IEEE SIGNAL PROCESSING LETTERS, VOL. 8, NO. 8, AUGUST 2001 231 Set-Membership Affine Projection Algorithm Stefan Werner and Paulo S. R. Diniz Abstract—This letter presents a new data selective adaptive filtering algorithm, the set-membership affine projection (SM-AP) algorithm. The algorithm generalizes the idea of the recently proposed set-membership NLMS (SM-NLMS) algorithm to include constraint sets constructed from the past input and desired signal pairs. The resulting algorithm can be seen as a set-membership version of the affine-projection (AP) algorithm with an optimized step size. Also, the SM-AP algorithm does not trade convergence speed with misadjustment and computational complexity as most adaptive filtering algorithms. Simulations show the good performance of the algorithm, especially for colored input signals, in terms of convergence, final misadjustment, and reduced computational complexity. Index Terms—Adaptive filter, affine projections, data selective, normalized data-reusing algorithms, set-membership filtering. I. INTRODUCTION F OR HIGHLY correlated input signals, the recursive least squares (RLS) algorithms are known to present faster con- vergence than the least mean square (LMS) algorithm and its normalized version, the NLMS algorithm [1]. This advantage comes at the expense of a higher computational complexity. Data-reusing algorithms [2], [3] are known to be a viable al- ternative to the RLS algorithm in terms of lower computational complexity in situations where the input signal is correlated. The penalty to be paid when increasing the number of data reuse is a slight increase in algorithm misadjustment. Tradeoff between final misadjustment and convergence speed is achieved through the introduction of a step-size, which is not the best solution. An alternative solution to this drawback is to employ the con- cept of set-membership filtering (SMF) [4] to data reusing algo- rithms. SMF specifies an upper bound on the estimation error and reduces computational complexity on the average due to its data-discerning property. The set-membership NLMS (SM- NLMS) algorithm proposed in [4] was shown to achieve both fast convergence and low misadjustment, and its data-selectivity and low computational complexity per update makes it very at- tractive in various applications [5], [6]. An early attempt in this direction was the introduction of the set-membership binormal- ized data-reusing LMS algorithm (SM-BNDRLMS) [7]. This paper generalizes the ideas in [7] by adopting past data-pairs. The resulting algorithms include the SM-NLMS and SM-BN- DRLMS as special cases, which correspond to choosing , and , respectively. The conventional affine-projection Manuscript received September 1, 2000. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. G. Ram- poni. S. Werner is with the Signal Processing Laboratory, Helsinki University of Technology, FIN-02015, Espoo, Finland (e-mail: stefan.werner@hut.fi). P. S. R. Diniz is with COPPE/Universidade Federal do Rio de Janeiro, Rio de Janeiro 21945-970, Brazil (e-mail: diniz@lps.ufrj.br). Publisher Item Identifier S 1070-9908(01)05397-4. (AP) algorithm [2] is also shown to be a particular limiting case of the new algorithms, when the predefined bound of the esti- mation error goes to zero. The paper is organized as follows. Section II briefly reviews the concept of set-membership filtering. The new algorithm, the set-membership affine projection (SM-AP) algorithm, is de- rived in Section III. Section IV contains the simulations and Sec- tion V the concluding remarks. II. SET-MEMBERSHIP FILTERING (SMF) SMF specifies an upper bound on the magnitude of the es- timation error . This bound is a design param- eter and can vary with the specific application. The SMF ex- tends the set-membership identification (SMI) problem [8] with its bounded noise assumption to include more general filtering problems. For a properly chosen bound , there are several valid estimates of . Let denote the set of all possible input-desired data pairs of interest and the set of vectors with estimation errors upper bounded in magnitude by whenever . The set is referred to as the feasibility set and is given by (1) Adaptive solutions try to find estimates belonging to this feasi- bility set. We define for an input-desired data pair at time instant the constraint set containing all vectors with estimation errors upper bounded in magnitude by (2) The membership set defined as (3) will contain and will be equal to if all data pairs belonging to are traversed up to time instant . Since in (3) is not easily computed, adaptive approaches are needed [4]. The sim- plest approach computes a point estimate using, for example, the information provided by the constraint set as in the set-mem- bership NLMS (SM-NLMS) algorithm, or and as is done in the set-membership binormalized data-reusing LMS (SM-BNDRLMS) algorithm. III. SET-MEMBERSHIP AFFINE PROJECTION ALGORITHM SM-AP The membership set defined in (3) suggests the use of more constraint-sets in the update. This section derives an al- gorithm whose updates belong to a set formed by constraint sets. 1070–9908/01$10.00 © 2001 IEEE