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    A light source starts to emit at time = 0 and the amplitude of the emission is e^{-t/ \tau} \sin (\omega_0 t) for t > 0. Show that the power spectrum is:

    \dfrac{\omega_0^{2} \tau^4}{[1 + (\omega - \omega_0)^2 \tau^2][1 + (\omega + \omega_0)^2 \tau^2]}

    As a hint, I'm told to FT the signal and then square it, which makes sense to me. But I am having trouble FTing the function they've given. As I understand it, my integral should look like

    F(\omega) = \displaystyle \int_0^{\infty} e^{-t/ \tau} e^{- i \omega t} \sin (\omega_0 t) dt

    but I get totally stumped here. Any tips, anyone?
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    (Original post by trm90)
    A light source starts to emit at time = 0 and the amplitude of the emission is e^{-t/ \tau} \sin (\omega_0 t) for t > 0. Show that the power spectrum is:

    \dfrac{\omega_0^{2} \tau^4}{[1 + (\omega - \omega_0)^2 \tau^2][1 + (\omega + \omega_0)^2 \tau^2]}

    As a hint, I'm told to FT the signal and then square it, which makes sense to me. But I am having trouble FTing the function they've given. As I understand it, my integral should look like

    F(\omega) = \displaystyle \int_0^{\infty} e^{t/ \tau} e^{- i \omega t} \sin (\omega_0 t) dt

    but I get totally stumped here. Any tips, anyone?
    I'd start by noting that

    e^ae^b = e^{a+b}
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    (Original post by EEngWillow)
    I'd start by noting that

    e^ae^b = e^{a+b}
    Ye I tried putting it in that form before, but the problem is the sin(wt) term... I know that the FT of \exp({-t(1/ \tau + i \omega)}) is \tau / (1 + i \omega \tau) but I don't feel I'm going to get anywhere with the trig term in there
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    2i sin wt = exp(iwt)-exp(-iwt).
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    (Original post by DFranklin)
    2i sin wt = exp(iwt)-exp(-iwt).
    Brilliant - thanks very much
 
 
 
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