Cramér-Rao Bounds for Estimation of Linear System Noise Covariances

The performance of Kalman filter depends directly on the noise covariances, which are usually not known and need to be estimated. Several estimation algorithms have been published in past decades, but the measure of estimation quality is missing. The Cramér-Rao bounds represent limitation of quality of parameter estimation that can be obtained from given data. In this article, The Cramér-Rao bounds for noise covariance estimation of linear time-invariant stochastic system will be derived. Two different linear system models will be considered. Further, the performance of earlier published methods will be discussed according to the Cramér-Rao bounds. The analogy between the Cramér-Rao bounds and the Riccati equation will be pointed out.