1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Fourier series change of variable

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

    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\}

    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:

    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)

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

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

    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?

    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)}

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?

Similar Discussions: Fourier series change of variable