Circuits Syst Signal Process (2011) 30: 439–462 DOI 10.1007/s00034-010-9219-z Convergence Performance of the Simplified Set-Membership Affine Projection Algorithm Paulo S.R. Diniz Received: 14 March 2010 / Revised: 25 June 2010 / Published online: 11 November 2010 © Springer Science+Business Media, LLC 2010 Abstract Set-membership (SM) adaptive filtering is appealing in many practical sit- uations, particularly those with inherent power and computational constraints. The main feature of the SM algorithms is their data-selective coefficient update leading to lower computational complexity and power consumption. The set-membership affine projection (SM-AP) algorithm does not trade convergence speed with misadjustment and computation complexity as many existing adaptive filtering algorithms. In this work analytical results related to the SM-AP algorithm are presented for the first time, providing tools to setup its parameters as well as some interpretation to its de- sirable features. The analysis results in expressions for the excess mean square error (MSE) in stationary environments and the transient behavior of the learning curves. Simulation results confirm the accuracy of the analysis and the good features of the SM-AP algorithms. Keywords Adaptive filter analysis · Analysis of the set-membership affine projection algorithm 1 Introduction In recent years the affine projection (AP) algorithm, first proposed in [12, 13], has gained increasing popularity due to its faster convergence than the stochastic gradient algorithms, such as the LMS, and its lower computational complexity than the RLS algorithm [3, 12, 13]. The AP algorithm reuses old data, leading to fast convergence The author would like to thank the financial support provided by CNPq and FAPERJ, national and state research councils, respectively. He also wants to thank Markus V.S. Lima for carefully reading the manuscript and for checking the simulations. P.S.R. Diniz ( ) LPS – Programa de Engenharia Elétrica, COPPE/Poli/UFRJ, Cx. P. 68504, Rio de Janeiro, RJ, Brazil e-mail: diniz@lps.ufrj.br