Widely Linear Kalman Filtering

For a zero mean, proper, complex random vector x, the Hermitian covariance Exx is a complete second-order characterization. However, if the vector x is improper, it is correlated with its complex conjugate, meaning Exx 6= 0. This improper or complementary covariance must be accounted for in a complete second-order characterization. The improper covariance has been exploited for widely linear (WL) Wiener filters and WL minimum mean squared error (MMSE) estimators, and the improvement in performance of the WLMMSE estimator over the LMMSE estimator has been quantified. In this paper we consider the design of the widely linear Kalman filter (WLKF). We analyze the WLKF, extended WLKF, and unscented WLKF. The key idea of this paper is to modify the error covariance matrices and the construction of effective sigma points in the WLKF in a systematic way that exploits the Hermitian and complementary covariance of improper states and noises.