Energy-Efficient Estimation of Clock Offset for Inactive ... - IEEE Xplore

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 1, JANUARY 2010

Energy-Efficient Estimation of Clock Offset for Inactive Nodes in Wireless Sensor Networks Qasim M. Chaudhari, Erchin Serpedin, and Jang-Sub Kim

Abstract—For a meaningful processing of the information sensed by a wireless sensor network (WSN), the clocks of the individual nodes need to be matched through some well-defined procedures. Extending the idea of having silent nodes in a WSN overhear the two-way timing message communication between two active (master and slave) nodes, this paper derives the maximum-likelihood estimator (MLE) for the clock offsets of the listening nodes located within the communication range of the active nodes by assuming an exponential link delay modeling, hence synchronizing with the reference node at a very low cost. A vital advantage for adopting such an approach is that the performance of sender–receiver protocols can be compared with receiver–receiver protocols on equal footings, because their main critical aspect was associated with the high-communication overhead induced by the point-to-point nature of communication links relative to broadcast communications. The MLE is also shown to be the minimum variance unbiased estimator (MVUE) of the clock offset when the mean of exponential link delays is known. Since it is attractive to know in advance the extent to which an estimator can perform through its lower bound, the Chapman–Robbins bound and the Barankin bound for the clock offset estimator are also derived. It is shown that for an exponential link delay model, the mean square error of the clock offset estimator is inversely proportional to the square of the number of observations, and hence its performance is on a similar scale, albeit slightly lesser, as compared to the usual sender–receiver clock offset estimator. In addition, a novel method referred to as the Gaussian mixture Kalman particle filter (GMKPF) is proposed herein to estimate the clock offsets of the listening nodes in a WSN. GMKPF represents a better and flexible alternative to the MLE for the clock offset estimation problem due to its improved performance and applicability in arbitrary and generalized non-Gaussian random delay models. Index Terms—Clock, estimation, sensor, synchronization, timing.

I. INTRODUCTION

W

IRELESS SENSOR NETWORKs (WSNs) represent the epitome of our future world, where the machines will assume microdimensions and the acquisition of informa-

Manuscript received February 15, 2008; revised February 25, 2009. Current version published December 23, 2009. Q. M. Chaudhari was with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843-3128 USA. He is now with the Department of Electrical Engineering, Iqra University, Islamabad 44000, Pakistan (e-mail: [email protected]). E. Serpedin and J. Kim are with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843-3128 USA (e-mail: [email protected]). Communicated by H. Bölcskei, Associate Editor for Detection and Estimation. Color versions of Figures 1 and 5–7 in this paper are available online at http:// ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2009.2034817

tion, communication, and control tasks will be distributed. WSNs will be playing a key role in many areas of human interest in the near future [1]. As the size of sensor nodes gets smaller, the power and hardware constraints become the driving factor in defining the mode and utility of their operation. In general, the lesser the nodes communicate with each other, the longer the network lives due to the reduction of energy costs. At the same time, many applications such as data fusion, localization, and object tracking put severe restrictions on the maximum synchronization deviation the clocks of the nodes in a sensornet can tolerate. In addition, the existing clock synchronization protocols proposed for Internet and wireless local area networks (WLANs) cannot be used due to energy, bandwidth, and hardware limitations [2], [3]. Researchers have proposed various protocols targeting the clock synchronization in WSNs mainly based on packet synchronization techniques (see [4]–[6] for alternative schemes), which are divided into three fundamental approaches: sender–receiver (SR) synchronization (e.g., [7]–[11]), receiver–receiver (RR) synchronization (e.g., [12]–[15]), or a hybrid of both (e.g., [16]). The difference between SR and RR approaches is the way in which they carry out a handshake to synchronize a pair of network nodes. In RR, two receiver nodes exchange timing information about the message that they have received from a common sender, while in SR, the two nodes exchange timing messages between themselves. The two opposite requirements of closely synchronizing the network with a minimum number of RF transmissions and with high accuracy can be efficiently addressed using the approach suggested by Maroti et al. [16], where multiple listening nodes can hear the synchronization messages transmitted by the master node in one-way timing cells exchange mechanism. Advancing the utility of this one-way mechanism, Maroti et al. [16] proposed the synchronization of nodes present in the communication range of the master node (broadcasting the timing beacons), where each node receiving the timing packets transmitted by the master node estimates its own clock parameters and synchronizes with the master node accordingly. However, the similar situation pertaining to the two-way timing exchange mechanism, i.e., the framework where the nodes, located in the common broadcast region of a master and slave node, can overhear the time synchronization packets between them and exploit the acquired information for achieving clock synchronization, largely remained unnoticed until Noh et al. [17] shed some light on it. Note that although the idea of SR synchronization is quite old and is most famously being used in network time protocol (NTP) [18] for a long time, it is due to the wireless nature of communication channels in sensornets that the technique of synchronization of silent nodes located

