Differentiating Fourier Series
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
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.
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COPYRIGHT © 1996 Cuthbert Nyack.