Method for Numerical Evaluation of the Coefficients of Fourier Series

Abstract

The coefficients a0, am, and bm of a Fourier series f(x) = a0 + Σmam cosmx + bm sinmx, where m = 1,2,3,⋯, ∞ is the harmonic order, can be evaluated by a simple formula if f(x) is a sequential trace of n straight lines. The formula makes use of the slopes Ai and the intercepts Bi of the 1 ≦i≦n lines contained within the period. If the line crossings are equidistant in the direction of the x axis, the formula will not only be more simplified by the direct use of these n sample values but the coefficients can be taken as the products am = Cmam* and bm = Cmbm*. The factor Cm decreases with increasing values of the integers m. The discrete values Cm appear within an envelope that is the picture of an attenuated square cosine wave. The harmonic order m and the subdivision number n of the period range are parameters in a tabulation of Cm. The second factors are introduced as periodic Fourier coefficients am* and bm*. These coefficients are an infinite repetition of the finite range 1 ≦ m ≦ n, if m runs from 1 to infinity. Assuming n is a multiple of 4, only n/4 periodic coefficients have to be evaluated. Multiplication with the tabulated values Cm result in an arbitrary series of coefficients am and bm. So far as any periodic function f(x)—given by an experimental observation, for instance—can be approximated by a sequential trace of straight lines, f(x) can be approximated by the Fourier series of the straight-line trace. The numerical evaluation of the coefficients of the trace is not restricted to a certain number. in the harmonic order, such as to the first 6 or first 12 harmonics, for instance. It can easily be programed for desk machine calculators or for computers. A simple schematic calculation can be made to yield either an arbitrary series or any particular order of coefficients. The method can be extended to numerical evaluation of the Fourier integral as well.