A binary communication system has t ransmitted sign al X,

Chapter 11, Problem 11.22

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A binary communication system has t ransmitted sign al X, t he Bernoulli (1/ 2) random variable. At the receiver, 've observe Y = V X + W, where V is a "fading factor" and W is additive noise. Note that V and Ware exponential (1) random variables and that X , V, and W are mutually independent . Given the observation Y, we inust guess whether X = 0 or X = 1 was transmitted. use a binary hypothesis test to determine t he rule that minimizes the probability Pe of a decoding error. For the optimum decision rule, calculate Pe.

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