Example of Computation of Fourier series of a CT SIGNAL

A practice problem on "Signals and Systems"


Fourier Transform

Let $ x(t)=sin(\pi t) + cos(2\pi t) $

Remember that the formula for CT Fourier Series are:

$ 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 $.

Solution

$ x(t)= \frac{e^{\pi jt}+e^{-\pi jt}}{2j} + \frac{e^{2\pi jt}+e^{-2\pi jt}}{2} $

$ \omega_0 = \pi $

$ a_1=\frac{1}{2j} $

$ a_2=\frac{1}{2} $

else $ a_k $ equals 0


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