Periodic CT Signal
$ x(t) = cos(3\pi t+\pi) \! $ with fundamental frequency of $ \pi $
$ x(t) = \frac{e^{j(3\pi t+\pi)}+e^{-j(3\pi t+\pi)}}{2} $
$ = \frac{e^{j3\pi t}e^{\pi}+e^{-j3\pi t}e^{\pi}}{2} $
$ = \frac{-e^{j3\pi t}-e^{-j3\pi t}}{2} $
$ x(t) = \frac{4\pi}{3} + \frac{1}{j2000}(e^{j1000\pi t}+e^{j-1000\pi t}) - \frac{1}{j1000}(e^{j1000\pi t}-e^{j-1000\pi t}) $
Fourier Series Coefficients
$ a_0 = \frac{4\pi}{3} $
$ a_1 = \frac{1}{1000} $
$ w_0 = 1000\pi\ $
