)asacrudeapproximation
towaterwaves,forexample],motors,rotatingmachines(harmonicmotion),
or some repetitive pattern of a driving force are described by periodic func-
tions. Fourier series are a basic tool for solving ordinary differential equa-
tions (ODEs) and partial differential equations (PDEs) with periodic bound-
aryconditions.Fourierintegralsfornonperiodicphenomenaaredevelopedin
Chapter15.ThecommonnameforthewholefieldisFourieranalysis.
AFourierseriesisdefinedasanexpansionofafunctionorrepresentation
ofafunctioninaseriesofsinesandcosines,suchas
(cid:2)∞ (cid:2)∞
a
f(x)= 0 + a cosnx+ b sinnx. (14.1)
n n
2
n=1 n=1
The coefficients a ,a , and b are related to the periodic function f(x) by
0 n n
definiteintegrals:
(cid:3)
1 2π
a = f(x)cosnxdx, (14.2)
n π
0
(cid:3)
1 2π
b = f(x)sinnxdx, n=0,1,2,.... (14.3)
n π
0
This result, of course, is subject to the requirement that the integrals exist.
Theydoif f(x)ispiecewisecontinuous(orsquareintegrable).Noticethata
0
issingledoutforspecialtreatmentbytheinclusionofthefactor 1.Thisisdone
2
sothatEq.(14.2)willapplytoalla ,n=0aswellasn>0.
n
The Sturm–Liouville theory of the harmonic oscillator in Example 9.1.4
guaranteesthevalidityofEqs.(14.2)and(14.3)and,byuseoftheorthogonality
relations (Example 9.2.1), allows us to compute the expansion coefficients.
663
664 Chapter14 FourierSeries
Anotherwayofdescribingwhatwearedoinghereistosaythat f(x)ispart
ofaninfinite-dimensionalHilbertspace,withtheorthogonalcosnxandsinnx
as the basis. The statement that cosnx and sinnx(n = 0,1,2,...) span this
Hilbertspaceisequivalenttosayingthattheyformacompleteset.Finally,the
expansioncoefficientsa andb correspondtotheprojectionsof f(x),
n n
withtheintegralinnerproducts[Eqs.(14.2)and(14.3)]playingtheroleofthe
dotproductofSection1.2.ThesepointsareoutlinedinSection9.4.
The conditions imposed on f(x) to make Eq. (14.1) valid, and the series
convergent,arethat f(x)hasonlyafinitenumberoffinitediscontinuitiesand
only a finite number of extreme values, maxima, and minima in the interval
[0,2π].1 Functionssatisfyingtheseconditionsarecalledpiecewiseregular.
The conditions are known as the Dirichlet conditions. Although there are
somefunctionsthatdonotobeytheseDirichletconditions,theymaywellbe
labeledpathologicalforpurposesofFourierexpansions.Inthevastmajorityof
physicalproblemsinvolvingaFourierseries,theseconditionswillbesatisfied.
Inmostphysicalproblemsweshallbeinterestedinfunctionsthataresquare
integrable, for which the sines and cosines form a complete orthogonal set.
ThisinturnmeansthatEq.(14.1)isvalidinthesenseofconvergenceinthe
mean(seeEq.9.63).
Completeness
The Fourier expansion and the completeness property may be expected be-
causethefunctionssinnx,cosnx,einxarealleigenfunctionsofaself-adjoint
linearODE,
y(cid:6)(cid:6)+n2y=0. (14.4)
Weobtainorthogonaleigenfunctionsfordifferentvaluesoftheeigenvaluen
fortheinterval[0,2π]tosatisfytheboundaryconditionsintheSturm–Liouville
theory(Chapter9).Thedifferenteigenfunctionsforthesameeigenvaluenare
orthogonal.Wehave
(cid:3) (cid:4)
2π πδ , m=(cid:7) 0,
sinmxsinnxdx= mn (14.5)
0, m=0,
0
(cid:3) (cid:4)
2π πδ , m=(cid:7) 0,
cosmxcosnxdx= mn (14.6)
2π, m=n=0,
0
(cid:3)
2π
sinmxcosnxdx=0, forallintegralmandn. (14.7)
0
Note that any interval x ≤ x ≤ x + 2π will be equally satisfactory. Fre-
0 0
quently,weusex =−π toobtaintheinterval−π ≤ x≤π.Forthecomplex
0
eigenfunctionse±inx,orthogonalityisusuallydefinedintermsofthecomplex
1Theseconditionsaresufficientbutnotnecessary.
