LMS Algorithm and Its Improved Variants

In this article, we focus on examining the differences between the LMS algorithm and its improved variants (such as the NLMS algorithm, variable step-size LMS algorithm, and transform-domain LMS algorithm), while introducing their practical applications. We conduct an in-depth investigation into the characteristics, advantages, limitations, and implementations of these algorithms across various domains including signal processing, image processing, and speech recognition. From a code perspective, the core LMS algorithm employs a gradient descent approach with a fixed step-size parameter μ for weight vector updates: w(n+1) = w(n) + μe(n)x(n). The NLMS enhancement normalizes the step size by the input power to improve stability. Variable step-size implementations dynamically adjust μ based on error signals to balance convergence speed and steady-state performance. Transform-domain variants apply orthogonal transformations (like FFT) to decorrelate input signals for faster convergence. Furthermore, we explore methods to enhance algorithm performance through modifications like adaptive filtering structures and discuss extending their applications to broader domains such as adaptive noise cancellation and system identification. Ultimately, this article aims to deliver a comprehensive and thorough exploration of LMS algorithms and their improvements, facilitating better understanding and implementation to provide robust support for research and development in related fields.