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Easy Fourier Transform

  1. Feb 27, 2012 #1
    1. The problem statement, all variables and given/known data

    Computer the Fourier transform of U(t), where U(t) = 1 for |t| < 1, and U(t) = 0 for |t| > 1.

    2. Relevant equations

    Fourier Transform: F(w) = ∫U(t)e-iwtdt (bounds: ∞, -∞)

    3. The attempt at a solution

    If |t| < 1, obviously F(w) = 0.

    If |t| > 1,
    F(w) = (-1/wt)*[cos(-wt) + i sin(-wt)] | - (-1/wt)*[cos(-wt) + i sin(-wt)] |-∞.

    How do I evaluate that? Obviously limt-->∞cos(-wt) and limt-->∞sin(-wt) don't exist. Or am I missing something important?
     
  2. jcsd
  3. Feb 27, 2012 #2

    jbunniii

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    I think you are missing something important.

    Take this statement for example:

    "If |t| < 1, obviously F(w) = 0."

    This makes no sense. "F(w) = 0" is an equation containing no "t", so why would "|t| < 1" make it true?

    Note that since U(t) = 0 unless -1 <= t <= 1, you can rewrite the integral as follows:

    [tex]\int_{-\infty}^{\infty} U(t) e^{-i w t} dt = \int_{-1}^{1} e^{-i w t} dt[/tex]
     
  4. Feb 27, 2012 #3

    lanedance

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    Not quite, F(w) is dependent on the value of w, with the integral carried out over all t.

    The effect of U(t) =0 for |t|>1 means you can change the interval of the integral to be [-1,1], as U(t) is zero outside this inetrval
     
  5. Feb 27, 2012 #4
    Ah, I see. I knew something was fishy. I guess that's what happens when it's 7 weeks into the quarter and I still haven't bought the textbook.
     
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