# Differentiating Fourier Series Applet.

Cuthbert Nyack
A differentiator has a transfer function whose magnitude is proportional to frequency f and has a constant phase advance of p/2. The effect of differentiation on a square and a triangle wave is shown below. The Fourier Series expansion of a square wave is shown below in blue. To show the effects of differentiation the parts in red was added. As 'a' varies from 0 to 1 the signal changes from a square wave to a differentiated square wave. As a changes, the effect can be seen in the applet below. Differentiating a square wave (shown as magenta) produces sharp "spikes" ( shown in red) at the "jumps". Because of the slow convergence a large number of terms must be added to see the result.

For a triangle wave, the equivalent equation and applet is shown below. When a triangle wave is differentiated, the result is a square wave. One result of differentiating a signal is to "sharpen" any discontinuities, which is another way of saying that it increases the high frequency content of the signal. After differentiation the Fourier series converges more slowly.

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