Toeplitz Operators on Locally Compact Abelian Groups
نویسندگان
چکیده
The problem of global optimization of M incoherent phase signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N 2, and the optimal signal sets are determined for M 2, 3, 4, 6 and 12. The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points sj, j 1, 2, ..., M, on the unit sphere $1 in CN. If Wjk is the half-space determined by sj and Sk and containing sj, i.e., Wjk {r CN:l(r, sj)l __> I(r, sk)l}, then {j f3 k Wk :j 1, 2, ..., M}, the maximum likelihood decision regions, partition $1. For additive complex Gaussian noise n and a received signal r sj e + n, where 0 is uniformly distributed over [0, 2hi, the probability of correct decoding for the signal-to-noise ratio A2 is where r2Ne-r2+AZ)U(r) dr, -Pc -U(r) -1 fg Io(2Ar[(s, sj)[) da(s), j=l Rj j {"I S For N 2, it is proved that U(r) Io(2Nrl(s, %)1) d(s) . h [M(C) (S)] where C {seS :l(s,s)l }, 2K is the total number of half-spaces that actually determine the decision regions, and h is the strictly increasing, strictly convex function of a(C, W) (where Wis a half-space not containing sj), given by h Io(2arl[) c C C W. Conditions for equality are established and these give rise to the globally optimal signal sets for M 2, 3, 4, 6 and 12.
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