Presentation on theme: "FOURIER SERIES Philip Hall Jan 2011"— Presentation transcript:
1FOURIER SERIES Philip Hall Jan 2011 Definition of a Fourier seriesA Fourier series may be defined as an expansion of a function in a seriesof sines and cosines such as(7.1)The coefficients are related to the periodic function f(x)by definite integrals: Eq.(7.11) and (7.12) to be mentioned later on.Henceforth we assume f satisfies the following (Dirichlet) conditions:(1) f(x) is a periodic function;(2) f(x) has only a finite number of finite discontinuities;(3) f(x) has only a finite number of extrem values, maxima and minima in theinterval [0,2p].Fourier series are named in honour of Joseph Fourier ( ), who made important contributions to the study of trigonometric series, in connection with the solution of the heat equation
2(7.3)Expressing cos nx and sin nx in exponential form, we may rewrite Eq.(7.1) as(7.4)in which(7.5)and(7.6)
3Completeness and orthogonality We can show Fourier series satisfycertain completeness properties- see next yearNow we need to find out how to determine the coefficientsin the Fourier series expansion and find out in what circumstances theFourier series and function coincide..
4We can easily check the orthogonal relation for different values of the eigenvalue n by choosing the interval(7.7)(7.8)for all integer m and n.(7.9)
5By use of these orthogonality, we are able to obtain the coefficients (7.9)Similarly(7.10)(7.11)(7.12)Substituting them into Eq.(7.1), we write
6EXAMPLE: Sawtooth wave (7.13)This equation offers one approach to the development of the Fourier integral andFourier transforms.EXAMPLE: Sawtooth waveLet us consider a sawtooth wave(7.14)For convenience, we shall shift our interval fromto. In this intervalwe have simply f(x)=x. Using Eqs.(7.11) and (7.12), we have
7So, the expansion of f(x) reads (7.15).Figure 7.1 shows f(x) for the sum of 4, 6, and 10 terms of the series.Three features deserve comment.There is a steady increase in the accuracy of the representation as the number ofterms included is increased.2.All the curves pass through the midpointat
8Figure 7.1 Fourier representation of sawtooth wave
9CONVERGENCE OF FOURIER SERIES W assume f satisfies the following (Dirichlet) conditions:(1) f(x) is a periodic function;(2) f(x) has only a finite number of finite discontinuities;(3) f(x) has only a finite number of extrem values, maxima and minima in theinterval [0,2p].We can then show that at a point where f(x) iscontinuous the Fourier series converges to f(x). Howeverat a point of discontinuity the Fourier series converges to½(sum of right and left hand limits of limits of f(x). Hence at any point x(7.18)where of course the right and left hand limits coincide where f is continuous.
107.2 ADVANTAGES, USES OF FOURIER SERIES Discontinuous FunctionOne of the advantages of a Fourier representation over some other representation,such as a Taylor series, is that it may represent a discontinuous function. An exampleid the sawtooth wave in the preceding section. Other examples are considered inSection 7.3 and in the exercises.Periodic FunctionsRelated to this advantage is the usefulness of a Fourier series representing a periodicfunctions . If f(x) has a period of, perhaps it is only natural that we expand it in,a series of functions with period,,This guarantees that ifour periodic f(x) is represented over one intervalortherepresentation holds for all finite x.
11is an even function of x. Hence , At this point we may conveniently consider the properties of symmetry. Using theinterval,is odd andis an even function of x. Hence ,by Eqs. (7.11) and (7.12), if f(x) is odd, allif f(x) is even all. Inother words,even, (7.21)odd (7.21)Frequently these properties are helpful in expanding a given function.We have noted that the Fourier series periodic. This is important in consideringwhether Eq. (7.1) holds outside the initial interval. Suppose we are given only that(7.23)and are asked to represent f(x) by a series expansion. Let us take three of theinfinite number of possible expansions.
121.If we assume a Taylor expansion, we have (7.24)a one-term series. This (one-term) series is defined for all finite x.2.Using the Fourier cosine series (Eq. (7.21)) we predict that(7.25)3.Finally, from the Fourier sine series (Eq. (7.22)), we have(7.26)
13Figure 7.2 Comparison of Fourier cosine series, Fourier sine series and Taylor series.
14These three possibilities, Taylor series, Fouries cosine series, and Fourier sine series, are each perfectly valid in the original interval. Outside, however, their behavioris strikingly different (compare Fig. 7.3). Which of the three, then, is correct? Thisquestion has no answer, unless we are given more information about f(x). It may beany of the three ot none of them. Our Fourier expansions are valid over the basicinterval. Unless the function f(x) is known to be periodic with a period equal to ourbasic interval, orth of our basic interval, there is no assurance whatever thatrepresentation (Eq. (7.1)) will have any meaning outside the basic interval.It should be noted that the set of functions,, forms acomplete orthogonal set over. Similarly, the set of functions,forms a complete orthogonal set over the same interval. Unless forcedby boundaryconditions or a symmetry restriction, the choice of which set to use is arbitrary.
15FOURIER SERIES OVER ARBITRARY INTERVALS So far attention has been restricted to an interval of length of. This restrictionmay easily be relaxed. If f(x) is periodic with a period, we may write(7.27)with(7.28)(7.29)
16replacing x in Eq. (7.1) with and t in Eq. (7.11) and (7.12) with(For convenience the interval in Eqs. (7.11) and (7.12) is shifted to. )The choice of the symmetric interval (-L, L) is not essential. For f(x) periodic witha period of 2L, any intervalwill do. The choice is a matter ofconvenience or literally personal preference.
