In a BPSK communication system, a source wishes to

Chapter 11, Problem 11.24

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In a BPSK communication system, a source wishes to communicate a random bit XE {-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]'. F ind P[X = llY = y]. Express your answer in terms of the likelihood ratio L( ) = fy1x(YI - 1) y fYIX (yll) . (b) Suppose after n, transmissions, the receiver observes Y = y and decides X * = {1 -1 P [X = l lY = y] > 1/2, otherwise. F ind the probability of error Pe = f>[X* # X] in terms of the 1 - E, the receiver guesses X * = 1. If Xn(Y) < -1 + E, the receiver guesses X * = -1. Following the receiver's guess, the transmitter starts sending a new bit. F ind upper and lower bounds to Pe = P[X* # X ]. That is, find E 1 and E 2 such that