Variable Explicit Regularization in Affine Projection Algorithm: Robustness Issues and Optimal Choice

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A variable regularized affine projection algorithm (VR-APA) is introduced, without requiring the classical step size. Its use is supported from different points of view. First, it has the property of being <emphasis><formula formulatype="inline"> <tex>$H^{\infty}$</tex></formula></emphasis> optimal and it satisfies certain error energy bounds. Second, the time-varying regularization parameter is obtained by maximizing the speed of convergence of the algorithm. Although we first derive the VR-APA for a linear time invariant (LTI) system, we show that the same expression holds if we consider a time-varying system following a first-order Markov model. We also find expressions for the power of the steady-state error vector for the VR-APA and the standard APA with no regularization parameter. Particularly, we obtain quite different results with and without using the independence assumption between the <emphasis emphasistype="bold"><emphasis emphasistype="italic">a priori</emphasis></emphasis> error vector and the measurement noise vector. Simulation results are presented to test the performance of the proposed algorithm and to compare it with other schemes under different situations. An important conclusion is that the former independence assumption can lead to very inaccurate steady-state results, especially when high values of the projection order are used. </para>