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CHAUDHARI et al.: ENERGY-EFFICIENT ESTIMATION OF CLOCK OFFSET FOR INACTIVE NODES IN WIRELESS SENSOR NETWORKS

in their common broadcast region can be exploited. Therefore, the clock synchronization requirements can be reasonably met without paying any price on the network lifetime (i.e., without exchanging additional messages for clock synchronization purposes and thereby reducing battery life) or nodes hardware (e.g., by improving the quality of the quartz crystals or by utilizing more expensive power efficient batteries). The main contributions of this paper are as follows. First, it derives the maximum-likelihood estimate (MLE) for the clock offset of the silent nodes, which are only receiving the timing packets exchanged by the master–slave pair, and proves the uniqueness of the MLE. One very important implication of this work is that the performance of the SR protocols (e.g., [7]), whose main disadvantage has always been categorized as the high communication overhead in WSN scenarios due to the point-to-point rather than the broadcast nature of these protocols, can be compared with that of RR or hybrid protocols on equal grounds. Second, it is proved that the MLE coincides with the minimum variance unbiased estimator (MVUE) when the mean link delay is known. Third, the Chapman–Robbins and the Barankin bounds for the clock offset of the listening nodes are derived to use as benchmarks to assess the performance of any estimators. Fourth, to extend our results to arbitrary network delay distributions, a novel clock offset estimation method, called the Gaussian mixture Kalman particle filter (GMKPF), has been proposed which has the merits of being robust and yielding very accurate clock offset estimates in the presence of unknown underlying random delay models. II. PROBLEM FORMULATION Consider a WSN consisting of several sensor nodes as shown in Fig. 1, which dynamically elect a master node through any master election algorithm proposed in the literature, and whose time is chosen as the reference time subsequently for the rest of that synchronization cycle. Depending on the SR synchronization protocol employed for operation, node chooses another node as the slave node at the start of the synchronization cycle. Let denote the clock offset of node with respect transmits timing to node . As illustrated in Fig. 1, node cell over the wireless channel to node , which responds by and transmitting timing cell to node . The timestamps are recorded by node at pretransmission and postreception of timing cells and , respectively. Similarly, node records and according to its own time reference by ) at postreception and pretransmis(offset from node sion of timing cells and , respectively. such timing cells is are exchanged between and and the first of them chosen as the initial reference time. Now observe from Fig. 1 that for any geometrical shape for the transmission range of sensor nodes, a few other nodes (e.g., is ) lie node , whose clock offset with respect to node within the intersection of the broadcast regions of nodes and . Node and other similar nodes can listen to the whole message exchange between nodes and , estimate their respective clock offsets, and save the energy that otherwise would have been spent on their individual two-way exchanges with their reference nodes. For this reason, let all the transmitted messages be

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m p

Fig. 1. WSN with two active nodes and exchanging timing cells with silent nodes like , located within the common broadcast region of the active nodes and .

m

p

q

represented by the transmitter’s index only without any referand in the above ence to the receiving node so that paragraph now change to and , respectively. As illustrated in Fig. 2, assume that node timestamps the timing cells coming and as and , respectively. Notice from nodes , sent by node that node is also receiving the packets and timestamped by node , along with because node is required to send this information back to node inside the packet containing . During the interval between the pretransmission and postreception records of a timing packet, there are different kinds of incurred link delay uncertainties in the radio message delivery, which might assume magnitudes greater than the required precision of time synchronization. Therefore, it is very important to dig deeper into the exact nature and significance of all the components comprising these sources of error. Taking into account even the minutest details, Maroti et al. [16] classified all the link delay uncertainties incurred by the message as either deterministic or nondeterministic. The sources of delays such as send time, channel access time, interrupt handling time, receive time, etc., are nondeterministic and can range from around 5 s to 500 ms. On the other hand, there are deterministic sources of delays such as encoding time, transmission time, propagation time, reception time, decoding time, byte alignment time, etc., which can range from 0 s to 20 ms. Besides [16], numerous other authors have divided the link delay uncertainties in deterministic and nondeterministic components such as [19]–[21]. Interested readers are also encouraged to go through [22] and [23] for a detailed study of network delays and their breakdown in detail. For the discussion in this paper, we have assumed that the deterministic part of link delays is unknown but the same for all the nodes receiving the messages from nodes and . This is because usually the nodes in a WSN share the same hardware specifications and characteristics and hence undergo similar transmission, reception, encoding, decoding, and byte alignment times. In addition, the propagation time of RF waveforms is less than 1 s for ranges under 300 m, which implies that for nodes lying close by at short distances from each other, the

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Fig. 2. Two-way timing cell exchange mechanism between nodes


m and p with node q overhearing them.

difference in the propagation time of the same message will be even less than a few nanoseconds. Therefore, instead of , or , the deterministic part of link delays is denoted as in this paper. Last, the nondeterministic or random link delays , and have been modeled as coming from an exponential distribution with similar means. Network delay modeling has always been an active research topic for the last decades. Out of the proposed probability density function models to capture the distribution of the network delays, the Weibull, exponential, Gamma, and log-normal distributions [24]–[26] have received the greatest attention. Among them, the exponential distribution fits quite well several applications queue can fittingly [27]. In addition, a single-server represent the cumulative link delay for point-to-point hypothetical reference connections, where the random delays are independently modeled as exponential random variables [21]. Moreover, Abdel-Ghaffar [21] proposed five different clock offset estimation algorithms such as the median round delay, the minimum round delay, the minimum link delay, the median phase, and the average phase, among which the minimum link delay algorithm has been experimentally demonstrated to be superior than the rest [28]. Jeske [19] later mathematically proved that this algorithm yields the MLE under exponential link delays. All these results confirm that the assumption of exponential distribution for network delays is a sufficiently adequate model for experimental observations. The following equations summarize the model depicted : above for (1) (2) (3) , and are independent and identically where distributed (i.i.d.) exponential random variables with the same mean . Having formulated the problem and the associated model completely, next we will present a procedure for estimating the clock offsets of these silent nodes based on the maximum-likelihood (ML) technique at an essentially negligible

