Phenomenon of Gibbs

At the time of the study of and transformed the Fourier series of Fourier, it appears sometimes a deformation of the signal, known under the name of phenomenon of Gibbs . This phenomenon is a Effect edge which occurs near a discontinuity, during the analysis of a derivable function per pieces.

History

The phenomenon was put for the first time in obviousness in 1848 by Wilbraham, but this discovery hardly knew echo.

In 1898, Albert Michelson developed a mechanical system able to calculate and summon the Fourier series of a signal given in entry. It then observed an effect of amplification of discontinuities, which persisted in spite of the increase in the number of calculated coefficients.

Whereas Michelson suspected a defect in manufacture of its machine, Gibbs showed that the phenomenon was of origin mathematical and occurred under very general conditions. In 1906, Maxime Bôcher gave the first satisfactory interpretation of the phenomenon to which it gave the name of phenomenon of Gibbs.

Description of the phenomenon

The phenomenon of Gibbs is, to some extent, a “defect of approximation” for a derivable function per pieces. For such a function F , the theorem of Dirichlet ensures that the Fourier series converge simply. In any point X of continuity the sum of the Fourier series is F (X) .

The trigonometrical Polynôme N - ième term of the Fourier series, $S_n (F)$ , is a continuous function, it is thus normal that it cannot approach uniformly the function at the points of discontinuity. Precisely, on a segment on which F is derivable, one observes a uniform convergence (it is the case of the zones of “plate” in the example of the function crenel).

On the level of the point of discontinuity X , $S_n (F)$ undergoes strong a Oscillation, a kind of “projection” which is measured by comparing the values in $x- \ frac {\ pi} {N}$ and $x+ \ frac {\ pi} {N}$ . When N becomes large, the amplitude of these oscillations tends towards a limit strictly larger than the amplitude of discontinuity, whereas the width of the zone of oscillation tends towards 0.

It is remarkable that the phenomenon is quantitatively expressed independently function considered. If the function has a discontinuity of amplitude $\ Delta y$ , then $S_n (F)$ , while remaining continuous, will know a “projection” in ordinate being worth about 18% of more in the vicinity of discontinuity.

Bonds with causality

The measuring devices behave like low-pass filters: they are unable to react to signals of too high frequency. Let us suppose that one measures a level of tension: it will be deformed and will reveal oscillations before and after the discontinuity of the real signal.

However, that would mean that the oscillations occurred before that the level is not measured, which would break the Causalité. Actually, that can have two interpretations: either the apparatuses are not perfect filters (what is comprehensible), or the level of tension cannot be perfectly discontinuous.

If one considers one or the other of these corrections, the transform of Fourier is simply shifted: the oscillations begin at the time when the level is sent , the level being observed only after one light delay.

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