1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Fourier Transform in the Form of Dirac-Delta Function

  1. Sep 16, 2016 #1
    1. The problem statement, all variables and given/known data
    Given [itex]x(t)=8cos(70\pi t)+4sin(132\pi t)+8cos(24\pi t)[/itex], find the Fourier transform [itex]X(f)[/itex] in the form of [itex]\delta[/itex] function.

    2. Relevant equations
    [itex]X(f)=\int ^{\infty}_{-\infty}x(t)e^{-j\omega _0t}dt[/itex]
    [itex]cos(\omega t)=\frac{e^{j\omega t}+e^{-j\omega t}}{2}[/itex]
    [itex]sin(\omega t)=\frac{e^{j\omega t}-e^{-j\omega t}}{2j}[/itex]
    [itex]\int ^{\infty}_{-\infty}cos(\omega _0t)e^{-j\omega t}dt=\frac{\pi}{2}(\delta (\omega +\omega _0)+\delta (\omega -\omega _0))[/itex]
    [itex]\int ^{\infty}_{-\infty}sin(\omega _0t)e^{-j\omega t}dt=\frac{\pi}{j2}(\delta (\omega +\omega _0)-\delta (\omega -\omega _0))[/itex]

    3. The attempt at a solution
    [itex]X(f)=\frac{8\pi}{2}(\delta (\omega +70\pi)+\delta (\omega -70\pi))+\frac{4\pi}{j2}(\delta (\omega +132\pi)-\delta (\omega -132\pi))+\frac{8\pi}{2}(\delta (\omega +24\pi)+\delta (\omega -24\pi))[/itex]

    Simplifying: [itex]X(f)=4\pi (\delta (\omega +70\pi)+\delta (\omega -70\pi))+\frac{2\pi}{j} (\delta (\omega +132\pi)-\delta (\omega -132\pi))+4\pi (\delta (\omega +24\pi)+\delta (\omega -24\pi))[/itex]
     
  2. jcsd
  3. Sep 17, 2016 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    All clear, Captain!
    (You could changew +1/2j to -j/2 but that would be quibbling!)
    Nice work.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted