(d) Fourier Series. (9 pts)
Consider an LTI system with frequency response
\begin{equation*}
H(j\omega) = \begin{cases}
-2, & -2 \le \omega \le 3 \\
0, & \text{otherwise}
\end{cases}
\end{equation*}
i. (3 pts) Suppose the input is a periodic signal $x(t)$ with the Fourier Series representation
$\omega_0 = \frac{1}{2}$, $a_0 = 1$, $a_2 = a_{-2} = e^{-j\frac{\pi}{2}}$, $a_4 = 2$, $a_{-4} = 1$, $a_7 = 5$, and $a_k = 0$ for $k$ otherwise.
Find the output Fourier coefficients $b_k$, and specify your answer for all $k$. Show that $b_0 = -2$,
$b_2 = -2e^{-j\frac{\pi}{2}}$, $b_{-2} = -2e^{j\frac{\pi}{2}}$, $b_4 = -4$, $b_{-4} = 2$, $b_7 = 0$, and $b_k = 0$ otherwise.
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ii. (3 pts) Using your results from (A), find the output $y(t)$. Simplify $y(t)$ as much as possible.
Show that
$y(t) = -2 - 4cos(t - \frac{\pi}{2}) - 4jsin(2t) - 2e^{j2t}$.
iii. (3 pts) If $x(t)$ is delayed by 2, then what are the output coefficients $b_k$ now? Show that
$b_0 = -2$, $b_2 = -2e^{-j(\frac{\pi}{2} + 2)}$, $b_{-2} = -2e^{j(\frac{\pi}{2} + 2)}$, $b_4 = -4e^{-j4}$, $b_{-4} = 2e^{j4}$, $b_k = 0$ otherwise.
