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Convolution math homework

  1. Apr 15, 2008 #1
    1. The problem statement, all variables and given/known data

    Find [tex] R(\tau) [/tex] if a) [tex]S(\omega) = \frac{1}{(4+\omega^2)^2} [/tex]

    2. Relevant equations

    I have given [tex] \frac{4}{4+\omega^2} [/tex] <==> [tex] e^{-2|\tau|} [/tex]

    3. The attempt at a solution
    So [tex] S(\omega) = \frac{1}{(4+\omega^2)^2}=
    \frac{1}{16}\frac{4}{(4+\omega^2)}\frac{4}{(4+\omega^2)} [/tex]


    [tex] R(\tau)= \frac{1}{16} e^{-2|\tau|} * e^{-2|\tau|} [/tex]

    Where * is convolution


    So

    [tex] R(\Tau) = \frac {1}{8}\int_{0}^{\infty} e^{-2(\tau-\alpha)} e^{-2\alpha} d\alpha [/tex]

    But that turns out to be infinite. Does anyone have any idea where I went wrong?
     
    Last edited: Apr 15, 2008
  2. jcsd
  3. Apr 16, 2008 #2

    benorin

    User Avatar
    Homework Helper

    The Laplace transform is [tex]
    \frac{a}{a^2+\omega^2} \Leftrightarrow \sin a\tau
    [/tex]
     
    Last edited: Apr 16, 2008
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