- Research Article
- Open Access
A Robust Approach for Clock Offset Estimation in Wireless Sensor Networks
© Jang-Sub Kim et al. 2010
- Received: 2 January 2009
- Accepted: 26 April 2010
- Published: 30 May 2010
The maximum likelihood estimators (MLEs) for the clock phase offset assuming a two-way message exchange mechanism between the nodes of a wireless sensor network were recently derived assuming Gaussian and exponential network delays. However, the MLE performs poorly in the presence of non-Gaussian or nonexponential network delay distributions. Currently, there is a need to develop clock synchronization algorithms that are robust to the distribution of network delays. This paper proposes a clock offset estimator based on the composite particle filter (CPF) to cope with the possible asymmetries and non-Gaussianity of the network delay distributions. Also, a variant of the CPF approach based on the bootstrap sampling (BS) is shown to exhibit good performance in the presence of reduced number of observations. Computer simulations illustrate that the basic CPF and its BS-based variant present superior performance than MLE under general random network delay distributions such as asymmetric Gaussian, exponential, Gamma, Weibull as well as various mixtures.
- Mean Square Error
- Wireless Sensor Network
- Gaussian Mixture Model
- Bootstrap Sampling
- Predictive Distribution
Wireless sensor networks (WSNs) have been recently proposed for observing and monitoring various aspects of the physical world. In WSNs, the basic operation is data fusion, where data from multiple sensors are integrated together to form a single meaningful result . The fusion of individual sensor readings is possible only by exchanging messages that are timestamped by each sensor's local clock. This mandates the need for a common notion of time among the sensors. Such a common notion of time is achieved through the clock synchronization task. In WSNs, clock synchronization is an important research area .
The two-way message exchange mechanism used in the Network Time Protocol (NTP)  and Timing Synch Protocol for Sensor Networks (TPSN)  is adopted herein as the clock synchronization approach between two nodes of the WSN. Due to the presence of nondeterministic and unbounded message delays, messages can get delayed arbitrarily, which makes the synchronization very difficult in WSNs . The most commonly widely used models to capture the non-deterministic delay distributions in WSNs are Gaussian, exponential, Gamma, and Weibull probability density functions (pdfs) [5–7]. However, in general, it is difficult to determine which delay model should be adopted in a given WSN at a certain time instant. Recently,  studied the performance of maximum likelihood estimators corresponding to symmetric Gaussian (GML) and exponential (EML) network delay distributions. Preliminary computer simulations  illustrated the fact that GML and EML are not robust to asymmetries or uncertainties in the network delay distributions.
In , the inference of general state-space models characterized by nonlinear process and observation equations is addressed via the concept of Gaussian sum particle filter (GSPF) which approximates the filtering and predictive distributions by weighted Gaussian mixtures, that is, banks of Gaussian particle filters (GPFs). With non-Gaussian noise approximated by Gaussian mixtures, the non-Gaussian noise models are approximated by banks of Gaussian noise models. However, in wireless sensor networks, the process and observation equations are linear functions. Therefore, we extend the use of a new filter, the composite particle filter (CPF), to encompass linear and additive non-Gaussian noise models. For a linear state-space model with additive non-Gaussian noise, CPF approximates the posterior distributions as Gaussian mixtures using banks of parallel Kalman filters (KFs). The main contribution of this paper is a novel clock offset estimation method, called the composite particle filter (CPF), which is shown to be robust to the unknown distribution of network delays. The CPF approximates the filtering and predictive distributions by using weighted Gaussian mixtures and is basically implemented via banks of Kalman filters (KFs) instead of Gaussian Particle filters (GPFs) . Thus, CPF appears as a variation of the Gaussian sum particle filter (GSPF) , fit for estimation of linear models perturbed by non-Gaussian random noise components.
There is another method capable of coping with nonlinear processes and observation equations, and non-Gaussian noise models, which is called the Gaussian mixture sigma point particle filter (GMSPPF) . Both GSPF and GMSSPF approximate the filtering and predictive distributions by weighted Gaussian mixtures. The big difference consists in the integrating sub-techniques. Specifically, the GMSPPF combines the Particle filter and Sigma Point filter with a Gaussian Mixture Model (GMM) whose features are estimated via the Expectation-Maximization (EM) algorithm, while the GSPF integrates the Gaussian Particle filter with GMM via a Gaussian sum filtering approach. In , the Gaussian mixture Kalman particle filter (GMKPF) was proposed, which is a slightly changed version of the GMSPPF and obtained by replacing the sigma point Kalman filter (SPKF) with a KF. The CPF proposed in this paper and the GMKPF  present a different updating mechanism of GMM parameters such as the weights. Also, the bootstrapping sampling technique is shown herein to be an effective mechanism to improve the performance of clock estimation schemes.
