A Factor Graph Approach to Clock Offset Estimation in Wireless Sensor Networks

Abstract

The problem of clock offset estimation in a two-way timing message exchange regime is considered when the likelihood function of the observation time stamps is Gaussian, exponential, or log-normally distributed. A parameterized solution to the maximum likelihood (ML) estimation of clock offset is analytically obtained, which differs from the earlier approaches where the likelihood function is maximized graphically. In order to capture the imperfections in node oscillators, which may render a time-varying nature to the clock offset, a novel Bayesian approach to the clock offset estimation is proposed by using a factor graph representation of the posterior density. Message passing using the max-product algorithm yields an exact expression for the Bayesian inference problem. Several lower bounds on the variance of an estimator are derived for arbitrary exponential family distributed likelihood functions which, while serving as stepping stones to benchmark the performance of the proposed clock offset estimators, can be useful in their own right in classical as well Bayesian parameter estimation theory. To corroborate the theoretical findings, extensive simulation results are discussed for classical as well as Bayesian estimators in various scenarios. It is observed that the performance of the proposed estimators is fairly close to the fundamental limits established by the lower bounds

Convergence Analysis of the Variance in Gaussian Belief Propagation

“…Applying shown in (18) into this inequality, it can be inferred that , that is, for all . Combining with shown in (18), for all , we have (19) It can be seen from (18) and (19) that is a monotonically non-increasing and lower bounded sequence. Thus, must converge to a vector , that is, .…”

“…Combining with the fact as proved above, we have . By induction, it can be derived that (18) Then, from the definition of in (15), the assumption is equivalent to for all , where are the elements of with arranged in the same order as . By rearranging the terms in , we obtain .…”

An Overview of a Class of Clock Synchronization Algorithms for Wireless Sensor Networks: A Statistical Signal Processing Perspective

“…However, in this case, the MLE is biased. Different from the graphical analysis used in [28], [29] analytically derived the MLE of clock offset under exponential delays using convex optimization tools. In particular, the system model in Equation (4) is re-written as:…”

“…The advantage of this analytical approach is that it provides a more general derivation of MLEs, which can by applied to obtain the MLEs for the delays under other types of distributions, which include Gaussian and log-normal as special cases [29].…”

Reliable clock estimation using linear weighted fusion based on pairwise broadcast synchronization

“…Chaudhari et al [6] derived the minimum variance unbiased estimator (MVUE) of clock offset under both symmetric and asymmetric exponential delay models. Ahmad et al [7] studied the time-varying clock offset estimation by using the factor graph theory and consider a class of delay models. Since the rate (speed) of the clock (crystal oscillator) is not exactly the same for every sensor [8], i.e., a clock skew inherently exists, the estimation of clock skew is necessary and even more crucial than that of clock offset.…”