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)=
$ 0, -2< t< -1 $ $ 1, -1\le t\ge 1 $ $ 0, 1< t< 2 $
Answer
Since it is CT, we need to find all $ a_k $'s.
$ a_k=\frac{1}{T}\int_{0}^{T}x(t)e^{-jk\omega t}dt $ $ =\frac{1}{4}\int_{-2}^{2}x(t)e^{-jk\omega t}dt $
$ \omega=\frac{2\pi}{T}=\frac{\pi}{2} $ so,
$ a_k=\frac{1}{4}\int_{-1}^{1}e^{-jk\frac{\pi}{2}t}dt $ $ =\frac{1}{4}\frac{e^{-jk\frac{\pi}{2}t}}{-jk\frac{\pi}{2}}|_{-1}^{1} $ $ =\frac{-1}{2\pi kj}[e^{-j\frac{\pi}{2}k}-e^{j\frac{\pi}{2}k}] $
$ a_k=\frac{1}{\pi k}[sin{\frac{k\pi}{2}}] $
