$4)$ A binary communication system transmits signals $\backslash ($ s$\_\{1\}=2\backslash )$ and $\backslash ($ s$\_\{2\}=4\backslash )$ to denote $\text{'}0\text{'}$ and $\text{'}1\text{'}$ respectively as shown in Figure Q$4.$ The middle of the pulses are sampled at the receiver and are corrupted by noise, which can be expressed as $\backslash ($ r$=$s$\_\{$i$\}+$n$($i$=1\backslash )$ or $2)$ where the noise $\backslash ($ n $\backslash )$ are uniformly distributed from $-2$ to $2.$

$($a$)$ What is the average transmitted signal power?

$[4$ marks$]$

$($b$)$ Draw the probability distributions of $\backslash ($ r $\backslash )$ given $\backslash ($ s$\_\{1\}\backslash )$ and $\backslash ($ r $\backslash )$ given $\backslash ($ s$\_\{2\}\backslash ).$ What should be the bit decision threshold?

$[4$ marks$]$

$($c$)$ What is the corresponding probability of detection error?

$[4$ marks$]$

$($d$)$ Instead of using the values $2$ and $4,$ is there a way to re$-$design the signals $\backslash ($ s$\_\{1\}\backslash )$ and $\backslash ($ s$\_\{2\}\backslash )$ so that the average signal power is lower while keeping the same detection error probability?

$[4$ marks$]$

$($e$)$ Design a PAM$-4$ system using NRZ pulse shape with the same average power and bit rate as the system in $($d$).$ Specify the different amplitude levels and pulse duration of such PAM$-4$ signal.

$[4$ marks$]$