On the Convergence Speed of the Normalized Subband Adaptive Filter: Some New Insights and Interpretations

In this paper we revisit the well known and popular Normalized Subband Adaptive Filter (NSAF). Based on an analysis of the algorithm in the mean and using an analysis strategy presented in [1], we find that the NSAF can be seen as a Richardson iteration applied to a preconditioned augmented Wiener-Hopf equation. This equation is formulated in such a way that its convergence speed can be predicted directly from a matrix Q(T)R, where Q(T) is a matrix formed as a weighted sum of Toeplitz matrices directly related to the filter banks used and R is a tall, rectangular, autocorrelation matrix. Since the Q(T)R itself is symmetric and Toeplitz, we can make quantitative statements about its convergence speed in much the same way as we can for the least mean square (LMS) algorithm with the important difference that in the NSAF case we can directly use Q(T) to control the convergence speed. Through some experiments we demonstrate the validity of our approach in the prediction of convergence speed. Also, and perhaps more important, we point out that our formalism holds the potential for providing an optimization problem such that the filter banks used in NSAF can be designed based on this rather than merely being selected/postulated as is done presently.