The QPSK system of Example 11.13 can be generalized to an Mary phase shift keying (MPSK) 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
{x| ||x - si|| ‰¤ d} Š‚ Ai
(c) Use d to find an upper bound for the probability of error.

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