An augmented complex-valued Lyapunov stability theory based adaptive filter algorithm

- A novel algorithm (ACLAF) is derived by using the LST and augmented statistics.
- The ACLAF algorithm provides a significant performance gain in noncircular signals.
- Rigorous analyses of the ACLAF algorithm are presented.
- Its performance is verified on both benchmark and real-world data.

A novel augmented complex-valued Lyapunov stability theory (LST) based adaptive filter (ACLAF) algorithm is proposed for the widely linear adaptive filtering of noncircular complex-valued signals. After a candidate Lyapunov function is determined, the design procedure is formulated as an inequality constrained optimization problem by using augmented statistics and LST. Thus, the proposed algorithm has improved the adaptive filtering of noncircular complex-valued signals by a unified framework of the LST and augmented complex statistics. Moreover, we statistically show that the ACLAF algorithm converges to the optimal Wiener solution under stationary environments, the required condition of the step size for the stability of the ACLAF algorithm is obtained by convergence in mean analysis and a new approach. In addition, the variance of the ACLAF algorithm is statically analysed in this study. The performance of the ACLAF algorithm is tested on circular and noncircular benchmark signals and on a real-world noncircular wind signal. Simulation results verify that the ACLAF algorithm outperforms complex-valued LST based adaptive filter (CLAF), complex-valued least mean square (CLMS), complex-valued normalized least mean square (CNLMS), augmented CLMS (ACLMS) and augmented CNLMS (ACNLMS) algorithms for adaptive prediction of noncircular signals in terms of prediction gain, convergence rate and mean square error (MSE). Also, the ACLAF algorithm enhances the prediction gain by more than 25% when compared to the other augmented algorithms.