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Fourier series, complex coefficents

  1. Sep 30, 2007 #1

    malawi_glenn

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

    Let f be a periodic function with period 2pi

    Let:

    [tex] g = e^{2it}f(t-3) [/tex]

    Find a relation between f and g's complex fourier coefficents.


    2. Relevant equations

    [tex]y(t) = \sum _{n-\infty}^{\infty} b_n e^{in\Omega t}[/tex]

    [tex]T \Omega = 2\pi[/tex]

    T is period

    [tex]b_n = \dfrac{1}{T}\int _0^{T} y(t)e^{-in\Omega t}dt[/tex]


    3. The attempt at a solution

    g is also periodic with 2pi, since e^{2it} is.

    [tex]f(t) = \sum _{n = -\infty}^{\infty} c_n e^{in t}[/tex]

    [tex]c_n = \dfrac{1}{2\pi}\int _0^{2\pi} f(t)e^{-int}dt[/tex]

    [tex]g(t) = \sum _{m = -\infty}^{\infty} d_m e^{im t}[/tex]

    [tex]d_m= \dfrac{1}{2\pi}\int _0^{2\pi}e^{2it}e^{-imt}f(t-3)dt[/tex]

    [tex]f(t-3) = \sum _{n = -\infty}^{\infty} \left( \dfrac{1}{2\pi}\int _0^{2\pi} f(t-3)e^{-int}dt \right) e^{int} [/tex]

    [tex]d_m= \dfrac{1}{2\pi}\int _0^{2\pi}e^{it(2-m)}\left( \sum _{n = -\infty}^{\infty} \left( \dfrac{1}{2\pi}\int _0^{2\pi} f(t-3)e^{-int}dt \right) e^{int} \right) dt[/tex]

    [tex]r = t-3[/tex]

    Then r = t

    [tex]d_m = \dfrac{1}{2\pi}\int _0^{2\pi}e^{it(2-m)}\left( \sum _{n = -\infty}^{\infty} \left( \dfrac{1}{2\pi}\int _{-3}^{2\pi -3} f(t)e^{-in(t+3)}dt \right) e^{int} \right) dt[/tex]

    We are still integrating over an integer multiple of periods, so that we can write:

    [tex]d_m = \dfrac{1}{2\pi}\int _0^{2\pi}e^{it(2-m)}\left( \sum _{n = -\infty}^{\infty} \left( \dfrac{1}{2\pi}\int _{0}^{2\pi} f(t)e^{-int}e^{-i3n}dt \right) e^{int} \right) dt[/tex]

    [tex]d_m= \dfrac{1}{2\pi}\int _0^{2\pi}e^{it(2-m)}\left( \sum _{n = -\infty}^{\infty}c_n e^{-i2n}dt \right) dt[/tex]

    Does this look right to you guys? I don't have an answer to this one, and we have no smiliar problems in our course book =/
     
    Last edited: Sep 30, 2007
  2. jcsd
  3. Sep 30, 2007 #2

    malawi_glenn

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    Do you want more info?
     
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