As mentioned before, RBS is based on the receiver-receiver synchronization (RRS) scheme. RRS is an approach synchronizing a set of children nodes which receive the beacon messages from the parent node. Reference [27] suggested the maximum likelihood estimator of the relative clock offset which is equivalent to the estimator presented in [25]. The estimation of clock offset and skew in [27] is performed in accordance with the following approach. Consider a parent node P and arbitrary nodes A and B, which are located within the communication range of the parent node P, assuming, as illustrated in Figure 4, that both Node A and Node B receive the i_th beacon from node P at time instants T 2 , i ( A ) and T 2 , i ( B ) of their local clocks, respectively. Nodes A and B record the arrival time of the broadcast packet according to their own timescales and then exchange their timestamps. Suppose that X i ( P A ) denotes the nondeterministic delay components and d^(PA) represents the deterministic delay component from Node P to Node A. Then T 2 , i ( A ) can be expressed as: (2) T 2 , i ( A ) = T 1 , i + d ( P A ) + X i ( P A ) + θ O ( P A ) + θ S ( P A ) ⋅ ( T 1 , i − T 1 , 1 ) ,where T₁,_i is the transmission time at the reference node, θ O ( P A ) and θ S ( P A ) are the clock offset and skew of Node A with respect to the reference node, respectively. Likewise, at Node B: (3) T 2 , i ( B ) = T 1 , i + d ( P B ) + X i ( P B ) + θ O ( P B ) + θ S ( P B ) ⋅ ( T 1 , i − T 1 , 1 ) .

Subtracting (3) from (2), we obtain the following equation: (4) T 2 , i ( A ) − T 2 , i ( B ) = d ( P A ) − d ( P B ) + X i ( P A ) − X i ( P B ) + θ O ( B A ) + θ S ( B A ) ⋅ ( T 1 , i − T 1 , 1 ) ,where θ O ( B A ) = θ O ( P A ) − θ O ( P B ) and θ S ( B A ) = θ S ( P A ) − θ S ( P B ) are the relative clock offset and skew between Node A and Node B at the time they receive the i^th broadcast packet from the reference node, respectively. Reference [27] assumes that the random delays, X i ( P A ) and X i ( P B ) are Gaussian random variables with mean μ and variance σ²/2.

Letting the noise component z [ i ] = μ ′ + X i ( P A ) − X i ( P B ), where μ′ = d^(PA) − d^(PB) and z[i] ∼ N(μ′, σ²), x [ i ] = T 2 , i ( A ) − T 2 , i ( B ) − μ ′, and w[i] = z[i] − μ′, we can write the set of observation data in matrix form as follows: (5) x = H θ + w ,where x = [x[1]x[2] ⋯ x[N]]^T, w = [w[1]w[2] ⋯ w[N]]^T, θ = [ θ O ( B A ) θ S ( B A ) ], and (6) H = [ 11 ⋯ 1 0 T 1 , 2 − T 1 , 1 ⋯ T 1 , N − T 1 , 1 ] T ⋅

By using standard results from estimation theory [28, Theorem 3.2, p. 44] and making some mathematical manipulations, the minimum variance unbiased (MVU) estimator for the relative clock offset and skew takes the expression [27]: (7) [ θ O ( B A ) θ S ( B A ) ] = 1 N ∑ i = 1 N D i 2 − [ ∑ i = 1 N D i ] 2 [ ∑ i = 1 N D i 2 ∑ i = 1 N x [ i ] − ∑ i = 1 N D i ∑ i = 1 N [ D i ⋅ x [ i ] ] N ∑ i = 1 N [ D i ⋅ x [ i ] ] − ∑ i = 1 N D i ∑ i = 1 N x [ i ] ] ,where D_i = T₁,_i − T_1,1. In case that there is no relative clock skew ( θ S ( B A ) = 0), the maximum likelihood estimator of the relative clock offset θ ^ O ( B A ) becomes (8) θ ^ O ( B A ) = 1 N ∑ i = 1 N [ T 2 , i ( A ) − T 2 , i ( B ) ] ,which is equivalent to the estimator presented in [25].