cost of a few computations. The rest of this paper is organized as follows. The complete derivation of the MLE of the clock offset is carried out and its equivalence to minimum variance unbiased estimate for known mean of link delays proved in Section III. Next, the Chapman–Robbins and the Barankin bounds for the clock offset are obtained in Sections IV and V, respectively. To deal with generalized delay distributions, Section VI proposes GMKPF which combines the importance sampling (IS)-based measurement update step with a Kalman filter (KF)-based Gaussian sum filter for the time update and proposal density generation. Section VII presents the computer simulations results which corroborate the superior performance of the proposed method and its robustness to exponential family network delay distributions. Finally, Section VII concludes the paper by summarizing its results and suggesting some potential future research problems. III. MAXIMUM-LIKELIHOOD ESTIMATION Based on (1)–(3), the likelihood function can be expressed as

(4) unit step functions are defined as being equal where the to if their argument is positive and otherwise, and represent the support constraints for the likelihood function. Now since these constraints do not depend on , the likelihood function by will be maximized by for all the fixed values of forcing the derivative of the log-likelihood function to be zero


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TABLE I INTERSECTIONS OF SURFACES (7)–(9)

us closely analyze this support region by writing the constraints in the following form: (6) (7) (8) (9) Fig. 3. Support region of the reduced likelihood function L (

; ; ).

which implies

Plugging the above value of back in (4) and exploiting the fact that the indexed values in the unit step functions are independent of the unknown parameters yields the reduced likelihood function

(5) where the subscript (1) denotes the minimum order statistics , and of the corresponding observations, i.e., are the minimum values of , and , respectively. . It is clear that the reOur goal is to estimate duced likelihood function can be maximized by , minimizing the expression . Since which subsequently becomes the cost function this cost function is linear in both and , the maximum cannot be found through its differentiation and hence must be searched over the boundary of its support region. Therefore, let

Fig. 3 draws the 3-D support region of the reduced likelihood function over which it has to be maximized, where is drawn as and . A 2-D aerial view of this support region a function of is drawn in Fig. 4, which illustrates the lines on the plane where the intersections of the curves (6)–(9) lie. It is further broken down into seven regions and highlights three sets of lines: solid, dashed, and dotted. Each of these three sets is explained in detail in the following discussion. • Solid Lines: Observe that the base of this support region is formed by the intersection of (6) with the surfaces (7)–(9), respectively. Hence, slicing horizontally this 3-D region in reveals the 2-D view of this base formed Fig. 3 at by (10) shown at the bottom of the page. The border of this base is illustrated as solid lines in Fig. 4, and is constrained to remain inside of it. • Dashed Lines: As explained above and shown in Fig. 3, the walls of the support region are formed by the three plane, distinct surfaces (7)–(9). The lines on the on which their respective intersections lie, are depicted as three dashed lines in Fig. 4 and summarized in Table I. Also explained by this table and shown in Fig. 4 is the point on plane, where all the above three surfaces meet each other, which is of paramount importance for the study considered herein. • Dotted Lines: For simplifying the derivation of the MLE and proving that it is unique, dotted lines are drawn in Fig. 4 in order to further break the base into easier-to-workwith geometrical figures. over the set Note that in maximizing , four different

(10)

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Fig. 4. Aerial 2-D view of the support region and its division in seven parts.

cases need to be considered with reference to point and the in Fig. 4. point 1) : This is the case drawn in Fig. 4 and it will suffice to derive the MLE considering it, since the other possible case is handled in a similar fashion. : 2) In this case, boundaries of the support region and the intersections of the surfaces are drawn by a mirror image or rotation of Fig. 4. The MLE remains exactly the same and its derivation follows similar arguments as in case 1). : 3) This case is not possible since implies , which is in contradiction with the first condition . : 4) This is also not possible due to a similar reason as mentioned in case 3). The following result is introduced to ease the derivation of the MLE. Theorem 1: The MLE lies on the edge of the support region, i.e., somewhere on the ceiling of any of the surfaces (7)–(9). Proof: Suppose that the MLE lies anywhere inside the support region at a point . Now for the same

can further be minimized by increasing until it touches the edge of the overlying surface. Having considered all the possibilities for the data and having into the regions , each of divided the base these regions will be individually analyzed to derive the MLE and prove its uniqueness with the help of Theorem 1. From here onward, to avoid labeling too many equations and hence keeping the presentation simple, in inequalities (7)–(9) will be denoted , and , respectively. by A. Region 1) Boundary Evaluation: As shown in Fig. 4, the base of the is a triangle formed by the vertices region , and . To find the surface marking the boundary of this region, consider any point in this region (shown in Fig. 4) and ordinate whose abscissa is at distance from abscissa . Therefore, is the point is at distance from ordinate . with coordinates Notice that and is always true since both of them are mere Euclidean distances. In addition, the relation always holds true within because the point lies between and . To satisfy the lines


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TABLE II BOUNDARY EVALUATION OF REGION R

the constraints (7)–(9) simultaneously, plug the coordinates of in them such that

