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Find the Fourier Transform of sin(pi*t) , |t|<t0

  1. Dec 19, 2011 #1
    1. The problem statement, all variables and given/known data

    Solve the above F.T.

    2. Relevant equations
    http://en.wikipedia.org/wiki/Euler's_formula

    http://en.wikipedia.org/wiki/Fourier_transform

    3. The attempt at a solution
    I use euler's formula and apply the definition of the F.S. and i get to zero, not surprisingly, as the sine is an odd function.
    At a "common F.T. pairs, there's an entry sin(ω0t) <-> jπ(δ(ω+0)-δ(ω-ω0))
    However the entry says nothing about t being contained.Any hints? Do i manipulate the signal being odd?
     
    Last edited: Dec 19, 2011
  2. jcsd
  3. Dec 19, 2011 #2

    vela

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    Are you trying to find the Fourier series, which is what I assume you mean by F.S., or the Fourier transform as in the title of this thread?
     
  4. Dec 19, 2011 #3
    The Fourier Transform.
    I edited the OP.
     
  5. Dec 19, 2011 #4

    vela

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    Show us the integral you did. You shouldn't get 0.
     
  6. Dec 19, 2011 #5
    1/2j * Integral from -1/3 to 1/3 of (ejπt-e-jπt) * e-jΩtdt

    you mean that one, right?
     
  7. Dec 19, 2011 #6
    Ah, hold on, i end up at
    [1/(jπ-jΩ)] * sin((π-Ω)/3) + [1/(jπ+jΩ)] * sin((π+Ω)/3)

    how does that look to you?
     
    Last edited: Dec 19, 2011
  8. Dec 19, 2011 #7

    vela

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    I didn't check every detail, but it looks right.
     
  9. Dec 20, 2011 #8
    Thank you.
    One last thing: From there, can you manipulate it to end up in a sinc function? (normalized or not)
    Or do you need to have it in a form of sin(a*Ω)/aΩ (meaning, having Ω in the argument as a common factor.
     
  10. Dec 20, 2011 #9

    vela

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    I think you made a sign error somewhere. You can write it as a difference of two sinc functions.
     
  11. Dec 20, 2011 #10
    I just used the property sin(-a) = - sin(a)

    what would the argument of the sinc function be? You can't factor out Ω.
     
  12. Dec 20, 2011 #11

    vela

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    The sinc function is defined as ##\text{sinc } x := \frac{\sin x}{x}##. The x is a dummy variable in this expression. You can replace it with anything you want as long as you do it everywhere, e.g.,
    $$\text{sinc }(meow) = \frac{\sin (meow)}{meow}$$
     
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