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Fourier Series: How do i simpligy this integral?

  1. Oct 9, 2009 #1
    Fourier Series: How do i simplify this integral?

    1. The problem statement, all variables and given/known data
    http://img18.imageshack.us/img18/4586/matlabfouriercomplex.jpg [Broken]


    2. Relevant equations
    [tex]
    g(t)=\sum_{n=-\infty}^{\infty}c_{n}e^{jn \omega t}[/tex]

    [tex]
    c_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}g(t)e^{-jn \omega T}
    [/tex]


    3. The attempt at a solution

    I need to find the complex fourier series of the above function extended as an odd function
    I Just want to know how to simplify the final solution i have. So far i have

    [tex]
    c_{n}=\frac{1}{T}\int_{t_{0}}^{t_{0}+T}g(t)e^{-jn \omega T}
    [/tex]
    [tex]
    c_{n}=\frac{1}{T}(\int_{-T/10}^{0}-Ae^{-jn \omega T}dt + \int_{0}^{T/10}Ae^{-jn \omega T}dt)[/tex]
    [tex]
    c_{n}=\frac{A}{T}( \frac{1}{jn \omega } - \frac{e^{(jn \omega T/10)}}{jn \omega } - \frac{e^{(-jn \omega T/10})}{jn \omega } + \frac{1}{jn \omega})[/tex]
    [tex] \omega = \frac{2 \pi }{T}[/tex]
    [tex]
    c_{n}=\frac{A}{jn2 \pi }(2 - e^{(jn2 \pi /10)} - e^{(-jn2 \pi /10)})[/tex]

    Now if i put this in matlab i will put

    for n = -N:???????:N, % loop over series index n
    cn = A/(j*n*2*pi)*(2-exp(j*n*(pi/5))-exp(-j*n*(pi/5))); % Fourier Series Coefficient
    yce = yce + cn*exp(j*n*wo*t); % Fourier Series computation
    end

    I dont know what to incriment N by.. All the other fourier examples i have done have been in the form:
    [tex]c_{n}=\frac{1}{jn \pi }(1 - e^{(jn \pi )})[/tex]

    Here i understand [tex](1 - e^{(jn \pi )})[/tex] will be 0 when n is even 2 when n is odd
    Hence the series becomes:

    [tex]
    \sum_{n=odd}^{\infty}\frac{2}{ \pi jn}e^{jn \omega t}[/tex]


    and the loop will go:
    for n = -N:2:N, % loop over series index n (odd)
    cn = 1/(j*n*pi)*(1-exp(j*n*pi)); % Fourier Series Coefficient
    yce = yce + cn*exp(j*n*wo*t); % Fourier Series computation
    end


    But what is it for my example??? It is not so straight forward. Can it be simplified like above? How do i progress from here?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
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