Complementary mean square deviation and stability analyses of the widely linear recursive least squares algorithm

Abstract

The widely linear recursive least squares (WL-RLS) algorithm is useful for dealing with noncircular signals. In this paper, we focus on the theoretical prediction and stability of the WL-RLS algorithm. First, the steady-state complementary mean square deviation (CMSD) behavior under a time-varying system is analyzed, which is different from the traditional standard mean square analysis that takes the form of real-valued evolution. Second, the Lyapunov stability and numerical stability are taken into account to provide an in-depth understanding of the stability issue. Specifically, by employing the Lyapunov stability theory (LST) approach, we perform the stability analysis of the WL-RLS algorithm for time-invariant system and noise-free output, illustrating that the WL-RLS algorithm is exponentially convergent in the sense of Lyapunov. Moreover, the numerical stability is analyzed by considering the finite word-length effects, and it is shown that the WL-RLS algorithm is numerically stable when the forgetting factor λ<1. Numerical simulations for system identification scenarios verify the accuracy of the steady-state theoretical prediction.