Consider the following sinusoidal carrier wave, X$($t$),$ with a carrier frequency of fc and a carrier phase $($t$)$: X$($t$)=$ cos$(2$fct $+($t$))$ The phase of this carrier is altered or modulated to encode a stream of binary data to be transmitted. Assume that the data stream is an I.I.D sequence of RVs that are distributed Bernoulli$($p$).$ Consider the n th bit $($bn in $\{0,1\})$ in the stream of binary data is assigned to $/2$ with probability p and $$$/2$ with probability $(1$ p$)$: $[$n$]=$ $2$ Inth bit is $1$ $2$ Inth bit is not $1$ The carrier wave applies this modulated phase associated with each bit of the data stream for T seconds, which is some multiple of the period of the carrier frequency fc$.($t$)=$ $[$n$]$ It in $[$nT $,($n$+1)$T$)-2-101200\mathrm{.}0020\mathrm{.}0040\mathrm{.}0060\mathrm{.}0080\mathrm{.}010\mathrm{.}0120\mathrm{.}0140\mathrm{.}01600\mathrm{.}0020\mathrm{.}0040\mathrm{.}0060\mathrm{.}0080\mathrm{.}010\mathrm{.}0120\mathrm{.}0140\mathrm{.}016-1-0\mathrm{.}500\mathrm{.}51$ Clearly X$($t$)$ is a random process and we want to answer the following questions: $($a$)(8$ points$)$ What is the mean function of the sinusoidal carrier wave? For notational simplicity one can define c $=2$fc SOLUTION $($b$)(16$ points$)$ What is the auto$-$covariance of the sinusoidal carrier wave?