Fourier Analysis: A new computing approach [Lecture Notes]

A Fourier expansion is a special case of signal decomposition that decomposes a signal into oscillatory components. In this method, the signal is represented as a linear combination of trigonometric or exponential basis functions. The expansion coefficients (or weights) are then computed by correlating the signal with the corresponding basis functions [1], [2]. The process of computing the coefficients is known as Fourier analysis. In real applications, we are interested in using a few terms of a Fourier expansion, or it may be impossible to use all of the terms to approximate the signal. Therefore, a truncated Fourier expansion is used instead [3]. However, when a truncated Fourier expansion is used to approximate a signal with a jump discontinuity, an overshoot/undershoot at the discontinuity occurs, which is known as the Gibbs phenomenon. The correct size of the overshoot and the undershoot of a truncated Fourier expansion near the point of discontinuity was computed by Gibbs; the size of the overshoot/undershoot is approximately 9% of the magnitude of the jump [4].