This phase occurs at the deployment of the network. The root node is assigned a level 0 and it initiates this phase by broadcasting a level_discovery packet. The level_discovery packet includes the identity and the level of the sender node. The immediate neighbors of the root node receive this packet and assign themselves a level, one greater than the level of the packet they just received, i.e., the level 1. After setting their own level, they broadcast a new level_discovery packet containing their own level. This procedure is continued and finally every node in the network is assigned a level. Once a node is assigned a level, it disregards any such future packets, which prevents flooding congestion from taking place in this phase. Therefore, a hierarchical structure is created with only the root node assigned to level 0. In general, a user node that acts as the gateway between the sensor network and the external world can act as the root node. The user node can be equipped with a GPS receiver, which enables the sensor nodes to be synchronized to the physical world.

Pair wise synchronization is performed in this phase along the edges of the hierarchical structure constructed in the level discovery phase. As mentioned above, the classical approach of sender-receiver synchronization for implementing the handshake between a pair of nodes is used along each edge of the hierarchical tree. Consider a two-way message exchange between node A and node B as shown in Figure 5. At time T₁,_i (according to its local clock), node A sends a synchronization_pulse packet, which contains the level of node A and the value of T₁,_i, to node B. Node B receives this packet at time T₂,_i = T₁,_i + d + θ_A, where θ_A and d represents the clock offset between the two nodes A and B, and the propagation delay, respectively. Node B sends back an acknowledgement packet to node A at time T₃,_i. This packet carries the level of node B and the values of T₁,_i, T₂,_i, and T₃,_i. Then node A receives the acknowledgement packet at time T₄,_i. Assuming that the clock offset and the propagation delay is constant in this small span of time, node A can calculate the clock offset and propagation delay as illustrates by the equation (9) below, and synchronize itself to the clock of node B. This represents a sender-initiated approach, where the sender synchronizes its clock to that of the receiver: (9) θ A = ( T 2 , i − T 1 , i ) − ( T 4 , i − T 3 , i ) 2 ; d = ( T 2 , i − T 1 , i ) + ( T 4 , i − T 3 , i ) 2

This message exchange begins with the root node's initiating the synchronization phase by broadcasting a time sync packet. As soon as receiving this packet, nodes with level 1 wait for some random time before initiating the two-way message exchange with the root node, so as to avoid the contention in medium access. After receiving back an acknowledgement, these nodes adjust their clocks to the clock of the root node. The nodes with level 2 overhear this message exchange, and then they back off for some random time in order to ensure that the nodes with level 1 have completed their synchronization, after which they initiate the message exchange with nodes with level 1. This process eventually enables all nodes to be synchronized to the root node.

The advantages of TPSN are as follows [21, 29]:

It is scalable and the synchronization precision does not deteriorate significantly as the size of the network increases.

On the other hand, TPSN has the following disadvantages [21, 29]:

Energy conservation is not so effective since a physical clock correction needs to be performed on the local clocks of sensors while achieving synchronization.

Now let us take a look at some methods for estimating the clock offset and clock skew in a two-way message exchange model.

Modeling of network delays in WSNs seems to be a challenging task [31]. Several probability distribution function (PDF) models for random queuing delays have been proposed so far, the most widely used being Gaussian, exponential, gamma, Weibull distributions [32, 33]. By the Central Limit Theorem (CLT), the PDF of the sum of a large number of independent and identically distributed (i.i.d.) approaches that of a Gaussian RV. This model is proper if the delays are thought to be the addition of numerous independent random processes. The Gaussian distribution for the clock offset errors was also reported by a few authors, such as [34], based on laboratory tests. On the other hand, a single-server M/M/1 queue can fittingly represent the cumulative link delay for point-to-point hypothetical reference connections, where the random delays are independently modeled as exponential random variables [35]. In this paper, we limit our presentation to mainly the situations where the portions of delays are Gaussian or exponential random variables.