As explained in [4, 12], energy conservation is a very important concern. Reference  pointed out that much less power is consumed in processing data than transmitting it. In fact,  showed that the energy required for a sensor node to transmit 1 Kbit over 100 meters (3 Joules) in a particular WSN was equivalent to the energy required to execute 3 millions of instructions. Therefore, the possibility of trading off computational power for more savings in energy consumption appears as a very feasible approach. Thus, one way to reduce the amount of energy spent on signal transmissions and implicitly on achieving clock synchronization is the usage of possibly more sophisticated signal processing algorithms with the goal of achieving more accurate clock offset estimates especially in operational regimes characterized by a reduced number of observations and unknown network delay distributions. In case that message exchange errors occur, a node will not retransmit the message to a neighbor node but will resample the observation data from the original observation data using the bootstrap sampling (BS) approach [14, 15]. The BS can be implemented by constructing a number of resamples of the observed data, each of which being obtained by random sampling with replacement from the original observed data. Notice that a node will then estimate the clock offset from the resampled observation data by using the CPF. The computer simulations highlight that the CPF with BS achieves better performances in various random delay models, and it aims at reducing the number of message exchanges. Therefore, the CPF with BS leads to less power consumption relative to the CPF, GML, and EML.
It is interesting also to remark that the clock synchronization literature for wireless sensor networks is quite scarce in terms of contributions addressing the robustness or improving the mean square error (MSE) performance of existing state-of-the-art clock synchronization algorithms in the presence of message errors, unknown and possibly time-varying network delay distributions, or reduced number of observations (data measurements). Thus far, it appears that only very few preliminary and straightforward applications of standard Kalman filtering or general adaptive signal processing techniques have been recently reported (see [16–18]) to improve the MSE performance of protocols such as RBS  or TPSN . However, no attempts have been made to address the problem of building clock synchronization algorithms that are robust to the unknown distribution of random network delays, message errors, or presence of reduced number of observations. This paper aims to answer these questions from the light of a composite particle filtering technique.
The rest of this paper is organized as follows. Section 2 introduces the state-space model that will be used throughout the paper and a description of problem formulation. Section 3 provides a description of the CPF and BS-based CPF approaches for estimating the clock offset in wireless sensor networks. The results of computer simulations are given in Section 4. Finally, Section 5 provides concluding remarks.
The two-way timing message exchange protocol is a recently proposed clock synchronization approach for WSNs [4, 5, 8]. Under this protocol, the synchronization between two generic nodes A and B is achieved by transmitting timing messages in both directions. The message exchanges between nodes A and B are organized in terms of cycles, and during each cycle a message exchange occurs in each direction. For example, during the th cycle, the Node A sends its time reading to Node B, which records the time of arrival of the message sent by node A as , according to its own time scale. Similarly, a timing message exchange is performed from Node B to Node A. At time node B transmits back to node A the timing information and . According to Node A's clock, the message transmitted by Node B arrives at node A at time . Therefore, at the end of the th cycle, node A has access to all the time information , , that prove to be sufficient for estimating the clock phase offset and deterministic propagation delay.
where the observation noise vector may assume any probability density function (pdf). Hence, it turns out that our initial problem is now casted as the estimation problem of a Gauss-Markov model with unknown state (see (2) and (3)).
The maximum likelihood clock offset estimator was reported in  for the two-way timing message exchange protocol such as TPSN and NTP under the assumption of Gaussian or exponential delay models. Herein we will derive a CPF for clock phase offset estimation assuming a general unknown distribution of network delays and then compare the CPF and existing maximum likelihood clock offset estimators that were derived for Gaussian (GML) and exponential (EML) delay models. Under the Bayesian framework, an emerging powerful technique for obtaining the posterior, predictive, and filtering probability density functions is referred to as the particle filtering (PF) (see, e.g., [20, 21]). The PF technique allows for a complete representation of the state posterior distribution, which approximates , by stochastic samples generated using a sequential importance sampling strategy. The most common employed PF strategy is to sample from the transition prior distribution due to its simplicity. Since the prior importance sampling distribution employs no information from observations in proposing new samples, its use is often ineffective and leads to poor filtering performance. To overcome these challenges, we will derive an extension of GSPF  applicable for linear non-Gaussian models.
