$($Total: $25$ points$)[$Na$]17\mathrm{.}19$: An LTI system is specified by the unit$-$sample response h$[$n$]$ and an input

x$[$n$].$

$($a$)(8$ points$)$ b Find the DFT$$s of h$[$n$]=\{1,1,1,0\}$ and x$[$n$]=\{1,2,2,1\}.$ Call them H$[$k$]$ and

X$[$k$],$ respectively.

$($b$)(6$ points$)$ b Let W $[$k$]=$ X$[$k$]$H$[$k$].$ Find w$[$n$],$ the inverse DFT of W $[$k$].$

$($c$)(6$ points$)$ b Find y$[$n$]=$ x$[$n$]$ h$[$n$].$ Why is it not equal to w$[$n$]?$

$($d$)(5$ points$)$ What must be done to x$[$n$]$ and$/$or h$[$n$]$ so that the inverse DFT of the product of their

DFT$$s produces the output of the LTI system? Do this and verify your solution with a computer.