Abstract
The paper presents a new review of parallel Kalman filtering for nonlinear channel equalization. A Network of Extended Kalman Filters (nekf) has already been suggested for this purpose. This equalizer gives recursively a minimum mean squared error (mmse) estimation of a sequence of transmitted symbols according to a state formulation of a digital communication scheme. It is essentially based on two mechanisms: the approximation of the non Gaussiana posteriori probability density function (pdf) of the symbol sequence by a Weighted Gaussian Sum (wgs); and the local linearization of the nonlinear channel function for each branch of the network. Since the linearization, bearing on scattered symbol states, is one of the major limitations of thenekf, a new Kalman filtering approach, the Unscented Kalman Filter (ukf) suggested by Julier and Uhlman is considered in this paper for an interesting adaptation to the equalization context. Theukf algorithm is based on the equations of a Kalman filter, as the optimal linear minimum variance estimator, and on determining conditional expectations based on a kind of deterministic Monte-Carlo simulations. The new equalizer referred to as the Network ofukf (nukf), thus combines density approximation by awgs and the Unscented Transformation (ut) principle to circumvent the linearization brought within eachekf and is shown to perform better than thenekf based equalizer for severe nonlinear channels. Also, an adaptive version of thenukf is developed using the k-means clustering algorithm for noise-free channel output identification, since thenukf-based algorithm does not require the knowledge of the channel nonlinearity model.
Résumé
Dans cet article, nous revisitons le thème d’égalisation par filtrage de Kalman parallèle pour les canaux non linéaires. Un Réseau de Filtres de Kalman Etendus (rfke) a déjà été suggéré dans ce but. Cet égaliseur fournit à chaque instant une estimation récursive, selon le critère d’erreur quadratique moyenne minimale (eqmm), d’une séquence de symboles transmis de longueur finie, en se basant sur une formulation d’état d’un système de communications numériques en bande de base. Il opère essentiellement selon deux mécanismes : l’approximation de la densité de probabilité (ddp) de la séquence de symboles par une Somme Pondérée de Gaussiennes (spg) et la linéarisation du canal non linéaire au niveau de chaque branche de réseau. En remarquant que cette linéarisation est menée autour d’états de symboles dispersés dans leur espace de représentation, ce qui constitue une vraie limitation pour l’utilisation durfke en tant qu’égaliseur, nous avons considéré une nouvelle approche par filtrage de Kalman, à savoir par “Unscented Kalman Filter” (ukf) proposée initialement par Julier et Uhlman pour une adaptation possible dans le contexte de l’égalisation. L’algorithme du filtreukf est basé sur les équations de récursions d’un filtre de Kalman, considéré comme l’estimateur linéaire optimal à minimum de variance, et sur la détermination des moyennes conditionnelles mises en jeu par une sorte de simulations de Monte-Carlo déterministes. L’égaliseur résultant, appelé l’égaliseur par Réseau de filtresukf (Rukf), combine ainsi l’approximation de ddp parspg et le principe de la transformation menée au niveau duukf afin de contourner les mauvaises linéarisations au niveau de chaque filtre de Kalman étendu. On montre par simulations l’amélioration des performances de l’égaliseur parrfke lorqu’il est transformé en unRukf. Une version adaptative duRukf est aussi développée en utilisant des algorithmes de regroupement par k-moyennes afin d’identifier les sorties non bruitées du canal non linéaire.
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Rim AMARA is born in 1974 in Kairouan, Tunisia. She obtained the engineer Diploma from the Ecole polytechnique de Tunisie in 1997, thedea of control and Signal processing from Paris XI university, Orsay, in 1998 and the PhD degree in signal processing from the same university in 2002 when she does her research work in the Laboratoire des Signaux et Systèmes(lss)-cnrs laboratory, Supélec. Nowadays, she is a professor assistant in the institute of applied sciences and technology in Tunis,insat. Her research work are in the field of signal processing for communications, particularily in linear and nonlinear channel equalization, optimal Bayesian filtering and estimation, nonlinear adaptive filtering.
Sylvie MARCOS is an engineer from École Centrale de Paris (Chatenay Malabry, 1984), a Doctor from Paris XI university (Orsay, 1987) and entitled of research (hdr, Orsay, 1995). She is charged with research incnrs since 1988 in laboratoire des signaux et systèmes (lss) in Supélec, Gif Sur Yvette. Her scope fields lay in general signal processing and especially in adaptive filtering, linear and nonlinear, antenna processing, equalization, multi-user detection. The related applications concern digital communications, radar and sonar. She is the author and co-author of many papers and communications related to the cited subjects and she directed the writing of some books on high resolution methods, on Spectral analysis and antenna processing, edited at Hermès in 1998. She is a lecturer in signal processing in thedea of control and signal processing in Paris XI university, Orsay.
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Amara, R., Marcos, S. Improving parallel EKF-based nonlinear channel equalization using unscented transformation. Ann. Télécommun. 59, 304–324 (2004). https://doi.org/10.1007/BF03179700
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DOI: https://doi.org/10.1007/BF03179700