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} $


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Ryne Rayburn