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Fourier transform pair

  1. Apr 23, 2007 #1
    1. The problem statement, all variables and given/known data

    I'd like to prove a F/T pair and to confim if they are correct.

    s(t) = A Sin[w0 t] * rect[t/T - T/2] ..... (1)

    it's Fourier transform is

    S(f) = exp(-j w T)*T/2*A* {Sinc[(w+w0)T/2/Pi] + Sinc[(w-w0)T/2/Pi]} ....(2)

    where rect is rectangular function

    2. Relevant equations

    I can prove rect[t/T] -> T Sinc[Pi f T]

    3. The attempt at a solution

    I tried to use mathematica, but it didn't give me as good results as (2)

    Somebody can prove it?
  2. jcsd
  3. Apr 23, 2007 #2
    You know what the FT of rect and sin(wt) is. Use convolution theorem to get the FT of rect*sin.
  4. Apr 23, 2007 #3
    Hi, thanks chistianjb. I was going to using convolution, but seems too much maths involved. Because rectangular has only value within a range, this will simplfy the integration lots.

    I worked out the range of t is between (T+T^2)/2 and (T^2-T)/2, am I right?

    sb can try to integrate[Sin[w0 t], {t, (T+T^2)/2, (T^2-T)/2}]?
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