2928 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 11, NOVEMBER 1999 BEACON: An Adaptive Set-Membership Filtering Technique with Sparse Updates Shirish Nagaraj, Student Member, IEEE, Sridhar Gollamudi, Student Member, IEEE, Samir Kapoor, Member, IEEE, and Yih-Fang Huang, Fellow, IEEE Abstract—This paper deals with adaptive solutions to the so- called set-membership filtering (SMF) problem. The SMF method- ology involves designing filters by imposing a deterministic con- straint on the output error sequence. A set-membership decision feedback equalizer (SM-DFE) for equalization of a communica- tions channel is derived, and connections with the minimum mean square error (MMSE) DFE are established. Further, an adaptive solution to the general SMF problem via a novel optimal bound- ing ellipsoid (OBE) algorithm called BEACON is presented. This algorithm features sparse updating, wherein it uses about 5–10% of the data to update the parameter estimates without any loss in mean-squared error performance, in comparison with the conventional recursive least-squares (RLS) algorithm. It is shown that the BEACON algorithm can also be derived as a solution to a certain constrained least-squares problem. Simulation results are presented for various adaptive signal processing examples, including estimation of a real communication channel. Further, it is shown that the algorithm can accurately track fast time variations in a nonstationary environment. This improvement is a result of incorporating an explicit test to check if an update is needed at every time instant as well as an optimal data- dependent assignment to the updating weights whenever an update is required. I. INTRODUCTION T HE PROBLEM of designing a linear-in-parameter filter, given knowledge of the input and the corresponding de- sired output, is studied in this paper. Traditional methodologies include the minimum mean square error (MMSE) filters and their deterministic counterparts [the least-squares error (LSE) filters], which seek a filter by minimizing the 2-norm of the error sequence [1]. Performance of the MMSE filters relies on the accurate knowledge of the statistics of the input and output observations, whereas an off-line LSE procedure requires data processing in batches, which is not computationally attractive. On-line, or recursive, methods to iteratively achieve the same goals include the least-mean squares (LMS) and the recursive least-squares (RLS) algorithms, which have been studied extensively in the literature; see, e.g., [1] and [2]. Manuscript received January 27, 1998; revised April 6, 1999. This work was presented in part at the 1997 Conference on Information Sciences and Systems, Baltimore, MD, and at the 1997 International Symposium on Nonlinear Theory and its Applications Hawaii. This work was supported in part by the National Science Foundation under Grant MIP-9705173, in part by the Center for Applied Mathematics, University of Notre Dame, and in part by the Tellabs Research Center, Mishawaka, IN. The associate editor coordinating the review of this paper and approving it for publication was Prof. Chi Chung Ko. S. Nagaraj, S. Gollamudi, and Y.-F. Huang are with the Laboratory for Image and Signal Analysis, Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA. S. Kapoor is with NEC USA, Inc., Princeton, NJ 08840 USA. Publisher Item Identifier S 1053-587X(99)08310-5. In conventional system identification problems, we have some a priori knowledge about the system to be estimated. Incorporating this in the form of constraints in the estimation procedure leads to solutions that are consistent with that knowledge. In a more general framework of filter design, constraints can be imposed in the estimation procedure if we need assurance of good performance on a deterministic (point- wise) basis. As before, these requirements then have to be incorporated in the design of the estimator/filter to ensure such an acceptable performance. Moreover, by taking the structure and constraints of the problem into account, it is likely that computationally attractive recursive algorithms emerge as possible solutions. With these motivations in mind, the authors have recently introduced, in [3], a methodology for the design of filters that bound the worst-case error achieved by the filter. This method is termed set-membership filtering (SMF), and the resulting filter is called an SM filter [3]. SMF owes its name to the so-called set-membership identification (SMI) technique [4]–[8], which is applicable only for identifying a linear-in- parameter plant with output corrupted by additive bounded noise. It is straightforward to show that SMI is a special case of the SMF problem. The objective of an SM filter is to estimate a member of a so-called feasibility set. This set defines the SM filter’s performance specification. SMF requires that this specification be met for every possible input-desired output pair of data that come from a certain design space. Any closed-form solution of an SMF problem requires accurate characterization of the design space over which the filter is required to meet the specification. Such a description of the design space would, in general, need knowledge of a functional relationship between the input and desired outputs. In problems where such knowledge is unknown or is not accurate enough, we require a tool to estimate a point in the feasibility set in a recursive fashion. It has been shown that one such method is given by the class of optimal bounding ellipsoid (OBE) algorithms. These algorithms were originally developed for SMI [9] and have gained much attention in the past decade [7], [8], [10] due to some of their attractive features. Among others, the OBE algorithms employ a discerning update rule, i.e., they use the data selectively in updating the parameter estimates. In this paper, we present a novel OBE algorithm called the Bounding Ellipsoidal Adaptive CONstrained least-squares (BEACON) algorithm, which shares many of the desirable fea- tures exhibited by the various OBE algorithms [5] developed to date. In addition, however, the proposed algorithm features 1053–587X/99$10.00  1999 IEEE