Problem
Let the signal $ x(t) = 5cos(3\pi t) + sin(\pi t)\, $
Find the Fourier Series coefficients
Analysis
First rewrite the signal as a sum of complex exponentials:
- $ x(t) = 5(\frac{e^{3\pi jt} + e^{-3\pi jt}}{2}) + \frac{e^{\pi jt} - e^{-\pi jt}}{2j} $
Simplifying gives:
- $ x(t) = \frac{5}{2} e^{3\pi jt} + \frac{5}{2} e^{-3\pi jt} + \frac{1}{2j} e^{\pi jt} - \frac{1}{2j} e^{-\pi jt} $
The fundamental period is:
- $ \omega_o = \frac{2\pi}{T} = \pi $, with $ T = 2\, $ being the period of the original signal.
From the fundamental period, it is easily seen that the fourier series coefficients are:
- $ a_{-3} = \frac{5}{2} $
- $ a_{-1} = -\frac{1}{2j} $
- $ a_{1} = \frac{1}{2j} $
- $ a_{3} = \frac{5}{2} $
- $ a_{k} = 0\, $, for all other k
