Example of Computation of Fourier series of a CT SIGNAL
A practice problem on "Signals and Systems"
Useful Info
$ x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0t} $
$ a_k=\frac{1}{T}\int_0^Tx(t)e^{-jk\omega_0t}dt $.
- Let
- $ x(t) = 2sin(2\pi t) + cos(\pi t). $
Solution
- $ x(t) = 2\frac{e^{2 \pi jt}+e^{-2 \pi jt}}{2j} + \frac{e^{\pi jt}+e^{-\pi jt}}{2} $
- $ x(t) = \frac{1}{j}(e^{2 \pi jt} + e^{-2 \pi jt}) + \frac{1}{2}(e^{\pi jt}+e^{-\pi jt}) $
- $ a_1 = \frac{1}{j} $
- $ a_2 = \frac{1}{2} $
- $ \omega_0 = \frac{2\pi}{T} = \frac{2\pi}{2\pi} = 1 $
- else
- $ a_k = 0 $
