Novel Clock Phase Offset and Skew Estimation Using Two ... - ECE/CIS

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Novel Clock Phase Offset and Skew Estimation Using Two-Way Timing Message Exchanges for Wireless Sensor Networks Kyoung-Lae Noh, Qasim Mahmood Chaudhari, Erchin Serpedin, and Bruce W. Suter, Senior Member, IEEE

Abstract—Recently, a few efficient timing synchronization protocols for wireless sensor networks (WSNs) have been proposed with the goal of maximizing the accuracy and minimizing the power utilization. This paper proposes novel clock skew estimators assuming different delay environments to achieve energy-efficient networkwide synchronization for WSNs. The proposed clock skew correction mechanism significantly increases the re-synchronization period, which is a critical factor in reducing the overall power consumption. The proposed synchronization scheme can be applied to the conventional protocols without additional overheads. Moreover, this paper derives the Cramer–Rao lower bounds and the maximum likelihood estimators under different delay models and assumptions. These analytical metrics serves as good benchmarks for the thus far reported experimental results. Index Terms—Clock synchronization, timing-sync protocol for sensor networks (TPSN), wireless sensor network (WSN).

I. INTRODUCTION ODAY the advancements in fabrication technology have enabled the development of tiny low-power devices capable of performing onboard sensing, computing, and communication tasks. Wireless sensor networks (WSNs) are a special type of ad hoc networks formed by networking these tiny devices in a certain area without any infrastructure [1], [2]. The limitations of the sensor devices result in WSNs having their peculiar characteristics such as cheap and unreliable sensor nodes, limited energy resources, etc. At the same time, WSNs are attracting considerable attention due to their wide range of applications, e.g., monitoring the status of industrial machines, emergency services, infrastructure monitoring, surveillance, tracking, etc. The efficient operation of WSNs hugely depends on the synchronized time among its nodes. The coordination among the nodes for power saving sleep/wake up modes, localization of sensor nodes and other sources, data fusion, object tracking, and

T

Paper approved by H. Minn, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received November 28, 2005; revised June 17, 2006. This work was supported in part by the National Science Foundation under Award CCR-0092901 and by the National Research Council/Air Force Office of Scientific Research under a fellowship. K.-L. Noh, Q. M. Chaudhari, and E. Serpedin are with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843-3128 USA (e-mail: [email protected]; [email protected]; [email protected]). B. W. Suter is with the Air Force Research Laboratory, Rome NY 13441 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2007.894102

distributive communication protocols require all the nodes running on a common time frame. In addition, the conventional network synchronization protocols for the Internet cannot be used due to the low-power and low-complexity requirements of WSNs [3]. The time synchronization problem in WSNs generally involves two steps, which are: 1) synchronizing the nodes in the network to one common absolute time by adjusting the clock time phase offset (clock offset) among the nodes and 2) correcting the clock frequency offset (clock skew) relative to a certain standard frequency. The second step is because the imperfections in the quartz crystal and the environmental conditions cause different clocks to run at slightly different frequencies. In fact, the effect of clock skew is the main reason why clock offsets keep drifting away. Hence, adjusting clock skew guarantees long-term reliability of synchronization and, therefore, reduces network-wide energy consumption in synchronization protocols. 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. Thus far, there have been proposed a few efficient algorithms such as timing synch protocol for sensor networks (TPSN), reference broadcast synchronization (RBS), and flooding time synch protocol (FTSP) [4]–[6],1 but there is a lack of analytical performance bounds and metrics to assess and compare the performance of these algorithms. At present, the accuracy achieved by these algorithms is reported based only on experimental basis without employing any statistical optimality criteria. II. RELATED WORK As mentioned before, a few protocols have been reported for synchronizing the nodes of WSNs. These protocols are subject to their own benefits, as well as limitations. For protocols that correct only the clock offset (such as TPSN [4]), synchronization has to be done more frequently than in protocols that correct both the clock offset and skew. This is due to the fact that the clocks start drifting apart in the presence of uncompensated clock skew. For example, re-synchronization must be performed every few minutes in TPSN for applications using Berkeley motes . On the other hand, protocols that correct both the clock offset and skew (such as RBS [5] and FTSP [6]) assume simultaneous reception of reference broadcasts, which is not applicable in some cases, e.g., in underwater acoustic sensor networks [7]. In [7], it has been asserted that for this type of 1TinyOS.

