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Fourier transform matrices

  1. Apr 30, 2014 #1
    1. The problem statement, all variables and given/known data
    (i) Verify that 5 is a primitive 4th root of unity in F13.
    (ii) Let F be the 4x4 matrix whose (i, j)th entry is 5ij in F13 for i, j = 0,1,2, 3.
    Compute F(hat) and verify that F(hat)F= I.


    2. Relevant equations
    The matrix F(hat) is called the inverse discrete Fourier transform of F.


    3. The attempt at a solution
    I have already solved part (i):
    Since 52 = 15 = -1 (mod 13) and 54 = (-1)2 = 1 (mod 13), we conclude that 5 is a primitive 4th root of unity in F13.

    But I do not know how to obtain matrix F for part (ii), but I understand that F(hat) is the inverse matrix of F, so if I can find matrix F then I can easily solve for matrix F(hat). If someone can please help me out I'd really appreciate it.
     
  2. jcsd
  3. Apr 30, 2014 #2

    LCKurtz

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    For those of us not familiar with this area and your notation, you need to give us definitions.

    1. What is ##F_{13}##? I'm guessing integers mod 13?

    2. What does ##5_{ij}## mean?

    3. What is the definition of ##\hat F##?
     
  4. Apr 30, 2014 #3

    Zondrina

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    Yes to your first question.

    ##F## is defined as the Discrete Fourier Transform, it looks like this:

    http://gyazo.com/8d9c1acfec21ff3a180cc0b94d43e706

    Notice the entries ##(ω)## are just the e'th root of primitive unity raised to powers.

    Also ##\hat F## is defined as the Inverse Discrete Fourier Transform. It satisfies ##F^{-1} = \frac{1}{e} \hat F## where the entries in ##\hat F## happen to be the inverses of the entries in ##F##.
     
  5. Apr 30, 2014 #4
    1. F_13 is a field of 13 elements.

    2. My apologies, I meant to write 5^(ij).

    3. I already defined that above. The matrix F(hat) is called the inverse discrete Fourier transform of matrix F.
     
  6. Apr 30, 2014 #5

    Zondrina

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    Start by writing down ##F##. It shouldn't be too difficult to find ##\hat F## afterwards.
     
  7. Apr 30, 2014 #6
    That was my original question stated above. I do not understand how to write down F and wanted to see if anyone knew how to come up with the matrix for F so then I can easily obtain F(hat).
     
  8. Apr 30, 2014 #7

    Zondrina

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    I posted it in my post above, but here it is again:

    http://gyazo.com/8d9c1acfec21ff3a180cc0b94d43e706

    Now, the question wants you to compute each matrix entry, namely:

    ##(5^{i \space \times \space j}) \mod 13## for ##i, j = 0, 1, 2, 3##.

    What do ##i## and ##j## equal for the first row, first column entry in your matrix?

    Now how about the first row, second column entry? Second row, first column?

    Etc. Notice ##5## is the 4th primitive root of unity.
     
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