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Calculate the Fourier Transform

  1. Jan 2, 2012 #1
    1. The problem statement, all variables and given/known data

    calculate the Fourier Transform of the following function:

    2. Relevant equations

    x(t) = e-|t| cos(2t)

    3. The attempt at a solution

    0-∞ et ((e2jt + e-2jt) / 2) e-jωt + ∫0 e-t ((e2jt + e-2jt) / 2) e-jωt

    The first integral is easy to calculate and equals: (1/2) * (1/(1+(2-ω)j) + 1/(1+(-2-ω)j))
    But how about the second integral?
     
  2. jcsd
  3. Jan 2, 2012 #2

    AlephZero

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    You did the first integral OK. Why do you think the second integral is harder than the first one?

    But if you sketch a graph of x(t), you might notice something interesting that saves you some work...
     
  4. Jan 2, 2012 #3
    My problem is with the infinite boundaries. The first integral is simple since is done from -∞ to 0 and it makes the limit of exponential part will be zero But I am confused about the second one since its boundaries are from 0 to ∞ and I can get it how the limit of exponential part of this function will be zero too!

    I sketched the graph. Do you mean that the function is asymmetric? so the doing the integration for one side will be equal to the another?
     
  5. Jan 2, 2012 #4

    AlephZero

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    The real part of the second integral is [itex]e^{-t}[/itex] which goes to 0 as t goes to [itex]+\infty[/itex]. That is similar to the first integral, where [itex]e^{+t}[/itex] goes to 0 as t goes to [itex]-\infty[/itex].

    It would be better to call it "an even function" not "asymmetric", but you got the point that the two integrals are equal.
     
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