Fourier series representation help

Discussion in 'Homework Help' started by fdsa, Jan 14, 2014.

  1. fdsa

    Thread Starter New Member

    Aug 16, 2011
    9
    0
    I'm trying to find the fourier series representation of the following periodic function: x(t)=cos(t)+sin(2t)+cos(3t-\frac{\pi}{3})

    So I've found:
    c_1 = c_{-1} = 1/2
    c_2 = -\frac{j}{2}
    c_{-2} = \frac{j}{2}

    c_3 = \frac{1}{2}\times e^{-j\pi / 3}
    c_{-3} =\frac{1}{2}\times e^{j\pi / 3}

    According to the solutions the answer is supposed to be:
    c_3 = c_{-3}= 1/2

    Did I do something wrong?
     
  2. anhnha

    Active Member

    Apr 19, 2012
    773
    47
    Here the formula of Fourier series:

    x(t) =  \sum_{- \propto }^ \propto   c_{n}  e^{jn  \omega _{0} t}

    And by using Euler formula:

    cos( \varphi ) =  \frac{e^{j \varphi } + e^{-j \varphi }}{2}

    sin( \varphi ) =  \frac{e^{j \varphi } - e^{-j \varphi }}{2j}

    Now we can rewrite x(t) as follows:

    x(t) = \frac{e^{jt} + e^{-jt}}{2} + \frac{e^{j2t} - e^{-j2t}}{2j} +  \frac{e^{j(3t -  \frac{ \Pi }{3}) } + e^{-j(3t -  \frac{ \Pi }{3} )}}{2}

    And according to that result, you are correct.
     
    fdsa likes this.
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