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Hilbert transform of Sinc

  1. May 3, 2016 #1
    1. The problem statement, all variables and given/known data

    Show that the Hilbert transform of ##\frac{\sin(at)}{at}## is given by

    $$\frac{\sin^2(at/2)}{at/2}.$$

    2. Relevant equations

    The analytic signal of a function is given by ##f_a(t) = 2 \int^\infty_0 F(\nu) \exp(j2 \pi \nu t) \ d\nu,## where ##F(\nu)## is the Fourier transform of the function. We have ##f_a (t) = f(t) +j\hat{f}(t),## where ##\hat{f}(t)## is the Hilbert transform.

    ##FT(sinc(at)) = \frac{1}{a} \Pi (\frac{\nu}{a})##, where ##\Pi## denotes the rectangular function.

    3. The attempt at a solution

    This is the analytic signal whose imaginary part would be the Hilbert transform:

    $$f_a(t) = 2 \int^\infty_0 FT \Big[ \frac{\sin(at)}{at} \Big] \exp(j2 \pi \nu t) \ d\nu$$

    I tried to rewrite this as:

    $$f_a(t) = 2 \int^\infty_{-\infty} u(t) \ FT \Big[ \frac{\sin(at)}{at} \Big] \exp(j2 \pi \nu t) \ d\nu$$

    So, it looks like an inverse Fourier transform, so this becomes ##\left( \frac{1}{j2 \pi \nu} + \frac{\delta(\nu)}{2} \right) \frac{\sin(at)}{at}.## So this is not the correct solution. So here is another approach:

    $$f_a(t) = 2 \int^\infty_0 FT \Big[ \frac{\sin(at)}{at} \Big] \exp(j2 \pi \nu t) \ d\nu = 2 \int^\infty_0 \frac{1}{|a|} \Pi \left( \frac{\nu}{a} \right) \exp(j2 \pi \nu t) \ d\nu$$

    So, how do I continue from here? What method should I use?
     
    Last edited: May 3, 2016
  2. jcsd
  3. May 8, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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