Error Performance of Multidimensional Lattice Constellations-Part I: A Parallelotope Geometry Based Approach for the AWGN Channel

TL;DR

This paper introduces a novel geometrical approach based on parallelotope geometry for evaluating the error performance of multidimensional lattice constellations in AWGN channels, providing exact and bound expressions.

Contribution

It presents the first analytical expression for symbol error probability and introduces new bounds using parallelotope geometry for multidimensional lattice constellations.

Findings

Exact SEP expression for multidimensional lattice constellations

Tight bounds (MLSB and MSUB) for error probability

Efficient performance evaluation and design tools

Abstract

Multidimensional lattice constellations which present signal space diversity (SSD) have been extensively studied for single-antenna transmission over fading channels, with focus on their optimal design for achieving high diversity gain. In this two-part series of papers we present a novel combinatorial geometrical approach based on parallelotope geometry, for the performance evaluation of multidimensional finite lattice constellations with arbitrary structure, dimension and rank. In Part I, we present an analytical expression for the exact symbol error probability (SEP) of multidimensional signal sets, and two novel closed-form bounds, named Multiple Sphere Lower Bound (MLSB) and Multiple Sphere Upper Bound (MSUB). Part II extends the analysis to the transmission over fading channels, where multidimensional signal sets are commonly used to combat fading degradation. Numerical and simulation results show that the proposed geometrical approach leads to accurate and tight expressions, which can be efficiently used for the performance evaluation and the design of multidimensional lattice constellations, both in Additive White Gaussian Noise (AWGN) and fading channels.