It is clear from above that since . Also, since here. Therefore, the surface forms the boundary of the support region in . The main points of the above discussion are summarized in Table II. 2) Parameter Estimation: To derive the MLE in re, consider the minimization of cost function gion . By virtue of Theorem 1 and the above boundary evaluation study, the MLE lies . To see the variation in on the surface on this surface, substitute to get the modified cost function , . It is clear that can be which depends only on minimized by choosing as small as possible on this particular surface, which corresponds to the point in . Hence, is given by the MLE in

is a function depending only on the data, and is a function of depending on the data only through . Therefore, according to Neyman–Fisher factorization theorem, is a sufficient statistics for . Let stand for the probability density function (pdf) of . To check the completeness of , let be a function of such that . Suppose that there exists another is also true. Then, function for which , where and the expectation is taken with respect to . Therefore

(11)

The expression on the left-hand side above is the two-sided . It follows from the Laplace transform of the function uniqueness theorem for two-sided Laplace transform that almost everywhere, resulting in and hence there is only one unbiased function of . This when proves that the statistic is complete for estimating is known. Finally, the complete sufficient statistic is also minimal owing to Bahadur’s theorem. According to Rao–Blackwell–Lehmann–Scheffé theorem, an unbiased estimator for as a function of is

To make the paper more concise and readable, the boundary evaluation and parameter estimation problems for the remaining have been discussed in detail in the regions for each region Appendix. In conclusion, the MLE is again given by the expression (11), and hence it is unique. . To prove that it is also the MVUE, let For a known , the likelihood function in (4) can then be factored as shown in the equation at the bottom of the page, where

Notice that the MVUE of and coincide with their corresponding MLE, whereas the bias from the MLE of has been from it due to the removed in its MVUE by subtracting fact that the MVUE has been derived assuming that is known. In the next section, we turn our attention to deriving the Chapman–Robbins bound for any unbiased estimator of the clock offset .

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IV. CHAPMAN–ROBBINS BOUND In practical applications, it is extremely useful to know in advance the best performance an estimator might achieve by deriving a lower bound for it. In addition to providing information on how well the estimator can perform, it helps the researchers in finding an unbiased estimator that has the minimum possible variance among all unbiased estimators. Also, it places a benchmark against which different estimators can be compared to rank the finest one(s), without undergoing an empirical procedure. Cramer–Rao lower bound (CRLB) is the most widely used lower bound in estimation theory, but it relies on some regularity conditions. Specifically, the domain of the pdf should be independent of the unknown parameter. But in the current scenario, the domain of the likelihood function depends on the undue to which the order of differentiaknown clock offset tion and integration in the regularity condition cannot be interchanged and hence CRLB cannot be found by employing the likelihood function. Therefore, we turn our attention towards finding the Chapman–Robbins bound, which is free from regularity assumptions. The Chapman–Robbins bound for a scalar parameter is given by

For an -component vector parameter , the vector form of the Chapman–Robbins bound can be expressed as

Taking the expectation

The positive-semidefinite property of the matrix on the lefthand side of (12) implies that all the diagonal elements are positive. Therefore, utilizing (12)

(12) where nite, and

is interpreted as the matrix being positive semidefi-

The minimizing solution to the above expression on the righthand side is found to be , which yields the desired Chapman–Robbins bound as (13) It can be observed from the above relation that the Chapman–Robbins bound is correctly decaying with the square of the number of observations. V. BARANKIN BOUND The Barankin bound is the greatest lower bound on the variance of any unbiased estimator and it is given by the expression [29], [30]

.. .

.. .

.. .

.. .

In the above relation, allows to vary over the whole domain represented by the indicator functions in (4). For the current problem, the elements of are all positive, since in (12) becomes infinite otherwise. and the likelihood function as in (4), With we get

(14) where

is the solution to the integral equation (15)

and (16)


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Now proceeding with plugging the likelihood function (4) in (16), we get

Since

where and and similar expressions hold for becomes

and

. Then, (15)

the final form of the Barankin bound for the clock offset of the listening node can be expressed as Differentiating both sides with respect to

implies

Rearranging yields

Differentiating again with respect to

Therefore

Now, plugging this value in (14)

and using

(17) The above expression for the Barankin bound sets the greatest benchmark up to which any unbiased estimator of the clock offset of the listening nodes can perform. However, the next section discusses a sequential Monte Carlo sampling method built within the Bayesian paradigm, which not only gives better results in the current scenario but also deals with the general delay distributions not restricted to the exponential delay model. VI. A COMPOSITE PARTICLE FILTERING APPROACH If the exponential delay model for the network delays does not hold true for some scenarios, it becomes incumbent to generalize these results and derive a clock offset estimator for the listening nodes, which is robust to any delay distribution. As explained in Section II, the most widely used random delay models (other than exponential) are the Gamma and the Weibull. The problem with these relatively complex models is that closedform expression for the clock offset estimator does not exist in such cases, since we have to simultaneously solve a number of nonlinear equations. Therefore, we turn our attention towards adopting a composite particle filtering approach, which is applicable to any non-Gaussian delay distribution. Let the vector observation model in (1)–(3) take the form

(18)

