Tuesday, October 16, 2012

Convolve n Square Pulses to Gaussian

The other day, I was perusing Wikipedia, and stumbled upon the convolution page.
Intrigued by the convolution of 2 square pulses, which is a triangular function, I set about convolving a square pulse with itself several times and soon empirically observed that it seems to converge to a Gaussian shape, and fast.

The graph below shows the result of such convolutions for n from 0 (initial square pulse) to 9.

After only 3 convolutions the resulting function is visually indistinguishable from a Gaussian shape.