Q$8)$ Take home $[20$ pts$]$: Consider the system shown bellow that can be considered as the ideal analog to digital converter $($ ADC $)$ and ideal digital to Analog $($DAC$)$ interpolator, with the input signal $x(t)=\frac{sin(1000t)}{t}$ and the sampler signal given by $s(t)={\displaystyle \underset{n=-}{\overset{}{}}}10(t-n{T}_s),$ where $T_s=\frac{}{2000}.$ When we multiply the input signal with the sampler, we end up with a sampled data signal $x_s(t)=x(t)s(t).$ The sampled signal is then used as an input to a LTI system described by the impulse response $h(t)=\frac{sin(6000t)}{t}.$ For this problem, Do the following

Determine the CTFT for $x(t).$

Determine the CTFS and CTFT for $s(t).$

Determine the CTFT for $x_s(t).$

Find the output signal $y(t).$

If we allow the sampling rate $T_s$ to change what is the minimum sampling rate required to recover the original signal $x(t)$ along with another appropriate LTI invariant system of the form $h(t)=\frac{sin(W_{B}t)}{t}$ and find the value of $W_B$ and $.$