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where

The nondeterministic link delays and have been modeled as coming from an exponential, Gamma and Weibull distributions, with similar features. and deterministic part of link Since clock offsets are assumed constant and unknown, the unknown state delay can be modeled as obeying a Gauss–Markov dynamic model of the from

where is the state transition matrix for unknown state (clock offsets and deterministic part of link delays). The additive can be modeled as Gaussian with process noise component . Since the state is zero mean and covariance constant, we set the as lower values . , our goal Given the observation samples is to find the minimum variance estimate of the unknown state . Thus, it turns out that we are looking to determine the estimator

where

denotes the set of observed samples up to time . Particle filtering is a sequential Monte Carlo sampling method built within the Bayesian paradigm. From a Bayesian is perspective, at time , the posterior distribution the main entity of interest. However, due to the non-Gaussianity of the model (18), the analytical expression of cannot be obtained in the closed-form expression, excepting for some special cases like Gaussian or exponential pdfs. Alternatively, particle filtering can be applied to approximate by stochastic samples generated using a sequential IS strategy. Since the particle filtering with the prior importance function employs no information from observations in proposing new samples, its use is often ineffective and leads to poor filtering performance. Herein, we implement a slightly changed version of the Gaussian mixture sigma point particle filter (GMSPPF) proposed in [31], and which will be referred to as a composite approach. This composite approach comes out from the utilization of another filtering technique producing a filtering probability density function used as importance function (IF) for the particle filtering. The GMSPPF is a family of methodologies that uses hybrid sequential Monte Carlo simulation and a Gaussian sum filter to efficiently estimate posterior distributions of unknown states in a nonlinear dynamic system. However, in our state–space modeling, because of the linear model, we do modify this method further. Following [31], we will next describe briefly the general framework assumed by the GMKPF method, obtained by


replacing the SPKF with a KF. We next outline the main features of the proposed approach. First, we remark that any probacan be approximated as closely as desired by bility density a Gaussian mixture model (GMM) of the following form [32]:

where stands for the number of mixing components, deis a normal distribunotes the mixing weights, and tion with mean and covariance . Thus, the predicted and updated Gaussian components, i.e., the means and covariances of the involved probability densities (posterior, importance, and so on) are calculated using the KF instead of the sigma point Kalman filter (SPKF) [31], [33]. Since the state and observation equations are linear, the KF was employed instead of the SPKF. Therefore, the resulting approach is called the GMKPF. In order to avoid the particle depletion problem in cases where the observation (measurement) likelihood is very peaked, the GMKPF represents the posterior density by a GMM, which is recovered from the resampled equally weighted particle set using the expectation–maximization (EM) algorithm. In general, for the particle filtering approach, the posterior , where and density , constitutes the complete solution to the sequential estimation problem. Our objective is to generate samples from . For this purpose, we have colthe distribution with weights lected sets of samples . The particles approximate . Finally, the conditional mean state and the corresponding error covariance can be calculated

At the end of each recursion, the particles are resampled to ensure they occur with the same probability as the weights. The GMKPF combines the IS-based measurement update step with a KF-based Gaussian sum filter for the time update and proposal density generation. In the time-update stage, GMKPF approximates the prior, proposal, and posterior density function as GMMs using banks of parallel KFs. The updated mean and covariance of each mixand follow from the KF updates. In the measurement update stage, the GMKPF uses a finite GMM representation of the posterior filtering density

where is the number of GMMs, are the mixing weights, and is a normal distribution determined from and positive-defithe th KF with predicted mean nite covariance . This is recovered from the weighted posterior particle set of the IS-based measurement update stage, by means of an EM step [34]. The EM algorithm can be used to obtain Gaussian mixture approximations from these particles and weights. Through this mechanism, the EM-based posterior GMM further mitigates the “sample depletion” problem through


its inherent “kernel smoothing” nature. The EM algorithm provides an iterative method to estimate via

with the Gaussian mixture specified by the parameter set . Specifically, the EM algorithm is a two-step iterative algorithm which works , it finds the next value via the as follows. Given a following: -step : ; • -step : . • The reader is directed to [34] for more detailed explanations of the EM algorithm for GMM. Finally, the conditional mean state estimate and the corresponding error covariance can be calculated as follows:

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• calculate their corresponding importance weights

• normalize the weights ; • approximate the state posterior distribution using the . EM algorithm 5) Infer the conditional mean and covariance: and • ; • or equivalently, upon fitting the posterior GMM, calculate the variables in (19).

VII. SIMULATION RESULTS

(19) Below we provide a pseudocode for a GMKPF algorithm that is fit for estimating clock offset in non-Gaussian delay models. Algorithm , initialize the densities: 1) At time • the posterior density is approximated by

• the process noise density is approximated by

• the observation noise density is approximated by

2) Preprediction step: • calculate the prepredictive state density using KF ; • calculate the preposterior state density using KF . 3) Prediction step: • calculate the predictive state density using GMM ; • calculate the posterior state density using GMM . 4) Observation update step: • draw samples from the importance density function

;

