Tutorial problems

$1.$ A discrete$-$time system is described by the following signal flow graph:

$($a$)$ Determine the difference equation relating output $()/(($ y$[$n$]()/()))$ to the input $()/(($ x$[$n$]()/()).)$

$($b$)$ Determine the impulse response of the system.

A discrete$-$time system is given by

H$($z$)=(1-2\mathrm{.}55$z$^(-1)+4\mathrm{.}4$z$^(-2)-5\mathrm{.}09$z$^(-3)+2\mathrm{.}41$z$^(-4))/(1+0\mathrm{.}26$z$^(-1)-0\mathrm{.}38$z$^(-2)-0\mathrm{.}45$z$^(-3)+0\mathrm{.}23$z$^(-4))$

Determine and draw each of the following structures:

$($a$)$ Cascade form with second$-$order sections in normal direct form I,

$($b$)$ Cascade form with second$-$order sections in transposed direct form I,

$($c$)$ Cascade form with second$-$order sections in normal direct form II$,$

$($d$)$ Cascade form with second$-$order sections in transposed direct form II$.$