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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