Extensions to the Theory of Widely Linear Complex Kalman Filtering

For an improper complex signal
x
, its complementary covariance Exx
^T
is not zero and thus it carries useful statistical information about
x
. Widely linear processing exploits Hermitian and complementary covariance to improve performance. In this paper, we extend the existing theory of widely linear complex Kalman filters (WLCKF) and unscented WLCKFs [D. P. Mandic and S. L. Goh,Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models (New York: Wiley, 2009)]. We propose a WLCKF which can deal with more general dynamical models of complex-valued states and measurements than the WLCKFs in Mandic and Goh. The proposed WLCKF has an equivalency with the corresponding dual channel real KF. Our analytical and numerical results show the performance improvement of a WLCKF over a complex Kalman filter (CKF) that does not exploit complementary covariance. We also develop an unscented WLCKF which uses modified complex sigma points. The modified complex sigma points preserve complete first and second moments of complex signals, while the sigma points in Mandic and Goh only carry the mean and Hermitian covariance, but not complementary covariance of complex signals.