MATLAB Implementation of Recursive Least Squares (RLS) Algorithm with Code Explanation

Recursive Least Squares (RLS) is an efficient adaptive filtering algorithm widely used in system identification, signal processing, and parameter estimation. Compared to conventional least squares methods, RLS avoids repetitive computations through recursive updates, significantly improving computational efficiency. Implementing the RLS algorithm in MATLAB centers around maintaining and updating a covariance matrix throughout iterations. The implementation typically begins by initializing the weight vector and covariance matrix - the weight vector is usually initialized to zeros or small random values, while the covariance matrix starts as a large identity matrix. In MATLAB code, this might appear as: weights = zeros(n,1); P = delta*eye(n); where delta is a large positive constant. During each iteration, the RLS algorithm calculates the gain vector and updates both the weights and covariance matrix. This process relies heavily on selecting an appropriate forgetting factor (λ), which determines the weight given to historical data. A larger forgetting factor means the algorithm maintains stronger memory of past data. The MATLAB implementation involves computing the gain vector: K = P*x/(lambda + x'*P*x); then updating weights: weights = weights + K*error; and finally adjusting the covariance matrix: P = (P - K*x'*P)/lambda; The MATLAB implementation of RLS can be broken down into these key steps: parameter initialization, gain vector calculation, weight estimation update, and covariance matrix adjustment. Compared to the LMS algorithm, RLS demonstrates faster convergence rates, making it particularly suitable for handling non-stationary signals. This algorithm performs exceptionally well in real-time systems such as channel estimation in communication systems or adaptive noise cancellation. By fine-tuning the forgetting factor and regularization parameters, developers can balance the algorithm's tracking capability against steady-state error performance.