# 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 sinc^{3} can be regarded as the product of a sinc and a sinc^{2} and from the
convolution theorem the IFT of the sinc^{3} can be obtained by convolving the IFT
of the sinc^{2} (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 sinc^{3} 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 sinc^{3}(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 sinc^{2} in blue
(transform of triangular pulse) and a sinc^{3} in magenta (transform of magenta
curve above,

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COPYRIGHT © 1999 Cuthbert A. Nyack.