$5\mathrm{.}50.($a$)$ Suppose we want to design a discrete$-$time LTI system which has the property that if the input is x$[$n$]$ u$[$n$]$ u$[$n $-1],$ then the output is y$$n$]=()"$ un$]($i$)$ Find the impulse response and frequency response of a discrete$-$time LTI system that has the foregoing property. $($ii$)$ Find a difference equation relating x$[$n$]$ and y$[$n$)$ that characterizes the system. $($b$)$ Suppose that a system has the response $(1/4)"$u$[$n$]$ to the input $($n$+2)(1/2)"$u$[$n$].$ If the output of this system is o$[$n$]-(-1/2)"$u$[$n$],$ what is the input?