Reference no: EM131083908
The QPSK system of Example 8.13 can be generalized to an M-ary phase shift keying (M-PSK) system with M > 4 equally likely signals. The signal vectors are {s0,...,sM-1} where and θi = 2πi/M. When the ith message is sent, the received signal is X = si +N where N is a Gaussian (0, σ2I) noise vector
(a) Sketch the acceptance set Ai for the hypothesis Hi that si was transmitted.
(b) Find the largest value of d such that
(c) Use d to find an upper bound for the probability of error.
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.
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The acceptance sets based on the receiver outputs
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