Fundamental limits on phase and frequency tracking and estimation in drifting oscillators

Distributed beamforming requires phase and frequency synchronization. As oscillators drift, through Brownian motion induced phase noise, their instantaneous phases must be tracked and compensated. Several papers and IEEE 1588 have proposed Kalman Filter (KF) based tracking algorithms using the unwrapped phase measurements. This paper quantifies the effect of Brownian Motion induced drift at two levels. First we derive Cramer-Rao Lower Bounds (CRLB) manifest in one shot estimation of frequency and phase from unwrapped phase observations, and reveal fundamental and illuminating differences with the existing frequency and phase estimation CRLBs in the literature derived in the absence of Brownian motion. Second, we consider a KF that tracks the instantaneous phase in intervals where there is no beamforming, and is switched off during beamforming. Bounds are derived relating the error growth as a function of the underlying duty cycle.