Fourier Series Convergence at Discontinuity, Gibbs Phenomenon

Cuthbert Nyack

The square wave has discontinuities at -T/4 and +T/4. At either discontinuity, the Fourier Series converges to the mid point of the "jump". About either side of the jump the series oscillate. The height of the peaks of the oscillation decreases away from the jump, but the height of peak1, peak2 etc remain the same as the number of terms summed increases. The effect is referred to as the Gibbs phenomenon and is illustrated by the applet below.

In the above applet 2 curves are shown summed for different number of terms. The center of the plot corresponds to +0.25T for a square wave. With the Horiz Gain = 1, the horizontal axis goes from +0.2T to +0.3T. The relation between the the curves can be seen by summing the red curve for a larger number of terms and then using the horizontal gain to superimpose the 2 curves.
eg parameters:-
99, 49, 2
199, 49, 4
199, 99, 2
201, 67, 3

Similar to how windows are used in FIR filters to reduce ripple in the transfer function then a similar approach can be used here. In the above applet the red curve is summed using a convergence factor which gradually reduces the components in the spectrum so they reach zero at the last harmonic used in the sum. The result is a reduction in the ripple with a reduction in the slope at the discontinuity.

eg parameters:-
49, 49, 1; 201, 201, 1 show the difference between the 2 cases with and without the convergence factor.

When activated the following gif image show how the applets should appear.

Return to main page
Return to page index