14.1 GeneralProperties 665
conjugateofoneofthetwofactors,
(cid:3)
2π
(eimx)∗einxdx=2πδ . (14.8)
mn
0
Thisagreeswiththetreatmentofthesphericalharmonics(Section11.5).
EXAMPLE14.1.1 Sawtooth Wave LetusapplyEqs.(14.2)and(14.3)tothesawtoothshape
showninFig.14.1toderiveitsFourierseries.Oursawtoothfunctioncanalso
beexpressedas
(cid:4)
x, 0≤ x<π,
f(x)=
x−2π, π ≤ x≤2π,
which is an odd function of the variable x. Hence, we expect a pure sine
expansion.Integratingbyparts,weindeedfind
(cid:3) (cid:5) (cid:3)
an = π1 π xcosnxdx= nxπ sinnx(cid:5)(cid:5)(cid:5)π − n1π π sinnxdx
−π −π −π
(cid:5)
(cid:5)π
= 1 cosnx(cid:5)(cid:5) = 1 [(−1)n−(−1)n]=0,
n2π n2π
−π
while
(cid:3) (cid:5) (cid:3)
bn = π1 π xsinnxdx=−nxπ cosnx(cid:5)(cid:5)(cid:5)π + n1π π sinnxdx
−π −π −π
(cid:5)
(cid:5)π
=−2(−1)n− 1 cosnx(cid:5)(cid:5) =−2(−1)n.
n n2π n
−π
ThisestablishestheFourierexpansion
(cid:6) (cid:7)
sin2x sin3x sinnx
f(x)=2 sinx− + −···+(−1)n+1 +···
2 3 n
(cid:2)∞
sinnx
=2 (−1)n+1 = x, −π < x<π, (14.9)
n
n=1
Figure14.1
SawtoothWave y
p
0 x
–2p –p p 2p
–p
666 Chapter14 FourierSeries
which converges only conditionally, not absolutely, because of the disconti-
nuity of f(x) at x = ±π. It makes no difference whether a discontinuity is
intheinterioroftheexpansionintervaloratitsends:Itwillgiverisetocon-
ditional convergence of the Fourier series. In terms of physical applications
withx=ωafrequency,conditionalconvergencemeansthatoursquarewave
isdominatedbyhigh-frequencycomponents. (cid:2)
BehaviorofDiscontinuities
The behavior at x = nπ is an example of a general rule that at a finite dis-
continuitytheseriesconvergestothearithmeticmean.Foradiscontinuityat
x= x theseriesyields
0
1
f(x )= [f(x +0)+ f(x −0)], (14.10)
0 0 0
2
wherethearithmeticmeanoftherightandleftapproachestox= x .Ageneral
0
proofusingpartialsumsisgivenbyJeffreysandbyCarslaw(seeAdditional
Reading).
AnideaoftheconvergenceofaFourierseriesandtheerrorinusingonly
a finite number of terms in the series may be obtained by considering the
expansionofthesawtoothshapeofFig.14.1.Figure14.2shows f(x)for0 ≤
x < π forthesumof4,6,and10termsoftheseries.Threefeaturesdeserve
comment:
1. There is a steady increase in the accuracy of the representation as the
numberoftermsincludedisincreased.
2. Allthecurvespassthroughthemidpoint f(x)=0atx=π.
3. In the vicinity of x = π there is an overshoot that persists and shows no
signofdiminishing.Thisovershoot(andundershoot)iscalledtheGibbs
phenomenon and is a typical feature of Fourier series. The inclusion of
moretermsdoesnothingtoremovetheovershoot(undershoot)butmerely
movesitclosertothepointofdiscontinuity.TheGibbsphenomenonisnot
Figure14.2
f(x)
Fourier 10 terms
6
Representationof 4
SawtoothWave
x
p
14.1 GeneralProperties 667
Figure14.3
f(t)
FullWaveRectifier
wt
–2p –p p 2p
limitedtoFourierseries.Itoccurswithothereigenfunctionexpansions.For
moredetails,seeW.J.Thomson,FourierseriesandtheGibbsphenomenon,
Am.J.Phys.60,425(1992).
