Adaptive Filters

CHAPTER 13 Affine Projection Algorithm

The LMS and ∈-NLMS algorithms were obtained by using simple instantaneous approximations for the covariance and cross-covariance quantities {R_du,R_u}. More involved algorithms, with better performance but at increased computational costs, can be obtained by resorting to more sophisticated approximations for {R_du, R_u}. We illustrate this situation by motivating the so-called affine projection algorithm.

13.1 INSTANTANEOUS APPROXIMATION

Just like ∈-NLMS, we again start from the regularized Newton’s recursion (10.8), namely,

(13.1)

albeit with a fixed step-size μ and a fixed regularization parameter . Now, however, we shall employ a better approximation for both the covariance matrix, R_u, and the cross-covariance vector, Rdu. Specifically, we choose a positive integer K (usually K ≤ M, where M × 1 is the size of the weight vector) and replace {R_du, R_u} by the following instantaneous approximations:

In other words, at each iteration i, we use the K most recent regressors and the K most recent observations,

to compute the approximate values for ...