(Solution) - In a BPSK communication system a source wishes to communicate -(2025 Original AI-Free Solution)

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In a BPSK communication system, a source wishes to communicate a random bit X ? {- 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]?. 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 ? function.
(c) Now suppose the system uses ARQ as follows. If |n(y)| < 1 - ?, the receiver requests that X be retransmitted; otherwise the receiver guesses what is transmitted. In particular, if n(y) > 1 - ?, the receiver guesses X* = 1. If n(y) < - 1 + ?, the receiver guesses X* = - 1. Following the receiver's guess, the transmitter starts sending a new bit. Find upper and lower bounds to Pe = P[X* ? X]. That is, find ?1 and ?2 such that
?1 ? Pe ? ?2