6.6 Gibbs phenomena

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The Fourier Series is the representation of continuous-time, periodic signals in terms of complex exponentials. The Dirichletconditions suggest that discontinuous signals may have a Fourier Series representation so long as there are a finite number ofdiscontinuities. This seems counter-intuitive, however, as complex exponentials are continuous functions. It does not seempossible to exactly reconstruct a discontinuous function from a set of continuous ones. In fact, it is not. However, it can beif we relax the condition of 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 suprisingly, occur atthe points of discontinuities.

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:

$\frac{d {f}_{N}(t)}{d }}=\sum_{n=-N}^{N} {c}_{n}e^{i{\omega }_{0}()nt}$
Albert Michelson (an extraordinary experimental physicist) built a machine in 1898 that could compute ${c}_{n}$ up to $n=±(79)$ , and he re-synthesized
$\frac{d {f}_{79}(t)}{d }}=\sum_{n=-79}^{79} {c}_{n}e^{i{\omega }_{0}()nt}$
The machine performed very well on all tests except thoseinvolving discontinuous functions . When a square wave, like that shown in [link] , was inputed into the machine, "wiggles" around the discontinuities appeared, and even as the numberof Fourier coefficients approached infinity, the wiggles never disappeared - these can be seen in the last plot in [link] . J. Willard Gibbs first explained this phenomenon in 1899, and therefore thesediscontinuous points are referred to as Gibbs Phenomenon .

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 .

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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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