P$3.$ Consider the SS DT LTI $($state$-$space discrete$-$time linear time$-$invariant$)$ system described by:

$x(k+1)=Ax(k)+Bu(k)$ State Eqn

$y(k)=Cx(k)+Du(k)$ Observation Eqn

Where, again, the state $($or system$)$ matrix $A,$ the input matrix $B,$ the output matrix $C,$ and feedforward matrix $D$ are all constant, and $x,u,y$ are the state, input and output variables respectively.

a$.$ Derive the Pulse TF $G(z)$ of this SS DT model.

$5$ pts

Instructions: derive the DT TF $G(z)$ in a similar manner to the derivation of $G(s)$ in CT$.$ Assume zero IC's $($initial conditions$).$ Hence, take the $z-$transform of both sides of the State Eqn first, express $x(z)$ in terms of $U(z),$ and then plug into the $z-$transformed Observation Eqn. The answer should parallel the CT TF$.$