In this section, computer simulation results are drawn to assess the performance of MLE and GMKPF for estimating the and silent ) in WSNs. The staclock offsets (both active tionary process is assumed to achieve a given constant variance , where denotes the diagonal matrix. The number of particles and the number of Gaussian component (GMM) are 500 and 5, respectively. Since the GMKPF require proper initialization, we can use the MLE for exponential random delay as the initial guess. This initial is near their true values and the GMKPF value is used to start the tracking algorithm fairly close to the true values. The convergence of GMKPF is achieved after some iterations depending on the number of observations. Figs. 5–7 show the MSE of the estimators assuming that the random delay models are exponential, Gamma, and Weibull, respectively. The MSEs are plotted against the number of obser, uses vations, ranging from 5 to 25. The first, GMKPF -component GMM for the state posterior, and -component GMM for the process noise density, and -component GMM for the measurement noise density. In this case, a - and -component GMM is used to approximate the exponential, Gamma, and Weibull distribution of measurement noise. The GMKPF is slightly better than most of results and the GMKPF estimator did adaptively fit the posterior probability function (likelihood function) using EM. The MSEs of both estimators shown in Fig. 5 decrease with the square of the number of observations, which is due to the positive only nature of the link delays justifiably modeled as exponential random variables. Had these delays been obeying a symmetric pdf like Gaussian, the MSE would have fallen proportional to the number of data points, instead of its square. In addition, we have compared the performance of GMKPF with the MLE for exponential random delay model in Fig. 5. It is evident that the performance of GMKPF is better than the MLE and hence it should be the preferred method of estimation for the clock synchronization problem. On the other hand, interpreting this result from another angle, it can be deduced that extremely simple computational form of the MLE can be employed for clock offset estimation in resource-constrained networks, where the total computational load on the silent node is equal to finding

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Fig. 5. MSEs of clock offsets estimators for exponential random delays ( = 1).


Fig. 7. MSEs of clock offsets estimators for Weibull random delays ( = 2; = 2).

of the GMKPF. Since the analytical closed-form expressions for the clock offsets for Gamma and Weibull random delay models cannot be derived, we have carried out the simulations for Gamma and Weibull random delay models where the performance of GMKPF is again compared with the MLE derived for exponential delays. In this scenario, it is clear that the proposed GMKPF method is robust, exhibits superior performance, and can be applied to deal with any delay distribution. VIII. CONCLUSION AND FUTURE WORK

Fig. 6. MSEs of clock offsets estimators for Gamma random delays ( = 2; = 1).

the minimum of three sets of observations (size each) plus two additions. Assuming that computing the minimum of two numbers can be accomplished in a single cycle, the workload for the listening node can be summarized as additions. In conclusion, from a practical implementation point of view, if a certain degree of belief holds for the network delays to be exponential, the MLE is the more viable candidate as compared to the GMKPF where the price paid is a certain sacrifice in performance. The choice between the two is also open to adaptive implementation in the sense that different sensor nodes can run different algorithms depending on the synchronization requirement of the application and the amount of battery life left in them. Now focusing on the issue of robustness to different delay distributions, Figs. 6 and 7 clearly demonstrate the superiority

This paper extends the idea based on silent nodes overhearing a two-way timing packet exchange mechanism between the reference node and a randomly selected node by deriving the MLE for the clock offset of the listening nodes with the link delays obeying an exponential distribution. It is further shown to coincide with the MVUE when mean of link delays is known. For the clock offset estimator, Chapman–Robbins and Barankin bounds have also been derived as a measure of its performance threshold. In addition, the paper presented a robust estimator based on the GMKPF that is capable of estimating the clock offset with superior performance and in non-Gaussian random delay models, a result which targets applications in numerous WSNs applications with tight synchronization requirements in addition to being robust to any network delay distributions. As a promising future direction, formulating a procedure through which the timing error accumulation over a series of hops encompassing the whole network could be quantified is an idea worth exploring. APPENDIX The uniqueness of the MLE will now be proved by showing in (11), maximizes the likelihood that the point , or function (4) for all the regions.


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TABLE III BOUNDARY EVALUATION OF REGION R

TABLE IV BOUNDARY EVALUATION OF REGION R

TABLE V BOUNDARY EVALUATION OF REGION R

A. Region The boundary surface of the support region is depicted in Table III. According to Theorem 1, the MLE lies on the surface . Substituting this into the cost function yields . varies on the boundary surface in Now in this case, with both and , where the minimum (due to the positive sign) corresponds to the point , but the maximum (due to the negative sign) corresponds to the point . anywhere in the For deriving the MLE, consider a point at a distance of from the point and with region

the coordinates . Now relating evident that within this region, to the point through the boundary surface yields

. It is

Since the maximum value can achieve in is , it implies . Hence, the minimization problem of

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TABLE VI BOUNDARY EVALUATION OF REGION R

TABLE VII BOUNDARY EVALUATION OF REGION R

is equivalent to minimization of which in turn is proportional to minimization of . It is clear from Fig. 4 that achieves its minimum value at point . It can also be verified by considering the region as a sum of vertical segments starting and ending on the line on the line , having infinitesimal distances between is constant on each such vertical line segment, them. Since can be minimized by the greatest possible , which . This gives coincides with the line a set of points on this line for which the minimum should be searched, which in turn can be found by noting that is proportional to on the line , which corresponds to the minimum , and hence the point . is the same as in given by exTherefore, the MLE in pression (11). B. Region Working along similar lines as before, Table IV summa. Finding rizes the boundary evaluation problem for region is straightforward. Due to the above the MLE in region is given boundary evaluation study and Theorem 1, by . Clearly, it as small as possible and as can be minimized by making large as possible, both of which conditions are satisfied by the is again given by (11). point . Hence, the MLE in

