傅里叶分析之一 傅里叶级数.

1、Fourier Series电子技术系：刘佳ContentlFourier Series and Fourier TransformlAnalysis and SynthesislPeriodic Phenomenon and FunctionlTrigonometric functionlFourier SerieslComplex Form of the Fourier SerieslDetail of Fourier SeriesFourier SeriesFourier Series and Fourier TransformFourierlFourier Series lAlmost

2、 periodic phenomenonlFourier TransformlNon-periodic phenomenonl一些概念上是通用的，一些则不通用Fourier SeriesAnalysis and SynthesisFourier analysislThe process of decomposing a musical instrument sound or any other periodic function into its constituent sine or cosine waves is called Fourier analysislF ( x ) = a /2

3、 + a 1 cos x + b 1 sin x + a 2 cos 2 x + b 2 sin 2 x + .+ an cos nx + bn sin nx + .Fourier synthesislFourier synthesis works by combining a sine wave signal and sine-wave or cosine-wave harmonics (signals at multiples of the lowest, or fundamental, frequency) in certain proportions.lF ( x ) = a /2 +

4、 a 1 cos x + b 1 sin x + a 2 cos 2 x + b 2 sin 2 x + .+ an cos nx + bn sin nx + .Linear OperationlFourier synthesis and analysis based on Linear Operation: Integration and series.lFourier Transform is part of linear systems.Fourier SeriesPeriodic Phenomenon & FunctionsPeriodic PhenomenonlGeneral

5、ly speaking we think about periodic phenomena according to whether they are periodic in time or periodic in space.Periodic Phenomenon in timelTime For example, you stand at a fixed point in the ocean wash over you with a regular, recurring pattern of crests and troughs. The height of the wave is a p

6、eriodic function of time.Periodic Phenomenon in space波具有波具有时间周期时间周期（T )otAyT(1) Fixed x= xo, corresponding to the oscillating curve (振动曲线振动曲线) of medium element at position xo , i.e. y(t, xo).盯住一点盯住一点拍电影拍电影Wave motion: Temporal and spatial periodicity come together.periodicity in time is measured by

7、 the frequency , with dimension 1/sec l(2) Fixed t =to, corresponding to the wave pattern curve (波形曲线波形曲线) at time to.oyx 2x1xA波具有波具有空间周期空间周期（ )广镜头广镜头拍照片拍照片periodicity in space is measured by the wave length and vlThe frequency and wavelength are related through the equation v = lwhere v is the spee

8、d of propagation this is nothing but the wave version of speed = distance/time. lThus the higher the frequency the shorter the wavelength, and the lower the frequency the longer the wavelength.More on spatial periodicityIts reasonable to say that one of the patterns is low frequency and that the oth

9、ers are high frequency, meaning roughly that there are fewer stripes per unit length in the one than in the others.The Mathematic FormulationlAny function that satisfies( )()f tf tTwhere T is a constant and is called the period of the function.Why mathematics come?l周期性是一种物理属性。l为什么能用数学描述呢？l因为有一种简单的函数

10、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。lsine and cosineFourier SeriesTrigonometric Function History of sine and cosinelsine（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:4cos3cos)(tttfFind

11、its period.)()(Ttftf)(41cos)(31cos4cos3cosTtTtttFact:)2cos(cosm mT23 nT24 mT6 nT8 24Tsmallest TExample1:tttf21coscos)(Find its period.)()(Ttftf)(cos)(coscoscos2121TtTtttmT21nT22nm2121must be a rational numberExample2:tttf)10cos(10cos)(Is this function a periodic one?101021not a rational numberExampl

12、e3：would you say it had frequency 1 Hz? I dont think so. It has one period but youd probably say that it has, or contains two frequencies, one cosine of frequency 1 Hz and one of frequency 2 Hz.Periodic of sine and cosineQuestion:lHow to use such simple function to build Complicated periodic functio

13、n?Answer：It All Adds UplWe can combine the basic function of period 1 such as sin2t and cos2t to form more complicated periodic functions.lIdea1:l One period, many frequencies.This is important！lOne period, many frequencies.Idea2: How complicate signal is?lHow general a periodic phenomena can this f

14、ormula express ? Alternative formula：lIts more common to write a general trigonometric sum as：if we include a constant term (n= 0), asNotes:lThe constant term with the fraction 1/2 is because it simplifies the computation.lIn electrical engineering the constant term is often referred to as the DC co

15、mponents in “direct current”.lThe other terms, being periodic, “alternate”, as in AC.Using Eulers FormulaComplex FormlIn this final form of the sum, the coefficients cn are complex numbers, and they satisfyTherefore the sum is real:Fourier SeriesFourier SeriesIntroductionlSuppose we have a complicat

