Question

In one sentence describe the Gibb's phenominon in approximating a square wave as a sum of sines.

Answer 1

1)As you add sinusoids waves of increasingly higher frequency, the approximation gets better and better.

2)The addition of higher frequencies better approximates the rapid changes, or details, (i.e., the discontinuity) of the original function (in this case, the square wave).

3)On either side of the discontinuity there is a small overshoot (called "Gibb's phenomenon" or "Gibb's overshoot"). This overshoot is always present in the Fourier representation of a signal with a discontinuity and has the same height for any number of harmonics greater than 1. The magnitude of the overshoot varies with the number of harmonics used but quickly converges to an amplitude of about 9% of the magnitude of the discontinuity for a square wave. In this case the discontinuity has an amplitude of 1 unit, so the Gibb's overshoot is about 0.09. Though the height of the overshoot is finite (at about 9%) as we add more harmonics, note that the width decreases, so the area of the overshoot (and hence the energy) decreases. As the area goes to zero, its effect in most systems of interest to engineers also goes to zero.