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Math Help - Complex Fourier series

  1. #1
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    Apr 2009
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    Complex Fourier series

    Hi there;

    I have a function;

    x(t) = cos (\displaystyle\frac{2\pi}{5}t) + 3sen (\displaystyle\frac{2\pi}{7}t)

    Where T=35 (period)

    First is asking me to develope de complex Fourier series. Because of the type of function, I know that I can directly substitude by the Euler formula,

    x(t)=\dfrac{1}{2}\left(e^{\frac{2\pi i}{5} t}+e^{-\frac{2\pi i}{5} t}\right)+\dfrac{3}{2i}\left(e^{\frac{2\pi i}{5} t}-e^{-\frac{2\pi i}{5} t}\right)=

    =\dfrac{1}{2}\left(e^{7\cdot \frac{2\pi i}{35} t}+e^{-7\cdot \frac{2\pi i}{35} t}\right)+\dfrac{3}{2i}\left(e^{7\cdot \frac{2\pi i}{35} t}-e^{-7\cdot \frac{2\pi i}{5} t}\right)=


    =\dfrac{1}{2}e^{7\cdot \frac{2 \pi i t}{35}}+\dfrac{1}{2}e^{-7\cdot \frac{2 \pi i t}{35}}-\dfrac{3i}{2}e^{5\cdot \frac{2 \pi i t}{35}}+\dfrac{3i}{2}e^{-5\cdot \frac{2 \pi i t}{35}}

    Second part of the problem is asking me to find de transfer function H(s) which represents the relation between the input X(t) and the output Y(t) represented by the ODE:

    y''(t)+4y'(t)+3y(t)=x(t)

    knowing that I have that H(s)=\displaystyle\frac{1}{s^2+4s+3}

    the last part of the problem is asking me to find the complex Fourier series of Y(t).

    and here is where I'm a bit lost, I'm not too sure if I have to follow this approach:

    L^{-1}(L(Y(t))=L^{-1}H(s)*L^{-1}(L(X(t))

    if I have to, I find my self having to operate complex exponentials and real ones, can I just operate them as normal?

    Any suggestions?

    Thank you
    Last edited by Kruiz; April 5th 2009 at 01:11 AM.
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