  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.
