Multidimensional constellations. II. Voronoi constellations

For pt.I see ibid., vol.7, no.6, p.877-92 (1989). Voronoi constellations, also called Voronoi codes are implementable N-dimensional constellations based on partitions of N-dimensional lattices (Λ) that can achieve good shape gains and that are inherently suited for use with coded modulation. Two methods are given for specifying Voronoi constellations on the basis of arbitrary lattice partitions Λ/Λ_s, where Λ_s, the shaping lattice, is an N-dimensional sublattice of Λ. One of the methods is conjectured to be optimum, and the other has desirable symmetries and naturally supports opportunistic secondary channels. When Λ and Λ_s are 2-D-symmetric, the constituent 2-D constellation is itself a Voronoi constellation. The shaping constellation expansion ratio and peak-to-average-power ratio are determined in general and for various Λ_s. Methods for labeling Voronoi constellations are given. Their complexity is shown to be dominated by that of decoding Λ_s. It is also shown that coding and shaping are separable and dual. Bounds on the shape gain of Voronoi constellations are given that depend on the depth and normalized informativity of Λ_s. These bounds suggest the use of lattices Λ with depth 2 and normalized informativity less than 1, which can achieve near-optimal shape gains with reduced constellation expansion and implementation complexity