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Nth derivative Fourier transform property

  1. Feb 26, 2015 #1

    ElijahRockers

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    Gold Member

    1. The problem statement, all variables and given/known data

    I am given f(t) = e^-|t| and I found that F(w) = ##\sqrt{\frac{2}{\pi}}\frac{1}{w^2 + 1}##

    The question says to use the nth derivative property of the Fourier transform to find the Fourier transform of sgn(t)f(t), and gives a hint: "take the derivative of e^-|t|"

    I also found the Fourier transform for t*f(t) using another property, but this part has me stumped.

    2. Relevant equations

    sgn(t) =
    1 for t>0
    0 for t=0
    -1 for t<0

    3. The attempt at a solution

    I took the derivative of e^-|t|, and got ##\frac{-te^{-|t|}}{|t|}##

    But i'm not quite sure how I can use that result, combined with the nth derivative property, to find the F.T. of sgn(t)f(t). I plotted f(t), f'(t) and sgn(t)f(t), but I'm struggling to see the link between them that can help me solve this one... any guidance would be welcome.
     
  2. jcsd
  3. Feb 26, 2015 #2

    Dick

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    Science Advisor
    Homework Helper

    You do know that ##\frac{t}{|t|}=sgn(t)##, right? Aside from the issue of the left hand side not being defined at ##t=0##, but that ambiguity doesn't matter for a fourier transform.
     
    Last edited: Feb 26, 2015
  4. Feb 26, 2015 #3

    ElijahRockers

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    Gold Member

    I definitely did not realize that.... derp. That simplifies things. Thank you!
     
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