Synchronization by Two-Way Message Exchanges: Cramér-Rao Bounds, Approximate Maximum Likelihood, and Offshore Submarine Positioning

Accurate clock synchronization is vital to many applications of wireless sensor networks (WSNs). The availability of a mathematical tool that at an early design stage can provide insight into the theoretically achievable performance of the clock synchronization may accordingly be valuable in the initial design phase of the network. Therefore, the achievable clock synchronization accuracy is examined in a WSN employing a two-way message exchange model under a Gaussian assumption. The Cramer-Rao bound for the estimation of the clock parameters is derived for four different parameterizations (i.e., different nuisance parameters), reflecting different levels of prior knowledge concerning the system parameters. The results on the Cramer-Rao bound are transformed into a lower bound on the mean square error of the clock offset, a figure of merit often more relevant, characterizing the system performance. Further, by introducing a set of artificial observations through a linear combination of the observations originally obtained in the two-way message exchange, an approximate maximum likelihood estimator for the clock parameters is proposed. The estimator is shown to be of low complexity and it obeys near-optimal performance, that is, a mean square error in the vicinity of the Cramer-Rao bound. The applicability of the derived results is shown through a simulation study of an offshore engineering scenario, where a remotely operated underwater vehicle is used for operations at the seabed. The position of the vehicle is tracked using a WSN.