C. Region The boundary evaluation of region is summarized in Table V. In region again, is proportional to . Although the maximum corresponds to the point , the minimum does not, requiring a closer look at the reanywhere in whose abscissa gion. Now consider a point and ordinate are and , and its neighborhood, respectively. Over the point is given by

Since in and hence the minimum value can . Therefore, minachieve is corresponds to minimization of imization of which subsequently requires minimization of . Recall that , consequently resulting in the coordinates of point being the MLE for . D. Regions

and

Tables VI and VII show that the surface is the envelope of the support region in both and . Notice that these two regions could have been combined as one


595

TABLE VIII BOUNDARY EVALUATION OF REGION R

larger region because both the boundary surface and the MLE and . This (as shown in the next section) are the same for has not been pursued due to the difference in the boundary evalin , but uation procedure, since in . In these two regions, the MLE lies on the surface , which is plugged into to yield . Again, the maximum (owing to the negative sign) in these two regions yields the as the solution. However, the minimum (owing to point and the positive sign) corresponds to the open areas of where . Therefore, consider a point somewhere in any of these two regions with the coordinates . Over this point and its vicinity, can be written as

Note that minimizing is now equivalent to mini. Using the relationship mizing the expression in these two regions, can achieve a minimum value of which implies . A positive coefficient with above implies that it should be chosen as small as possible, which is achieved on point . Therefore, the MLE in these two cases is also given by relation (11). E. Region The boundary evaluation of region is described in Table VIII. In this region, the modified cost function is again proportional to the expression . It is evident should be maximized and should be minimized for that the minimization of , both of which can be accomplished by choosing the point . REFERENCES [1] J. Kahn, R. Katz, and K. Pister, “Next century challenges: Mobile networking for smart dust,” in Proc. 5th Annu. Int. Conf. Mobile Comput. Netw., Seattle, WA, Aug. 1999, pp. 271–278. [2] B. Sundararaman, U. Buy, and A. Kshemkalyani, “Clock synchronization for wireless sensor networks: A survey,” Ad Hoc Netw. (Elsevier), vol. 3, no. 3, pp. 281–323, May 2005.

[3] B. Sadler and A. Swami, “Synchronization in sensor networks: An overview,” in Proc. IEEE Military Commun. Conf., Washington, DC, 2006, pp. 1–3. [4] O. Simeone and U. Spagnolini, “Distributed time synchronization in wireless sensor networks with coupled discrete-time oscillators,” EURASIP J. Wireless Commun. Netw., vol. 2007, 2007, Article ID 57054. [5] N. Khajehnouri and A. H. Sayed, “A distributed broadcasting timesynchronization scheme for wireless sensor networks,” in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., Philadelphia, PA, Mar. 2005, vol. 5, pp. 1053–1056. [6] Y.-W. Hong and A. Scaglione, “A scalable synchronization protocol for large scale sensor networks and its applications,” IEEE J. Sel. Areas Commun., vol. 23, no. 5, pp. 1085–1099, May 2005. [7] S. Ganeriwal, R. Kumar, and M. B. Srivastava, “Timing Synch protocol for sensor networks,” in Proc. 1st Int. Conf. Embedded Netw. Sensor Syst., Nov. 2003, pp. 138–149. [8] Q. Li and D. Rus, “Global clock synchronization in sensor networks,” in Proc. IEEE Conf. Comput. Commun., Hong Kong, Mar. 2004, vol. 1, pp. 564–574. [9] S. Ping, “Delay measurement time synchronization for wireless sensor networks,” Intel Res., Berkeley Lab., Berkeley, CA, IRB-TR-03-013, Jun. 2003. [10] K. Romer, “Time synchronization in ad hoc networks,” in Proc. ACM Symp. Mobile Ad Hoc Netw. Comput., Oct. 2001, pp. 173–182. [11] M. Sichitiu and C. Veerarittiphan, “Simple, accurate time synchronization for wireless sensor networks,” in Proc. IEEE Wireless Commun. Netw. Conf., 2003, pp. 1266–1273. [12] J. Elson, L. Girod, and D. Estrin, “Fine-grained network time synchronization using reference broadcasts,” in Proc. 5th Symp. Operat. Syst. Design Implement., Dec. 2002, pp. 147–163. [13] M. Mock, R. Frings, E. Nett, and S. Trikaliotis, “Continuous clock synchronization in wireless real-time applications,” in Proc. 19th IEEE Symp. Reliable Distrib. Syst., Oct. 2000, p. 125. [14] S. PalChaudhuri, A. Saha, and D. B. Johnson, “Adaptive clock synchronization in sensor networks,” in Proc. Inf. Process. Sensor Netw., Berkeley, CA, Apr. 2004, pp. 340–348. [15] W. Su and I. F. Akyildiz, “Time-diffusion synchronization protocol for wireless sensor networks,” IEEE/ACM Trans. Netw., vol. 13, no. 2, pp. 384–397, Apr. 2005. [16] M. Maroti, B. Kusy, G. Simon, and A. Ledeczi, “The flooding time synchronization protocol,” in Proc. 2nd Int. Conf. Embedded Netw. Sensor Syst., 2004, pp. 39–49. [17] K.-L. Noh, E. Serpedin, and K. Qaraqe, “A new approach for time synchronization in wireless sensor networks: Pairwise broadcast synchronization,” IEEE Trans. Wireless Commun., vol. 7, no. 9, pp. 3318–3322, Sep. 2008. [18] D. Mills, “Internet time synchronization: The network time protocol,” Internet Request for Comments, no. RFC 1129, pp. 1–27, Oct. 1989. [19] D. R. Jeske, “On the maximum likelihood estimation of clock offset,” IEEE Trans. Commun., vol. 53, no. 1, pp. 53–54, Jan. 2005. [20] K.-L. Noh, Q. Chaudhari, E. Serpedin, and B. Suter, “Novel clock phase offset and skew estimation using two-way timing message exchanges for wireless sensor networks,” IEEE Trans. Commun., vol. 55, no. 4, pp. 766–777, Apr. 2007.

