Adaptive noise reduction using numerically stable fast recursive least squares algorithm (original) (raw)
Related papers
2005
In this paper, we present a new version of numerically stable fast recursive least squares (NS-FRLS) algorithm. This new version is obtained by using some redundant formulas of the fast recursive least squares FRLS algorithms. Numerical stabilization is realized by using a propagation model of first order of the numerical errors. An advanced comparative method is used to study the efficiency of this new version. The simulation over very long
Analysis of fast recursive least squares algorithms for adaptive filtering
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.
Stability Analysis of Fast Recursive Least Squares Algorithm: Application to Adaptive Filtering
2000
In this paper, we introduce a new numerically stable version for the Fast Recursive Least Squares (FRLS) algorithm. As an additional contribution, we present an analysis of the FRLS algorithm instability problems. An experimental study is conducted in order to determine the origin of the numerical instability. The originality of our investigation lies mainly in finding the relation between the
Journal of Algorithms, 2000
In this paper, a new multichannel recursive least squares (MRLS) adaptive algorithm is presented which has a number of very interesting properties. The proposed computational scheme performs adaptive filtering via the use of a finite window, where the burdening past information is dropped directly by means of a generalized inversion lemma; consequently, the proposed algorithm has excellent tracking abilities and very low misjudgment. Moreover, the scheme presented here, due to its particular structure and to the proper choice of mathematical definitions behind it, is very robust; i.e., it is less sensitive in the finite precision numerical error generation and propagation. Also, the new algorithm can be parallelized via a simple technique and its parallel form and, when executed with four processors, is faster than all the already existing schemes that perform both infinite and finite window multichannel adaptive filtering. Finally, due to the particular structure of this scheme and to the intrinsic flexibility in the choice of the window length, the proposed algorithm can act as a full substitute of the infinite window MRLS ones.
An efficient recursive total least squares algorithm for FIR adaptive filtering
IEEE Transactions on Signal Processing, 1994
Absbuct-An algorithm for recursively computing the total least squares (TLS) solution to the adaptive filtering problem is described. This algorithm requires O( N ) multiplications per iteration to effectively track the N-dimensional eigenvector associated with the minimum eigenvalue of an augmented sample covariance matrix. It is shown that the recursive least squares (RLS) algorithm generates biased adaptive filter coefficients when the filter input vector contains additive noise. The TLS solution on the other hand, is seen to produce unbiased solutions. Examples of standard adaptive filtering applications that result in noise being added to the adaptive filter input vector are cited. Computer simulations comparing the relative performance of RLS and recursive TLS are described.
IEEE Transactions on Signal Processing, 2003
This paper presents a numerically stable fast Newton-type adaptive filter algorithm. Two problems are dealt with in the paper. First, we derive the proposed algorithm from an order-recursive least squares algorithm. The result of the proposed algorithm is equivalent to that of the fast Newton transversal filter (FNTF) algorithm. However, the derivation process is different. Instead of extending a covariance matrix of the input based on the min-max and the max-min criteria, the derivation shown in this paper is to solve an optimum extension problem of the gain vector based on the information of the th-order forward or backward predictor. The derivation provides an intuitive explanation of the FNTF algorithm, which may be easier to understand. Second, we present stability analysis of the proposed algorithm using a linear time-variant state-space method. We show that the proposed algorithm has a well-analyzable stability structure, which is indicated by a transition matrix. The eigenvalues of the ensemble average of the transition matrix are proved all to be asymptotically less than unity. This results in a much-improved numerical performance of the proposed algorithm compared with the combination of the stabilized fast recursive least squares (SFRLS) and the FNTF algorithms. Computer simulations implemented by using a finite-precision arithmetic have confirmed the validity of our analysis.
Signal Processing, 1994
In this paper, we derive a new fast algorithm for Recursive Least-Squares RLS adaptive ltering. This algorithm is especially suited for adapting very long lters such as in the acoustic echo cancellation problem. The starting point is to introduce subsampled updating SU in the RLS algorithm. In the SU RLS algorithm, the Kalman gain and the likelihood variable are matrices. Due to the shift invariance of the adaptive FIR ltering problem, these matrices exhibit a low displacement rank. This leads to a representation of these quantities in terms of sums of products of triangular Toeplitz matrices. Finally, the product of these Toeplitz matrices with a vector can be computed e ciently by using the Fast Fourier Transform FFT. Zusammenfassung Dieser Artikel beschreibt die Herleitung eines neuen Algorithmus zur schnellen adaptiven Recursive Least Square RLS Filterung. Dieser Algorithmus eignet sich besonders f uer aufwendige Filter, wie sie zum Beispiel zur akkustischen Echounterdr ueckung benutzt werden. Im Zentrum dieses Algorithmus steht die Einf uehrung von unterabgetastetem Updating SU. Der Kalman Gewinn und die Likelihood Variable treten im SU RLS Algorithmus als Matrizen auf. Aufgrund der Verschiebungsinvarianz in der adaptiven FIR Filterung zeigen diese Matrizen einen niedrigen Verschiebungsrang. Dies f uehrt zu einer Darstellung dieser Gr oessen als Summe von Produkten von triangul aeren Toeplitz Matrizen. Das Produkt dieser Matrizen mit einem Vektor k ann auf sehr e ziente Weise mit der Fast Fourier Transform FFT berechnet werden. R esum e Dans ce papier, nous pr esentons un nouvel algorithme des moindres carr es r ecursif rapide. Cet algorithme pr esente un int erêt certain pour l'adaptation de ltres tr es longs comme ceux utilis es dans les probl emes d'annulation d' echo acoustique. L'id ee de d epart est d'utiliser l'algorithme RLS avec une mise a jour sous-echantillonn ee" du ltre. Dans cet algorithme le SU RLS le gain de Kalman et la variable de vraisemblance sont des matrices qui ont des rangs de d eplacement faibles. Ces quantit es sont alors repr esent ees et mises a jour par le biais de leurs g en erateurs, sous forme de sommes de produits de matrices de Toeplitz triangulaires. Le produit de l'une de ces quantit es avec un vecteur peut alorsêtre calcul e en utilisant la transform ee de Fourier rapide FFT.
New filtered-x recursive least square algorithm for active noise control
A new multichannel filtered-x recursive least square algorithm for active noise control systems is proposed. It is shown that the use of the filtered-x structure, instead of the commonly used modified filtered-x structure lead to a more efficient implementation and similar convergence performance and stability. The paper is also focused on examining the benefits of auxiliary normal equations solving methods needed by the formulation of the RLS problem. The behavior of the proposed algorithm is investigated in cases of ideal and nonideal acoustic plants.
Fast Recursive Least Squares Algorithm for Acoustic Echo Cancellation Application
2007
Adaptive filtering is used in a wide range of applications including echo cancellation, noise cancellation and equalization. In these applications, the environment in which the adaptive filter operates is often non-stationary. For satisfactory performance under non-stationary conditions, an adaptive filtering is required to follow the statistical variations of the environment. Tracking analysis provides insight into the ability of an adaptive filtering algorithm to track the changes in surrounding environment. The tracking behavior of an algorithm is quite different from its convergences behavior. While convergence is a transient phenomenon, tracking is a steady-state phenomenon. Over the last decade a class of equivalent algorithms such as the Normalized Least Mean Squares algorithm (NLMS) and the Fast Recursive Least Squares algorithm (FRLS) has been developed to accelerate the convergence speed. In acoustic echo cancellation context, we propose in this paper to use numerically stable Fast Recursive Least Squares algorithm to improve the quality and the intelligibility of the enhanced speech.