Gibbs phenomenon

English

A partial sum (to 5 terms) of the Fourier series of a square wave, showing Gibbs phenomenon peaks beside each jump discontinuity

Alternative forms

• Gibbs' phenomenon

Etymology

After J. Willard Gibbs, who identified the behaviour in 1899, unaware of its previous discovery in 1848 by Henry Wilbraham.

A widespread myth has it that the phenomenon was observed in a device developed in 1898 by Albert A. Michelson to compute and synthesise Fourier series, but that it was assumed due to physical imperfections in the device. In fact, the graphs produced were not precise enough for the phenomenon to be clearly observed, and Michelson made no mention of it in a paper describing the device.

Proper noun

Gibbs phenomenon

1. (mathematics) A behaviour of the Fourier series approximation at a jump discontinuity of a piecewise continuously differentiable periodic function, such that partial sums exhibit an oscillation peak adjacent the discontinuity that may overshoot the function maximum (or minimum) itself and does not disappear as more terms are calculated, but rather approaches a finite limit.
   • 1974, K. F. Riley, Mathematical Methods for the Physical Sciences, Cambridge University Press, page 197:

     The Gibbs phenomenon is characteristic of Fourier series at a discontinuity, its size being proportional to the magnitude of the discontinuity.

   • 1999, Werner S. Weiglhofer, Kenneth A. Lindsay, Ordinary Differential Equations and Applications, page 121:

     At a point of discontinuity, the oscillations accompanying the Gibbs phenomenon have an overshoot of approximately 18% of the amplitude of the discontinuity.

   • 2007, Uwe Meyer-Baese, Digital Signal Processing with Field Programmable Gate Arrays, 3rd edition, Springer, page 173:

     The observed “ringing” is due to the Gibbs phenomenon, which relates to the inability of a finite Fourier spectrum to reproduce sharp edges.