Analysis of Fast Recursive Least Squares Algorithms for Adaptive Filtering M.AREZKI 1, 2 , A.BENALLAL 1 , P.MEYRUEIS 2 , A.GUESSOUM 1 , D.BERKANI 3 1 LATSI Laboratory - Department of Electronic, University of BLIDA, ALGERIA md_arezki@hotmail.com; a_benallal@hotmail.com; guessouma@hotmail.com 2 Laboratoire des Systèmes Photoniques - ENSPS Université Louis Pasteur Boulevard Sébastien Brant -BP 10413 ILLKIRCH 67412 FRANCE. meyrueis@sphot.u-strasbg.fr 3 Signal &Communications Laboratory – Department of Electrical Engineering (ENP), ALGIERS, ALGERIA. dberkani@hotmail.com yxwv Abstract—In this paper, we present new version of numerically stable fast recursive least squares (NS-FRLS) algorithm. This new version is obtained by using some redundant formulae of the fast recursive least squares (FRLS) algorithms. Numerical stabilization is achieved by using a propagation model of first order of the numerical errors. A theoretical justification for this version is presented by formulating new conditions on the forgetting factor. An advanced comparative method is used to study the efficiency of this new version relatively to RLS algorithm by calculating their normalized square norm gain error (NGE). We provide a theoretical justification for this version by formulating new conditions on forgetting factor. It will be followed by an analytical analyze of the convergence of this version and we show, both theoretically and experimentally, their robustness. The simulation over a very long duration for a stationary signal did not reveal any tendency to divergence. Key-Words: Fast RLS, Estimation, Adaptive Filtering, Propagation of Errors, Numerical Stability. 1 Introduction In general the problem of system identification involves constructing an estimate of an unknown system given only two signals, the input signal and a reference signal. Typically the unknown system is modelled linearly with a finite impulse response (FIR), and adaptive filtering algorithms are employed to iteratively converge upon an estimate of the response. If the system is time-varying, then the problem expands to include tracking the unknown system as it changes over time. The system identification problem has numerous applications in control systems, digital communications, and signal processing, and a recent survey of adaptive filtering algorithms highlights the rich diversity of techniques available in the literature [1]. Adaptive filtering has been, and still is, an area of active research, playing important roles in an ever increasing number of applications [1], [2]. Numerous algorithms for the solution of the adaptive filtering problem have been proposed over the years. The recursive least squares (RLS) algorithms are used in a broad class of applications. The RLS algorithm solves this problem, but at the expense of increased computational complexity. A large number of fast RLS (FRLS) algorithms have been developed over the years, but, unfortunately, it seems that the better a FRLS algorithm is in terms of computational efficiency, the more severe is its problems related to numerical stability [3]. Several numerical solutions of stabilization, with stationary signals, are proposed in the literature [5]–[10]. In the following section, we propose a new version of numerically stable FRLS (NS-FRLS) algorithm. This new version is obtained by using some redundant formulae of the fast recursive least squares FRLS algorithms. Numerical stabilization is achieved by using a propagation model of first order of the numerical errors [5], [8]. We provide a theoretical justification for this version by formulating new conditions on forgetting factor. It will be followed by an analytical analyze of the convergence of this version and we show, both theoretically and experimentally, their robustness. Proceedings of the 11th WSEAS International Conference on SYSTEMS, Agios Nikolaos, Crete Island, Greece, July 23-25, 2007 474