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 an 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.