Given the filter

$H(z)=\frac{1-0\mathrm{.}9z^{-1}-0\mathrm{.}1z^{-2}}{1+0\mathrm{.}3z^{-1}-0\mathrm{.}04z^{-2}}$

A$.$ Realize the $H(z)$ and develop the difference equations using cascade form the first order sections

b$.$ Determine the poles and zeros and check the stability of the filter

c$.$ Determine the output $y(n)$ if the input $x(n)=(0\mathrm{.}25{)}^{n}u(n)$

d$.$ Determine the input if the output is unit impulse response

C$.$ Assume the filter length is $5.$ Determine the optimal Coefficients of the filter