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Fourier transformation

  1. May 23, 2005 #1
    For an experiment about Holograms I have to find a couple of Fourier transformations. The one I'm having troubles with is the following:

    Find the Fouriertransform of a rectangular wave:
    thus: f(x)=1 from (-5b,-3b), (-b,b) and (3b,5b) and (7b,9b) etc.
    and f(x) = -1 from (-3b,-b), (b,3b), (5b,7b) etc.


    Could someone give me a hint?
     
  2. jcsd
  3. May 23, 2005 #2
    I would do this by breaking it up into simpler pieces that you know the transforms of. Let your rectangular wave centered at zero be f(x), and say

    f(x) = A(x) + B(x)

    where A(x) is a the rectangular wave shifted up by one (so it goes from zero to two), and B(x) = -1. This helps because F(f(x)) = F(A(x)+B(x)) = F(A(x)) + F(B(x)), where F( ) is the Fourier transform. Now your rectangular wave is a train of boxes, which can also be thought of as a single box convolved with an impusle train, or A(x) = A1(x)[tex]\star[/tex]A2(x). And since convlolution in one domain is multiplication in the other, you now have

    F(f(x)) = F( A1(x)[tex]\star[/tex]A2(x) ) + F(B(x)) = F(A1(x))F(A2(x)) + F(B(x))

    Now you are left with all simple transforms, and it looks like you will end up with a sampled sinc function with an impulse somewhere.
     
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