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Calculating DFT for specific polynomial

  1. Apr 29, 2015 #1
    Hello everyone,

    This seems like a simple problem but I get the impression that I'm missing something.

    1. The problem statement, all variables and given/known data

    Given the values ## v_1,v_2,...,v_n ## such that DFTn ## (P(x)) = (v_1, v_2, \ldots, v_n) ## and ##deg(P(x)) < n##, find DFT2n## P(x^2)##

    2. Relevant equations


    3. The attempt at a solution

    My attempt:
    Let [itex]P(x) = \sum\limits_{i=0}^{k} a_ix^i[/itex] .
    First, expand ##P(x^2)## :
    [tex]P(x^2) = a_0 + a_1x^2 + a_2x^4 + \ldots + a_kx^{2k}[/tex]
    Let's denote by ##W^j_n## the j'th root of unity of order n.
    We have:
    [tex]W^j_{2n} = e^{\frac{2\pi i j}{2n}} = e^{\frac{\pi i j}{n }}[/tex]
    Using this information, let's expand a term p of ##P(x^2)## on ##W^j_{2n}##:
    [tex]a_p(x^2) = a_p(e^\frac{\pi ij}{n})^{2p} = a_p(e^{2\pi i}) ^\frac{jp}{n} = a_p[/tex] (since ##e^{2\pi i} = 1## ).
    So, when evaluating any term of the compound polynomial on some unity root of order 2n we get ##\sum\limits_{i=0} ^k ak## and therefore the final answer is:
    [tex](\sum\limits_{i=0} ^k ak, \sum\limits_{i=0} ^k ak,\ldots, \sum\limits_{i=0} ^k ak)[/tex].

    This doesn't seem right, I didn't even use the information about ## (v_1, \ldots, v_n) ## (answer should be in terms of those?).

    Any help would be appreciated.
     
    Last edited by a moderator: May 2, 2015
  2. jcsd
  3. May 4, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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