The stationary phase error distribution of a digital phase-locked loop

The stationary properties of a first-order digital phase-locked loop based on the extended Kalman filter (EKF-PLL) are investigated. A discrete Markov chain approximation of the phase error process is used to derive the asymptotic distribution of the phase error as well as the distribution of the time at which the EKF-PLL is first "out of lock", given that it is in a steady state to begin with. These approximations are compared with large computer simulations.