A multilevel QPSK communications system transmits three bits every unit of time. For each possible sequence ijk of three bits, one of eight symbols, {s000,s001,...,s111}, is transmitted. When signal sijk is transmitted, the receiver output is
Where N is a Gaussian (0, σ2I) random vector The 2-dimensional signal vectors s000,...,s111 are
Let Hijk denote the hypothesis that sijk was transmitted. The receiver output X = [X1 X2] is used to decide the acceptance sets {A000,..., A111}. If all eight symbols are equally likely, sketch the acceptance sets.