# Thread: Inverse Fourier Transform using convolutions

1. ## Inverse Fourier Transform using convolutions

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

2. Originally Posted by thelostchild
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

3. I realised that - just I was having trouble evaluating the convolution integral