We now turn our attention to a more accurate model defining the relationship between two clocks by the addition of clock skew. In practice, the time synchronization problem in WSNs generally involves two steps: synchronizing the nodes in the network to one common absolute time by adjusting clock phase offset (clock offset) among the nodes, and correcting the clock frequency offset (clock skew) relative to a certain standard frequency. The second step is required because the imperfections in quartz crystals and environmental conditions induce different clocks to run at slightly different frequencies. Actually, the effect of clock skew is the main reason why clock offset keeps drifting apart. Hence, adjusting clock skew guarantees long-term reliability of synchronization, and therefore reduces network-wide energy consumption in synchronization procedures. Indeed, developing long-term and network-wide time synchronization protocols that are energy-efficient represents one of the key strategies for the successful deployment of long-lived WSNs.

The main topics in this chapter are as follows. First, the MLE and the corresponding CRLB for the conventional clock offset model in a general sender–receiver protocol assuming a Gaussian model for the noise are derived. Second, the joint MLE and corresponding CRLB using a more realistic linear clock offset and skew model assuming Gaussian random delays are obtained. Third, the CRLB for the clock offset for the exponential delay model is derived as a performance threshold. Fourth, the joint MLE for the clock offset and skew under the exponential delay model is obtained and the corresponding algorithms to find these estimators are described in detail.