Trellis detection for random lattices

Abstract

In general, lattice problems are simple to describe but rather hard to solve optimally. Several suboptimal solutions have been proposed for the closest vector problem (CVP), which is central in multiple-input multiple-output (MIMO) communication systems. It is known that some lattices have a trellis representation, however, those lattices require very particular geometries that are not found in lattices randomly generated. In this paper we show that for the typical number of dimensions used in MIMO communication, with high probability, there exists a synthetic lattice that is a member of the family of lattices that have a trellis representation and which is sufficiently close to any given random lattice. For that purpose we present a method to find a trellis-oriented basis for a given random lattice. The basis vectors of the synthetic lattice and the basis vectors of the original lattice are close and for finite alphabets the two lattices are roughly the same in the region of interest. Therefore, the optimal decision (Voronoi) regions of both lattices chiefly overlap. A linear transformation then focuses the original lattice onto the synthetic one, known to have a trellis representation. This minimizes the distortion of the Voronoi regions associated with maximum-likelihood detection and therefore the performance attained in the MIMO-CVP is close to optimal.