Normalized Least Mean Square (NLMS) and Recursive Least Squares (RLS) Algorithms for Adaptive Filtering Solutions

This article explores two prominent approaches for implementing adaptive filters: the Normalized Least Mean Square (NLMS) algorithm and the Recursive Least Squares (RLS) algorithm. We provide comprehensive MATLAB source code implementations for both methods, designed to offer clear guidance and support for researchers and engineers working in this field. The NLMS algorithm implementation demonstrates how step-size normalization prevents gradient noise amplification while maintaining computational efficiency through weight update equations that adapt to input signal power variations. The RLS implementation showcases exponential weighting of past data samples using a forgetting factor, along with efficient matrix inversion through the Sherman-Morrison formula for real-time covariance matrix updates. We conduct an in-depth comparative analysis of both methods' advantages and limitations, including convergence speed, computational complexity, and numerical stability considerations. Practical application scenarios illustrate how to deploy these algorithms effectively for real-world signal processing challenges such as system identification, acoustic echo cancellation, and channel equalization. Through this comprehensive examination, readers will gain deep insights into adaptive filtering principles and learn how to optimize these algorithms to enhance filter performance and operational efficiency in various engineering applications.