4. Compute the coefficients $ a_k $ of the Fourier series of the signal $ x(t) $ periodic with period $ T=4 $ defined by
$ \,x(t)=\left\{\begin{array}{cc} 0, & -2<t<-1 \\ 1, & -1\leq t\leq 1 \\ 0, & 1<t\leq 2 \end{array} \right. \, $
(Simplify your answer as much as possible.)
Answer
$ \,a_k=\frac{1}{4}\int_{-2}^{2}x(t)e^{-jk\frac{\pi}{2}t}\,dt\, $
$ \,a_k=\frac{1}{4}\int_{-1}^{1}e^{-jk\frac{\pi}{2}t}\,dt\, $
$ \,a_k=\frac{1}{4}\left. \frac{1}{-jk\frac{\pi}{2}}e^{-jk\frac{\pi}{2}t}\right|_{-1}^{1}\, $
$ \,a_k=\frac{-1}{jk2\pi}(e^{-jk\frac{\pi}{2}} - e^{jk\frac{\pi}{2}})\, $