One of the advantages of a Fourier representation over some other
representation, such as a Taylor series, is that it can represent a dis-
continuous function. An example is the sawtooth wave in the preceding
sectionandExample14.1.3.Otherexamplesareconsideredintheexercises.
EXAMPLE14.1.2 Full-WaveRectifier ConsiderthecaseofanabsolutelyconvergentFourier
seriesrepresentingacontinuousperiodicfunction,displayedinFig.14.3.Let
us ask how well the output of a full-wave rectifier approaches pure direct
current. Our rectifier may be thought of as having passed the positive peaks
ofanincomingsinewaveandinvertingthenegativepeaks.Thisyields
f(t)=sinωt, 0<ωt <π,
(14.11)
f(t)=−sinωt, −π <ωt <0.
Since f(t)definedhereiseven,notermsoftheformsinnωtwillappear.Again,
fromEqs.(14.2)and(14.3),wehave
(cid:3) (cid:3)
1 0 1 π
a =− sinωtd(ωt)+ sinωtd(ωt)
0 π π
(cid:3) −π 0
2 π 4
= sinωtd(ωt)= , (14.12)
π π
0
(cid:3)
2 π
a = sinωtcosnωtd(ωt)
n π
0
2 2
=− , neven,
π n2−1
=0, nodd. (14.13)
668 Chapter14 FourierSeries
Figure14.4
f(x)
SquareWave
h
x
–3p –2p –p p 2p 3p
Note that (0,π) is not an orthogonality interval for both sines and cosines
togetherandwedonotgetzeroforevenn.Theresultingseriesis
2 4 (cid:2)∞ cosnωt
f(t)= − . (14.14)
π π n2−1
n=2,4,6,...
Theoriginalfrequencyωhasbeeneliminated.Thelowestfrequencyoscillation
is2ω.Thehigh-frequencycomponentsfalloffasn−2,showingthatthefull-wave
rectifierdoesafairlygoodjobofapproximatingdirectcurrent.Whetherthis
goodapproximationisadequatedependsontheparticularapplication.Ifthe
remainingaccomponentsareobjectionable,theymaybefurthersuppressed
byappropriatefiltercircuits.Thesetwoexampleshighlighttwofeatureschar-
acteristicofFourierexpansions:2
•
If f(x)hasdiscontinuities(asinthesawtoothwaveinExample14.1.1),we
canexpectthenthcoefficienttobedecreasingasO(1/n).Convergenceis
conditionalonly.3
•
If f(x)iscontinuous(althoughpossiblywithdiscontinuousderivativesasin
thefull-waverectifierofExample14.1.2),wecanexpectthenthcoefficient
tobedecreasingas1/n2,thatis,absoluteconvergence. (cid:2)
EXAMPLE14.1.3 Square Wave–High Frequencies One application of Fourier series, the
analysis of a “square” wave (Fig. 14.4) in terms of its Fourier components,
occurs in electronic circuits designed to handle sharply rising pulses. This
exampleexplainsthephysicalmeaningofconditionalconvergence.Suppose
thatourwaveisdefinedby
f(x)=0, −π < x<0,
(14.15)
f(x)= h, 0< x<π.
2Raisbeek,G.(1955).OrderofmagnitudeofFouriercoefficients.Am.Math.Mon. 62,149–155.
3AtechniqueforimprovingtherateofconvergenceisdevelopedintheexercisesofSection5.9.