Noh et al. proposed the maximum likelihood estimator (MLE) of clock offset in a two-way message exchange model, which will be deeply discussed below [36]. The authors suppose that the clock offsets of two nodes remain equal during the synchronization period, and the delays at i_th nodes X_i and Y_i are Gaussian random variables with mean μ and variance σ²/2. From Figure 5, T₂,_i and T₄,_i can be expressed as: (10) T 2 , i = T 1 , i + d + θ A + X i , (11) T 4 , i = T 3 , i + d − θ A + Y i ,where the variables θ_A, d, X_i, and Y_i denote the clock offset between the two nodes, the propagation delay, and the variable portions of delays, respectively. After some mathematical manipulations, the likelihood function based on the observations { U i } i = 1 N and { V i } i = 1 N is given by: (12) L ( θ A , μ , σ 2 ) = ( π σ 2 ) − N 2 e − 1 σ 2 [ ∑ i = 1 N ( U i − d − θ A − μ ) 2 + ∑ i = 1 N ( V i − d + θ A − μ ) 2 ] ,where N stands for the number of message exchanges, U_i = T₂,_i − T₁,_i, and V_i = T₄,_i − T₃,_i. Differentiating the log-likelihood function gives: (13) ∂ ln L ( θ A ) ∂ θ A = − 2 σ 2 ∑ i = 1 N [ 2 θ A − ( U i − V i ) ] .

Therefore, the MLE of clock offset is given as follows [36] under the assumption that there is no clock skew is given by [36]: (14) θ ^ A = arg max θ A [ ln L ( θ A ) ] = ∑ i = 1 N ( U i − V i ) 2 N = U ¯ − V ¯ 2 .

Thus, Node A can be synchronized to the node B by simply taking the difference of the average observations Ū and V̄. Noh et al. also proposed the joint MLE of clock offset and clock skew under the assumption of Gaussian random delays [36], a result which will not be detailed herein.

For exponential random delays X_i and Y_i, Jeske proved in [37] that the maximum likelihood estimator of clock offset, θ_A exists when d is unknown and is the same form as the estimator proposed in [34], namely: (15) θ ^ A = min 1 ≤ i ≤ N U i − min 1 ≤ i ≤ N V i 2 .

In case of one round of message exchange (N = 1), the MLE of clock offset for both Gaussian and exponential delay models is θ̂_A (U − V)/2, which is exactly the same as the estimator presented in [29]. Notice further that the extension of MLE for joint estimation of clock phase offset and skew in networks with exponential delays was recently reported by Chaudhari et al. in [53].

In general, the delay distribution in the upstream, F_X is not equal to that in the downstream F_Y, because the node A → node B and node B → node A transmission paths through the network typically present different traffic characteristics, and thus the network delays in each path are potentially different. Equation (15) fits well the symmetric exponential delay model where both the uplink and downlink have the same exponential delay distributions. However, if this MLE is used in the asymmetric exponential delay model, there will be a bias in the clock offset. Therefore, it is necessary to achieve a more accurate estimate of the clock offset by using an alternate approach.

In [38], Lee et al. proposed a clock offset estimator using the bootstrap technique and bias correction method, which gives better performance than Jeske's MLE in the asymmetric exponential delay model. Specifically, bias-corrected estimators through non-parametric and parametric bootstrap were proposed. The procedures of bootstrap bias correction follow [39, 40]. These two bootstrap bias corrected estimators require Monte Carlo resampling of the empirical distribution functions. As far as the bootstrap bias correction method is concerned, in [54], Jeske had proposed a closed-form expression of the clock offset estimator by bootstrap bias correction approach based on the nonparametric technique and had compared analytically the estimator with Paxson's estimator [34] which was proved to be the MLE in [37]. Additionally, the effectiveness of bootstrap bias correction in the context of clock offset estimation was reported in [55] within the context of Pareto distribution, which was suggested in recent internet traffic modeling research.

At first, let us take a look at the clock offset estimation using bootstrap bias correction based on the nonparametric bootstrap method and the parametric bootstrap method, and then we will consider the estimation of the clock data offset using the particle filtering approach.

In Figure 5, the k_th up and down link delay observations corresponding to the k_th timing message exchange are assumed to be given by U_k = T₂,_k − T₁,_k = d + θ_A + X_k and V_k = T₄,_k − T₃,_k = d + θ_A + Y_k, respectively. The fixed value θ_A denotes the clock offset between the two nodes. X_k and Y_k denote the variable portions of U_k = T₂,_k − T₁,_k = d + θ_A + X_k delays, which are assumed to be any distributions such as Gaussian, exponential, Gamma, and Weibull. Given the observation samples z_k = [U_k, V_k]^T, the goal is to find minimum variance estimates of the unknown clock offset θ_A. For convenience, we adopt the new notation x_k = θ_A. Thus, we are looking to determine: (22) x ^ k = E { x k ∣ Z K } ,where Z^K denotes the set of observed samples up to time K, Z^K = {z₀, z₁,⋯,z_K}.

