Example of Computation of Fourier series of a CT SIGNAL

A practice problem on "Signals and Systems"


Periodic CT Signal

The first signal that comes to mind when i think of a periodic CT signal is one involving sines and cosines, so let's work with one of those.

Let $ x(t) = sin(14t)+(1+3j)cos(2t) $

This can also be expressed as

$ x(t) = \frac{e^{14jt}}{2j}-\frac{e^{-14jt}}{2j} + \frac{(1+3j)*e^{2jt}}{2} + \frac{(1+3j)e^{-2jt}}{2} $

I contend that the $ \omega_0=2 $ since both functions are periodic based on it.

$ a_7=\frac{1}{2j} $

$ a_{-7} = \frac{-1}{2j} $

$ a_1 = \frac{1+3j}{2} $

$ a_{-1} = \frac{1+3j}{2} $

We can write this as a sum:

$ x(t)=\sum^{\infty}_{k = -\infty} a_k e^{j2k}\, $

Where

$ a_1=a_{-1}=\frac{1+3j}{2} $

$ a_7 = -a_{-7} = \frac{1}{2j} $

$ a_k = 0, k \neq 1, -1, 7, -7 $


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