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Inverse Fourier Transform of cos(4ω + pi/3)

  1. May 29, 2013 #1
    1. The problem statement, all variables and given/known data
    Find the inverse fourier transform of F(jω) = cos(4ω + pi/3)

    2. Relevant equations
    δ(t) <--> 1
    δ(t - to) <--> exp(-j*ωo*t)
    cos(x) = 1/2 (exp(jx) + exp(-jx))

    3. The attempt at a solution
    So first I turned the given equation into its complex form using Euler's Formula.

    F(jω) = 1/2 (exp(j*4*ω + j*pi/3) + exp(-j*4*ω - j*pi/3))

    And using the relevant equation above, I get..

    exp(j*4*ω) <--> δ(t + 4) and exp(j*4*ω) <--> δ(t - 4)

    I'm not exactly sure how to do inverse fourier transformation on exp(j*pi/3) and exp(-j*pi/3). My guess is you simply get simply exp(j*pi/3)*δ(t) and exp(-j*pi/3)*δ(t).

    Assuming I'm correct in the above step, I now multiply the resulting expressions.

    1/2 ( exp(j*pi/3)*δ(t)*δ(t + 4) + exp(-j*pi/3)*δ(t)*δ(t - 4) )

    Is my solution correct? I feel like I would have to simply the final answer I got but I'm not really sure how.
    Last edited: May 29, 2013
  2. jcsd
  3. May 29, 2013 #2
    Well...I don't think so. Are you saying that ##\cos(4\omega + \frac{\pi}{3})## is the Fourier transform and you are attempting to find ##\mathcal{F}^{-1}[\cos(4\omega + \frac{\pi}{3})](t)##?
  4. May 30, 2013 #3


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    Homework Helper

    Your factors of ##\exp(\pm i \pi/3)## are constants multiplying the factors ##\exp(\pm i 4\omega)##.

    What is the inverse fourier transform ##\mathcal F^{-1}[c \hat{F}(\omega)]##, where ##c## is a constant (or any function, really)? It's not ##\mathcal F^{-1}[c]\mathcal F^{-1}[ \hat{F}(\omega)]##, but that's what you implicitly assumed in your final answer.
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