$(35$ points$)$ Consider the following continuous$-$time $($CT$)$ linear time$-$invariant $($LTI$)$ system with overall impulse response $h(t)$ shown in Figure $2.$ The system is represented with three linear time$-$invariant interconnected systems with corresponding impulse response $h_1(t),h_2(t),$ and $h_3(t),$ respectively.

$($a$)(10$pts$)$ Find frequency response $H()$ for the overall system in terms of $H_1(),H_2(),$ and $H_3()$ of the interconnected LTI systems.

$($b$)(10$pts$)$ Assume the Fourier transform of the input $x()=(-5).$ Find the output $y(t).$

$($c$)(15$pts$)$ Assume $h_1(t)=cos(\frac{}{3}t),h_2(t)=(t-3),$ and $h_3(t)=e^{-2t}u(t).$ Use properties of Fourier transform to find the overall impulse response $h(t).$ To receive full credit do not use convolution and simplify.

HINT: $L\{e^{-a(t-t_0)}u(t-t_0)\}=\frac{e^{-s{t}_0}}{s-a}.$

Answer: