Transversal adaptive filtering

This paper presents a tutorial on the gradient (G) and recursive least squares (RLS) algorithme, both commonly used in adaptive transversal filters for estimating a linear model . It is shown how the algorithms utilize recursivity ta realize at the saine time the averaging involved in the optimisation criterion as well as the minimization. In steady state both algorithms can be viewed as minimizing the saure mean square error criterion. Then they have equivalent performances : the saure residual fluctuations up to an equivalence between the adaptation step-size t of (G) and the forgetting rate (1-w) of (RLS) . Due to the criterion itself emphasizing adaptation speed, it is the transient period which exhibits the (RLS) superiority . This is essentially at the price of a four times higher complexity and of difficulties with controling numerical effects, related (in part) to the presence of divisions that are not in (G) . A general methodology for decoupling the transient effects and permanent fluctuations is given ; the latter are due to the measurement noise disturbing the model . The more general problem of tracking the model variations is treated with that methodology and we emphasize the différence between the problems of tracking and of squeezing the transient period for a fixed model : the corresponding adaptation step-sizes differ significantly .Synthèse sur les algorithmes du gradient et des moindres carrés rapides. En régime permanent, ils ont des performances équivalentes. C'est en régime transitoire que les moindres carrés sont supérieurs; ceci au prix d'une complexité environ quatre fois supérieure et de difficultés de contrôle des variables numérique