Inverse Fourier Transform using convolutions

May 2008
106
29
Hi I'm having a little problem with this question - its asking me to find the inverse fourier transform of using the convolution therem.

\(\displaystyle F(k) = \frac{1}{(1+k^2)^2} \)

Looking at it by inspection it seems pretty obvious to me that the approach they want me to follow is to consider

\(\displaystyle h(x) = \exp(-|x|) \)

This has a fourier transform

\(\displaystyle \tilde{h}(k) = \frac{2}{1+k^2} \)

and hence the inverse fourier transform will be a quater of the convolution of h with itself.

The problem I'm having however is with evaluating this

\(\displaystyle \int_{-\infty}^{\infty} h(y) h(x-y) dy \)

any hints?

Cheers

Simon
 

CaptainBlack

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Hi I'm having a little problem with this question - its asking me to find the inverse fourier transform of using the convolution therem.

\(\displaystyle F(k) = \frac{1}{(1+k^2)^2} \)

Looking at it by inspection it seems pretty obvious to me that the approach they want me to follow is to consider

\(\displaystyle h(x) = \exp(-|x|) \)

This has a fourier transform

\(\displaystyle \tilde{h}(k) = \frac{2}{1+k^2} \)

and hence the inverse fourier transform will be a quater of the convolution of h with itself.

The problem I'm having however is with evaluating this

\(\displaystyle \int_{-\infty}^{\infty} h(y) h(x-y) dy \)

any hints?

Cheers

Simon
You want the inverse FT for \(\displaystyle (h(t)/2)^2\) so you want the convolution of \(\displaystyle e^{-|x|}/2\) with itself

CB
 
May 2008
106
29
I realised that - just I was having trouble evaluating the convolution integral (Crying)