Q$\mathrm{.}5($a$)$ Consider a discrete$-$time system with input, x$[$n$],$ and output, y$[$n$].$ The input$-$output relationship for this system is

y$[$n$]=$ x$2[$n$]2($ii$)$ Is the system time$-$invariant?

$($i$)$ Is the system linear? Justify your answers.

$[8$ marks$]$

$($b$)$ Let X$1($z$)$ denote the z$-$transform of the sequence x$1[$n$]=2[$n $+1]+3[$n $1]$

where $[$n$]$ is the unit$-$sample sequence.

Determine X$1($z$)$ and specify the region of convergence for X$1($z$).$

$($c$)$ Let x$2[$n$]=(4)$nu$[$n $2]$ where u$[$n$]$ is the unit$-$step sequence. Show that X$2($z$),$ the z$-$transform of x$2[$n$],$ is given by

$16$z$2$

X$2($z$)=14$z$1,|$z$|>4.$

$[4$ marks$]$

$[8$ marks$]$