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NOH et al.: NOVEL CLOCK PHASE OFFSET AND SKEW ESTIMATION USING TWO-WAY TIMING MESSAGE EXCHANGES FOR WSNs

sensor networks, there are large variations in packet delay between nodes resulting in significant synchronization error. Thus, an adequate solution would be the TPSN protocol, provided that the clock skew can also be estimated along with the clock offset. This paper analyzes the clock sync protocols relying on two-way message exchanges between the nodes, a setup similar to TPSN. A thorough analysis of two-way message exchange between two nodes under the symmetric exponential noise model is carried out in [8]. Assuming that the exponential noise are known, parameter and the fixed portion of the delays [8] has argued that the maximum likelihood estimator (MLE) of does not exist because the likelihood the clock time offset function does not possess a unique maximum with respect to . It has been recently shown in [9] that the MLE of exists when is unknown, and it coincides to the estimator proposed in [10]. The contributions of this paper are as follows. First, we analyze and derive the MLEs and corresponding Cramer–Rao lower bounds (CRLBs) for the conventional clock offset model, as used in [4], assuming Gaussian and exponential models for the noise, respectively. Second, we derive the joint MLE and corresponding CRLB using a more realistic linear clock skew model assuming Gaussian random delays. Third, novel and practical clock skew estimators, which do not require to know the fixed portion of delays, are proposed. The introduction of a clock skew correction mechanism prolongs the re-synchronization period significantly and, therefore, far less power resources will be required in the synchronization process. In fact, the proposed clock synchronization mechanism can be directly applied to the conventional protocols using simple and low complexity modifications, a feature which is strongly demanding for WSNs consisting of cheap and small nodes. The rest of this paper is organized as follows. In Section III, the MLEs of clock offset are analyzed and the corresponding CRLB are derived for exponential and Gaussian random delays, respectively. Section IV presents the clock skew model adopted in this paper and derives the corresponding joint maximum likelihood (ML) clock offset and skew estimator for the Gaussian random delay model. Section V proposes practical and robust clock skew and offset estimators for both exponential and Gaussian random delays, respectively. In Section VI, various computer simulation results are provided for performance evaluations, and finally, Section VII concludes this paper.

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Fig. 1. Two-way timing message exchange between master–slave nodes that assume only clock offset.

the most widely deployed of which are Gamma, exponential, and Weibull PDFs [11], [12]. As explained in [8], a single-server M/M/1 queue can fittingly represent the cumulative link delay for point-to-point hypothetical reference connection, where the random delays are independently modeled as exponential random variables (RVs). The reason for adopting Gaussian PDF is due to the central limit theorem, which asserts that the PDF of the sum of a large number of independent and identically distributed (iid) RVs approaches that of a Gaussian RV. This model will be appropriate if the delays are thought to be the addition of numerous independent random processes. The Gaussian distribution for the phase offset errors is reported by a few authors, such as in [5], based on laboratory tests. The th up and down link delay observations corresponding to the th timing message exchange are given by and , respectively (using similar notations as in [8]), and are graphically repredenotes the clock offset sented in Fig. 1. The fixed value between two nodes, and denote the variable portions of delays, which are assumed to be either exponentially distributed RVs with means and or normal distributed RVs with mean and variance , respectively.

