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,...,s_M-1} where and θ_i = 2πi/M. When the ith message is sent, the received signal is X = s_i +N where N is a Gaussian (0, σ²I) noise vector

(a) Sketch the acceptance set A_i for the hypothesis H_i that s_i 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 s_i. 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.