傅里葉分析之一 傅里葉級數(shù)

FourierSeries電子技術(shù)系：劉佳liujia1022@ContentFourierSeriesandFourierTransformAnalysisandSynthesisPeriodicPhenomenonandFunctionTrigonometricfunctionFourierSeriesComplexFormoftheFourierSeriesDetailofFourierSeriesFourierSeriesFourierSeriesandFourierTransformFourierFourierSeriesAlmostperiodicphenomenonFourierTransformNon-periodicphenomenon一些概念上是通用的，一些則不通用FourierSeriesAnalysisandSynthesisFourieranalysisTheprocessofdecomposingamusicalinstrumentsoundoranyotherperiodicfunctionintoitsconstituentsineorcosinewavesiscalledFourieranalysisF

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x

)=

a

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a

1

cos

x

+

b

1

sin

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+

a

2

cos2

x

+

b

2

sin2

x

+...

+

a

n

cos

nx

+

b

n

sin

nx

+...FouriersynthesisFouriersynthesisworksbycombininga

sinewave

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

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x

)=

a

/2+

a

1

cos

x

+

b

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sin

x

+

a

2

cos2

x

+

b

2

sin2

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+

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n

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

波具有時間周期（T)T(1)

Fixedx=xo,correspondingtotheoscillatingcurve

(振動曲線)

ofmediumelementatpositionxo,i.e.y(t,xo).盯住一點拍電影Wavemotion:Temporalandspatialperiodicitycometogether.periodicityintimeismeasuredbythefrequencyν,withdimension1/sec(2)Fixedt=to,correspondingtothewavepatterncurve

(波形曲線)attimeto.波具有空間周期（

)廣鏡頭拍照片periodicityinspaceismeasuredbythewavelengthλλandv

Thefrequencyandwavelengtharerelatedthroughtheequationv=λνwherevisthespeedofpropagation—thisisnothingbutthewaveversionofspeed=distance/time.Thusthehigherthefrequencytheshorterthewavelength,andthelowerthefrequencythelongerthewavelength.MoreonspatialperiodicityIt’sreasonabletosaythatoneofthepatternsislowfrequencyandthattheothersarehighfrequency,meaningroughlythattherearefewerstripesperunitlengthintheonethanintheothers.TheMathematicFormulationAnyfunctionthatsatisfies whereTisaconstantandiscalledtheperiodofthefunction.Whymathematicscome?周期性是一種物理屬性。為什么能用數(shù)學(xué)描述呢？因為有一種簡單的函數(shù)能表示周期的性質(zhì)，利用這種簡單的函數(shù)，就可以對周期性進行建模。sineandcosineFourierSeriesTrigonometricFunctionHistoryofsineandcosinesine（正弦）一詞始于阿拉伯人雷基奧蒙坦。他是十五世紀(jì)西歐數(shù)學(xué)界的領(lǐng)導(dǎo)人物，他于1464年完成的著作《論各種三角形》，1533年開始發(fā)行，這是一本純?nèi)菍W(xué)的書，使三角學(xué)脫離天文學(xué)，獨立成為一門數(shù)學(xué)分科。cosine（余弦）及cotangent（余切）為英國人根日爾首先使用，最早在1620年倫敦出版的他所著的《炮兵測量學(xué)》中出現(xiàn)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不能用若干個連續(xù)現(xiàn)象來表示一個離散的現(xiàn)象.Afinitesumofcontinuousfunctionsiscontinuousandthesquarewavehasjumpdiscontinuities.Trianglewave不能用有限個可微分函數(shù)的和表示表示一個不可微分的函數(shù)Howgoodajobdothefinitesumsdoinapproximatingthetrianglewave?IttakeshighfrequenciestomakesharpcornersNotes:Filteringmeanscuttingoff.CuttingoffmeanssharpcornersSharpcornersmeanshighfrequenciesConclusion如果一個函數(shù)高階導(dǎo)數(shù)中存在不連續(xù)的情況(anydiscontinuityinanyderivative)，無論這個函數(shù)看起來有多平滑，都不能將函數(shù)f(t)表示成有限項的和。Therefore，weshouldthusconsidertheinfiniteFourierseries.Ittakeshighfrequenciestomakesharpcorners.Example:cutoffthesignalintroducehighfrequenciesTheinfiniteFourierseries

Any

non-smoothphenomenonsignalwillgenerateinfinitelymanyFouriercoeffients.TheinfiniteFourierseries

Torepresentthegeneralperiodicphenomenainfiniteseriesmayberequiredandthenconvergence

iscertainlyanissue.TheinfiniteFourierseries

Ifwecutitoffafterafinitenumberofterms,howacurateitwillbe?Iftheseriesisconverging,

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

meansquareconvergence

L2etc.Weneedunderstandthemeanoftheseterms.Continuescase:f(t)convergeforeachttothevaluef(t).逐點收斂:選擇一個時刻t0,將在這個點的級數(shù)加起來，即一系列常量的和,則可以保證級數(shù)收斂到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?。?/p>

WewantTheintegralofthesquareofthedifferencebetweenafunctionanditsfiniteFourierseriesapproximation:Convergenceinmeansquare：此時：Watchtheequal！等號不意味著：取出一個值t0

這個級數(shù)就會收斂到這個函數(shù)值f(t0).而是：如果你計算一個有限和的積分，同時讓

K趨于無窮，則均方誤差會趨近于0.

收斂和等號的概念在這里全變了！這里你要知道是前人花了幾個世紀(jì)才得到的結(jié)果FourierSeriesMoreDetailandtheFinisFundamentalResult周期為1的函數(shù)f(t),可以寫作：滿足的條件是functioninL2([0,1])andtheconvergenceinsquaremeanGeneralinintegralRiemann

integral:對函數(shù)在給定區(qū)間上的積分給出了一個精確定義。Lebesgue

integral:勒貝格積分是現(xiàn)代數(shù)學(xué)中的一個積分概念，它將積分運算擴展到任何測度空間中。測度（Measure）是一個函數(shù)，它對一個給定集合的某些子集指定一個數(shù)，這個數(shù)可以比作大小、體積、概率等等。傳統(tǒng)的積分是在區(qū)間上進行的，后來人們希望把積分推廣到任意的集合上，就發(fā)展出測度的概念MoreNotesonL2[0,1]Innerspace紹線性空間、度量空間、賦范空間、內(nèi)積空間BanachSpaceHilbertSpaceLesbesgueSpace是一種HilbertSpaceL2Space:所有在幾乎處處（almostverywhere）意義下平方可積（square-integrable）的復(fù)值的可測函數(shù)的集合OrthogonalityInordertocomputecoefficientckforseriesweuse:Thissimplecalculusisthecornerstoneforunderstandingthespaceofsquareintegralfunction.(Geometry!)InnerProductvectorsinRnasn-tuplesofrealnumbers：Thelength,ornormofvisInnerProductIfv=(v1,v2,...,vn)andw=(w1,w2,...,wn)thentheinnerproductisAgeometricapproachtotheinnerproducttheprojectionofvontotheunitvectorw/