36 Chapter 37 Infinite Series Eercise 5 Fourier Series Write seven terms of the Fourier series given the following coefficients.. a 4, a 3, a, a 3 ; b 4, b 3, b 3. a.6, a 5., a 3., a 3.4; b 7.5, b 5.3, b 3. Write a Fourier series for the indicated waveforms in Table 37 6, first writing the functions that describe the waveform. Check our Fourier series against those given in the table. 3. Waveform 3 4. Waveform 4 5. Waveform 6 6. Waveform 7 7. Waveform. Waveform 9 9. Waveform 37 6 Waveform Smmetries Kinds of Smmetr We have seen that for the square wave, all of the a n coefficients of the Fourier series were zero, and thus the final series had no cosine terms. If we could tell in advance that all of the cosine terms would vanish, we could simplif our work b not even concerning ourselves with them. And, in fact, we can tell in advance which terms vanish b noticing the kind of smmetr the given function has. The three kinds of smmetr that are most useful here are as follows:. If a function is smmetric about the origin, it is called an odd function, and its Fourier series has onl sine terms.. If a function is smmetric about the ais, it is called an even function, and its Fourier series has no sine terms. 3. If a function has half-wave smmetr, its Fourier series has onl odd harmonics. We now cover each tpe of smmetr. Odd Functions We noted in Chapter that a function f () is smmetric about the origin if it remains unchanged when we substitute both for and for. Another wa of saing this is that f () f (). Such functions are called odd functions, as shown in Fig. 37 6(a). = m = 3 = sin (a) Odd functions = = 4 = cos (b) Even functions FIGURE 37 6
Section 37 6 Waveform Smmetries 37 Eample : The function 3 is odd because () 3 3. Similarl,, 5, 7,... are odd, with the name odd coinciding with whether the eponent is odd or even. Even Functions We noted earlier that a function is smmetric about the ais [Fig. 37 6(b)] if it remains unchanged when we substitute for. In other words, f () f (). These are called even functions. Eample : The function is even because (). Similarl, other even powers of, such as 4, 6,..., are even. Fourier Epansion of Odd and Even Functions In the power series epansion for the sine function sin 3 6 5 5 ever term is an odd function, and thus sin itself is an odd function. In the power series epansion for the cosine function cos 4 4 6 ever term is an even function, so cos, made up of all even functions, is itself an even function. Thus, we conclude the following: Odd and Even Functions (a) Odd functions have Fourier series with onl sine terms (and no constant term). (b) Even functions have Fourier series with onl cosine terms (and ma have a constant term). 456 Eample 3: In Table 37 6, Waveforms, 4, 7, and are odd. When deriving their Fourier series from scratch, we would save work b not even looking for cosine terms. Waveforms,, and are even, while the others are neither odd nor even. Shift of Aes A tpical problem is one in which a periodic waveform is given and we are to write a Fourier series for it. When writing a Fourier series for a periodic waveform, we are often free to choose the position of the ais. Our choice of ais can change the function from even to odd, or vice versa. Eample 4: In Table 37 6, notice that Waveforms and are the same waveform but with the ais shifted b radians. Thus, Waveform is an odd function, while Waveform is even. A similar shift is seen with the half-wave rectifier waveforms ( and ). A vertical shift of the ais will affect the constant term in the series. Eample 5: Waveform 3 is the same as Waveform, ecept for a vertical shift of unit. Thus, the series for Waveform 3 has a constant term of unit. The same vertical shift can be seen with Waveforms 6 and 7.
3 Chapter 37 Infinite Series Half-Wave Smmetr When the negative half-ccle of a periodic wave has the same shape as the positive half-ccle (ecept that it is, of course, inverted), we sa that the wave has half-wave smmetr. The sine wave, the cosine wave, as well as Waveforms,, and 4, have half-wave smmetr. A quick graphical wa to test for half-wave smmetr is to measure the ordinates on the wave at two points spaced half a ccle apart. The two ordinates should be equal in magnitude but opposite in sign. Fundamental Second harmonic Third harmonic Fourth harmonic FIGURE 37 7 The fundamental and the third harmonic have half-wave smmetr. Figure 37 7 shows a sine wave and second, third, and fourth harmonics. For each, one arrow shows the ordinate at an arbitraril chosen point, and the other arrow shows the ordinate half a ccle awa ( radians for the sine wave). For half-wave smmetr, the arrows on a given waveform should be equal in length but opposite in direction. Note that the fundamental and the odd harmonics have half-wave smmetr, while the even harmonics do not. This is also true for harmonics higher than those shown. Thus, for the Fourier series representing a waveform, we conclude the following: Half-Wave Smmetr A waveform that has half-wave smmetr has onl odd harmonics in its Fourier series. 457 Common Error Don t confuse odd functions with odd harmonics. Thus, sin 4 is an even harmonic but an odd function. B seeing if a waveform is an odd or even function or if it has half-wave smmetr, we can reduce the work of computation. Or, when possible, we can choose our aes to create smmetr, as in the following eample.
Section 37 6 Waveform Smmetries 39 Eample 6: Choose the ais and write a Fourier series for the triangular wave in Fig. 37 (a). 3 (a) π π (b) f() sin 3 f() sin 3 (c) FIGURE 37 Snthesis of a triangular wave. Solution: The given waveform has no smmetr, so let us temporaril shift the ais up units, which will eliminate the constant term from our series. We will add to our final series to account for this shift. We are free to locate the ais, so let us place it as in Fig. 37 (b). Our waveform is now an odd function and has half-wave smmetr, so we epect that the series we write will be composed of odd-harmonic sine terms. b sin b 3 sin 3 b 5 sin 5 From Eq. 45, b n 3 f () sin n d From to our waveform is a straight line through the origin with a slope of, so f () p q The equation of the waveform is different elsewhere in the interval to, but we ll see that we need onl the portion from to.
4 Chapter 37 Infinite Series If we graph f () sin n for an odd value of n, sa, 3, we get a curve such as in Fig. 37 (c). The integral of f () sin n corresponds to the shaded portion under the curve. Note that this area repeats and that the area under the curve between and is one-fourth the area from to. Thus, we need onl integrate from to and multipl the result b 4. Therefore b n 3 4 3 [ [ f () sin n d 4 3 f () sin n d sin n d sin n d 3 n sin n n ] cos n n sin n n cos n ] We have onl odd harmonics, so n, 3, 5, 7,.... For these values of n, cos(n), so b n p n sin n q from which b b 5 p p 5 q b 3 q () b 7 5 p 3 p 7 q () 9 q () 49 Our final series, remembering to add a constant term of units for the vertical shift of ais, is then p sin q sin 3 sin 5 sin 7 9 5 49 This is the same as Waveform 4, ecept for the -unit shift. Eercise 6 Waveform Smmetries Label each function as odd, even, or neither.. Fig. 37 9(a). Fig. 37 9(b) 3. Fig. 37 9(c) 4. Fig. 37 9(d) 5. Fig. 37 9(e) 6. Fig. 37 9(f) Which functions have half-wave smmetr? 7. Fig. 37 9(a). Fig. 37 9(b) 9. Fig. 37 9(c). Fig. 37 9(d). Fig. 37 9(e). Fig. 37 9(f) Using smmetr to simplif our work, write a Fourier series for the indicated waveforms in Table 37 6. As before, start b writing the functions that describe the waveform. Check our series against those given in the table. 3. Waveform 4. Waveform 5 5. Waveform 6. Waveform
Section 37 7 Waveforms with Period of L 4 (a) (b) (c) (d) FIGURE 37 9 (e) (f) 37 7 Waveforms with Period of L So far we have written Fourier series onl for functions with a period of. But a waveform could have some other period, sa,. Here we modif our formulas so that functions with an period can be represented. Figure 37 (a) shows a function g(z) with a period of. The Fourier coefficients for this function are a 3 g(z) dz () π = g(z) a n 3 g(z) cos nz dz () (a) = f() and b n 3 g(z) sin nz dz (3) Figure 37 (b) shows a function f() with a period L, where and z are related b the proportion L z or z L. Substituting into Eq. (), with g(z) f() and dz d(l) (L) d, we obtain a 3 g(z) dz L 3L f() L d L L3L f() d L FIGURE 37 (b)