Convolution and Inverse Fourier Transform.

Cuthbert Nyack
Sometimes it is possible to find the Inverse Fourier Transform(IFT) of a frequency spectrum by using convolutions. Consider the spectrum shown below. The inverse of F(omega) cannot be found by the inverse transform formula but can readily be found by convolution.
The sinc3 can be regarded as the product of a sinc and a sinc2 and from the convolution theorem the IFT of the sinc3 can be obtained by convolving the IFT of the sinc2 (triangular pulse) with the IFT of a sinc (rectangular pulse). Since the triangular pulse is obtained by convolving the rectangular pulse with itself, then the IFT of the sinc3 can be obtained by convolving the rectangular pulse with itself and then again with the result.
In the above plot the red curve shows the initial rectangular pulse, (f(t) = (1/(2*a)) for |t| < a) blue line shows the triangular pulse (f(t) = -|t|/(4*a*a) + 1/(2*a)) resulting from convolving the rectangular pulse with itself and the magenta line shows the result of convolving the triangular pulse with the rectangular one. The inverse transform of sinc3(a omega) is the magenta curve above. Function for magenta curve is:-
For |t| < a; f(t) = (1/(4*a*a))(-t*t/(2*a) + 3*a/2)
For 3a < |t| < a; f(t) = (1/(4*a*a))(t*t/(4*a) - 3*|t|/2 + 9*a/4)
As this process is continued the result tends to a gaussian. This is so even if the starting pulse was not a rectangular one.
Plot above shows a sinc in red(transform of rectangular pulse), a sinc2 in blue (transform of triangular pulse) and a sinc3 in magenta (transform of magenta curve above,
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COPYRIGHT 1999 Cuthbert A. Nyack.