Abstract
We apply the random coding argument to coded modulation. The well-known union bound on the error probability of general signaling schemes is revisited. The random coded modulation idea is introduced and a simple bound on the average performance of coded constellations is presented. A relationship between the union bound and the cut-off rate is exhibited by introducing the concept of N-dimensional partial cut-off rate. We define finite theta series for bounded finite-dimensional constellations and their related transfer functions. Bounds on the block and symbol error probability based on the transfer function are derived. The discussion is then focused on the squared Euclidean distance distribution. The evaluation of such parameters as its first two moments (average squared distance and squared distance variance) is considered by either finite theta series or transfer function of the bounded signal set. The Euclidean distance spectra of a few multidimensional coded modulation schemes based on square/cross constituent two-dimensional constellations are presented. Their respective partial cut-off rates are computed. We discuss the asymptotic behaviour and we show that almost all very long coded constellations are good (actually, they tend to become quasi-identical in a certain sense). Finally, we examine how to extend the initial union bound to Gallager-type bounds.
Résumé
Le concept de codage aléatoire est appliqué à la modulation codée. On revient d’abord sur la borne par réunion, bien connue, de la probabilité d’erreur d’un système de communication quelconque. L’idée de codage aléatoire est introduite et une borne simple sur les performances moyennes des constellations codées est présentée. Une relation entre la borne par réunion et le débit de coupure (cutoff rate) est obtenue en introduisant le concept de débit de coupure partiel N-dimensionnel. La série thêta et la fonction de transfert correspondante sont introduites pour une constellation bornée dans un espace à nombre de dimensions fini. Des bornes de la probabilité d’erreur par bloc et par symbole, basées sur cette fonction de transfert, sont établies. On examine ensuite la distribution des distances euclidiennes carrées. Des paramètres comme ses deux premiers moments (moyenne et variance du carré de la distance) sont évalués au moyen soit de la série thêta, soit de la fonction de transfert de la constellation. Les spectres des distances euclidiennes de quelques schémas de modulation codée multidimensionnelle basés sur des constellations planes carrées ou en forme de croix sont présentés. Leurs débits de coupure partiels sont calculés. On considère le comportement asymptotique de constellations codées très longues et on montre qu’elles sont presque toutes bonnes; elles tendent d’ailleurs à devenir quasi identiques en un certain sens. Finalement, on examine comment la borne par réunion initiale peut être généralisée à des bornes du type de Gallager.
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References
Shannon (C. E.). Probability of error for optimal codes in a Gaussian channel.BSTJ (May 1959), 38, no 3, pp. 611–656.
Ungerboeck (G.). Channel coding with multilevel/phase signals.IEEE Trans, on Info. Theory (1982), 28, no 1, pp. 55–67.
Cheung (K. M.). Identities and approximations for the weight distributions of Q-ary codes.IEEE Trans, on Info. Theory (sep. 1990), 36, no 5, pp. 1149–1153.
Battail (G.), MagalhÃes de Oliveira (H.), Zhang Weidong. Coding and modulation for the Gaussian channel, in the absence or in the presence of fluctuations.Proc. Int. Colloquium on Coding Theory and Applications, Eurocode’90, Udine, Italy (5-9 Nov. 1990);Lecture Notes in Computer Science, Springer Verlag, no 514, pp. 337–349.
Viterbi (A. J.). Convolutional codes and their performance in communications systems.IEEE Trans. Commun. Technol. (1971), 19, no 5, pp. 751–772.
Zehavi (E.), Wolf (J. K.). On the performance evaluation of trellis codes.IEEE Trans, on Info. Theory (mars 1987), 33, no 2, pp. 196–202.
Battail (G.). Coding for the Gaussian channel: the promise of weighted-output decoding.Int. J. Satellite Commun. (1989), 7, pp. 183–192.
Battail (G.). Construction explicite de bons codes longs.Ann. Télécommunic. (juil.-oct. 1989), 44, no 7-8, pp. 392–404.
Magalhâes de Oliveira (H.), Battail (G.). A capacity theorem for lattice codes on Gaussian channels. Proc. IEEEISBT Int. Telecomm. Symposium, rrs’90 (sep. 1990), Rio de Janeiro, Brazil, pp. 1.2.1–1.2.5.
Wozencraft (J. M), Jacobs (I. M.). Principles of communication engineering.Wiley (1965).
Viterbi (A. J.), Omura (J. K.). Principles of digital communication and coding.McGraw-Hill (1979).
Forney (G. D.), Wei (L. E). Multidimensional constellations. Part I: Introduction, figures of merit and generalized cross constellations.IEEE J. Select. Areas Commun, (août 1989), 7, no 6, pp. 877–892.
Magalhâes de Oliveira (H.), Battail (G.). On generalized constellations and the opportunistic secondary channel.Ann. Télécommunic.
Sloane (N. J. A.). Tables of sphere packings and spherical codes.IEEE Trans, on Info. Theory (1981), 27, no 3, pp. 327–338.
Conway (J. H.), Sloane (N. J. A.). Sphere packings, lattices and groups.Springer (1988).
Beckenbach (E. E) (Ed). Applied combinatorial mathematicsWiley (1964).
Kullback (S.). Information theory and statisticsWiley (1959).
Pierce (J. N.). Limit distribution of the minimum distance of random linear codes.IEEE Trans, on Info. Theory (oct. 1967), 13, no 4, pp. 595–599.
Elias (P.). Coding for noisy channels.IRE Conv. Rec. (mars 1955), pp. 37–46.
Gallager (R. G.). Information theory and reliable communication.Wiley (1968).
Hughes (B.). On the error probability of signals in additive white gaussian noise.IEEE Trans, on Info. Theory (jan. 1991), 37, no 1, pp. 151–155.
Magalhâes de Oliveira (H.), Battail (G.). Performance of lattice codes over the Gaussian channel.Ann. Télécommunic, à paraître.
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This work was presented in part at the IEEE Int. Symp. on Info. Theory, Budapest, Hungary, June 23-28, 1991. This paper is a part of the Doctoral dissertation of the first author, upheld on Feb. 14, 1992, at Telecom Paris.
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Oliveira, H.M.D., Battail, G. The random coded modulation: performance and euclidean distance spectrum evaluation. Ann. Télécommun. 47, 107–124 (1992). https://doi.org/10.1007/BF02999683
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DOI: https://doi.org/10.1007/BF02999683
Key words
- Random coding
- Error probability
- Signal set
- Code distance
- Quadrature modulation
- M-ary modulation
- Asymptotic behavior
- Upper bound