LMS Algorithm and RLS Algorithm with Comparative Analysis

In this paper, we provide an in-depth examination of two fundamental digital signal processing algorithms: the LMS (Least Mean Squares) algorithm and the RLS (Recursive Least Squares) algorithm, along with their comparative analysis. The LMS algorithm serves as an adaptive filter that estimates and filters input signals to minimize error through gradient descent optimization. Its implementation typically involves iterative weight updates using a step-size parameter μ, with core operations including error calculation (e(n) = d(n) - y(n)) and coefficient adjustment (w(n+1) = w(n) + μ·e(n)·x(n)). The RLS algorithm represents a recursive least squares filter that updates filter parameters based on previous input signals and measurements to enhance estimation accuracy through matrix inversion lemma operations. Key implementation aspects include recursive computation of the inverse correlation matrix P(n) and Kalman gain vector K(n), enabling faster convergence than LMS at the cost of higher computational complexity O(N²). We thoroughly discuss both algorithms' working principles, advantages/disadvantages, and application domains in areas such as system identification, noise cancellation, and channel equalization. Furthermore, we provide annotated explanations highlighting critical implementation considerations, including MATLAB code segments demonstrating filter initialization, iteration loops, and convergence behavior monitoring to assist readers in better understanding algorithm implementation and practical usage scenarios.