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
![](https://test.transtutors.com/qimg/27db726b-6175-496e-86a6-d02d90295e69.png)
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
![](https://test.transtutors.com/qimg/26eff783-7ff6-4667-a4cd-a509b494759c.png)
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.