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Performance of lattice codes over the Gaussian channel

Performances des codes en treillis sur le canal gaussien

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Abstract

We derive an upper bound on the error probability of lattice codes combined with Quadrature Amplitude Modulation (qam) over the additive white Gaussian noise channel. This bound depends on a lattice figure of merit and is readily put in exponential form by using Chernoff bound. An interesting lower bound is derived by a similar reasoning. We also examine the estimation of the average information rate based upon the continuous approximation of the average power normalized to two dimensions, and suggest to improve it by using the sphere packing idea. Examples of performance evaluation are given for a few lattices. Finally, we present upper and lower bounds on the best fundamental coding gains per dimension (due to both density and thickness) for an arbitrarily large number of dimensions. It is shown in the Appendix that, as the Ungerboeck codes, the lattice codes do not shape the signal power spectrum.

Résumé

On établit une borne supérieure de la probabilité ďerreur ďun code défini par un treillis de points combiné à la modulation ďamplitude en quadrature, sur le canal à bruit additif gaussien et blanc. Cette borne dépend ďun facteur de mérite du treillis et peut être facilement mise sous la forme exponentielle de Chernoff. Une intéressante borne inférieure est obtenue par un raisonnement similaire. On examine aussi ľestimation du débit ďinformation basée sur ľapproximation continue de la puissance moyenne normalisée è deux dimensions. On suggère de ľaméliorer en utilisant ľidée ďempilement de sphères. Des exemples ďévaluation des performances sont donnés pour quelques treillis. On présente finalement des bornes inférieure et supérieure des meilleurs gains de codage fondamentaux par dimension (déduits de la densité et de ľépaisseur) pour un nombre de dimensions arbitraitement grand. On montre dans ľannexe que les codes en treillis, comme ceux ďUngerboeck, n’ont pas ďeffet sur la forme du spectre de puissance des signaux.

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Authors and Affiliations

  1. Department of electronics and systems, communication research group — CODEC, Federal University of Pernambuco, Cid. Universitària, 50.741, Recife, Brazil

    Hélio MagalhÃes De Oliveira

  2. Département communications, Ecole nationale supérieure des télécommunications (Telecom Paris), 46, rue Barrault, F-75634, Paris Cedex 13, France

    Gérard Battail

Authors
  1. Hélio MagalhÃes De Oliveira
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  2. Gérard Battail
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Additional information

This paper is based on a section of the Doctoral dissertation of the first author, upheld on Feb. 14, 1992, at Telecom Parìs. Parts of this work will be presented at ISSSE’92, Paris, France, September 1–4, 1992 and Eurocode ’92, Udine, Italy, October 26–30, 1992.

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MagalhÃes De Oliveira, H., Battail, G. Performance of lattice codes over the Gaussian channel. Ann. Télécommun. 47, 293–305 (1992). https://doi.org/10.1007/BF03004043

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