Example of Computation of Fourier series of a CT SIGNAL
A practice problem on "Signals and Systems"
Lets define the signal
$ \ x(t) = (1+2j)cos(t)+5sin(4t) $
Knowing that its Fourier series is
$ \ x(t) = (1+2j)(\frac{e^{jt} + e^{-jt}}{2}) + 5(\frac{e^{j4t} - e^{-j4t}}{2j}) $
We simplify
$ \ x(t) = \frac{1+2j}{2} e^{jt} + \frac{1+2j}{2} e^{-jt} + \frac{5}{2j}e^{j4t} - \frac{5}{2j}e^{-j4t} $
So, we get the coefficients:
$ \ a_{0} = 0 $
$ \ a_{1} = \frac{1+2j}{2} $
$ \ a_{-1} = \frac{1+2j}{2} $
$ \ a_{2} = \frac{5}{2j} $
$ \ a_{-2} = - \frac{5}{2j} $
