Exam 1: Problem 4
4. Compute the coefficients $ a\ _k $ of the Fourier series of the signal $ x\ (t) $ periodic with period $ T\ = 4 $ defined by:
$ x(t) = \begin{cases} 0, & \mbox{if } -2 < t < -1\mbox{ } \\ 1, & \mbox{if } -1 \le t \le 1\mbox{ } \\ 0, & \mbox{if } 1 < t \le 2\mbox{ } \end{cases} $
Answer
$ a\ _0 = $ average of the signal over period $ = \frac{2}{4} = \frac{1}{2} $
$ a\ _k = \frac{1}{T} \int_{-\infty}^{\infty} x\ (t) e^{-jk(\frac{2\pi}{T})t} \, dt = \frac{1}{4} \int_{-1}^{1} e^{-j\frac{\pi}{2}t} \, dt = \frac{1}{4} \left [ \frac{e^{-jk\frac{\pi}{2}}t}{-jk\frac{\pi}{2}} \right ]_{-1}^{1} = \frac{1}{2jk\pi} \left [ e^{-jk\frac{\pi}{2}} - e^{jk\frac{\pi}{2}} \right ] = \frac{sin(k\frac{\pi}{2})}{k\pi} $
