$7\mathrm{.}21$ Consider a causal LTI system with

impulse response $\backslash$root$(([$n$]))(),$ and input

x$[$n$]=$x$\_(1)[$n$]-$x$\_(1)[$n$-2]+$x$\_(1)[$n$-4]$

where x$\_(1)[$n$]=$u$[$n$]-$u$[$n$-2].$ The impulse

response of the system is

$()/($bar$)($h$)[$n$]=$u$[$n$]-$u$[$n$-2].$

$($a$)$ Use the convolution sum to find

the output of the system, $.$

$($b$)$ If the given system is cascaded

with another causal LTI system

with impulse response $,$ we

know the overall impulse response

of the two cascaded systems is

h$\_($T$)[$n$]=\{(1,$n$=0,3),(0$ otherwise $)$:$\}$

Find the impulse response g$[$n$].$

Answers: If y$\_(1)[$n$]=($x$\_(1)*$h$)[$n$]$ then

y$[$n$]=$y$\_(1)[$n$]-$y$\_(1)[$n$-2]+$y$\_(1)[$n$-4].$