$($a$)$ Consider a non$-$periodic signal x$($t$)$ and prove that $()=1\{()\}$ where $()=$ lim $$ $1|()|2($b$)$ Consider an ergodic stochastic process $()$ and prove that $\{()(+)\}=1\{()\}$ and $\{2()\}=()($c$)$ The white noise is an ergodic random process with zero mean and power spectral density $02.$ Use the results in $($b$)$ and $-$ find the average power of a noise sample. $-$ find the correlation between two AWGN noise samples that are separated by a delay $.($d$)$ Consider the additive noise defined by $()=()()$ where $()$ is an AWGN with specral density equal to $02$ and $()=$ sin$(390)5202|$ P a g e $-$ Find the average power of a noise sample $-$ Find the correlation between two AWGN noise samples separated by a delay $$ Problem $4\mathrm{.}2(25$ points$)$ Consider an analog modulated signal. The transmitted signal has an average power of $$ and is subject to AWGN complex noise $()$ with power spectral density $()=02.($a$)$ The noise can be written as: $()=()$ cos$()+()$ sin$()$ Derive an expression of the power spectral density of $()$ and $()$ as a function of $0($Proof of the results is necessary$)($b$)$ Derive the expression of the SNR for the case of DSB$-$SC$,$ SSB and AM received signal. $($Proof of the results is necessary$)$