Gibbs phenomenon

Approximation of square wave in 5 steps

Approximation of square wave in 25 steps

Approximation of square wave in 125 steps

The Gibbs phenomenon (also known as ringing artefacts) is the peculiar mode in which the Fourier series of a piecewise differentiable periodic function f behaves at a jump discontinuity: the nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as the frequency increases, but approaches a finite limit.

The three pictures on the right demonstrate this for a square wave, whose Fourier series is

$\sin(t)+{\frac {1}{3}}\sin(3t)+{\frac {1}{5}}\sin(5t)+\dotsb$

As can be seen, as the number of terms rises, the error of the approximation is reduced in width and energy, but converges to a fixed height.

The Gibbs phenomenon was first observed by Albert Michelson via a mechanical graphing machine. Michelson developed a device in 1898 that could compute and re-synthesize the Fourier series. When a square wave was input into the machine, the graph would move to and from around the discontinuities. This would occur, and continue to occur, as the number of Fourier coefficients approached infinity.

The phenomenon was first explained by J. Willard Gibbs in 1899.

See also

• List of mathematical topics
• Square wave
• Scientific phenomena named after people
• Yale University
• Compare with Runge's phenomenon for polynomial approximations

Publication

• Gibbs, J. W., "Fourier Series". Nature 59, 200 and 606, 1899.

External links and references

• Braennlund, Johan, "Why are sine waves fundamental".
• Weisstein, Eric W., "Gibbs Phenomenon". From MathWorld--A Wolfram Web Resource.
• Prandoni, Paolo, "Gibbs Phenomenon".
• Radaelli-Sanchez, Ricardo, and Richard Baraniuk, "Gibbs Phenomenon". The Connexions Project. (Creative Commons Attribution License)
• Pavel, "Gibbs phenomenon". math.mit.edu. (Java applet)