In a BPSK communication system, a source wishes to communicate a random bit X {- 1, 1} to a receiver. Inputs X = 1 and

In a BPSK communication system, a source wishes to communicate a random bit X ˆˆ {- 1, 1} to a receiver. Inputs X = 1 and X = - 1 are equally likely. In this system, the source transmits X multiple times. In the ith transmission, the receiver observes Yi = X + Wi, where the Wi are iid Gaussian (0, 1) noises, independent of X.
(a) After n transmissions of X, you observe Y = y = [y1 ... yn]ʹ. Find P[X = 1|Y = y]. Express your answer in terms of the likelihood ratio

(b) Suppose after n transmissions, the receiver observes Y = y, and decides

Find the probability of error Pe = P[X* ‰  X] in terms of the Φ function.
(c) Now suppose the system uses ARQ as follows. If |n(y)| n(y) > 1 - ϵ, the receiver guesses X* = 1. If n(y) ϵ1 ‰¤ Pe ‰¤ ϵ2

fyix(y1-1) fyix (y1) X* ={-1 otherwise-yl > 1/2,