Since the clock offset value is constant, the clock offset is assumed to obey a Gauss-Markov dynamic state-space channel model [47] of the form: (23) x k = F x k − 1 + v k − 1 ,where F stands for the state transition matrix of clock offset. Since the clock offset is constant, we set F = 1. The noise vector v_k is Gaussian random vector with zero mean and covariance E [ v k v k T ] = Q.

The vector observation model follows from the observed samples and assumes the expression: (24) z k = [ d + x k + X k d − x k + Y k ] = [ 1 1 ] d + [ 1 − 1 ] x k + n k , = A d + H x k + n kwhere A = [1,1]^T, H = [1,−1]^T and the observation noise vector n_k = [X_k, Y_k] might assume any distribution. In symmetric Gaussian delay models, n_k is a zero-mean Gaussian random vector with covariance R = σ²I. From the above discussion, the problem of estimating the clock offset can be formulated as a Gauss-Markov model with unknown states as depicted by the equations (23) and (24).

Under the Bayesian framework, an emergent technique for obtaining the posterior probability density function (PDF) is known as particle filtering (PF). PF is based on Monte Carlo simulations with sequential importance sampling (SIS). These methods allow for a complete representation of the posterior distribution of the states using sequential importance sampling and resampling [48] for the various probability densities. Since the true posterior PDF embodies all the available statistical information about the channel estimates, PF is optimal in the sense that all the available information has been used.

The posterior density p(x_0:k÷z_1:k), where x_0:k = {x₀, ⋯, x_k} and z_1:k = {z₁, ⋯, z_k}, constitutes the complete solution to the sequential estimation problem. In many applications, such as tracking, it is of interest to estimate one of its marginals, namely the filtering density p(x_k÷z_1:k). By computing the filtering density recursively, we do not need to keep track of the complete history of the states. Therefore, from a storage point of view, the filtering density is more parsimonious than the full posterior density function. If we know the filtering density, we can easily derive various estimates of the system's states including means, modes, medians and confidence intervals. We show how the filtering density may be approximated using sequential importance sampling techniques.

The filtering density is estimated recursively in two stages: prediction and update (correction), as illustrated in Figure 7. In the prediction step, the filtering density is propagated into the future via the transition density as follows, (25) p ( x k ∣ z 0 : k − 1 ) = ∫ p ( x k ∣ x k − 1 ) p ( x k − 1 ∣ z 0 : k − 1 ) d x k − 1 .

The transition density is defined in terms of the probabilistic model governing the states' evolution (23) and the process noise statistics.

The update stage involves the application of Bayes' rule when new data is observed [49, 50] (26) p ( x k ∣ z 0 : k ) = p ( z k ∣ x k ) p ( x k ∣ z 0 : k − 1 ) p ( z k ∣ z 0 : k − 1 ) ,where the normalizing constant, p(z_k÷z₀:_k−1), depends on the likelihood function p(z_k÷x_k) that is defined in terms of the measurements model (24), and p(x_k÷z₀:_k−1) is the prior information. This is a central issue in Bayesian inference. In order to recursively evaluate the Bayesian equation (26), we utilize the Sequential Importance Sampling (SIS) algorithm. Again, the key idea is to represent the required posterior density function by a set of particles with associated weights and to compute estimates based on these samples and weights. As the number of particles becomes very large, this Monte Carlo characterization becomes an equivalent representation to the usual functional description of the posterior PDF, and the PF approaches the optimal Bayesian estimate. If we know the posterior density function, we can easily derive various estimates of the system's states including means, modes, medians and confidence intervals. Thus the posterior density at k can be approximated as: (27) p ( x 0 : k ∣ z 1 : k ) ≈ ∑ i = 1 N w k ( i ) δ ( x 0 : k − x 0 : k ( i ) ) ,where { x 0 : k ( i ) } i = 1 N are a set of particles drawn from the posterior distribution and δ(●) is the Dirac delta function. The weights ( w k ( i )) themselves can be shown to be updated as [50]: (28) w k ( i ) = w k − 1 ( i ) p ( z k ∣ x k ( i ) ) p ( x k ( i ) ∣ x k − 1 ( i ) ) q ( x k ( i ) ∣ x k − 1 ( i ) , z k ) ,where the proposal distribution q ( x k ( i ) ∣ x k − 1 ( i ) , z k ) represents all the a priori knowledge. It is, however, usually difficult to sample directly from a given posterior distribution. Thus we choose what can be called a proposal distribution that is a probability distribution from which we can easily sample [48]. The selection of the proposal function is one of the most critical design issues in importance sampling algorithms and is the source of the main concern. The more accurate the proposal is to the true posterior, the better the performance of the particle filter is. It is often convenient to choose the proposal distribution to be the prior [51]: (29) q ( x k ∣ x k − 1 ( i ) , z k ) = p ( x k ∣ x k − 1 ( i ) ) .

