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Homework Help: Fourier series change of variable

  1. Jul 4, 2006 #1
    Hi, I'm working on the (odd) square wave function

    [tex]
    f\left( t \right) = \left\{ {\begin{array}{*{20}c}
    { - 1, - \frac{T}{2} \le t < 0} \\
    { + 1,0 \le t < \frac{T}{2}} \\
    \end{array}} \right\}
    [/tex]

    The question says to move the origin of t to the centre of an interval in which f(t) = +1 (ie. consider the even square wave function). A part of the question says to express the square wave function as a cosine series by making the change of variable [itex]t' = t - \frac{T}{4}[/itex].

    Calculate the Fourier coefficients involved by making the suggested change of variables in the result 12.8

    Result 12.8 is:

    [tex]
    f\left( t \right) = \frac{4}{\pi }\left( {\sin \omega t + \frac{{\sin 3\omega t}}{3} + \frac{{\sin 5\omega t}}{5} + ...} \right)...(12.8)
    [/tex]

    This isn't apart of result 12.8 but the sine Fourier coefficients are:

    [tex]
    b_r = \frac{2}{{\pi r}}\left[ {1 - \left( { - 1} \right)^r } \right]
    [/tex]

    I don't know how to calculate the coefficients for the cosine series using (12.8) and the change of variables t' = t - (T/4). Using the sin(A+B) expansion for each sine term in result 12.8 is tedious so should I write f(t) as follows and then use the sine addition formula on the general sine term?

    [tex]
    f\left( t \right) = \sum\limits_{r = 1}^\infty {\frac{2}{{\pi r}}\left[ {1 - \left( { - 1} \right)^r } \right]\sin \left( {\frac{{2\pi rt}}{T}} \right)}
    [/tex]

    I probably haven't analysed the information carefully enough but I'm not sure how to start this. I've only just started on Fourier series. Any help would be good thanks.
     
    Last edited: Jul 5, 2006
  2. jcsd
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