$[40$p$]$ Consider an LTI system with the following frequency response

$H(e^j)=\frac{100-110e^{-j}}{100-99e^{-j}}$

a$)$ Specify the difference equation that is satisfied by the input and the output of the system.

b$)$ Determine the impulse response $h[n]$ of the system.

c$)$ Determine the z transform $H(z)$ of the impulse response in part $($b$).$ Draw the pole$-$zero plot with ROC.

d$)$ Inspect the system whether it is i$)$ stable ii$)$ causal.