Widely linear complex extended Kalman filters

Complex signals are generally second order noncircular (improper), that is, their probability distributions are rotation dependent, and conventional algorithms that assume second order circular distributions are generally inadequate. Recently the widely linear (augmented) complex extended Kalman filter (ACEKF), which utilises augmented complex statistics, has been proposed for dealing with the generality of complex signals, both second order circular and noncircular. In this paper, we analyse the ACEKF and show that it has an equivalent (dual) real valued extended Kalman filter, and that this duality can be used to reduce its computational complexity. We also provide a mean square analysis of the linear conventional complex Kalman filter (CCKF) and the augmented complex Kalman filter (ACKF), and show that the ACKF has superior performance for second order noncircular signals. Simulations using both synthetic and real world proper and improper signals support the analysis.