Convergence analysis results for the class of affine projection algorithms

Over the last decade, a class of equivalent algorithms called the affine projection class of algorithms, which accelerate the convergence of the normalized LMS (NLMS) algorithm, has been discovered independently. The APA algorithms update weight estimates on the basis of multiple input signal vectors. In this paper, we present the results of the convergence analysis of the APA class of algorithms using a simple model for the input signal vectors. Conditions for convergence of the algorithms are presented. The convergence rate of APA is exponential, and it improves as the number of input signal vectors used for adaptation is increased. However, the rate of improvement in performance (time-to-steady-state) diminishes as the number of input signal vectors increases. For a given convergence rate, APA algorithms exhibit less misadjustment (steady state error) than NLMS. Simulation results are provided to corroborate the analytical results