Least-Squares Adaptation of Affine Combinations of Multiple Adaptive Filters Luis A. Azpicueta-Ruiz ∗ , Marcus Zeller † , Aníbal R. Figueiras-Vidal ∗ , and Jerónimo Arenas-García ∗ ∗ Department of Signal Theory and Communications Universidad Carlos III de Madrid, 28911 Leganés-Madrid, Spain Email: {azpicueta, arfv, jarenas}@tsc.uc3m.es † Multimedia Communications and Signal Processing University of Erlangen-Nuremberg, Cauerstr. 7, 91058 Erlangen, Germany Email: zeller@LNT.de Abstract— Adaptive combinations of adaptive filters are gaining pop- ularity as a flexible and versatile solution to improve adaptive filters performance. In the recent years, combination schemes have focused on two different approaches: Convex and affine combinations, developing principally practical implementations with just two component filters. However, combinations of a higher number of adaptive filters can offer additional advantages, mainly in tracking environments. In this paper, we introduce a practical adaptation scheme for the affine combination of an arbitrary number of filters, including a steady-state analysis where the proposed rule is compared with the optimal combination. Several experiments both in tracking and stationary scenarios serve to demonstrate the appropriate performance of this approach. I. I NTRODUCTION Adaptive combinations of adaptive filters with different properties is becoming a very useful and flexible approach in order to alleviate the different compromises that condition the operation of adaptive filters [1]–[3]. Variable step size schemes have been traditionally used with this purpose, but they normally introduce several parameters whose appropiate tunning needs some a priori knowledge about the statistics of the filtering scenario. Recently, algorithms based on com- binations of filters have found application in several areas of signal processing, including echo cancellation [4] and distributed estimation [5], among others. The key concept of combination schemes is that the overall filter behaves as well as the best of the contributing filters, and, under certain circumstances, even better [1]. Different schemes can be applied to mix the outputs of the component filters, including convex and affine linear combinations. Recently, theoretical comparisons between both kinds of combination schemes, in the case of two filter components, have shown that the affine combination can present some advantages in terms of overall filter performance in certain situations [2], [3]. These results demonstrate that the optimal combiner exhibits a negative mixing weight in a stationary scenario, allowing the affine combination to outperform the best of its components. The relation between affine and convex combinations has also been investigated theoretically in [6], extending the previous results and developing a formulation for the case of a combination of three filters. Up to now, several adaptation schemes have been proposed for the affine case, with particular focus on the case of just two contributing filters. For instance, in [2] a least-mean-square (LMS) adaptation scheme was proposed, and two different normalized versions of such approach were presented in [3]. We developed a different adaptation scheme for the mixing parameter based on the solution to a least- squares (LS) problem [7]. However, combination of more than two component filters can provide additional advantages with respect to the combination of just two elements, mainly in tracking situations, where, depending on the speed of changes, one of the components can clearly outperform the others. In this paper, a new practical adaptation scheme for the affine combination of multiple filters is presented, which can be considered as a generalization of [7]. It should be noted that convex combinations of multiple filters have already been considered in [10]–[11]. The paper is organized as follows: a description of the algorithm is provided in Sec. 2, as well as an analysis of the adaptation rule for the combination. Sec. 3 includes several experiments that show the performance of the proposed scheme both in stationary and tracking scenarios. The last section summarizes the conclusions of our work. II. LEAST- SQUARES ADAPTATION OF A COMBINATION OF SEVERAL ADAPTIVE FILTERS A. Algorithm description Consider an affine combination of K adaptive filters which obtains the overall filter output at time n as y(n)= K  k=1 λ k (n)y k (n) (1) where y k (n), k =1, ..., K, are the outputs of the component filters and λ k (n) are the mixing parameters satisfying ∑ K k=1 λ k (n)=1, that are adapted at each iteration to optimize the overall performance. In order to incorporate the affine constraint into the scheme, we will simply adapt the first K - 1 weights and consider that λK(n)= 1 - ∑ K-1 k=1 λ k (n). For the adaptation of the K - 1 mixing parameters we will minimize the LS cost function J (n)= n  i=1 β(n, i)e 2 (n, i), (2) where β(n, i) is a temporal weighting window and e(n, i)= d(i) - y(n, i), d(i) being the desired signal at time instant i and y(n, i) = K-1  k=1 λ k (n)y k (i)+  1 - K-1  k=1 λ k (n)  yK(i) = yK(i)+ K-1  k=1 λ k (n)[y k (i) - yK(i)] (3) represents the output of the overall filter when the outputs of the con- stituent filters at time i are combined by means of the weights at time n. Although an exponentially-weigthed window β(n, i) facilitates the formulation as a recursive LS (RLS) problem, allowing savings in terms of computational cost, the use of rectangular windows can provide some advantages as discussed in [7]. 978-1-4244-5309-2/10/$26.00 ©2010 IEEE 2976