14.1 GeneralProperties 669
FromEqs.(14.2)and(14.3)wefind
(cid:3)
1 π
a = hdt =h, (14.16)
0 π
0
(cid:3)
1 π
a = hcosntdt =0, n=1,2,3,..., (14.17)
n π
0
(cid:3)
1 π h
b = hsinntdt = (1−cosnπ); (14.18)
n π nπ
0
2h
b = , nodd, (14.19)
n nπ
b =0, neven. (14.20)
n
Theresultingseriesis
(cid:8) (cid:9)
h 2h sinx sin3x sin5x
f(x)= + + + +··· . (14.21)
2 π 1 3 5
Except for the first term, which represents an average of f(x) over the in-
terval [−π,π], all the cosine terms have vanished. Since f(x)−h/2 is odd,
wehaveaFouriersineseries.Althoughonlytheoddtermsinthesineseries
occur,theyfallonlyasn−1.Thisconditionalconvergenceislikethatofthe
harmonicseries.Physically,thismeansthatoursquarewavecontainsalotof
high-frequencycomponents.Iftheelectronicapparatuswillnotpassthese
components, our square wave input will emerge more or less rounded off,
perhapsasanamorphousblob. (cid:2)
BiographicalData
Fourier,JeanBaptisteJoseph,Baron. Fourier,aFrenchmathematician,
wasbornin1768inAuxerre,France,anddiedinParisin1830.Afterhisgrad-
uationfromamilitaryschoolinParis,hebecameaprofessorattheschoolin
1795.In1808,afterhisgreatmathematicaldiscoveriesinvolvingtheseries
andintegralsnamedafterhim,hewasmadeabaronbyNapoleon.Earlier,
hehadsurvivedRobespierreandtheFrenchRevolution.WhenNapoleonre-
turnedtoFrancein1815afterhisabdicationandfirstexiletoElba,Fourier
rejoined him and, after Waterloo, fell out of favor for a while. In 1822, his
book on the Analytic Theory of Heat appeared and inspired Ohm to new
thoughtsontheflowofelectricity.
Fourier series are finite or infinite sums of sines and cosines that describe
SUMMARY
periodic functions that can have discontinuities and thus represent a wider
classoffunctionsthanwehaveconsideredsofar.Becausesinnx,cosnxare
eigenfunctions of a self-adjoint ODE, the classical harmonic oscillator equa-
tion, the Hilbert space properties of Fourier series are consequences of the
Sturm–Liouvilletheory.
670 Chapter14 FourierSeries
EXERCISES
14.1.1 A function f(x) (quadratically integrable) is to be represented by a
finite Fourier series. A convenient measure of the accuracy of the
seriesisgivenbytheintegratedsquareofthedeviation
(cid:10) (cid:11)
(cid:3)
2π a (cid:2)p 2
(cid:10) = f(x)− 0 − (a cosnx+b sinnx) dx.
p n n
2
0 n=1
Showthattherequirementthat(cid:10) beminimized,thatis,
p
∂(cid:10) ∂(cid:10)
p =0, p =0,
∂a ∂b
n n
foralln,leadstochoosinga andb ,asgiveninEqs.(14.2)and(14.3).
n n
Note.Yourcoefficientsa andb areindependentof p.Thisindepen-
n n
denceisaconsequenceoforthogonalityandwouldnotholdforpowers
ofx,fittingacurvewithpolynomials.
14.1.2 Intheanalysisofacomplexwaveform(oceantides,earthquakes,mu-
sicaltones,etc.)itmightbemoreconvenienttohavetheFourierseries
writtenas
(cid:2)∞
a
f(x)= 0 + α cos(nx−θ ).
n n
2
n=1
ShowthatthisisequivalenttoEq.(14.1)with
a =α cosθ, α2 =a2 +b2,
n n n n n
b =α sinθ, tanθ =b /a .
n n n n n
Note. The coefficients α2 as a function of ndefine what is called the
n
power spectrum.Theimportanceofα2 liesinitsinvarianceundera
n
shiftinthephaseθ .
n
(cid:12)
14.1.3 Assumingthat π [f(x)]2dxisfinite,showthat
−π
lim a =0, lim b =0.
m m
m→∞ m→∞
Hint.Integrate[f(x)−s (x)]2,wheres (x)isthenthpartialsum,and
n n
useBessel’sinequality(Section9.4)(cid:12).Forourfiniteintervaltheassump-
t(cid:12)ionthat f(x)issquareintegrable( −ππ|f(x)|2dxisfinite)impliesthat
π |f(x)|dxisalsofinite.Theconversedoesnothold.
