Finite Dimensional Infinite Constellations

TL;DR

This paper investigates the performance limits of infinite constellations in Gaussian channels without power constraints, providing new bounds, asymptotic analysis, and demonstrating the tightness of results through numerical comparisons.

Contribution

It introduces two new achievability bounds, extends the sphere bound to finite dimensions, and offers asymptotic expansions that improve understanding of performance near capacity.

Findings

Asymptotic expansions are significantly tighter than previous error exponent results.

The gap to capacity decreases inversely with the square root of blocklength n.

The dispersion of infinite constellations is 1/2 times the square of the inverse Q-function of the error probability.

Abstract

In the setting of a Gaussian channel without power constraints, proposed by Poltyrev, the codewords are points in an n-dimensional Euclidean space (an infinite constellation) and the tradeoff between their density and the error probability is considered. The capacity in this setting is the highest achievable normalized log density (NLD) with vanishing error probability. This capacity as well as error exponent bounds for this setting are known. In this work we consider the optimal performance achievable in the fixed blocklength (dimension) regime. We provide two new achievability bounds, and extend the validity of the sphere bound to finite dimensional infinite constellations. We also provide asymptotic analysis of the bounds: When the NLD is fixed, we provide asymptotic expansions for the bounds that are significantly tighter than the previously known error exponent results. When the error probability is fixed, we show that as n grows, the gap to capacity is inversely proportional (up to the first order) to the square-root of n where the proportion constant is given by the inverse Q-function of the allowed error probability, times the square root of 1/2. In an analogy to similar result in channel coding, the dispersion of infinite constellations is 1/2nat^2 per channel use. All our achievability results use lattices and therefore hold for the maximal error probability as well. Connections to the error exponent of the power constrained Gaussian channel and to the volume-to-noise ratio as a figure of merit are discussed. In addition, we demonstrate the tightness of the results numerically and compare to state-of-the-art coding schemes.