Please solve problem $7$ and $9$ all parts

Problem $7$

Consider the following four systems where $\backslash$alpha $!=0.$

a$-[1$pt$]$ Which of these systems is time invariant? Which of these systems is linear? $($no

justifications are required here$)$

b $-[0\mathrm{.}5$ pt$]$ Which of these systems has a global frequency response equal to H$(\backslash$omega $)+\backslash$alpha $?$ Are

there several of them?

c $-[1\mathrm{.}5$ pts$]$ Assume that H$(\backslash$omega $)$ is as follows.

Can one of these systems cancel cos$((\backslash$pi $)/(3)$n$)?$

Which one and for what value of $\backslash$alpha $?$

Problem $9$

For a given sequence f$[$n$],$ let g$[$n$]=(-1)^($n$)$f$[$n$]$ and h$[$n$]=\backslash$delta $[$n$]-$

a $-[1$ pt$]$ Derive G$(\backslash$omega $)$ and H$(\backslash$omega $)$ in terms of F$(\backslash$omega $).$

b $-1$pt Assume from now on that F$(\backslash$omega $)$ is as follows.

Plot G$(\backslash$omega $)$ and H$(\backslash$omega $)$ for $\backslash$omega in$[-\backslash$pi $,\backslash$pi $].$ Can you see a relation between G$(\backslash$omega $)$ and H$(\backslash$omega $)?$

c $-1$pt Derive f$[$n$]$ in terms of n by separating the cases n$=0$ and n$!=0.$ Show the details

of your derivations.

d $-[1\mathrm{.}5$ pts$]$ Transform the relation you found between G$(\backslash$omega $)$ and H$(\backslash$omega $)$ in question b into a

relation between g$[$n$]$ and h$[$n$].$ Prove this relation using the expression of f$[$n$]$ you found in

question c$.$