On the equivalence between steady-state Kalman filter and DPLL

Fundamental results in the literature showed that a class of Kalman filters (KF) converges to a digital phase lock loop (DPLL) structure. Results were proved for second- and third-order KFs. We extend these results to KFs of any order and derive in closed form the equivalent loop filter constants as a linear combination of the steady-state Kalman gains. We also give the inverse relation. Both closed-form relations involve Stirling numbers of the first or second kind. We implement both filters in a global navigation satellite system (GNSS) receiver and illustrate their equivalence with real-world data. These extended theoretical results provide a deeper understanding of the equivalence between KF and DPLL and may be of practical interest in highly dynamic scenarios to design and tune tracking filters. •We prove a class of Kalman filter (KF) converges to a phase lock loop (PLL) for any order•Loop filter constants are sums of KF gains based on Stirling numbers of second kind•We prove a class of PLL has a steady-state KF architecture for any order•KF gains are sums of loop filter constants based on Stirling numbers of first kind