A. Exponential Delay Model 1) CRLB: It was proven in [9] that the MLE of exists when is unknown and exhibits the same form as the estimator proposed in [10], which is given by

III. ML CLOCK OFFSET ESTIMATION (1) Assuming no clock skew at this stage, we compute the MLE and CRLB for the clock offset using the two-way timing message exchange model. This scenario is depicted in Fig. 1, where Node A sends its time reading to Node B, which records its according to its own time scale. A similar time of arrival timing message exchange is performed from Node B to Node A, as shown in Fig. 1. Thus far, several probability density function (PDF) models have been proposed for modeling random queuing delays,

where denotes the number of observations of delay measurements. For simpler notations and further analysis, let and denote the order statistics of the sequences of and , respectively. Equadelay observations tion (1) can then be rewritten as

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where

and denote the corresponding order statistics of and , respectively. Let , then from the result in Appendix-A, the PDF of is given by

(2)

Let is given by

, then the PDF of

as a function of

(3)

Note that the estimate will be biased when uplink and . downlink delays are asymmetrically distributed, i.e., Thus, to derive the CRLB for the estimator, the delays are as. Equation sumed to be symmetric, which yields (3) can now be rewritten as

Differentiating the logarithm of (3) with respect to

Fig. 2. CRLB and MSE of the MLE of clock offset for the exponential delay model ( = 1).

Differentiating the log-likelihood function gives

gives (4)

where the regularity condition of the CRLB [13, p. 30] holds since (4) is finite and the expected value of (4) is 0. Calculating the expected value of the square of (4) gives

(6) Hence, the MLE of clock offset is given by

(7)

Therefore, the CRLB of clock offset

is given by (5)

Fig. 2 shows the simulation results corresponding to the variance and CRLB of the MLE when is 1. It can be seen that the variance of estimate goes to zero as increases (quadratic dependence), and is proportional to .

Consequently, the MLE of clock offset can be obtained by finding the means of observations and . 2) CRLB: The regularity condition [13, p. 30] holds for the given estimate since the expected value of (6) is 0. Thus, the CRLB for the MLE can be obtained by differentiating (6) w.r.t. , which gives

Hence, the CRLB for the MLE is given by

B. Gaussian Delay Model 1) ML Estimate: Assuming the set of delay observations and are independently and normally distributed with the same mean and variance , the likelihood and is function based on the observations given by

(8) Fig. 3 shows the result of the computer simulation when is 1. It can be seen that the variance of estimate is proportional to and inversely proportional to . In Fig. 4, the variances of both MLEs are compared in exponential and normal random delay channels, respectively. It can be seen that the performance of the ML clock offset estimator is strongly dependent on the type of random delay models.

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Fig. 3. CRLB and MSE of the MLE of clock offset for the Gaussian delay model ( = 1).

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find the joint MLE and CRLB for a more general clock model. and skew Fig. 5 shows the effect of clock offset on timing message exchanges between two nodes (using the same notations as in [8]). Here, the time stamps in the th message exchange and are measured by the local clock and are measured by the local clock of Node A, and of Node B, respectively. Node A transmits a synchronization packet, containing the level and ID of Node A and the value of , to Node B. Node B receives it at and transtime stamp . This packet mits an acknowledgement packet to Node A at contains the level and ID of Node B and the value of time stamps , , and . Finally, Node A receives the packet at . be the reference time, which is 0 at the real local Here, let , and and time denote the relative time stamps based on the local time observaand ) at Node A. Besides, let tions ( denote the reference clock offset at time , where denotes the real clock offset between the two nodes. Note that the can be obtained by estimating the referreal clock offset ence clock offset . The time stamp at Node is then given by B in the th uplink message

(9) where the additional term is due to the effect of clock skew. Similarly, the time stamp at Node B in the th downlink message is represented by

Fig. 4. MSEs of both MLEs of clock offset for exponential and Gaussian delays ( = 1 and  = 0:5).

IV. ML CLOCK SKEW ESTIMATION Since every oscillator has its unique clock frequency, the clock offset between two nodes generally keeps increasing. Therefore, a fixed value model for clock time difference, as in Fig. 1, is not sufficient for practical situations. Hence, estimating the difference of clock frequencies between two nodes (i.e., clock skew) increases synchronization accuracy and guarantees long-term reliability. Here, we derive the joint MLE for clock offset and skew based on the two-way timing message exchange model with Gaussian delays.

