6.6 Gibbs phenomena  (Page 6/2)

Introduction

The Fourier Series is the representation of continuous-time, periodic signals in terms of complex exponentials. The Dirichlet conditions suggest that discontinuous signals may have a Fourier Series representationso long as there are a finite number of discontinuities. This seems counter-intuitive, however, as complex exponentials are continuous functions. It does not seem possible to exactly reconstruct adiscontinuous function from a set of continuous ones. In fact, it is not. However, it can be if we relax the conditionof 'exactly' and replace it with the idea of 'almost everywhere'. This is to say that the reconstruction isexactly the same as the original signal except at a finite number of points. These points, not necessarily surprisingly,occur at the points of discontinuities.

History

In the late 1800s, many machines were built to calculate Fourier coefficients and re-synthesize:

Explanation

We begin this discussion by taking a signal with a finite number of discontinuities (like a square pulse ) and finding its Fourier Series representation. We thenattempt to reconstruct it from these Fourier coefficients. What we find is that the more coefficients we use, the morethe signal begins to resemble the original. However, around the discontinuities, we observe rippling that does not seem tosubside. As we consider even more coefficients, we notice that the ripples narrow, but do not shorten. As we approachan infinite number of coefficients, this rippling still does not go away. This is when we apply the idea of almosteverywhere. While these ripples remain (never dropping below 9% of the pulse height), the area inside them tends to zero,meaning that the energy of this ripple goes to zero. This means that their width is approaching zero and we can assertthat the reconstruction is exactly the original except at the points of discontinuity. Since the Dirichlet conditionsassert that there may only be a finite number of discontinuities, we can conclude that the principle of almosteverywhere is met. This phenomenon is a specific case of nonuniform convergence .