16、ed looking periodic signal f(t).lDecompose a periodic input signal into primitiveperiodiccomponents. Can we?A periodic sequenceT2T3Ttf(t)lQuestion is Solving for these coefficients.lA direct approach:Another idea is needed, and that idea is integrating both sides from 0 to 1.lSince the integral of t

17、he sum is the sum of the integrals, and the coefficients cn come out of each integral, all of the terms in the sum integrate to zero and we have a formula for the k-th coefficient:otherwise00 if21 if1)cos()cos(1knkndxkxnxotherwise01 if1)sin()sin(1kndxkxnxn,kdxkxnx integers allfor 0)sin()cos(1 Simila

18、r to the following integral relations:lThe cn are called the Fourier coefficients of f(t). They also denoted by :lThe sum is called a (finite) Fourier series.lAlso note that because of periodicity of f(t), any interval of length 1 will do to calculate f(n)Question: What if the period isnt 1?lHomewor

19、k !Warning!lThat is,given a periodic function can we expect to write it as a sum of exponentials in the way we have described? square wavel不能用若干个连续现象来表示一个离散的现象不能用若干个连续现象来表示一个离散的现象.lA finite sum of continuous functions is continuous and the square wave has jump discontinuities.Triangle wave不能用有限个可微分函

20、数的和表示表示一个不可微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分的函数How good a job do the finite sums do in approximating the triangle wave?lIt takes high frequencies to make sharp cornersNotes:lFiltering means cutting off.lCutting off means sharp cornerslSharp corners means high frequenciesConclusionl如果一个函数高阶导数中存在不连续的情况(any

21、 discontinuity in any derivative)，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。lTherefore，we should thus consider the infinite Fourier series. It takes high frequencies to make sharp corners.lExample: cut off the signal introduce high frequenciesThe infinite Fourier serieslAny non-smooth phenomenon signal will

22、generate infinitely many Fourier coeffients.The infinite Fourier serieslTo represent the general periodic phenomena infinite series may be required and then convergence is certainly an issue.The infinite Fourier serieslIf we cut it off after a finite number of terms, how acurate it will be?lIf the s

23、eries is converging, we have confidence that we will get a good approximation. Convergence is very hard！lconspiracy of cancellations. loscillation(震荡)lNo need for mathematical details. lUndersdant: Hard parts the answers are.Convergence in gernerallNeed Fundmantal change in perspective.lTerm: orthog

24、onality l mean square convergencel L2 etc.lWe need understand the mean of these terms.Continues case : f(t)lconverge for each t to the value f(t). l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).Smooth case: f(x)lThe Fourier series converges to f(x).lEstimate the erros will be useful.lThis conv

25、erges is more rigorous, we call it Uniform convergence (include pointwise) :l“Uniformly” means that the rate at which the series converges is the same for all points in 0, 1.lA sequence of functions fn(t) converges uniformly to a function f(t) if the graphs of the fn(t) get uniformly close to the gr

26、aph of f(t). More Detail:Pointwise Convergence vs. Uniform ConvergencelPointwise Convergence :lFor every value of t as n but the graphs of the fn(t) do not ultimately look like the graph of f(t).Discontinuity Case:lJump!converges to f(t-1)+f(t+1)/2=1/2General Case: lFourier said any function can be

27、represented by such the infinite series.lWe must learn not to ask the convergence of at a particular point. We must learn to ask for the convergence in the mean（average, energy）sense.Not completely general lNot all the periodic functions.lSuppose f(t) has Period 1,and lThe function that come up most

28、 often satisfied this condition.（Finite Energy！） We want lThe integral of the square of the difference between a function and its finite Fourier series approximation:lConvergence in mean square：此时：此时：lWatch the equal！l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).l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

29、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Fourier SeriesMore Detail and the FinisFundamental Resultl周期为1的函数f(t), 可以写作：l满足的条件是function in L2 (0,1) and the convergence in square meanGeneral in integrallRiemann integral:对函数在给定区间上的积分给出了一个精确定义。lLebesgue integral:勒貝格積分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。测度测度（Measure

30、）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More Notes on L20,1lInner spacel绍线性空间、度量空间、赋范空间、内积空间lBanach SpacelHilbert SpacelLesbesgue Space 是一种Hilbert SpacelL2 Space :所有在几乎处处（almost verywhere）意义下平方可积（square-integrable）的复值的可测函数的集合OrthogonalitylIn order to com

31、pute coefficient ck for series we use :lThis simple calculus is the cornerstone for understanding the space of square integral function.(Geometry!)Inner Productlvectors in Rn as n-tuples of real numbers：The length, or norm of v isInner ProductlIf v=(v1,v2,.,vn) and w=(w1,w2,.,wn) then the inner product islA geometric approach to the inner productlt