Analysis of the LMS and NLMS algorithms using the misalignment norm

• Published in: Signal, image and video processing Vol. 17; no. 7; pp. 3623 - 3628
• Main Author: Lopes, Paulo A. C.
• Format: Journal Article
• Language: English
• Published: London Springer London 01.10.2023; Springer Nature B.V
• Subjects: Algorithms; Approximation; Background noise; Computer Imaging; Computer Science; Expected values; Image Processing and Computer Vision; Kurtosis; Mean square errors; Misalignment; Multimedia Information Systems; Original Paper; Pattern Recognition and Graphics; Signal,Image and Speech Processing; Vision; NLMS; LMS; Stability; Misalignment; Analysis; Misadjustment
• ISSN: 1863-1703; 1863-1711; 1863-1711
• DOI: 10.1007/s11760-023-02588-x

This work describes the convergence of the misalignment square norm (MSN) of the NLMS and LMS algorithms. It is shown that the MSN decrease is almost proportional to the mean square error (MSE). This allows obtaining simple expressions for the steady-state MSE. Also, it allows limiting the amount of time that the MSE takes large values and a curve that limits the MSE of LMS at any given time, independent of the input and background noise signals’ properties. Finally, it is also shown that many complications in the analysis of the LMS and NLMS algorithms can come from variations in the input vector square norm. The proposed analysis becomes very simple for long filters or constant power signals.