In the GSPF, the filtering and predictive distributions are recursively represented as finite Gaussian mixtures (GMs) using Gaussian Particle Filtering (GPF) . One set of methods approximates the mixture components of the predictive and filtering distributions as Gaussian. The approximation can be implemented by the GSPF, resulting in a parallel bank of GPFs. Since Gaussian mixture models are increasingly used for modeling non-Gaussian densities [23, 24], herein we plan extending the use of the GSPF to linear non-Gaussian models. The resulting new approach will be referred to as CPF. Notice that in the measurement and time-update equations of CPF, the updated mean and covariance of each mix and follow from the KF. The CPF is implemented by means of parallel KFs, and the weights are adjusted according to the given update equations. Notice also that the CPF approach comes out of the utilization of another filtering technique (KF) producing a filtering probability density function used as importance function (IF) for the particle filtering.
for appropriate , and and .
Next, we introduce another clock estimation scheme obtained through the integration of the CPF technique with the bootstrap sampling (BS) approach. The reader is directed to [14, 15] for more detailed explanations about bootstrap sampling. In order to provide a consistent amount of observation data in the presence of errors during timing message transmissions, new sampled observation data from the original observation data are generated via the BS. Then, the clock offset is estimated based on the CPF. Notice that even in the presence of corrupted or lost data packets, BS can create additional samples to the original sample set, by drawing at random with replacement from , and without being necessary for additional retransmissions. Each of the bootstrap samples is considered as new data. Based on the additional sampled observation data, we can then approximate the clock offset by using the CPF. The major steps of the CPF approach with bootstrap sampling are summarized by the following pseudocode
Algorithm: CPF with BS
( ) Conduct the experiment to obtain the random sample and calculate the estimate from the sample .
( ) Construct the empirical distribution , which puts equal mass at each observation .
( ) From , draw a sample , operation called bootstrap resampling.
( ) From the bootstrap resample , estimate the clock offset by CPF.
The calculation of the computational cost of CPF is very complex, compared to GML and EML. In general, the computational cost of CPF is a function of the number of particles and the number of measurements. However, GML and EML are a function of the number of measurements and do not use the particle filtering method. Hence, it is difficult explicitly to compare the complexities of GML/EML and CPF. However, we will use the big notation to express the computational complexities of GML/EML and CPF in terms of flops by evaluating only the most computationally demanding steps. Letting , , and denote the dimension of the state vector , the number of particles, and the number of GMM, respectively, the GML is approximately , and the CPF is approximately which is the maximum complexity and occurs in the posterior pdf step. This shows that the CPF is approximately times slower than GML in an application with , , and .
In this section, extensive computer simulation results are presented to illustrate the performance of the CPF, CPF with BS, GML , and EML  approaches for estimating the clock offset in wireless sensor networks, assuming a variety of random network delay models such as asymmetric Gaussian, exponential, Gamma, and Weibull as well as a mixture of Gamma and Weibull, respectively. These computer simulations and numerous other simulations not shown herein due to space limitations corroborate the conclusion that the proposed method can be widely and flexibly applied for any delay distribution. The stationary process assumes the constant variance , while the number of particles and GMMs are set to 100 and 3, respectively. The bootstrap samples are twice the number of measurements.
This paper provided novel methods such as CPF and BS for estimating the clock offset in wireless sensor networks. The benefits are in terms of improved performance and applicability to any random delay models such as asymmetric Gaussian, exponential, Gamma, and Weibull, as well as mixtures of these delay models. In addition, the proposed CPF approaches are robust to the presence of a small number of observations, message exchange errors, and unknown network delay distributions. Also, the proposed iterative clock phase estimation algorithms can track time-varying clock phase offsets, which represents a notable improvement relative to the existing state-of-the-art GML and EML estimators. Possible disadvantages of the proposed composite particle filtering-based approaches are the facts that they present high computational complexity and require good initializations; analytical closed form expressions do not seem to exist for the clock estimators and the computation of the lower bound performance bounds appears difficult due to the non-Gaussian nature of involved distributions. In addition, the CPF with BS and CPF achieve excellent performance compared to GML and EML in environments which manifest in message exchange errors and time-varying network delay distributions.
This paper was supported in part by Qtel and QNRF.
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