Theoretical Performance of Timescale Steering Algorithm Based on Kalman Filter with Adjustment

Timescale steering algorithms are vital for generating time references. Among them, the “Kalman filter (KF) with adjustment” timescale steering algorithm has been widely adopted. Previous studies have analyzed the experimental performances of these algorithms, but their theoretical performances remain unclear. To fill this gap, based on several key previous research achievements, the transfer functions of the “three state KF with adjustment” steering algorithm are derived when the KF is in a steady state, making this algorithm equivalent to a digital phase locked loop (DPLL). Based on the transfer functions of the equivalent DPLL, the frequency stability and time synchronization error performance of this algorithm can be theoretically determined for the steady state KF. The experimental results confirm the theoretical conclusions. These findings provide theoretical support for previously reported experimental phenomena and subsequent research, indicating that in the future, for a steady state KF, the frequency stability and time synchronization error performance of the algorithm can be theoretically determined for any given free running timescale and reference timescale with arbitrary noise characteristics, rather than relying solely on experimental results.