$($a$)$ Consider the input and output pairs listed below. For each case, determine the system transfer function $H(z)$ along with its ROC, and indicate if the system considered is stable and$/$or causal.

i$)x[n]=(\frac{1}{2}{)}^{n}u[n],y[n]=3(\frac{1}{2}{)}^{n}u[n]+2(\frac{3}{4}{)}^{n}u[n]$;

ii$)x[n]=1\mathrm{.}25[n]-0\mathrm{.}25(0\mathrm{.}8{)}^{n}u[n],y[n]=(0\mathrm{.}8{)}^{n}u[n].$

$($b$)$ Consider a causal LTI system described by the following LCCDE:

$y[n]+0\mathrm{.}2y[n-1]-0\mathrm{.}24y[n-2]=x[n]+x[n-1]$

i$)$ Obtain the system transfer function $H(z)$; $(3\text{'})$

ii$)$ Discuss the ROC and stability of the system; $(2\text{'})$

iii$)$ Find the system output to a unit step function $u[n].$

$(4^{\text{'}})$