Finding Fourier Series Coefficients 2.7 2000/08/09 2002/04/08 00:00:00.007 GMT-5 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Doug Daniels rainking@alumni.rice.edu average coefficients fourier orthogonality series signal square value wave This module outlines the procedure for determining Fourier series coefficients. Assume for the moment that the Fourier series works. To find the Fourier coefficients, we note the following orthogonality properties of sinusoids. Orthogonality k l t 0 T 2 k t T 2 l t T 0 t 0 T 2 k t T 2 l t T T 2 k l k 0 l 0 0 k l k 0 l t 0 T 2 k t T 2 l t T T 2 k 1 k 0 l 0 T k 0 l 0 k l To use these, let's multiply the Fourier series for a square wave ( s t a 0 k 1 a k 2 k t T k 1 b k 2 k t T ) by 2 l t T and integrate. The idea is that, because integration is linear, the integration will sift out all but the term involving a l . t 0 T s t 2 l t T t 0 T a 0 2 l t T k 1 a k t 0 T 2 k t T 2 l t T k 1 b k t 0 T 2 k t T 2 l t T The first and third terms are zero; in the second, the only non-zero term in the sum results when the indices k and l are equal (but not zero), in which case we obtain a l T 2 . If k 0 l , we obtain a 0 T . Consequently, l l 0 a l 2 T t 0 T s t 2 l t T All of the Fourier coefficients can be found similarly. a 0 1 T t 0 T s t k k 0 a k 2 T t 0 T s t 2 k t T b k 2 T t 0 T s t 2 k t T The expression for a 0 is referred to as the average value of s t . Why? The average of a set of numbers is the sum divided by the number of terms. Viewing signal integration as the limit of a Riemann sum, the integral corresponds to the average.