Example of Computation of Fourier series of a CT SIGNAL
A practice problem on "Signals and Systems"
The Signal
mmm lets randomly take...
$ \sin4\pi t + \cos3\pi t + e^{j2\pi t} $
The Coefficients
Remeber... $ x(t) = \sum^{\infty}_{k = -\infty} a_k e^{jk\pi t}\, $
$ a_k=\frac{1}{T} \int_0^Tx(t)e^{-jk\omega_ot}dt $
Going to conver the equation into signal that is all in exponentials.
$ \frac{1}{2j}(e^{j4\pi t}-e^{-j4\pi t}) + \frac{1}{2}(e^{j3\pi t} + e^{-j3\pi t}) + e^{j2\pi t} $
The terms come out to be
$ 4, -4, 3, -3, and 2 $
$ a_4 = \frac{1}{2j} $ $ a_-4 = \frac{1}{2j} $ $ a_3 = \frac{1}{2} $ $ a_-3 = \frac{1}{2} $ $ a_2 = 1 $
$ a_k = 0, k \ne 4, -4, 3, -3, 2 $
