Analysis of the widely linear complex Kalman filter

The augmented complex Kalman filter (ACKF) has been recently proposed for the modeling of noncircular complex-valued signals for which widely linear modelling is more suitable than a strictly linear model. This has been achieved in the context of neural network training, however, the extent to which the ACKF outperforms the conventional complex Kalman filter (CCKF) in standard adaptive filtering applications remains unclear. In this paper, we show analytically that the ACKF algorithm achieves a lower mean squared error than the CCKF algorithm for noncircular signals. The analysis is supported by illustrative simulations.