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Math Help - Inverse Fourier Transform using convolutions

  1. #1
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    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.

     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

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

    This has a fourier transform

     \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

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

    any hints?

    Cheers

    Simon
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by thelostchild View Post
    Hi I'm having a little problem with this question - its asking me to find the inverse fourier transform of using the convolution therem.

     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

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

    This has a fourier transform

     \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

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

    any hints?

    Cheers

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

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