A continuous$-$time signal is sampled and then transmitted as a digital signal $g(t)$ with bit rate $\frac{1}{T}$ bit$/$s$.$ The receiver structure is shown in the following figure. Assuming the use of NRZ pulse shape with amplitude A denoting the bit $"1"$ and amplitude $-$A denoting the bit $"0".$ The random noise $n(kT)$ at the input of the decision device in the receiver has a noise

$($a$)$ Draw the waveform of $g(t)$ if the bit sequence set is $\text{'}01101011\text{'}$

$($b$)$What should the value of $g(kT)$ be for bit $"1"$ and $"0"?$

$($c$)$ What should the decision threshold be set at$?$

$($d$)$ What is the average power of $g(t)?$

$($e$)$ If there are additional distortion generated from certain parts of the communication system such that the BER in part $($d$)$ is increased to ${10}^{-3},$ what is the variance of the distortion ${?}_{\text{d}\text{i}\text{s}\text{t}\text{o}\text{r}\text{t}\text{i}\text{o}\text{n}}^2$Hint: you can assume that the system has an overall noise with variance $2,2,2$

total distortion

$($f$)$ If $4-$PAM is used to represent the bit sequence in part $($a$),$ draw the resulting transmitted waveform $p(t)$ while keeping the same average signal power and bit rate as $g(t)$ in part $($a$).$

Please use the graph below for answering parts of this question.