Mathematical analysis of bias in the extended Kalman filter

The state estimate measurement update process for the extended Kalman filter (EKF) as used in bearings-only estimation is investigated. Using a simplifying assumption it is shown mathematically that the gain and innovation sequences are correlated due to their joint dependence on the a priori cross-range estimation error. This correlation causes the range and range-rate estimates to be biased. Furthermore, it is shown that the modified gain EKF has the same state estimate update equation but that the gain is different due to a slightly different covariance update equation. Since the correlation between the gain and innovation sequences is not directly related to the covariance update, the claim that the modified gain EKF is unbiased is not substantiated. The difference in the covariance update equation does cause an alteration of the statistical properties of the gain sequence, but does not remove the correlation with the innovation sequence