$(3)$ Consider the following continuous$-$time signal $x_a(t)=x_{a1}(t)+x_{a2}(t)+x_{a3}(t),$ where,

$x_{a1}(t)=10sin(2f_{1}t)+cos(2f_{2}t),$

$x_{a2}(t)=8cos(2f_{3}t)+3cos(2f_{4}t),$

$x_{a3}(t)=8cos(2f_{5}t)+cos(2f_{6}t),$

and

$f_1=5kHz,f_2=35kHz,f_3=15kHz,f_4=25kHz,f_5=2kHz,$ and $f_6=30kHz$

It is required to design a digital signal processing$-$based system to separated the two signals $x_{a1}(t)$ and $x_{a2}(t),$ and remove the third signal. Thus, the system will have one input and $2$ outputs corresponding to $x_{a1}(t)$ and $x_{a2}(t).$ The requirements of the system are as follows:

The anti$-$aliasing analog filter has an order of $6,$ with ${}_p=0\mathrm{.}2dB,$ and ${}_{s}60dB.$

Concerning the IIR filters used, the order is $n=4$ and ${}_s=45dB.$

$($a$)$ Draw the line spectrum of the signals using different colors to distinguish them from each other.

$($b$)$ Sketch the overall block diagram of the system.

$($c$)$ Specify the requirements of the anti$-$aliasing analog LPF needed.

$($d$)$ From the above requirements, determine the required sampling frequency.

$($e$)$ For each digital filter, specify the requirements, hence design and write the difference equation.

$($f$)$ Plot the attenuation $($in dB $)$ response of each filter to verify your design.

$($g$)$ Write expressions for the filters steady$-$state outputs. Hence, sketch the spectra of the input and output spectra.