Recursive Least Squares Algorithm, LMS Algorithm, and Kalman Algorithm Implementations

This article discusses MATLAB implementations of the Recursive Least Squares (LMS) algorithm and the Kalman filter algorithm. These algorithms are widely used in engineering and scientific fields, particularly in signal processing, control systems, and machine learning applications. The Recursive Least Squares algorithm serves as an adaptive filtering technique that incrementally updates filter coefficients to minimize the mean square error, making it effective for noise reduction and interference cancellation in signal enhancement. In MATLAB implementations, this typically involves initializing filter weights, calculating the gain vector, and recursively updating the covariance matrix and filter parameters using real-time data samples. The Kalman filter is an optimal recursive estimation algorithm designed for tracking unknown system states in dynamic environments. It operates through a two-step process: prediction (based on system dynamics) and correction (using measurement updates). Common MATLAB implementations include defining state transition matrices, measurement matrices, process noise covariance, and measurement noise covariance, followed by iterative prediction and update cycles. Key functions often involve matrix operations for covariance propagation and Kalman gain calculation. When studying these algorithms, it's essential to understand their underlying principles, implementation approaches, application scenarios, and relative advantages/disadvantages to effectively apply them in practical engineering solutions. The recursive nature of both algorithms makes them particularly suitable for real-time applications where computational efficiency and memory management are critical considerations.