傅里叶分析之一 傅里叶级数

FourierSeries电子技术系：刘佳liujia1022@ContentFourierSeriesandFourierTransformAnalysisandSynthesisPeriodicPhenomenonandFunctionTrigonometricfunctionFourierSeriesComplexFormoftheFourierSeriesDetailofFourierSeriesFourierSeriesFourierSeriesandFourierTransformFourierFourierSeriesAlmostperiodicphenomenonFourierTransformNon-periodicphenomenon一些概念上是通用的，一些则不通用FourierSeriesAnalysisandSynthesisFourieranalysisTheprocessofdecomposingamusicalinstrumentsoundoranyotherperiodicfunctionintoitsconstituentsineorcosinewavesiscalledFourieranalysisF

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b

2

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+

b

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sinewave

signalandsine-waveorcosine-waveharmonics(signalsatmultiplesofthelowest,orfundamental,frequency)incertainproportions.F

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a

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+...LinearOperationFouriersynthesisandanalysisbasedonLinearOperation:Integrationandseries.FourierTransformispartoflinearsystems.FourierSeriesPeriodicPhenomenon&FunctionsPeriodicPhenomenonGenerallyspeakingwethinkaboutperiodicphenomenaaccordingtowhethertheyareperiodicintimeorperiodicinspace.PeriodicPhenomenonintimeTimeForexample,youstandatafixedpointintheoceanwashoveryouwitharegular,recurringpatternofcrestsandtroughs.Theheightofthewaveisaperiodicfunctionoftime.PeriodicPhenomenoninspace

波具有时间周期（T)T(1)

Fixedx=xo,correspondingtotheoscillatingcurve

(振动曲线)

ofmediumelementatpositionxo,i.e.y(t,xo).盯住一点拍电影Wavemotion:Temporalandspatialperiodicitycometogether.periodicityintimeismeasuredbythefrequencyν,withdimension1/sec(2)Fixedt=to,correspondingtothewavepatterncurve

(波形曲线)attimeto.波具有空间周期（

)广镜头拍照片periodicityinspaceismeasuredbythewavelengthλλandv

Thefrequencyandwavelengtharerelatedthroughtheequationv=λνwherevisthespeedofpropagation—thisisnothingbutthewaveversionofspeed=distance/time.Thusthehigherthefrequencytheshorterthewavelength,andthelowerthefrequencythelongerthewavelength.MoreonspatialperiodicityIt’sreasonabletosaythatoneofthepatternsislowfrequencyandthattheothersarehighfrequency,meaningroughlythattherearefewerstripesperunitlengthintheonethanintheothers.TheMathematicFormulationAnyfunctionthatsatisfies whereTisaconstantandiscalledtheperiodofthefunction.Whymathematicscome?周期性是一种物理属性。为什么能用数学描述呢？因为有一种简单的函数能表示周期的性质，利用这种简单的函数，就可以对周期性进行建模。sineandcosineFourierSeriesTrigonometricFunctionHistoryofsineandcosinesine（正弦）一词始于阿拉伯人雷基奥蒙坦。他是十五世纪西欧数学界的领导人物，他于1464年完成的著作《论各种三角形》，1533年开始发行，这是一本纯三角学的书，使三角学脱离天文学，独立成为一门数学分科。cosine（余弦）及cotangent（余切）为英国人根日尔首先使用，最早在1620年伦敦出版的他所著的《炮兵测量学》中出现Example:Finditsperiod.Fact:smallestTExample1:Finditsperiod.mustbearationalnumberExample2:Isthisfunctionaperiodicone?notarationalnumberExample3：wouldyousayithadfrequency1Hz?Idon’tthinkso.Ithasoneperiodbutyou’dprobablysaythatithas,orcontainstwofrequencies,onecosineoffrequency1Hzandoneoffrequency2Hz.Periodicofsineandcosine

Question:Howtousesuchsimplefunctionto

buildComplicatedperiodicfunction?Answer：ItAllAddsUpWecancombinethebasicfunctionofperiod1suchassin2πtandcos2πttoformmorecomplicatedperiodicfunctions.Idea1:Oneperiod,manyfrequencies.Thisisimportant！Oneperiod,manyfrequencies.Idea2:

