For the QPSK communications system of Example 8.13, identify the acceptance sets for the MAP hypothesis test when the symbols are not equally likely. Sketch the acceptance sets when σ = 0.8, E = 1, P[H0] = 1/2, and P[H1] = P[H2] = P[H3] = 1/6.
Example 8.13
In a quaternary phase shift keying (QPSK) communications system, the transmitter sends one of four equally likely symbols {s0,s1,s2,s3}. Let Hi denote the hypothesis that the transmitted signal was si. When si is transmitted, a QPSK receiver produces the vector X = [X1 X2] such that
Where N1 and N2 are iid Gaussian (0,σ) random variables that characterize the receiver noise and E is the average energy per symbol. Based on the receiver output X, the receiver must decide which symbol was transmitted. Design a hypothesis test that maximizes the probability of correctly deciding which symbol was sent.