Thèse de doctorat en Automatique et traitement du signal
Sous la direction de Bernard Picinbono.
Soutenue en 1987
à Paris 11 , en partenariat avec Université de Paris-Sud. Faculté des Sciences d'Orsay (Essonne) (autre partenaire) .
Estimation adaptative des coefficients d'une régression linéaire pouvant varier lentement au cours du temps. Pour cela on utilisera un algorithme récursif, rapide, adaptatif et stable avec une bonne vitesse de convergence initiale. Le développement de cet algorithme est obtenu par extension des équations du type Chandrasekhar à des problèmes de moindres carrés
Regularized fast recursive least squares algorithms for adaptive filtering and spectral analysis
We adress here the problem of adaptively estimating the coefficients of a linear regression with the practical and contradictory constraints of low numerical complexity, adaptivity, numerical stability and fast initial convergence. This work is motivated by the fact that Least-Mean-Squares (LMS) algorithms suffer from low initial convergence and that Fast Recursive Least-Squares (FRLS) algorithms may present numerical stability problems which are agravated by the use of an exponential data weighting. Our basic idea is that these instabilities are a reflect of an ill-posedness of the initial numerical problem and then may be faced by using regularization methods which are equivalent to the use of a priori information about the solution. But, in an adaptive context, an effective regularization requires to hold an appropriate balance between priors and data information. From this point of view, finite memory appears as the best remedy to stability problems. Regularized Fast Recursive Least-Squares (RFRLS) algorithms were developed by extending the Chandrasekhar type equations to apply to adaptive linear regression estimation. Partitioned matrices inversion techniques are also used. The second part of this work deals with the adaptive spectrum analysis problem. A method is presented here based on a Bayesian approach and using long AR models. The solution is then computed by a RFRLS algorithm, with O(p) numerical complexity, p being the solution order. The interest of the method is illustrated by simulation results and an application to actual Doppler signals.