Problem $2$: Consider a CT$-$LTI system described by:

$($dy$($t$))/($dt$)+2$y$($t$)=3($dx$($t$))/($dt$)+$x$($t$)$

where the system input is x$($t$)$ and the output is y$($t$).$ We also consider two related sub$-$systems:

system $2$ and $3.$ In particular, system $2$ is described by

$($dy$($t$))/($dt$)+2$y$($t$)=$z$($t$)$

where z$($t$)$ is system input and y$($t$)$ is its output. System $3$ is described by:

z$($t$)=3($dx$($t$))/($dt$)+$x$($t$)$

where x$($t$)$ is input, and z$($t$)$ is output.

$($a$)(10$ pts$)$ Based on the differential equation itself, prove that system $1$ is equivalent to the

series cascade configuration shown in Fig. $($c$).$

$($b$)(10$ pts$)$ Calculate the frequency response function H$\_($a$)($j$\backslash$omega $)$ of system $1,$ i$.$e$.,$ Fig. $($a$).$

$($c$)(10$ pts$)$ Calculate the frequency response H$\_($b$)($j$\backslash$omega $)$ of the system configuration in Fig. $($b$),$

as well as the frequency response function H$\_($c$)($j$\backslash$omega $)$ of the system configuration in Fig. $($c$).$

Demonstrate H$\_($a$)($j$\backslash$omega $)=$H$\_($b$)($j$\backslash$omega $)=$H$\_($c$)($j$\backslash$omega $).$

$($d$)(10$pts$)$ If the input signal for the system in Fig. $($a$)$ is x$($t$)=\backslash$delta $($t$),$ calculate its Fourier

transform x$($j$\backslash$omega $)$ and the Fourier transform of the output Y$($j$\backslash$omega $).$

$($e$)(10$ pts$)$ Based on the inverse Fourier transform of x$($t$)=\backslash$delta $($t$).$ Calculate the Fourier

transform of w$\_(1)($t$)$ and y$($t$).$

$($g$)(10$pts$)$ The input signal for the system in Fig. $($c$)$ is x$($t$)=\backslash$delta $($t$).$Calculate the Fourier

transform of w$\_(2)($t$)$ and y$($t$).$