# 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 Well-Known Member

Apr 19, 2012
776
48
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.

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