Time Synchronization in Photon-Limited Deep Space Optical Communications

Random jitter or offset between the transmitter/receiver clocks is an important parameter that has to be accurately estimated for optimal detection of pulse position modulation (PPM) symbols for high-data-rate optical communications. This parameter, in general, is modeled as an unknown random quantity that depends on the clock drift between the transmitter/receiver clocks and the random motion between the transmitter and receiver stations. In this paper, we have modeled the time jitter for two scenarios - phase modulation jitter and frequency modulation jitter. The phase modulation jitter is modeled as a Gaussian random variable which is estimated with the help of a maximum a posteriori probability (MAP) estimator. The frequency modulation jitter is characterized as a random walk, and this leads to the modeling of the jitter as a state space variable in the context of a dynamical system. Since the observations are the photon counts in each slot of a PPM symbol (for both MAP estimation and tracking), the resulting dynamical model is highly nonlinear, and particle filters are employed for tracking the frequency modulation jitter. We evaluate the performance of both the maximum a posteriori estimators and the particle filters in terms of the relative mean-square error and probability of error. We conclude that with MAP estimation and particle filters that estimate/track the time offset, we achieve a significant performance gain in terms of probability of error as compared to systems that do not have a time synchronization system in place.