Variable step size partial update LMS algorithm for fast convergence

Partial Update LMS (PU LMS) algorithms started to play an important role in an adaptive processing of sound. Due to reduction of the computational power demands, these algorithms allow to use longer adaptive filters, and therefore achieve better results in adaptive filtering applications, e.g. identification, adaptive line enhancement (ALE), and active noise control. However, adaptation of only a subset of all filter weights (in an extreme situation of only one) invalidates classical LMS algorithm stability and convergence analysis. Moreover, intuitive conclusions concerning convergence may be wrong, e.g. some PU LMS algorithms allow for faster convergence than the full update LMS algorithm. Such behavior may be observed if input signals with specific properties are involved, like signals with large autocorrelation matrix eigenvalue spread. This paper presents a control-theory based approach to the stability analysis of some PU LMS algorithms. New, weak stability sufficient condition is proposed, which is derived based on the stability matrix and the principle of the contraction mapping. Based on this condition, new variable step-size PU LMS algorithm is derived, which allows for fast adaptation and stable operation. The computational complexity overhead of this new algorithm is comparable with the overhead introduced by the normalization in the NLMS algorithm. Simulations with a system identification and with an adaptive line enhancer are presented to confirm good performance of the proposed algorithm.