In a ternary amplitude shift keying (ASK) communications system, there are three equally likely transmitted signals {s0, s1, s2}. These signals are distinguished by their amplitudes such that if signal Si is transmitted, the receiver output will be
X = a(i - 1) + N,
where a is a positive constant and N is a Gaussian (0, σN) random variable. Based on the output X, the receiver must decode which symbol si was transmitted.
(a) What are the acceptance sets Ai for the hypotheses Hi that Si was transmitted?
(b) What is P[DE], the probability that the receiver decodes the wrong symbol?