$3-$ Consider a binary communication system that transmits either a zero or a one with probability p$\_(1)$ and p$\_(2)$ respectively. Transmitting probabilities have a ratio of p$\_(1)//$p$\_(2)=3//4.$ Assume, Vo $>$ V$\_($I$)$ and non$-$zero threshold value alpha$=0\mathrm{.}13$ where Vo is the required voltage level for transmitting zero, V$\_($I$)$ is the required voltage level for transmitting one. Because of channel noise $($ SNR$\_$dB$=5$dB$),$ a transmitted zero is received as a one with probability $($p$),$ and a transmitted one is received as a zero with probability $(2$p$).$

a$)$ For AWGN $($Additive White Gaussian Noise$)$ communication channel, make a DERIVATION to obtain the formula for p $($in terms of SNR and alpha $)$ for 'a transmitted zero is received as a one'.

b$)$ Compute the probability of receiving zero.

c$)$ Suppose a zero is received, what is the probability that a zero was transmitted?

d$)$ Determine the overall error rate of the system.

SNR: signal to noise ratio

SNR$\_$dB: $10$log$($SNR$)$