In a BPSK amplify-and-for,vard relay system, a source

Chapter 11, Problem 11.23

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In a BPSK amplify-and-for,vard relay system, a source transmits a random bit '/ E {-1, 1} every T seconds to a destination receiver via a set of n, relay transmitters. V = 1 and '/ = -1 are equally likely. In this communication system, the source transmits during the time period (0, T / 2) such that relay i receives i = 1, 2, ... , n,, \V here the wi are iid Gaussian ( 0' 1) random variables representing relay i receiver noise. In the time interval (7,/ 2, T), each relay node amplifies and forwards the received source signal. The destination receiver obtains the vector Y = [Y1 Y'.;1 J' such that i= l ,2, ... ,ri, \Vhere the zi are also iid Gaussian (0, 1) random variables. In the following, assume that the parameters ai and f3i are all nonnegative. Also, let H0 denote the hypothesis that V = -1 and H1 the hypothesis v = l. (a) Suppose you build a suboptimal detector based on the sum Y = I:: 1 1 Yi. If Y > 0, the receiver guesses H1; otherwise the receiver guesses H0 . \i\fhat is the probability of error Pe for this receiver? (b) Based on the observation Y , now suppose the destination receiver detector performs a l\/IAI> test f o:r hypotheses H 0 or H1. \l\fhat is the Mi\.P detector rule? Simplify your answer as much as possible. Hint: First find the likelihood functions fYIHi(y). ( c) \i\fhat is the probabilitJr of bit error P; of the MAP detector? ( d) Compare the t\vo detectors when n = 4 and (a1, f31) = (1, 1), ( 0'.2 ) (32) = ( 10' 1) ' ( 0'.3' (33) = ( 1, 10) ' ( 0'.4) (34) = (10, 10). In general, what's bad about the suboptimal detector?

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