Maximum likelihood sequence estimation from the lattice viewpoint

Abstract

Dedicated to my mother and the memory of my father ii Acknowledgement I would like to acknowledge my supervisor, Dr. C. P. Kwong, who introduced me the sequence estimation problem and encouraged me to look at it from new viewpoints. It is also him who taught me the right attitude, which I believe every true researcher ought to have, towards a difficult problem.iii Abstract It is well-known that the use of the Viterbi algorithm to implement a sequence estimator is an optimal way to remove the effect of intersymbol interference for digital transmission systems. However, such an implementation usually results in a very complicated receiver. In this thesis, we transform the problem of maximum likelihood sequence estimation into the problem of finding the closest lattice point. Some related lattice algorithms such as the basis reduction algorithms and the enumeration algorithms are analyzed and some improved versions are suggested. Then efficient algorithms finding the nearest lattice point are derived. Based on these lattice algorithms, simple but effective sequence estimators are proposed for the PAM systems and their complexities are analyzed. Under some mild assumptions, our algorithms have both polynomial space and time complexities, and are therefore much superior to the conventional Viterbi detectors. Simulation results on three different channels show that the performance of the new sequence estimators depend on the distance spectrum of the channel. But, general speaking, the performance approaches optimal as the size of the signal set and the signal-to-noise ratio increase. Finally, the extensions to other lattice-type modulation schemes and the impacts of the lattice viewpoint on the design of bandlimited transmission systems are discussed.iv

Analyse numérique et réduction de réseaux

“…Cette question soulevée pour la première fois par Buchmann (Buchmann, 1994) est importante quand le réseau à réduire ne peut être connu qu'approximativement. Cela est le cas par exemple en théorie des communications (Mow, 1994;Hassibi et al, 1998), mais aussi pour certains problèmes en théorie algorithmique des nombres, dont le calcul du groupe des unités de l'anneau des entiers d'un corps de nombre (Fieker et al, 2006).…”

“…Citons par exemple la factorisation des polynômes à coefficients entiers (Lenstra et al, 1982), améliorée récemment par van Hoeij (van Hoeij, 2001) puis par Novocin dans sa thèse de doctorat (Novocin, 2008) ; la résolution d'équations diophantiennes (Hanrot, 2008) ; plusieurs cryptanalyses de variantes du système de chiffrement à clé publique RSA (May, 2008). L'algorithme LLL est aussi utilisé fréquem-ment en théorie algorithmique des nombres (Cohen, 1995) et en théorie des télécom-munications (Mow, 1994;Hassibi et al, 1998). Ainsi, toute amélioration algorithmique de LLL permet de résoudre plus efficacement de nombreux problèmes.…”

Equalization of Time-Varying Channels

“…Tree-search algorithms can be partitioned into optimal and suboptimal approaches. Optimal tree search methods are capable of implementing MLSD with a complexity that is on average much less than that of brute-force search [Mow94]. Though this average complexity has been claimed to grow as roughly O(K 3 ) at sufficiently high SNR [HV05], a careful analysis shows that, in fact, the average complexity of optimal tree search is exponential in K [JO05].…”

Rigorous and Efficient Short Lattice Vectors Enumeration

“…We refer to [24] for more details. Besides cryptology, lattice reduction and in particular KFP is used in many areas, including number theory [6] and communications theory [25,15], in which the present results may prove useful as well.…”