Gibbs Phenomenon

Gibbs phenomenon is the oscillatory behavior observed when a square wave is reconstructed from a finite number of harmonics.

The unit cell of the square wave is given by

Its Fourier series representation is

Matlab function

A Matlab function for summing a finite number (-M to M) of terms is

function y = gibbs(x,M,duty)
% GIBBS
%
%      Finite Fourier Series evaluation of square wave
%
%   y = Gibbs(x,M,duty)
%
%      where x are the values to evaluate (0<x<0.5)
%      M is the number of terms to use
%      duty is the duty cycle (0-1)
%
if (nargin<3)
   duty = 0.5;
end

p = (-M:M);

for (j=1:length(x))
   y(j) = sum(duty*sinc(duty*p).*exp(-i*2*pi*p*abs(x(j))));
end

Normal Reconstruction

The reconstruction covers the full range of the unit cell

s = linspace(-0.5,0.5,200);
plot(s,gibbs(s,M));

M=5

M=25

Expanded Reconstructions

The reconstruction is expanded over the domain -3/M to 3/M about the edge

s = linspace(-3,3,200);
plot(s,gibbs(s/M+0.25,M));

M=9

M=25

M=75

Conclusion

As M increases, the amplitude of the oscillations does not go to zero. Instead, the oscillations compress in location toward the edge.

Maintained by John Loomis, last updated 30 Aug 2003