Consider an LTI system described by the following difference equation $($assume $x[n]$ is causal and $y[n]$ has zero initial condition $y[-1]=y[-2]=0)$

$y[n]-0\mathrm{.}75y[n-1]+0\mathrm{.}125y[n-2]=x[n]$

$($a$)$ Find the transfer function for the above difference equation using z $-$transform.

$($b$)$ Draw a block diagram in time domain for the given difference equation using Direct Form II$.$ Hint: Recall lecture $08$

$($c$)$ Find the transfer function using the block diagram in z$-$domain. Hint: delay block in time domain means multiplying $z^{-1}$ in $z-$domain. Name the signal before the first delay block $V(z).$ Then write equations for the adders.

$($d$)$ Find the unit step response for the system using z $-$transform.