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Fourier transform of rect(x/2)*comb(x) + sketch

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Take the Fourier Transform of f(x)=rect(x/2)*comb(x) where rect is the rectangle function and comb is the Dirac comb. Sketch the results.

    2. Relevant equations
    The FT of a convolution is the product of the individual FTs.


    3. The attempt at a solution
    Taking the FT is pretty simple, but it's the graph that I'm a little confused about. If we take the FT of f(x)<->F(u)

    F(u)=2sinc(2u)comb(u), if we define sinc(u)=sin(pi u)/(pi u), rather than the other way which leaves out pi.

    The graph of such a function is simply the sampling of points along 2sinc(2u) at the period of comb(u), which should just be 1. Therefore at u=0, you'd have a point of 2 (although 2sinc(2u) isn't defined at u=0, though I believe it's commonly considered to be defined as its limit, which in this case is 2). All of the other integer values of 2sinc(2u) are simply 0.

    Am I correct in assuming that the graph of 2sinc(2u)comb(u) is just {2,0,0,...,0} for all n>=0?

    This seems like a silly answer for a problem, but maybe that's the point. I just have a tendency to second guess myself when given a stupid problem with a simple answer.
     
  2. jcsd
  3. Mar 22, 2009 #2
    Sorry, this thread should be in Calculus & Beyond. I thought I was in that forum before I posted. If a mod could move it, I'd appreciate not having to re-post it.
     
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