Variable Tap-Length LMS Algorithm with Adaptive Step Size

The fractional tap-length least mean square adaptive algorithm exhibits robustness and low complexity in adaptive filter design. This algorithm employs one parameter γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}, the tap-length adaptation step size, to balance convergence rate and steady-state tap-length fluctuation. From the viewpoint of threshold parameter, we justify that a time-varying γ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (n)$$\end{document} with large value at transient stage and small value at steady stage, instead of a fixed one, can provide both fast convergence rate and small fluctuation. Then, one time-varying strategy for γ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (n)$$\end{document} is suggested, where the parameter is adjusted by the difference between squared output error and squared segmented estimation error, and is limited by a sigmoid function. This strategy is motivated by the recognition that such difference indicates transient or steady state.