Gibbs Phenomenon, Effects
Table of Contents
What is GIBBS Phenomenon?
J. Willard Gibbs rediscovered the GIBBS phenomenon in 1899 after Henry Wilbraham had first discovered it in 1848.
If the Fourier series is added to a periodic signal with discontinuities in order to reconstruct it, overshoots at the edges will result. These overshoots dissipate away from the edges in a damped oscillatory fashion. The figure below illustrates this phenomenon, referred to as the GIBBS phenomenon.
The amount of overshoots at discontinuities varies depending on the number of terms in the Fourier series, but according to Gibbs, they are found to be approximately 9% of the height of the discontinuity. The Wilbraham-Gibbs Constant provides the precise ratio.
The frequency rises and the overshoots become sharper as more terms are added to the series,. But the amplitude of the adjacent oscillation decreases. In other words, as n increases, the error between the original signal x(t) and the truncated signal xn(t) decreases overall, with the exception of the edges. As a result, the truncated Fourier series gets closer to the original signal x(t) the more terms there are in the approximation.
Effects
Following are some consequences of the GIBBS phenomenon −
- The phenomenon in signal processing is undesirable because it results in clipping from overshoots and ringing artifacts from oscillations.
- When adjacent regions have signal intensities that significantly differ from one another, the GIBBS phenomenon in MRI results in artifacts.
- The phenomenon exhibits a cross-pattern artifact in the discrete Fourier transform of an image,. Where the boundaries at the top-bottom and left-right of the image have a sharper discontinuity.