Again, we choose the stochastic model given by (23) as our model for the proposal distribution. As a result of not incorporating the most recent observations, this would seem to be the most common choice of proposal distribution since it is intuitive and can be implemented easily. This has the effect of simplifying (28) to: (30) w k ( i ) = w k − 1 ( i ) p ( z k ∣ x k ( i ) ) .

The update weights are based on the likelihood function. New estimates of the posterior are then computed based on the previous samples.

A common problem with the SIS particle filter is the degeneracy phenomenon, where after a few iterations, all but one particle will have negligible weights. It has been shown [52] that the variance of the importance weights can only increase over time, and thus it is impossible to avoid the degeneracy phenomenon. A large number of samples are thus effectively removed from the sample set because their importance weights become numerically insignificant. To avoid this degeneracy, a resampling stage may be used to eliminate samples with low importance weights and multiply samples with high importance weights. A common heuristic used to maintain an appropriate number of particles is to first calculate the effective sample size N_eff introduced in [48, 51], and defined as: (31) N ^ e f f = 1 / ( ∑ i = 1 N ( w k ( i ) ) 2 ) .

A threshold number of particles N_th is then defined such that N_th ≤ N. Multiple resamplings of particles then becomes necessary whenever N_eff ≤ N_th.

We have so far explained how to compute the importance weights sequentially and how to improve the sample set by resampling. The essential structure of the PF to clock offset estimation using the proposal function (29) can now be presented in terms of the following pseudo-code.

Finally, we now introduce the PF with Bootstrap Sampling (BS) approach that integrates the PF with the BS for estimating the clock offset. The basic idea is quiet straightforward. In order to provide a large amount of observation data, we generate sampled observation data from the original observation data set by using the BS procedure. Then, we estimate the clock offset based on the PF. The important thing to check is how close the PDF of sampled data is to the true PDF. However, in case of less observation data, the performance's limitation is related to the finite number of observation data. Therefore, the solution is to overcome this limitation in the presence of reduced number of observation data. BS assumes additional data samples relative to the original data samples; these additional samples are defined by drawing at random with replacement. Each of the bootstrap samples is considered as new data. Based on the BS, we will increase the observation data set. Given a large number of new observation data, we can then approximate the clock offset by using PF. The following pseudo-code describes the procedure for estimating the clock offset via the nonparametric bootstrap sampling method.

Figure 8 shows the MSE (Mean Squared Error) of several estimators and the Cramer-Rao Lower Bound (CRLB) assuming the random delay models are symmetric Gaussian delays. GML and EML denote the MSEs of Gaussian maximum likelihood and exponential maximum likelihood approaches, respectively. Note that the PF with BS performs much better with over 50% increase in performance when compared to the GML, EML, and even PF. Moreover, the PF with BS exhibits the best performance in the presence of small amount of data. As shown in Figure 8, it is notable that the performance limitation is related to the reduced number of observation data available. In the absence of measurement samples, BS offers the additional gain of pseudo measurement sampling in symmetric Gaussian delay models. There are 10 measurement samples, but from the bootstrap sampling, we can get a total of 20 measurement samples. Thus, due to the pseudo sampling samples, it turns out that the performance is improved. However, in the case of a large number of measurement samples, the gain becomes smaller. This means that the impacting factor on performance may be related to the bootstrap sampling error as well as to the number of measurement samples.