Locally Diverse Constellations from the Special Orthogonal Group

TL;DR

This paper introduces a new family of rotation matrices for multidimensional constellations, optimizing their performance over Rayleigh fading channels, and proposes the concept of local diversity to explain their behavior.

Contribution

It develops a one-parameter subgroup of rotation matrices optimized via gradient descent, and introduces local diversity as a new measure for constellation performance.

Findings

Rotations outperform algebraic rotations at low SNR in dimension 4.

A new local diversity measure better predicts constellation performance.

Optimal rotation parameters are explicitly computed for low SNR.

Abstract

To optimize rotated, multidimensional constellations over a single-input, single-output Rayleigh fading channel, a family of rotation matrices is constructed for all dimensions which are a power of 2. This family is a one-parameter subgroup of the group of rotation matrices, and is located using a gradient descent scheme on this Lie group. The parameter defining the family is chosen to optimize the cutoff rate of the constellation. The optimal rotation parameter is computed explicitly for low signal-to-noise ratios. These rotations outperform full-diversity algebraic rotations in terms of cutoff rate at low SNR (signal-to-noise ratio) and bit error rate at high SNR in dimension $n = 4$. However, a QAM (quadrature amplitude modulation) constellation rotated by such a matrix lacks full diversity, in contrast with the conventional wisdom that good signal sets exhibit full diversity. A new notion of diversity, referred to as local diversity, is introduced to attempt to account for this behavior. Roughly, a locally fully diverse constellation is fully diverse only in small neighborhoods. A local variant of the minimum product distance is also introduced and is shown experimentally to be a superior predictor of constellation performance than the minimum product distance in dimension $n = 4$.