Fourier Series of 2 Segment Function.

Cuthbert Nyack

The diagram opposite shows a periodic function consisting of 2 segments ab and cd. The height and slope of the segments can be changed by a,b,c,d and the width of each segment can be changed by e and f.
For a flat symmetric pulse, the spectrum falls within a sinc envelope. The frequency of the first zero of the sinc increases as the width of the pulse reduces.
The Fourier spectrum of the pulse can be investigated by the applet below.

In the above Applet, the width of the pulse can be varied by changing e or f and the height by a,b,c,d. The important parameters are e/T and f/T. Here T = 4. Vertical scale is -1 to +1 when gain = 1. The applet illustrates how the Fourier Spectrum is affected by the width of the pulse relative to the period. The even part of the spectrum is in red and the odd part is in green. The even and odd parts for n are also shown.

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

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