Magno T. M. Silva

Improving the Tracking Capability of Adaptive Filters via Convex Combination

As is well known, Hessian-based adaptive filters (such as the recursive-least squares algorithm (RLS) for supervised adaptive filtering, or the Shalvi-Weinstein algorithm (SWA) for blind equalization) converge much faster than gradient-based algorithms [such as the least-mean-squares algorithm (LMS) or the constant-modulus algorithm (CMA)]. However, when the problem is tracking a time-variant filter, the issue is not so clear-cut: there are environments for which each family presents better performance. Given this, we propose the use of a convex combination of algorithms of different families to obtain an algorithm with superior tracking capability. We show the potential of this combination and provide a unified theoretical model for the steady-state excess mean-square error for convex combinations of gradient- and Hessian-based algorithms, assuming a random-walk model for the parameter variations. The proposed model is valid for algorithms of the same or different families, and for supervised (LMS and RLS) or blind (CMA and SWA) algorithms.

Convex Combination of Adaptive Filters with Different Tracking Capabilities

Recently, an adaptive convex combination of two LMS (least mean-square) filters was proposed and its tracking performance analyzed. Motivated by the performance of such scheme and by the differences between the tracking capabilities of the RLS (recursive least-squares) and LMS algorithms, we propose a convex combination of one LMS and one RLS filter. The resulting combination should profit of the best tracking behavior of each component filter. A steady-state analysis via energy conservation relation is also presented for stationary and non-stationary environments.

Transient and Steady-State Analysis of the Affine Combination of Two Adaptive Filters

In this paper, we propose an approach to the transient and steady-state analysis of the affine combination of one fast and one slow adaptive filters. The theoretical models are based on expressions for the excess mean-square error (EMSE) and cross-EMSE of the component filters, which allows their application to different combinations of algorithms, such as least mean-squares (LMS), normalized LMS (NLMS), and constant modulus algorithm (CMA), considering white or colored inputs and stationary or nonstationary environments. Since the desired universal behavior of the combination depends on the correct estimation of the mixing parameter at every instant, its adaptation is also taken into account in the transient analysis. Furthermore, we propose normalized algorithms for the adaptation of the mixing parameter that exhibit good performance. Good agreement between analysis and simulation results is always observed.

Tracking issues of some blind equalization algorithms

Due to the growing demand for mobile communications, blind adaptive algorithms have an important role in improving data transmission efficiency. In this context, the convergence and tracking analysis of such algorithms is a problem of interest. Recently, a tracking analysis of the Constant Modulus Algorithm was presented based on an energy conservation relation. In this letter we extend that analysis to blind quasi-Newton algorithms that minimize the Constant Modulus cost function. Under certain conditions, the considered algorithms can reach the same steady-state mean-square error. Close agreement between analytical and simulation results is shown.

Combinations of Adaptive Filters: Performance and convergence properties

Adaptive filters are at the core of many signal processing applications, ranging from acoustic noise supression to echo cancelation [1], array beamforming [2], channel equalization [3], to more recent sensor network applications in surveillance, target localization, and tracking. A trending approach in this direction is to recur to in-network distributed processing in which individual nodes implement adaptation rules and diffuse their estimation to the network [4], [5].

A transient analysis for the convex combination of two adaptive filters with transfer of coefficients

This paper proposes an improved model for the transient of convex combinations of adaptive filters. A previous model, based on a first-order Taylor series approximation of the nonlinear functions that appear in convex combinations, tended to overestimate the variance of the auxiliary variable used to estimate the mixing parameter. In this paper, we apply a second-order Taylor approximation that improves these estimates, and obtains better agreement with simulations. In addition, we also extend the model to include a simple mechanism for the transfer of coefficients between the constituent filters, a procedure that greatly improves the convergence of the overall filter, and provide an expression to select the free parameter used in such a scheme.

A transient analysis for the convex combination of adaptive filters

Combination schemes are gaining attention as an interesting way to improve adaptive filter performance. In this paper we pay attention to a particular convex combination scheme with nonlinear adaptation that has recently been shown to be universal -i.e., to perform at least as the best component filter- in steady-state; however, no theoretical model for the transient has been provided yet. By relying on Taylor Series approximations of the nonlinearities, we propose a theoretical model for the transient behavior of such convex combinations. In particular, we provide expressions for the time evolution of the mean and the variance of the mixing parameter, as well as for the mean square overall filter convergence. The accuracy of the model is analyzed for the particular case of a combination of two LMS filters with different step sizes, explaining also how our results can help the designer to adjust the free parameters of the scheme.

On the tracking performance of combinations of least mean squares and recursive least squares adaptive filters

Combinations of adaptive filters have attracted attention as a simple solution to improve filter performance, including tracking properties. In this paper, we consider combinations of LMS and RLS filters, and study their performance for tracking time-varying solutions. We show that a combination of two filters from the same family (i.e., two LMS or two RLS filters) cannot improve the performance over that of a single filter of the same type with optimal selection of the step size (or forgetting factor). However, combining LMS and RLS filters it is possible to simultaneously outperform the optimum LMS and RLS filters. In other words, combination schemes can achieve smaller errors than optimally adjusted individual filters. Experimental work in a plant identification setup corroborates the validity of our results.

Affine combinations of adaptive filters

We extend the analysis presented in for the affine combination of two least mean-square (LMS) filters to allow for colored inputs and nonstationary environments. Our theoretical model deals, in a unified way, with any combinations based on the following algorithms: LMS, normalized LMS (NLMS), and recursive-least squares (RLS). Through the analysis, we observe that the affine combination of two algorithms of the same family with close adaptation parameters (step-sizes or forgetting factors) provides a 3 dB gain in relation to its best component filter. We study this behavior in stationary and nonstationary environments. Good agreement between analytical and simulation results is always observed. Furthermore, a simple geometrical interpretation of the affine combination is investigated. A model for the transient and steady-state behavior of two possible algorithms for estimation of the mixing parameter is proposed. The model explains situations in which adaptive combination algorithms may achieve good performance.

A Soft-Switching Blind Equalization Scheme via Convex Combination of Adaptive Filters

Blind equalizers avoid the transmission of pilot sequences, allowing a more efficient use of the channel bandwidth. Normally, after a first rough equalization is achieved, it is necessary to switch these equalizers to a decision-directed (DD) mode to reduce the steady-state mean-square error (MSE) to acceptable levels. The selection of an appropriate MSE threshold for switching between the blind and the DD modes is critical to obtain a good overall performance; however, this is not an easy task, since it depends on several factors such as the signal constellation, the communication channel, or the signal-to-noise ratio. In this paper, we propose an equalization scheme that adaptively combines a blind and a DD equalizers running in parallel. The combination is itself adapted in a blind manner, and as a result the overall scheme can automatically switch between the component filters, avoiding the need to set the transition MSE level a priori. The performance of our proposal is illustrated both analytically and through an extensive set of simulations, where we show its advantages with respect to existing hard- and soft-switching equalization schemes.