On the robustness of LMS algorithms with time-variant diagonal matrix step-size

The Proportionate Normalized Least Mean Squares (PNLMS) algorithm has been quite successful in combining higher convergence rates with low to moderate complexity that at the same time avoids numerical difficulties in fixed-point implementations. While the algorithm is stable in the mean square and l₂-sense for time-invariant matrices, the treatment of time-variant matrices requires additional approximations. These approximations are discarded in this paper which allows us to analyse the robustness in terms of l₂-stability for actually time-variant matrix step-sizes. This provides important results, as the algorithm in its variants also occurs in other fields of adaptive filtering such as cascaded filter structures. By simulations as well as by theoretical analysis, we demonstrate that in general, even small variations of the matrix step-size are sufficient for the algorithm to loose its robustness. Only in special cases, where specific constraints are imposed additionally, robustness can be guaranteed.