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Fourier transform of sinusoidal functions

  1. Feb 23, 2014 #1
    1. The problem statement, all variables and given/known data

    image.png
    2. Relevant equations
    sinc(x) = [itex]\frac{sin(x)}{x}[/itex]

    3. The attempt at a solution
    bit unsure how to get started?? i know transform of rectangular pulse pτ(t)=τ*sinc(τω/2∏)

    also that sin(ωt)= ejωt-e-jωt / (2)

    I could also probably sketch sinc(t/2∏), if that helps.
     
    Last edited: Feb 23, 2014
  2. jcsd
  3. Feb 24, 2014 #2
    OK, so I guess I wasn't really thinking. There is duality property listed in my book, can I use that?

    since x(t) ⇔ X(ω) then pτ(t)=τ*sinc([itex]\frac{τω}{2\pi}[/itex])

    by duality X(t) ⇔ 2[itex]\pi[/itex]*x(-ω) then τ*sinc([itex]\frac{τt}{2\pi}[/itex])=2[itex]\pi[/itex]pτ(ω)

    so for a) it would be 2[itex]\pi[/itex]p1(ω). got this right at least? and how would i sketch this. would i be able to swap ω with t and just sketch the rectangular function p1(t) with amplitude 2[itex]\pi[/itex]??
     
    Last edited: Feb 24, 2014
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