SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course MODULE 23 FOURIER SERIES Module Topics 1. Periodic signals 2. Whole-range Fourier series 3. Even and odd functions A chapter on Fourier series appears in JAMES, but not in STROUD. Although some reference will be made here to that chapter, we will keep this module self-contained since we shall include a number of simpler examples and exercises. Work Scheme 1. Fourier series are used in many areas of engineering and the basic ideas are developed in this module. Most of you will discuss the method again in your second year mathematics units. Fourier series expansions of periodic functions are developed in this module. Periodic functions have waveforms which repeat themselves exactly at regular intervals, and two examples are shown in Figures 1(a) and (b). ...... ...... ..... ...... .... .... .... .... .... ..... ..... ..... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. .. .. .. .. .. ... ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... ...................................................................................................................................................................................................................................................................................................................................................................................................... .. .. .. .. .. .. .. ......... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... ... .. .. ... .. .. .. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .... .... ... ... ... ... .... ... ............ ............ ............ ............ Figure 1(a) .. .. .. .. .. .. ...... ...... ...... ...... ...... ...... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ..... ...... ...... ...... ...... ...... .. ........................................................................................................................................................................................................................................................................................................................................................... .......... t Figure 1(b) In both cases the behaviour of the wave repeats itself at regular intervals. Let us now be more precise. Df. A function f is periodic of period T (T>0) if and only if f(t + T )=f(t) for all t. Therefore the period T is defined as the time interval required for one complete fluctuation. It follows that the function f(t)=cost is periodic with period 2π since f(t +2π)=cos(t+2π)=cost=f(t) for all t: You should note that if f is periodic with period T , then it is clear from the graphs, or from repeated use of the definition, that f is also periodic with periods 2T;3T;:::. You should choose the minimum period of the function to be its period. –1– Example 1 Determine in each case whether the following functions are periodic and, if so, determine the period:- √ (i) f(t) = sin(2t); (ii) f(t)=cos( 3t); (iii) f(t)=cost+ sin(2t): (i) The function f(t) = sin(2t) is periodic with period π since f(t + π)=sin2(t+π)=sin(2t+2π)=sin2t=f(t) for all t: √ √ (ii) The function f(t)=cos( 3t) is periodic with period 2π/ 3 since 2π √ 2π √ √ f t + √ =cos 3 t+√ =cos( 3t+2π)=cos( 3t)=f(t) for all t: 3 3 (iii) Determining the period of more complicated functions is less straightforward. When f(t)=cost+sin2t it follows from the above that cos t has periods 2π, 4π, 6π;::: sin 2t has periods π, 2π, 3π;::: Clearly the minimum period for the sum of these quantities is the smallest number that appears in both lists of periods. In this Example clearly T =2π. √ The function f(t)=cost+cos( 2t) is more difficult. Here cos t has periods 2π, 4π, 6π;::: √ 2π 4π 6π cos( 2 t) has periods √ ; √ ; √ ;::: 2 2 2 Since the√ multiplicative factors of π in the periods√ for cos t are whole numbers but the corresponding factors for cos( 2 t) are irrational√ (always involving 2) there is no number that appears in both lists. Hence the function f(t)=cost+cos( 2t) is NOT periodic (it never repeats itself), despite its comparatively simple form. More generally, you can say that the sum of two or more cosine waves will be periodic only when the ratios of all pairs of frequencies√ (or all pairs of periods) form rational numbers (ratios of integers). In the last example the ratio is 1= 2, which is irrational. ***Do Exercise A: Determine in each case whether the following functions are periodic and, if so, determine the period:- √ (i) f(t)=cos(7t); (ii) f(t) = 1 + sin( 3 t); (iii) f(t)=cos2t: 2. Let us recall the definitions of the properties associated with a single sine wave (see Figure 2). ... ..... f ....... ..................... ....... ....... ..... ..... ..... ..... ...... ..... 0 ....................................................................................................................................................................... f sin φ..... .... .... .... .... .... .... .... ... .... .... .... .... .... .... .... .... .... ... .... .... .... ... ... .... .... .... ... .... ... ... ... ... ......... .................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .... .... .... ... − . 0 .... .... t φ/! . .... .... .2π/! . .... .... .... .... .... .... .... .... .... .... ..... ..... ..... ..... ..... ..... ..... ..... ...... ...... ........ ........ ................ Figure 2 –2– Consider the function f(t)=f0sin(!t + φ)=f0sin(2πνt + φ), where f0 is the amplitude, ! is the circular (or angular) frequency in radians/unit time, ν is the frequency in cycles/unit time (Hertz)and φis the phase angle with respect to the time origin in radians. The period of this sine wave is 1/ν =2π/! seconds. A positive phase angle φ shifts the waveform to the left (a lead) and a negative phase angle to the right (a lag). When φ = π/2 the wave becomes a cosine wave. 3. The basis of the Fourier analysis of periodic signals is the decomposition of a ‘complicated’ periodic wave shape into a sum of sine and cosine waves of appropriate amplitude and relative phases. Suppose that a periodic function f(t) can be represented by the trigonometric series: ∞ 1 X 2πnt 2πnt f(t)= a0+ an cos + bn sin ; (1) 2 n=1 T T where T is the period and a0;a1;a2 :::, b1;b2;::: are constants. It will be shown in section 7 that an and bn,theFourier coefficients of f(t), are given by Z +T=2 2 2πnt an = f(t)cos dt ; n =0;1;2::: ; (2) T −T=2 T Z 2 +T=2 2πnt bn = f(t)sin dt ; n =1;2;3::: : (3) T −T=2 T Equations (1), (2) and (3) appear on the Formula Sheet. Equations (2 ) and (3) follow immediately from the more general ones stated in J. (as equations 12.11 and 12.12 on p.817) on choosing d = −T=2 and introducing ! =2π/T. Equations (1)–(3) also appear in the Data Book but in a slightly different form, because there the functions are assumed to have period 2L. You must get familiar with using the equations in the form appropriate for your department. Let us now look at the situation for the general Fourier series in a slightly different way. Given a periodic function f(t) suppose that we formally calculate its Fourier coefficients using the integrals (2) and (3). Then you can write ∞ 1 X 2πnt 2πnt f(t) ∼ a0 + an cos + bn sin : (4) 2 n=1 T T The series on the right-hand side of (4) is called the whole-range Fourier series of f(t). The symbol ∼ is used to show that f(t) is not necessarily equal to the series on the right. Indeed, it could be that the series on the right is divergent, or that if it is convergent it could converge to a sum other than f(t). It can be proved that the series on the right of equation (4) does converge, under certain conditions, for a wide variety of functions. Sufficient conditions, known as the Dirichlet conditions, are stated below (you can also find a discussion of convergence in section 12.2.8 starting on p.831 of J.): If f(t) is a periodic function with period T and (i) is piecewise continuous in the interval −T=2 <t<+T=2 (i.e. it is bounded in the interval and is continuous in the interval except at a finite number of points), (ii) has only a finite number of maxima and minima in the interval, –3– " # 1 then the Fourier series converges at a point t0 to the value lim f(t) + lim f(t) . + − 2 t→t0 t→t0 The value written above represents the average of the left and right hand limits of f at t0 (see Figure 3). At all points t at which the function f(t) is continuous, the left and right-hand limits of f are the same, so the Fourier series converges to f(t). .... f ..... ..... .. ....... ...... ...... ....... ....... ....... ........ ......... ........... ............... ............................................ .... value of Fourier series at t0 . .... .... ... ............ ×. ............. ..........