−π
14.1.4 Applythesummationtechniqueofthissectiontoshowthat
(cid:13)
(cid:2)∞ sinnx 1(π −x), 0< x≤π
= 2
n=1 n −12(π +x), −π ≤ x<0
(Fig.14.5).
14.2 AdvantagesandUsesofFourierSeries 671
Figure14.5
f(x)
ReverseSawtooth
Wave
∞
Σ
f(x) = 1 sin nx
p n
n=1
2
x
–p p
p
2
14.1.5 Sumthetrigonometricseries
(cid:2)∞ sin(2n+1)x
,
2n+1
n=0
andshowthatitequals
(cid:4)
π/4, 0< x<π
−π/4, −π < x<0.
14.2 AdvantagesandUsesofFourierSeries
PeriodicFunctions
Relatedtotheadvantageofdescribingdiscontinuousfunctionsistheuseful-
nessofaFourierseriesinrepresentingaperiodicfunction.If f(x)hasaperiod
of2π,perhapsitisonlynaturalthatweexpanditinaseriesoffunctionswith
period2π,2π/2,2π/3,....Thisguaranteesthatifourperiodic f(x)isrepre-
sented over one interval [0,2π] or [−π,π], the representation holds for all
finitex.
Atthispoint,wemayconvenientlyconsiderthepropertiesofsymmetry.
Using the interval [−π,π], sinx is odd and cosx is an even function of x.
Hence,byEqs.(14.2)and(14.3),4if f(x)isodd,alla =0, andif f(x)iseven,
n
4Withtherangeofintegration−π ≤x≤π.
672 Chapter14 FourierSeries
allb =0.Inotherwords,
n
(cid:2)∞
a
f(x)= 0 + a cosnx, f(x)even, (14.22)
n
2
n=1
(cid:2)∞
f(x)= b sinnx, f(x)odd. (14.23)
n
n=1
Frequently,thesepropertiesarehelpfulinexpandingagivenfunction.
We have noted that the Fourier series is periodic. This is important in
consideringwhetherEq.(14.1)holdsoutsidetheinitialinterval.Supposewe
aregivenonlythat
f(x)= x, 0≤ x<π (14.24)
andareaskedtorepresent f(x)byaseriesexpansion.Letustakethreeofthe
infinitenumberofpossibleexpansions:
1. IfweassumeaTaylorexpansion,wehave
f(x)= x, (14.25)
aone-termseries.This(one-term)seriesisdefinedforallfinitex.
2. UsingtheFouriercosineseries[Eq.(14.22)],wepredictthat
f(x)=−x, −π<x≤0,
(14.26)
f(x)=2π −x, π<x≤2π.
3. Finally,fromtheFouriersineseries[Eq.(14.23)],wehave
f(x)= x, −π<x≤0,
(14.27)
f(x)= x−2π, π<x≤2π.
Thesethreepossibilities—Taylorseries,Fouriercosineseries,andFourier
sine series—are each perfectly valid in the original interval [0,π]. Outside,
however, their behavior is strikingly different (compare Fig. 14.6). Which
of the three, then, is correct? This question has no answer, unless we are
given more information about f(x). It may be any of the three or none of
them. Our Fourier expansions are valid over the basic interval. Unless the
function f(x) is known to be periodic, with a period equal to our basic in-
terval, or (1/n)th of our basic interval, there is no assurance whatever that
therepresentation[Eq.(14.1)]willhaveanymeaningoutsidethebasicinter-
val. Clearly, the interval of length 2π, which defines the expansion, makes a
realdifferenceforanonperiodicfunctionbecausetheFourierseriesrepeats
thepatternofthebasicintervalinadjacentintervals.Thisalsofollowsfrom
Fig.14.6.
Inadditiontotheadvantagesofrepresentingdiscontinuousandperiodic
functions,thereisathirdveryrealadvantageinusingaFourierseries.Suppose
thatwearesolvingtheequationofmotionofanoscillatingparticle,subjectto