# Fourier Transform N Pulses

Cuthbert Nyack

One way to see the connection between the Fourier Series and Transform is to
consider a set of pulses. A single pulse is of width 2t
and height 1/2t. The Fourier Transform F(w) of a single pulse and the Fourier Series coefficients
a_{n} of an infinite number of pulses separated by T is shown below.
The Fourier Transform of 2N pulses with the origin between 2 pulses is
shown below. The factor 1/2N is not part of the Fourier Transform but is
inserted for comparison between the Series and Transform.
For 2N+1 pulses with the origin in the middle of one pulse, the
Fourier Transform is shown below.

The following applet shows that as the number of pulses is increased, the
energy in the Fourier Transform becomes concentrated at the same frequencies given
by the Fourier Series.

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COPYRIGHT © 2005 Cuthbert Nyack.