On convergence and optimality of maximum-likelihood APA

TL;DR

This paper introduces ML-APA, a maximum likelihood-based adaptive parameter setting method for the affine projection algorithm, providing theoretical convergence analysis and demonstrating superior performance over existing methods.

Contribution

The paper derives a new ML-based approach for adaptively tuning APA parameters and fully characterizes its convergence and optimality properties for memoryless Gaussian inputs.

Findings

ML-APA converges to zero expected misalignment error at rate O(1/t)

ML-APA achieves asymptotic optimality in error performance

Incremental ML-APA outperforms ML-APA in simulations

Abstract

Affine projection algorithm (APA) is a well-known algorithm in adaptive filtering applications such as audio echo cancellation. APA relies on three parameters: $P$ (projection order), $\mu$ (step size) and $\delta$ (regularization parameter). It is known that running APA for a fixed set of parameters leads to a tradeoff between convergence speed and accuracy. Therefore, various methods for adaptively setting the parameters have been proposed in the literature. Inspired by maximum likelihood (ML) estimation, we derive a new ML-based approach for adaptively setting the parameters of APA, which we refer to as ML-APA. For memoryless Gaussian inputs, we fully characterize the expected misalignment error of ML-APA as a function of iteration number and show that it converges to zero as $O({1\over t})$. We further prove that the achieved error is asymptotically optimal. ML-APA updates its estimate once every $P$ samples. We also propose incremental ML-APA (IML-APA), which updates the coefficients at every time step and outperforms ML-APA in our simulations results. Our simulation results verify the analytical conclusions for memoryless inputs and show that the new algorithms also perform well for strongly correlated input signals.