Fourier Transform of Quadratic Pulse.

Cuthbert Nyack
A Rectangular pulse of width t and height 1/t has spectrum sinc(wt/2).
If this pulse is convolved with itself, the result is a triangular pulse with spectuum sinc(wt/2)2.
This proceedure can be repeated to produce pulses with Fourier Transform which reduces more rapidly with frequency. If the rectangular pulse is convolved with the triangular pulse, the result is a quadratic pulse with spectrum sinc(wt/2)3.
The effect is shown in the applet below.
The quadratic pulse is shown as the magenta curve and its spectrum as the cyan curve. The yellow curve shows the pulse which is reconstructed from a finite width of the spectrum. The result shows that the pulse shape is retained when passed through a filter of relatively narrow bandwidth.

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COPYRIGHT 2005 Cuthbert Nyack.