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Fourier Transform of a Gaussian Pulse

  1. Feb 3, 2013 #1
    1. The problem statement, all variables and given/known data
    Consider a Gaussian pulse exp[-(t/Δt)^2/2]exp(i*w*t), where Δt is its approximate pulse width in time. Use the Fourier transform to find its spectrum.


    2. Relevant equations
    The Fourier transform of a Gaussian is a Gaussian. If a Gaussian is given by

    f(t) = exp(-t^2/2)

    then its Fourier transform is

    F(w) = sqrt(2 * pi)exp(-w^2/2)

    3. The attempt at a solution

    I'm confused by the relevant equations (given by professor) because the exponents are not dimensionless... Nonetheless I just plugged and chugged to get

    sqrt(Δt)exp^(-(pi * Δt * w))^2

    since this creates a dimensionless exponent. I'm not sure where to include the 2 from the original Gaussian though...
     
  2. jcsd
  3. Feb 4, 2013 #2

    rude man

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    Convolve your Gaussian transform with the transform of exp(iwt).
     
  4. Feb 4, 2013 #3

    vela

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    Can you show your work? It looks like you made some algebra errors along the way.
     
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