A theory on the convergence behavior of the affine projection algorithm

In this paper, we present a theoretical convergence analysis of the affine projection algorithm (APA) based on the arguments of energy conservation. Although the APA and its convergence analysis have been widely studied, the dependency of weight-error vector on past noise is usually neglected for simplicity. To obtain accurate theoretical results for the APA, we here consider the dependency between the weight-error vector and past noise in the mean-square analysis presented by Shin and Sayed in ["Mean-square performance of a family of affine projection algorithms," IEEE Transactions on Signal Processing, vol. 52, no. 1, pp. 90-102, January 2004]. Through this work, we can also theoretically analyze the behavior of the periodic APA, which updates its weights periodically. Simulation results show that our theoretical results coincide closely with simulations.