Gibbs phenomenon

As Gibbs phenomenon is called in mathematics the behavior that so-called overshoots occur in Fourier series and the Fourier transform of piecewise continuous, derivable functions in the vicinity of discontinuities. These overshoots do not disappear even if the finite number of terms for the approximation and the band width is raised to any desired height, but finite values ​​, but have a constant relative displacement of about 9 % in the maximum deflection on.

The effect was named after the American physicist Josiah Willard Gibbs, which dealt in 1898 with the analysis of relaxation oscillations. The name comes from the mathematician Bôcher maxim, which in 1906 formulated the practically motivated work of Gibbs mathematically correct. However, initial work on the effect of dating on the 50 years before making English mathematician Henry Wilbraham, the 1848 published work at the time but took no further attention.

The Gibbs phenomenon is in the field of signal processing of a plurality of effects, which are also referred to as ringing. The specific Gibbs phenomenon should not be confused with the general overshooting of signals.

Description

If we develop a Fourier series of a discontinuous periodic function, square function shown such as on the pictures, so result at the points of discontinuity typical over-and undershoot, which is also not decrease when one tries to approximate the function by further sum of elements as in the representations with 5, 25 and 125 harmonics shown. It can be seen that, while increasing the frequency of the vibration and the duration is decreased, the maximum deflection of the vibration just before or after the jump point, however, remains constant.

Analogously to the Gibbs phenomenon occurs also in the Fourier transform of discontinuities, wherein the case does not need to be periodic function to be approximated.

Physically, the meaning is that each real existing system also has the characteristic of a low-pass filter and a signal limited in its bandwidth. Discontinuities which "infinitely many" have frequency components can not occur in real systems.

Calculation

The relative height of the overshoot in one direction, based on the step height, can be infinite in the limit of many Fourier sum members to determine:

Thus resulting in a percentage error of about 9 % of the step height. The term in the integral is also referred to as cardinal sine or sinc function. The value of the integral

Is called Wilbraham - Gibbs constant.