17HALF RANGE FOURIER SERIES We have seen above that even and odd functions have Fourier seriesWith either or zero.Now suppose we have a function defined on an interval [0,K], thenwe extend it as an even or odd function of period K so as to produce a Fouriercosine or sine series. But note the period is now 2K, it is always wiseto sketch the extended function.Exercise: extend the functions given below as odd and even functions,sketch the extended functions and give the formulae for the Fourier coefficients:(7.4-5)Note that extending a function to produce either an odd or even functioncan lead to discontinuities.
187.3 APPLICATION OF FOURIER SERIES )7.3 APPLICATION OF FOURIER SERIESExample Square Wave ——High FrequencyOne simple application of Fourier series, the analysis of a “square” wave (Fig. (7.5))in terms of its Fourier components, may occur in electronic circuits designed tohandle sharply rising pulses. Suppose that our wave is designed by
19(7.30)From Eqs. (7.11) and (7.12) we find(7.31)(7.32)(7.33)
20Example 7.3.2 Full Wave Rectifier The resulting series is(7.36)Except for the first term which represents an average of f(x) over the intervalall the cosine terms have vanished. Sinceis odd, we have a Fourier sineseries. Although only the odd terms in the sine series occur, they fall only asThis is similar to the convergence (or lack of convergence ) of harmonic series.Physically this means that our square wave contains a lot of high-frequencycomponents. If the electronic apparatus will not pass these components, our squarewave input will emerge more or less rounded off, perhaps as an amorphous blob.Example Full Wave RectifierAs a second example, let us ask how the output of a full wave rectifier approachespure direct current (Fig. 7.6). Our rectifier may be thought of as having passed thepositive peaks of an incoming sine and inverting the negative peaks. This yields
21Since f(t) defined here is even, no terms of the form will appear. (7.37)Since f(t) defined here is even, no terms of the formwill appear.Again, from Eqs. (7.11) and (7.12), we have(7.38)
22is not an orthogonality interval for both sines and cosines (7.39)Note carefully thatis not an orthogonality interval for both sines and cosinestogether and we do not get zero for even n. The resulting series is(7.40)The original frequencyhas been eliminated. The lowest frequency oscillation isThe high-frequency components fall off as, showing that the full waverectifier does a fairly good job of approximating direct current. Whether this goodapproximation is adequate depends on the particular application. If the remaining ac components are objectionable, they may be further suppressed by appropriate filter circuits.
23These two examples bring out two features characteristic of Fourier expansion. 1. If f(x) has discontinuities (as in the square wave in Example 7.3.1), we can expectthe nth coefficient to be decreasing as. Convergence is relatively slow.If f(x) is continuous (although possibly with discontinuous derivatives as in theFull wave rectifier of example 7.3.2), we can expect the nth coefficient to bedecreasing as3. More generally if f and its first r derivatives are continuous, but the r+1 is notthen the nth coefficient will be decreasing asExample Infinite Series, Riemann Zeta FunctionAs a final example, we consider the purely mathematical problem ofexpanding(7.41)
24by symmetry allFor the’s we have(7.42)(7.43)From this we obtain(7.44)
25As it stands, Eq. (7.44) is of no particular importance, but if we set (7.45)and Eq. (7.44) becomes(7.46)or(7.47)thus yielding the Riemann zeta function,, in closed form. From ourexpansion ofand expansions of other powers of x numerous otherinfinite series can be evaluated.
277.4 Properties of Fourier Series ConvergenceIt might be noted, first that our Fourier series should not be expected to beuniformly convergent if it represents a discontinuous function. A uniformlyconvergent series of continuous function (sinnx, cosnx) always yields a continuousfunction. If, however,(a) f(x) is continuous,(b)(c)is sectionally continuous,the Fourier series for f(x) will converge uniformly. These restrictions do not demandthat f(x) be periodic, but they will satisfied by continuous, differentiable, periodicfunction (period of)
28IntegrationTerm-by-term integration of the series(7.60)yields(7.61)Clearly, the effect of integration is to place an additional power of n in thedenominator of each coefficient. This results in more rapid convergence thanbefore. Consequently, a convergent Fourier series may always be integratedterm by term, the resulting series converging uniformly to the integral of theoriginal function. Indeed, term-by-term integration may be valid even if theoriginal series (Eq. (7.60)) is not itself convergent! The function f(x) need onlybe integrable.
29Strictly speaking, Eq. (7.61) may be a Fourier series; that is , if there will be a term. However,(7.62)will still be a Fourier series.DifferentiationThe situation regarding differentiation is quite different from that of integration.Here thee word is caution. Consider the series for(7.63)We readily find that the Fourier series is(7.64)
30Differentiating term by term, we obtain (7.65)which is not convergent ! Warning. Check your derivativeFor a triangular wave which the convergence is more rapid (and uniform)(7.66)Differentiating term by term(7.67)
31which is the Fourier expansion of a square wave (7.68)As the inverse of integration, the operation of differentiation has placed anadditional factor n in the numerator of each term. This reduces the rate ofconvergence and may, as in the first case mentioned, render the differentiatedseries divergent.In general, term-by-term differentiation is permissible under the same conditionslisted for uniform convergence.
32COMPLEX FORM OF FOURIER SERIES Recall from earlier that we can write a Fourier series in the complex formin whichandFrom the earlier definitions of we can show that
33The multiplication theorem and Parsevals theorem From the complex form of the definition of Fourier series for functionsf(x), g(x) of period 2T we can show thatWhere the c,d’s are the Fourier coefficients of f,g respectively.If we use the repalce the c’d’s by their real form and take f=gwe deduce Paresevals theorem,,