Define a Periodic CT Signal and Compute its Fourier Series Coefficients
Let's start this process by defining our signal. For simplicities sake lets use the the signal
$ x(t) = 4sin(3t) + 8cos(7t) $
The Fourier Series can be easily found by treating
$ Asin(\omega_0t) = \frac{A*(e^{j\omega_0t} - e^{-j\omega_0t})}{2j} $
and
$ Acos(\omega_0t) = \frac{A*(e^{j\omega_0t} + e^{-j\omega_0t})}{2} $
This alows us to to put x(t) in the form of
$ x(t) = \sum_{k=- \infty }^ \infty a_ke^{jk\omega_0t} $
which gives us
$ x(t) = \frac{4*(e^{j3t} - e^{-j3t})}{2j} + \frac{8*(e^{j7t} + e^{-j7t})}{2} $
Simplifying and distributing
$ x(t) = \frac{2*e^{j3t} }{j}- \frac{2*e^{j3t} }{j} + 4e^{j7t} + 4e^{-j7t} $
$ \ a_{-3} = \frac{2}{j} $
$ \ a_{3}= \frac{-2}{j} $
$ \ a_{-7} = 4 $
$ \ a{_7} = 4 $
all other $ \ a_k = 0 $
