Problem #$4(25$ Points$)$: Show all work to receive full credit.

Consider the ideal sampling$/$reconstruction scheme shown below.

This system achieves filtering of a CT signal by first sampling, then performing a filtering operation in the DT domain, and finally by converting the filtered DT signal back to CT$.$ Assume that the sampling is ideal and that the reconstruction filter is also ideal, with cutoff frequency $\frac{F_s}{2}$ and gain T $.$

Suppose that the DT LTI system is represented by a causal transfer function

$H(z)=\frac{Y(z)}{x(z)}=\frac{z^3+z}{(z+0\mathrm{.}5)(z^2+z+0\mathrm{.}5)}=\frac{z(z-e^{j\frac{}{2}})(z-e^{-j\frac{}{2}})}{(z+0\mathrm{.}5)(z-0\mathrm{.}707e^{j3\frac{}{4}})(z-0\mathrm{.}707e^{-j3\frac{}{4}})}$

a$)$ Sketch the poles and zeros of $H(z)$ in the complex plane. $(5$ Points$)$

b$)$ Sketch the magnitude response of this system approximately in the frequency range of $0,$ using the "graphical" approach. Make sure to mark all key frequencies. $(10$ Points$)$

c$)$ Find $CT$ steady$-$state output $y_{ss}(t)$ if a CT input $,x(t)=2cos(2500t-\frac{}{4}),$ is applied to this system and the sampling frequency of the system is $F_s=1000Hz.(10$ Points$)$