LMS Algorithm and Its Improved Variants in Adaptive Filtering

This paper conducts a comparative analysis of the standard LMS algorithm and its enhanced versions such as the Normalized LMS (NLMS), Variable Step-Size LMS, and Transform-Domain LMS algorithms, focusing on their structural differences and implementation approaches. The NLMS algorithm improves stability by normalizing the step size using input signal power estimation, typically implemented through a sliding window or recursive power calculation. Variable Step-Size LMS dynamically adjusts the convergence rate using error-dependent step size functions, requiring additional logic for step size adaptation. Transform-Domain LMS applies orthogonal transformations (like FFT) to decorrelate input signals, enhancing convergence performance through frequency-domain processing. Building upon the traditional LMS framework, the study further explores advanced applications of these algorithms in real-time signal processing systems. Additionally, through analytical examination of the RLS algorithm's matrix operations and exponential weighting mechanisms, the paper contrasts its convergence properties and computational demands with LMS-based approaches, particularly noting RLS's faster convergence but higher O(n²) complexity versus LMS's O(n) operations per iteration. This comparative analysis provides deeper insights into algorithm selection criteria for adaptive filtering applications.