A New Variable Forgetting Factor and Variable Regularized Square Root Extended Instrumental Variable PAST Algorithm With Applications

Abstract

This article proposes a square-root (SR) extended instrumental variable (EIV) projection approximation subspace tracking (PAST) algorithm with variable forgetting factor (VFF) and variable regularization (VR). A new local polynomial modeling (LPM) based VFF is proposed by minimizing the mean squares deviation of the EIV linear model and the IV-PAST algorithm. A new variable $\ell 2$ regularization term is also derived to reduce the variance of the estimator resulting from possibly ill conditioned covariance matrix at low input signal level. An SR version of the algorithm is developed to improve the numerical stability of the algorithm and avoid the problem of loss of positive definiteness of the inverse covariance matrix. The proposed LOFF-VR-SREIV-PAST algorithm can be implemented by both the conventional EIV-PAST algorithm and numerically more stable hyperbolic rotations. Furthermore, the convergence of the proposed VFF-EIV-PAST algorithm using the ordinary differential equation method is analyzed. Its application to the estimation and tracking of direction of arrival under spatial color sensor noise in both stationary and nonstationary scenarios are presented. Simulations demonstrate that the proposed algorithm yields improved performance over the conventional PAST and EIV-PAST algorithms, especially at medium to low signal-to-noise ratio, which is more frequently encountered in practical situations.