P$-11\mathrm{.}14$ The input to an LTI system such as Fig. P$-11\mathrm{.}12$ is defined by the equation

$x(t)={\displaystyle \underset{k=-}{\overset{}{}}}{a}_k{e}^{jk200t}$ where $a_k=\{\begin{array}{cc}\frac{1}{k^2},& k0\\ 1,& k=0\end{array}$

$($a$)$ Determine the Fourier transform of $x(t)$ and plot it on a graph that spans the frequency range $-1000in1000.$

$($b$)$ Suppose that the frequency response of the LTI system is given by the following equation:

$H(j)=\{\begin{array}{cc}10,& ||<mn>3</mn><mn>0</mn><mn>0</mn><mi></mi>\\ 0,& ||>300\end{array}$

Plot this function on the graph that you constructed in part$($a$).$

$($c$)$ Determine the $($DC$)$ value of the output signal; i$.$e$.,$ determine $b_0=\frac{1}{T_0}{\displaystyle \underset{0}{\overset{T_0}{}}}y(t)dt,$ where $T_0$ is the period of the output signal.

$($d$)$ Write an equation for $y(t).$ If possible, simplify it to include only cosine functions.