Example of Computation of Fourier series of a CT SIGNAL
A practice problem on "Signals and Systems"
CT signal
The CT signal I will use is:
$ x(t) = 4cos(2t) + (3j)sin(3t)\! $
The fundamental period is 2*pi
we know that Wo=T/(2*pi), so:
Wo=2*pi
Solution
Equation to find signal coeffiecients is: $ a_k=\frac{1}{T}\int_0^Tx(t)e^{-jk\omega_0t}dt $.
The Equation to find a fourier series is:
$ x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0t} $
By substituting signal into the first equation we get:
$ a_0=\frac{1}{2\pi}\int_0^{2\pi}[4cos(2t) + (3j)sin(3t)]e^{0}dt $
Solving this we get:
$ a_0=\frac{2}{\pi}\int_0^{2\pi}cos(2t)dt+\frac{3j}{2\pi}\int_0^{2\pi}sin(3t)dt $
$ a_0=\frac{1}{\pi}[sin(2t)]_0^{2\pi}-\frac{j}{2\pi}[cos(3t)]_0^{2\pi} $
$ a_0=\frac{1}{\pi}[sin(4\pi)-sin(0)]+\frac{j}{2\pi}[(cos(6\pi)-cos(0)] $
$ a_0=\frac{1}{\pi}[0]+\frac{j}{2\pi}[0] $
$ a_0=0\! $
