$(32$ points$)$ Frequency Response

$($a$)(18$ points$)$ Consider the LTI system depicted in figure $1$ whose response to an unknown input, $x(t),$ is

$y(t)=(4e^{-t}-2e^{-2t})u(t)$

Figure $1$: System for Problem $1.$

We know that for the same unknown input $x(t),$ the intermediate signal, $y_1(t),$ is given by:

$y_1(t)=2e^{-t}u(t)$

The overall LTI system is described by the following differential equation:

$\frac{d^2}{d{t}^2}y(t)+5\frac{d}{dt}y(t)+6y(t)=3x(t)$

i$.$ Find the frequency response, $H(j),$ of the overall system $h(t).$

ii$.$ Find the frequency responses $H_1(j)$ of the first LTI system and $H_2(j)$ of the second LTI system.

iii. Find the impulse responses $h(t),h_1(t)$ and $h_2(t).$

$($b$)(6$ points$)$ Assume $x(t)$ a real signal that is baseband, i$.$e$.,$ its Fourier transform $x(j)$ is non$-$zero for $||{}_0.$ We process this signal through an LTI system. Let $y(t)$ denote the corresponding output and let $Y(j)$ denote the Fourier transform of $y(t).$ Does $y(t)$ have frequency components different than those of $x(t)?$ i$.$e$.,$ is $Y(j)0$ for some $||>{}_0?$ What if we process $x(t)$ through a non$-$LTI system?

$($c$)(8$ points$)$ Consider the following two LTI systems with impulse responses:

$h_1(t)=sinc(\frac{t}{2})cos(t)$

$1$

and

$h_2(t)=2sinc(2t)$

Find the output of each system to the following input $x(t)=cos(3t)cos(4t).$ If we are given an input$-$output pair of an unknown LTI system, can we always identify this system?

Hint: Recall the input output relationship. If $h(t)$ is real,

$x(t)=cos({}_{0}t)y(t)=|H(j{}_0)|cos({}_{0}t\frac{+}{{?}_H}(j{}_0))$