FOURIER SERIES EVEN AND ODD FUNCTION AND APPLICATION SIGMA INSTITUTE OF ENGINEERING FOURIER SERIES EVEN AND ODD FUNCTION AND APPLICATION PREPARED BY: VAIBHAV BARIYA:130500119004 DIPAK BHARTI:130500119015 JAY PATEL:130500119027 ARPAN KOLI:130500119030 AKASH PATEL:130500119061 GUIDED BY: DIPTI MAM

10.4. Even and Odd Functions. Half-Range Expansions - If a function is even or odd  more compact form of Fourier series Even and Odd Functions Even function y=g(x): g(-x) = g(x) for all x (symmetric w.r.t. y-axis) Odd function y=g(x): g(-x) = -g(x) for all x Three Key Facts (1) For even function, g(x), (2) For odd function, h(x), (3) Production of an even and an odd function  odd function let q(x) = g(x)h(x), then q(-x) = g(-x)h(-x) = -g(x)h(x) = -q(x) Odd function Even function x g(x) g(x) x

In the Fourier series, f(x) even  f(x)sin(nx/L) odd, then bn=0 f(x) odd  f(x)cos(nx/L) odd, then a0 & an=0 Theorem 1: Fourier cosine series, Fourier sine series (1) Fourier cosine series for even function with period 2L (2) Fourier sine series for odd function with period 2L For even function with period 2 For odd function with period 2

Theorem 2: Sum of functions - Fourier coefficients of a sum of f1 + f2  sums of the corresponding Fourier coefficients of f1 and f2. - Fourier coefficients of cf  c times the corresponding coefficients of f. Ex. 1) Rectangular pulse Ex. 2) Sawtooth wave: f(x) = x +  (- < x < ) and f(x + ) = f(x) for f2 = :   for odd f1 = x  2k -  x f(x) + k previous result by Ex.1 in 10.2 -  x f(x)

APPLICATION OF FOURIER SERIES Example 1. Square Wave ——High Frequency One 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 to handle sharply rising pulses. Suppose that our wave is designed by

(7.30) From Eqs. (7.11) and (7.12) we find (7.31) (7.32) (7.33)

Example 2 Full Wave Rectifier The resulting series is (7.36) Except for the first term which represents an average of f(x) over the interval all the cosine terms have vanished. Since is odd, we have a Fourier sine series. Although only the odd terms in the sine series occur, they fall only as This is similar to the convergence (or lack of convergence ) of harmonic series. Physically this means that our square wave contains a lot of high-frequency components. If the electronic apparatus will not pass these components, our square wave input will emerge more or less rounded off, perhaps as an amorphous blob. Example 2 Full Wave Rectifier As a second example, let us ask how the output of a full wave rectifier approaches pure direct current (Fig. 7.6). Our rectifier may be thought of as having passed the positive peaks of an incoming sine and inverting the negative peaks. This yields

Since 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 form will appear. Again, from Eqs. (7.11) and (7.12), we have (7.38)

is not an orthogonality interval for both sines and cosines (7.39) Note carefully that is not an orthogonality interval for both sines and cosines together and we do not get zero for even n. The resulting series is (7.40) The original frequency has been eliminated. The lowest frequency oscillation is The high-frequency components fall off as , showing that the full wave rectifier does a fairly good job of approximating direct current. Whether this good approximation is adequate depends on the particular application. If the remaining ac components are objectionable, they may be further suppressed by appropriate filter circuits.

These 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 expect the nth coefficient to be decreasing as . Convergence is relatively slow. If f(x) is continuous (although possibly with discontinuous derivatives as in the Full wave rectifier of example 7.3.2), we can expect the nth coefficient to be decreasing as 3. More generally if f and its first r derivatives are continuous, but the r+1 is not then the nth coefficient will be decreasing as Example 3 Infinite Series, Riemann Zeta Function As a final example, we consider the purely mathematical problem of expanding (7.41)

by symmetry all For the ’s we have (7.42) (7.43) From this we obtain (7.44)

As 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 our expansion of and expansions of other powers of x numerous other infinite series can be evaluated.

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