An M-ary quadrature amplitude modulation (QAM) communications system can be viewed as a generalization of the QPSK system described in Example 8.13. In the QAM system, one of M equally likely symbols s0,...,sm-1 is transmitted every unit of time. When symbol si is transmitted, the receiver produces the 2-dimensional vector output
Where N has iid Gaussian (0, σ2) components Based on the output X, the receiver must decide which symbol was transmitted. Design a hypothesis test that maximizes the probability of correctly deciding what symbol was sent.
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.