The Global Optimization of Phase-Incoherent Signals

Author: Schaffner, Charles Albert

Year: 1968

Degree: Dissertation (Ph.D.)

Advisor: Grettenberg, Thomas L.

Committee Member: Unknown, Unknown

Option: Electrical Engineering

DOI: 10.7907/t799-rn87

Abstract

The problem of global optimization of M phase-incoherent 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 ṡi, i = 1, 2, …, M, on the unit sphere S1 in CN. If Wik is the halfspace determined by ṡi and ṡk and containing ṡi, i.e. Wik = {ṙϵCN:| ≥ | ˂ṙ, ṡk˃|}, then the Ʀi = ∩/k≠i Wik, i = 1, 2, …, M, the maximum likelihood decision regions, partition S1. For additive complex Gaussian noise ṅ and a received signal ṙ = ṡiejϴ + ṅ, where ϴ is uniformly distributed over [0, 2π], the probability of correct decoding is PC = 1/πN ∞/ʃ/0 r2N-1e-(r2+1)U(r)dr, where U(r) = 1/M M/Ʃ/i=1 Ʀi ʃ/∩ S1 I0(2r | ˂ṡ, ṡi˃|)dσ(ṡ), and r = ǁṙǁ.

For N = 2, it is proved that U(r) ≤ ʃ/Cα I0(2r|˂ṡ, ṡi˃|)dσ(ṡ) – 2K/M. h(1/2K [Mσ(Cα)-σ(S1)]), where Cα = {ṡϵS1:|˂ṡ, ṡi˃| ≥ α}, K is the total number of boundaries of the net on S1 determined by the decision regions, and h is the strictly increasing strictly convex function of σ(Cα∩W), (where W is a halfspace not containing ṡi), given by h = ʃ/Cα∩W I0 (2r|˂ṡ, ṡi˃|)dσ(ṡ). 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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