Gibbs phenomenon

In mathematics, Gibbs' phenomenon is the behavior that so-called overshoots occur in the vicinity of discontinuous Fourier series and when Fourier transforms piecewise continuous, differentiable functions . These overshoots do not disappear even if the finite number of terms for approximation or the bandwidth is increased to arbitrarily high but finite values, but rather show a constant, relative deflection of approx. 9% in the maximum deflection.

The effect was named after the American physicist Josiah Willard Gibbs , who dealt with the analysis of breakover vibrations around 1898 . The name comes from the mathematician Maxime Bôcher , who in 1906 formulated the practically motivated work of Gibbs mathematically correct. However, the first work on the effect dates back to the English mathematician Henry Wilbraham (1825–1883), who worked 50 years earlier, but whose work, published in 1848, received no further attention at the time.

In the field of signal processing, Gibbs' phenomenon is one of several effects that are also known as ringing . The specific Gibbs phenomenon should not be confused with the general overshoot of signals.

description

Gibbs' phenomenon for a periodic square wave signal, approximated with 5 harmonics

Square-wave signal with 25 harmonics approximated

Square-wave signal approximated with 125 harmonics

If you develop a Fourier series of a discontinuous , periodic function, such as the rectangular function , typical overshoots and undershoots result at the discontinuity points, which do not decrease even if you try to approximate the function using additional summation terms as in the representations with 5 , 25 and 125 harmonics demonstrated. It can be seen that although the frequency of the overshoot increases and the duration decreases, the maximum deflection of the overshoot shortly before or after the jump point remains constant.

Similarly, the Gibbs phenomenon also occurs in the case of Fourier transformation at jump points, whereby the function to be approximated does not have to be periodic.

From a physical point of view, the meaning lies in the fact that every system that actually exists also has the property of a low-pass filter and limits a signal in its bandwidth . Jump points that have an “infinite number” of frequency components cannot occur in real systems.

calculation

The relative height of the overshoot in one direction, based on half the jump height, can be determined in the limit value of an infinite number of Fourier summation terms:

${\displaystyle {\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\sin t}{t}}\,\mathrm {d} t-{\frac {1}{2}}\approx 0{,}08949}$(Follow A243268 in OEIS ),

This results in a percentage error of about 18% of the jump height. The integrand is also known as the cardinal sine or the si function. The value of the integral

${\displaystyle \int _{0}^{\pi }{\frac {\sin t}{t}}\ \mathrm {d} t=(1{,}851937052\dots )={\frac {\pi }{2}}+\pi \cdot (0{,}089490\dots )}$(Follow A036792 in OEIS )

is called the Wilbraham – Gibbs constant .

literature

• Edwin Hewitt , Robert E. Hewitt: The Gibbs-Wilbraham phenomenon: An episode in fourier analysis. Archive History Exact Sciences, Volume 21, 1979, pp. 129-160.
• Fernando Puente León, Uwe Kiencke, Holger Jäkel: Signals and Systems . 5th edition. Oldenbourg, 2011, ISBN 978-3-486-59748-6 .

Web links

• Eric W. Weisstein : Gibbs Phenomenon . In: MathWorld (English).

Individual evidence

1. ^ HS Carslaw: Introduction to the theory of Fourier's series and integrals . Third ed. Dover Publications Inc., New York 1930, Chapter IX ( google.com ).
2. ^ Edwin Hewitt, Robert E. Hewitt, The Gibbs-Wilbraham phenomenon: An episode in fourier analysis . In: Archive for History of Exact Sciences . 21, No. 2, 1979, pp. 129-160. doi : 10.1007 / BF00330404 .
3. ↑ Eric W. Weisstein : Gibbs Phenomenon . In: MathWorld (English).