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Fourier Transform (Numerical Analysis)

  1. Sep 25, 2011 #1
    1. Calculate the finite Fourier transform of order m of the following sequences:

    a) uk = 1, 0[itex]\leq[/itex]k[itex]\leq[/itex]N-1
    b) uk = (-1)k, 0[itex]\leq[/itex]k[itex]\leq[/itex]N-1 N even
    c) uk = k, 0[itex]\leq[/itex]k[itex]\leq[/itex]N-1




    2. Relevant equations
    Uk = (1/N)[itex]\sum[/itex]uke-2pi*i*k*j/N from j=0 to N-1 ; 0<=k<=N-1




    Attempt:
    a) First thing that I tried is that [itex]\sum[/itex]x^k = [itex]\frac{1}{1-x}[/itex] but that doesn't seem to get where I want. For example, I know that for a), we get 1 for k=0 and 0 for k[itex]\neq[/itex]0 , but this would give 1/N (1) for k = 0 and I don't know what for any other k.
    So, I found a formula that says:
    [itex]\sum[/itex]ei*k*j from j = 0 to N-1
    ={ 0 if e[itex]\neq[/itex]1,
    { N-1 else.

    b) wouldn't we get (1/N)[itex]\sum[/itex]-e-2pi*i*k*j/N from j=0 to N-1 ; 0<=k<=N-1 which is the same as part a) ?
    c) I think I need to use the idea that [itex]\sum[/itex]k*x^k = [itex]\frac{x}{(1-x)2}[/itex]

    Obviously, if this formula is valid (I have no idea), then it would give 1 for 0 and 0 for other k for part a) which is correct

    Any ideas?
     
    Last edited: Sep 25, 2011
  2. jcsd
  3. Sep 26, 2011 #2
    Just a bump
     
  4. Sep 26, 2011 #3
    Nevermind I got it
     
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