Set-membership affine projection algorithm with variable data-reuse factor Stefan Werner Signal Processing Laboratory Helsinki University of Technology Email: stefan.werner@hut.fi Paulo S. R. Diniz LPS-Signal Processing Laboratory COPPE/Poli/Univ. Fed. Rio de Janeiro Email: diniz@lps.ufrj.br Jos´ e E. W. Moreira Amazonia Celular Email: Ednelson.Wesen@AmazoniaCelular.com.br Abstract— This paper proposes a new data-selective affine projection algorithm. The algorithm generalizes the ideas of the conventional set-membership affine-projection (SM-AP) al- gorithm to include a variable data-reuse factor. By utilizing the information provided by the data-dependent step size, we propose an assignment rule that automatically adjust the number of data reuses. A particular reduced-complexity implementation of the proposed algorithm is also considered in order to reduce the dimensions of the matrix inversions involved in the computation of update. Simulations show that a significant reduction in the overall complexity can be obtained with the new algorithm as compared with the conventional SM-AP algorithm. In addition, the proposed algorithm retains the fast convergence of the conventional SM-AP algorithm, and the low steady-state error of the SM-NLMS algorithm. I. I NTRODUCTION Set-membership filtering (SMF) [1]–[4] is an approach to reduce computational complexity in adaptive filtering, where the filter is designed such that the output estimation error is upper bounded by a pre-determined threshold. Set-membership adaptive filters (SMAFs) feature reduced complexity imple- mentations due to a data-selective (sparse in time) updating, and a time-varying data-dependent step size that provides fast convergence to a low steady-state error. SMAFs with low computational complexity per update are the set-membership NLMS (SM-NLMS) [2], the set- membership binormalized data-reusing (SM-BNDRLMS) [3], and the set-membership affine projection (SM-AP) [4] algo- rithms. The SM-AP algorithm generalizes the ideas of the SM- NLMS and the SM-BNDRLMS algorithms to reuse constraint sets from a fixed number of past input and desired signal pairs. The resulting algorithm can be seen as a set-membership version of the affine-projection (AP) algorithm [8] with an optimized step size. As with the conventional AP algorithm, a faster convergence of the SM-AP algorithm comes at the expense of an increase in the computational complexity per update that is directly linked to the amount of reuses employed, or data-reuse factor. The idea of data reuse was also exploited in the context of OBE algorithms in [5]–[7]. The algorithm considered in this paper is essentially different from those of [5]–[7] as we use a point-wise approach to derive a data-selective algorithm with a computational complexity per update in the order of O(N ), N being the number of coefficients. In the following we propose a novel SM-AP algorithm that employs a variable data-reuse factor to lower the overall complexity of the conventional SM-AP algorithm. By quan- tizing the range of the data-dependent step size in the SM-AP algorithm, a proper data-reuse factor is adaptively assigned. The introduction of the variable data-reuse factor results in an algorithm that close to convergence take the form of the simple SM-NLMS algorithm. Thereafter, we consider an efficient implementation of the new SM-AP algorithm that reduces the dimensions of the matrices involved in the update. Finally, the performance of the proposed algorithm is evaluated through simulations which are followed by conclusions. II. SMF Set-membership filtering is a framework applicable to filter- ing problems that are linear in parameters. A specification on the filter parameters w ∈ C N is achieved by constraining the magnitude of the output estimation error, e = d k −w H x k , to be smaller than a deterministic threshold γ , where x k ∈ C N and d k ∈ C denote the input vector and the desired output signal, respectively. As a result of the bounded error constraint, there will exist a set of filters rather than a single estimate. Adaptive SMF algorithms seek solutions that belong to the exact membership set ψ k constructed from the observed input- signal and desired signal pairs, ψ k = k  i=1 H i (1) where H k is referred to as the constraint set containing all the vectors w for which the associated output error at time instant k is upper bounded in magnitude by γ : H k = {w ∈ C N : |d k − w H x k |≤ γ } (2) Adaptive approaches leading to algorithms with low peak complexity compute a point estimate through projections using information provided by a fixed number of past constraint sets [2]–[4]. Next section, we derive an algorithm that uses a time-varying number of past constraint sets for the update. 261 ISCAS 2006 0-7803-9390-2/06/$20.00 ©2006 IEEE