MATLAB Implementation of QR-RLS Algorithm for Adaptive Filtering

The QR-RLS (QR-Decomposition-based Recursive Least Squares) algorithm is an efficient adaptive filtering technique commonly used in predistortion systems to compensate for nonlinear distortion. This algorithm enhances numerical stability by employing QR decomposition instead of direct matrix inversion, making it suitable for real-time applications. The implementation approach involves several critical steps: During initialization, parameters such as the forgetting factor and filter order must be configured, and the initial input vector needs to be constructed. In MATLAB, this typically involves initializing matrices like the upper-triangular matrix R and the transformation vector using zeros() or eye() functions. The core computation process utilizes Givens rotations to perform QR decomposition, which avoids numerical instability issues common in conventional RLS algorithms. The implementation requires careful handling of rotation matrices and efficient updates to the decomposition. Each iteration updates the weighted error and filter coefficients, enabling real-time adaptation to channel variations. In predistortion applications, the QR-RLS algorithm continuously adjusts predistorter parameters to make the power amplifier output signal closely approximate the ideal linearly amplified signal. Key implementation aspects include maintaining orthogonal matrices and implementing efficient rotation updates using functions like planerot() for Givens rotations. Compared to traditional RLS algorithms, the QR-RLS version demonstrates superior numerical properties, particularly suitable for long-running real-time systems. Implementation considerations include proper selection of regularization parameters and balancing tracking speed with steady-state performance. Through appropriate setting of the forgetting factor, developers can achieve a balance between system change tracking and noise suppression. The algorithm excels in applications requiring high-precision linearization, such as digital predistortion systems, where MATLAB's matrix operations and built-in functions facilitate efficient implementation of the QR decomposition and coefficient updates.