# Inverse Fourier Transform using convolutions

#### 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

#### CaptainBlack

MHF Hall of Fame
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

#### thelostchild

I realised that - just I was having trouble evaluating the convolution integral (Crying)