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Fourier Transform of sin(wt+phi)

  1. Oct 1, 2012 #1
    1. The problem statement, all variables and given/known data

    I've been stuck on this for a while:

    Find the Fourier transform of f(t)=sin([itex]\omega0[/itex]t+[itex]\phi[/itex])

    2. Relevant equations

    I know that I have to use F(ω)=[itex]\int[/itex]f(x)e^-iωt dt (between - and + infinity) to solve this

    3. The attempt at a solution

    So far I have: F(ω)=[itex]\int[/itex]Sin(ω0t+[itex]\phi[/itex])e^-i[itex]\omega[/itex]tdt
    =[itex]\int[/itex](e^i[itex]\omega[/itex]0t+[itex]\phi[/itex] - e^-it[itex]\omega[/itex]0+[itex]\phi[/itex])/2i * e^-i[itex]\omega[/itex]t dt

    (Both evaluated between - and +infinity, and ω0 means ω subscript 0)

    But I'm really not sure where to go with this next. Any help or pointers would be really appreciated! Also sorry for my equation writing but I'm completely new to LATEX. Thank you.
  2. jcsd
  3. Oct 1, 2012 #2


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    Science Advisor
    Homework Helper

    The answer is going to involve a sum of Dirac delta functions. You might want to look up how the integral of a complex exponential is related to the Dirac delta.
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