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Homework Help: Fourier transform of a compicated function

  1. Sep 30, 2010 #1
    Hi

    Could someone help me to calculate the fourier transform of the following function:

    rect(x/d)exp(2ipia|x|)
     
  2. jcsd
  3. Sep 30, 2010 #2

    mathman

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    What is rect(x/d)?
     
  4. Oct 5, 2010 #3
    rect(x/d) is a rectangle function.
    rect(x/d)=1 if -d/2<x<d/2;
    rect(x/d)=0 if x<-d/2 or x>d/2.
     
  5. Oct 5, 2010 #4

    mathman

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    It looks like a straightforward integration.
    ∫(x=-d/2,0)exp(-2πiax+itx) dx + ∫(x=0,d/2)exp(2πiax+itx) dx
     
  6. Oct 5, 2010 #5
    Instead of numerical solution, is there an analytical solution for this problem?
     
  7. Oct 5, 2010 #6

    vela

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    That's not a numerical solution.

    You could try using the convolution theorem to find the Fourier transform, but that seems like even more work.
     
  8. Oct 5, 2010 #7
    Can you please give me the solution using the convolution theorem?
     
  9. Oct 6, 2010 #8

    vela

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    No, that's against the forum rules. Conceptually, it's straightforward. The convolution theorem tells you

    [tex]\mathcal{F}[f(x)g(x)]=\mathcal{F}[f(x)]*\mathcal{F}[g(x)][/tex]

    so you just have to find the transforms of the rectangle and exponential functions individually and convolve the results.
     
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