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[21] H. S. Abdel-Ghaffar, “Analysis of synchronization algorithm with time-out control over networks with exponentially symmetric delays,” IEEE Trans. Commun., vol. 50, no. 10, pp. 1652–1661, Oct. 2002. [22] H. Kopetz and W. Schwabl, “Global time in distributed real-time systems,” Technische Universitat Wien, Wien, Austria, Tech. Rep. 15/89, 1989. [23] M. Horauer, U. Schmid, K. Schossmaier, R. Holler, and N. Kero, “PSynUTCevaluation of a high precision time synchronization prototype system for ethernet LANs,” in Proc. 34th Annu. Precise Time Time Interval Meeting, Reston, VA, Dec. 2002, pp. 428–432. [24] A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [25] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 2nd ed. Reading, MA: Addison-Wesley, 1993. [26] C. Bovy, H. Mertodimedjo, G. Hooghiemstra, H. Uijterwaal, and P. Mieghem, “Analysis of end-to-end delay measurements in internet,” in Proc. Passive Active Meas. Workshop, Fort Collins, CO, Mar. 2002, pp. 26–33. [27] S. Moon, P. Skelley, and D. Towsley, “Estimation and removal of clock skew from network delay measurements,” in Proc. IEEE INFOCOM Conf. Comput. Commun., New York, Mar. 1999, pp. 227–234. [28] V. Paxson, “On calibrating measurements of packet transit times,” in Proc. 7th ACM Sigmetrics Conf., Jun. 1998, vol. 26, pp. 11–21. [29] P. Forster and P. Larzabal, “On lower bounds for deterministic parameter estimation,” in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., Orlando, FL, May 2002, pp. 1137–1140. [30] T. Marzetta, “Computing the Barankin bound, by solving an unconstrained quadratic optimization problem,” in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., Munich, Germany, Apr. 1997, pp. 3829–3832. [31] R. van der Merwe and E. Wan, “Gaussian mixture sigma-point particle filters for sequential probabilistic inference in dynamic state-space models,” in Proc. Int. Conf. Acoust. Speech Signal Process., Apr. 2003, pp. 1–4. [32] B. D. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [33] S. J. Julier and J. K. Uhlmann, “A general method for approximating nonlinear transformations of probability distributions,” RRG, Dept. Eng. Sci., Univ. Oxford, Oxford, U.K., Nov. 1996. [34] F. Pernkopf and D. Bouchaffra, “Genetic-based EM algorithm for learning Gaussian mixture models,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 27, no. 8, pp. 1344–1348, Aug. 2005.

Qasim M. Chaudhari received the B.E. degree from National University of Sciences and Technology, Rawalpindi, Pakistan, in 2001, the M.Sc. degree from the University of Southern California, Los Angeles, in 2004, and the Ph.D. degree from Texas A&M University, College Station, in 2008, all in electrical engineering. He worked at the Communications Enabling Technologies from 2001 to 2002 in the SoC Tools Group. He worked as an intern at Qualcomm Inc. in the HSDPA Performance-Test Team in 2005–2006. Currently, he holds an Assistant Professor position at the Department of Electrical Engineering, Iqra University, Islamabad, Pakistan. He coauthored the research monograph Synchronization in Wireless Sensor Networks (Cambridge, U.K.: Cambridge Univ. Press, 2009). His research interests include estimation and detection theory in general and synchronization in wireless networks in particular.

Erchin Serpedin received (with highest distinction) the Diploma of Electrical Engineering from the Polytechnic Institute of Bucharest, Bucharest, Romania, in 1991, the specialization degree in signal processing and transmission of information from Ecole Superiure DElectricit, Paris, France, in 1992, the M.Sc. degree from Georgia Institute of Technology, Atlanta, in 1992, and the Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, in January 1999. In July 1999, he joined Texas A&M University, College Station, as an Assistant Professor, and currently holds the position of Associate Professor. His research interests lie in the area of signal processing and its applications in wireless communications, bioinformatics, and genomics. He is the author of 67 journal papers and 90 conference papers. He coauthored with Dr. Q. Chaudhari the research monograph Synchronization in Wireless Sensor Networks (Cambridge, U.K.: Cambridge Univ. Press, July 2009). Dr. Serpedin is currently serving as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON INFORMATION THEORY, Signal Processing—Elsevier, EURASIP Journal on Advances in Signal Processing, and EURASIP Journal on Bioinformatics and Systems Biology.

Jang-Sub Kim received the M.Sc. and Ph.D. degrees from the School of Electrical and Computer Engineering, Sungkyunkwan University, Jangan Gu, Suwon, Korea, in 1999 and 2005, respectively. He is currently a Visiting Researcher at Texas A&M University, College Station. His research interests lie in the fields of signal processing, wireless communications, and telecommunications networks.