Howcomplicatesignalis?Howgeneralaperiodicphenomenacanthisformulaexpress?Alternativeformula：It’smorecommontowriteageneraltrigonometricsumas：ifweincludeaconstantterm(n=0),asNotes:Theconstanttermwiththefraction1/2isbecauseitsimplifiesthecomputation.InelectricalengineeringtheconstanttermisoftenreferredtoastheDCcomponentsin“directcurrent”.Theotherterms,beingperiodic,“alternate”,asinAC.UsingEuler’sFormulaComplexFormInthisfinalformofthesum,thecoefficientscnarecomplexnumbers,andtheysatisfyThereforethesumisreal:FourierSeriesFourierSeriesIntroductionSupposewehaveacomplicatedlookingperiodicsignalf(t).Decomposeaperiodicinputsignalintoprimitiveperiodiccomponents.Canwe?AperiodicsequenceT2T3Ttf(t)QuestionisSolvingforthesecoefficients.Adirectapproach:Anotherideaisneeded,andthatideaisintegratingbothsidesfrom0to1.Sincetheintegralofthesumisthesumoftheintegrals,andthecoefficientscncomeoutofeachintegral,allofthetermsinthesumintegratetozeroandwehaveaformulaforthek-thcoefficient:Similartothefollowingintegralrelations:ThecnarecalledtheFouriercoefficientsoff(t).Theyalsodenotedby:Thesumiscalleda(finite)Fourierseries.Alsonotethatbecauseofperiodicityoff(t),anyintervaloflength1willdotocalculatef^(n)Question:Whatiftheperiodisn’t1?Homework!Warning!Thatis,givenaperiodicfunctioncanweexpecttowriteitasasumofexponentialsinthewaywehavedescribed?squarewave不能用若干个连续现象来表示一个离散的现象.Afinitesumofcontinuousfunctionsiscontinuousandthesquarewavehasjumpdiscontinuities.Trianglewave不能用有限个可微分函数的和表示表示一个不可微分的函数Howgoodajobdothefinitesumsdoinapproximatingthetrianglewave?IttakeshighfrequenciestomakesharpcornersNotes:Filteringmeanscuttingoff.CuttingoffmeanssharpcornersSharpcornersmeanshighfrequenciesConclusion如果一个函数高阶导数中存在不连续的情况(anydiscontinuityinanyderivative)，无论这个函数看起来有多平滑，都不能将函数f(t)表示成有限项的和。Therefore，weshouldthusconsidertheinfiniteFourierseries.Ittakeshighfrequenciestomakesharpcorners.Example:cutoffthesignalintroducehighfrequenciesTheinfiniteFourierseries

Any

non-smoothphenomenonsignalwillgenerateinfinitelymanyFouriercoeffients.TheinfiniteFourierseries

Torepresentthegeneralperiodicphenomenainfiniteseriesmayberequiredandthenconvergence

iscertainlyanissue.TheinfiniteFourierseries

Ifwecutitoffafterafinitenumberofterms,howacurateitwillbe?Iftheseriesisconverging,

wehaveconfidencethatwewillgetagoodapproximation.Convergenceisveryhard！conspiracyofcancellations.oscillation(震荡)Noneedformathematicaldetails.Undersdant:Hardpartstheanswersare.ConvergenceingerneralNeedFundmantalchangeinperspective.Term:orthogonality

meansquareconvergence

L2etc.Weneedunderstandthemeanoftheseterms.Continuescase:f(t)convergeforeachttothevaluef(t).逐点收敛:选择一个时刻t0,将在这个点的级数加起来，即一系列常量的和,则可以保证级数收敛到f(t).Smoothcase:f(x)TheFourierseriesconvergestof(x).Estimatetheerroswillbeuseful.Thisconvergesismorerigorous,wecallit

Uniformconvergence(includepointwise):“Uniformly”meansthattherateatwhichtheseriesconvergesisthesameforallpointsin[0,1].Asequenceoffunctionsfn(t)convergesuniformlytoafunctionf(t)ifthegraphsofthefn(t)getuniformlyclosetothegraphoff(t).MoreDetail:

PointwiseConvergencevs.UniformConvergencePointwiseConvergence:Foreveryvalueoftasn→∞butthegraphsofthefn(t)donotultimatelylooklikethegraphoff(t).DiscontinuityCase:Jump!convergesto[f(t-1)+f(t+1)]/2=1/2GeneralCase:Fouriersaidanyfunctioncanberepresentedbysuchtheinfiniteseries.Wemustlearnnottoasktheconvergenceofataparticularpoint.

Wemustlearntoaskfortheconvergencein

themean（average,energy）sense.NotcompletelygeneralNotalltheperiodicfunctions.Supposef(t)hasPeriod1,andThefunctionthatcomeupmostoftensatisfiedthiscondition.（FiniteEnergy！）

WewantTheintegralofthesquareofthedifferencebetweenafunctionanditsfiniteFourierseriesapproximation:Convergenceinmeansquare：此时：Watchtheequal！等号不意味着：取出一个值t0

这个级数就会收敛到这个函数值f(t0).而是：如果你计算一个有限和的积分，同时让

K趋于无穷，则均方误差会趋近于0.

收敛和等号的概念在这里全变了！这里你要知道是前人花了几个世纪才得到的结果FourierSeriesMoreDetailandtheFinisFundamentalResult周期为1的函数f(t),可以写作：满足的条件是functioninL2([0,1])andtheconvergenceinsquaremeanGeneralinintegralRiemann

integral:对函数在给定区间上的积分给出了一个精确定义。Lebesgue

integral:勒貝格積分是现代数学中的一个积分概念，它将积分运算扩展到任何测度空间中。测度（Measure）是一个函数，它对一个给定集合的某些子集指定一个数，这个数可以比作大小、体积、概率等等。传统的积分是在区间上进行的，后来人们希望把积分推广到任意的集合上，就发展出测度的概念MoreNotesonL2[0,1]Innerspace绍线性空间、度量空间、赋范空间、内积空间BanachSpaceHilbertSpaceLesbesgueSpace是一种HilbertSpaceL2Space:所有在几乎处处（almostverywhere）意义下平方可积（square-integrable）的复值的可测函数的集合OrthogonalityInordertocomputecoefficientckforseriesweuse:Thissimplecalculusisthecornerstoneforunderstandingthespaceofsquareintegralfunction.(Geometry!)InnerProductvectorsinRnasn-tuplesofrealnumbers：Thelength,ornormofvisInnerProductIfv=(v1,v2,...,vn)andw=(w1,w2,...,wn)thentheinnerproductisAgeometricapproachtotheinnerproducttheprojectionofvontotheunitvectorw/