Determination of Theoretical Kalman Filter Performance for Atomic Clock Estimation Through Equivalent Linear Time-Invariant Systems

Kalman filters (KFs) are widely used in atomic clock estimation. However, their theoretical performances are not strictly determined by traditional methods when the observation models and process (
Q
) and observation (
R
) noise variances are not consistent with actual observations. This article proposes a new method of studying the classic two-state KF in these cases. The core idea is that a KF in the steady state can be treated as equivalent to three linear time-invariant (LTI) systems. Then, the theoretical performance of each term in the KF estimates can be accurately and independently determined by using the superposition and homogeneity of the LTI systems, whose structures and coefficients are completely determined by the KF model and chosen
Q
and
R
values and independent of actual observations. An example is used in which three kinds of terms are involved in the observation, and three submethods are proposed. For the polynomial and periodic fluctuation terms, the analytical expressions of corresponding terms in the KF estimates are derived. For the noise term, the noise filtering performance is accurately determined in the frequency domain. By combining the new and traditional methods, the complete conclusions are obtained. The experimental results validate the theoretical conclusions. The research indicates that this method can effectively determine the theoretical performances of KFs when the KF model and
Q
and
R
values are not consistent with actual observations.