(10) where the term is again due to the effect of clock skew. Assuming and are zero mean independent Gaussian distributed RVs with variance , then the joint PDF of and is given by the first equation shown at the bottom of the next page. Further assuming that the fixed portion of delay is known and , then the likelihood function for , based , , , and , on observations is given by the second equation shown at the bottom of the next page. Differentiating the log-likelihood function with respect to gives

A. Joint ML Estimation of Clock Offset and Skew The theory applied thus far for finding the MLE and CRLB for the clock offset (assuming no clock skew) can be extended to

(11)

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Fig. 5. Two-way timing message exchange model that assume clock offset and skew.

Hence, in the given clock skew model, the joint MLE of clock can be expressed as offset

Thus, the estimate is given by

maximizing the log-likelihood function

Hence, the joint MLE of clock skew

is given by

(12) where denotes the average value of . Note that the clock offset estimate (12) in the case of the clock skew model with Gaussian random delays presents an additional term, which depends on , and this expression reduces to (7) is zero. Similarly, differentiating the log-likelihood when gives function with respect to

(13)

(14) and can be obtained by In the sequel, the joint MLE of [see (12)] into that of [see plugging the expression of and (14)]. From the result in Appendix-B, the joint MLE of can be expressed as (15) and (16), shown at the bottom of the next page, where . Note that the joint MLE depends on the value of the fixed portion of delays , which is assumed to be known in this section. Although estimating is an achievable task, we do not consider as another unknown (nuisance) parameter due to the inherent highly nonlinear and complex operations required for estimating . B. CRLB for the Joint MLE can be The CRLB for the vector parameter by taking derived from the 2 2 Fisher information matrix its inverse. From (11) and (13), the second-order derivatives of

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the log-likelihood function with respect to as

and

are found

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where and are due to and . Therefore, the Fisher information matrix becomes (17), shown at the bottom of the page. From [13, p. 40], the CRLB can be obtained by taking the th element of the Fisher information matrix inverse of the (i.e., ), and the inverse is given by

Taking the negative expectations yields (18) . Consequently, where from the result in [13, p. 37], the CRLBs of clock offset and skew for the Gaussian delay model are given, respectively, by (19)

(20)

V. PROPOSED CLOCK SKEW ESTIMATORS The joint MLE of clock offset and skew for Gaussian delays has been derived in Section IV. However, for exponentially distributed delays, the joint PDF does not possess local maxima

(15)

(16)

(17)

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with respect to either plex expression

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or

, and it assumes the highly com-

A. Exponential Delay Model , , , and are assumed to For exponential delays, be i.i.d. exponentially distributed RVs with mean . and then become zero mean Laplace distributed RVs with variance , respectively. Thus, the joint PDF of and is given by

The likelihood function becomes where denotes the indicator function (i.e., is 1 whenever its inner condition holds, otherwise being equal to 0). Thus, an alternative estimator is required for the exponential skew model. Besides, even for the Gaussian delay model, finding the joint MLE of clock skew requires some computations, as shown in (16), and the fixed portion of delays must be known (or estimated), which might not be applicable for WSNs consisting of low-end terminals. In practice, it requires an additional estimation procedure, which might deteriorate the robustness of the joint MLE. For these reasons, this paper proposes simple and robust clock skew estimators for the exponential and Gaussian delay models, respectively, which do not require prior knowledge of . Since the clock difference between two wireless terminals is monotonically increasing (or temporary decreasing, then increasing) based on the linear clock skew model adopted in this paper, the clock difference will be maximized between the first and last time stamps. From this intuition, novel and practical clock skew estimators can be developed by using the first and last observations of timing message exchanges. In this regard, this paper proposes an ML-like estimator (MLLE) that maximizes the likelihood function obtained based on a reduced subset of observations (the first and last timing stamps). from leads to From (9), subtracting

(21) Similarly, from (10), subtracting

from

Substituting rewritten as

into

, the likelihood function can be

where and . The estimate maximizing the likelihood function is given by

(23) where the order statistics , and observations either or . Let

are generated from the given represents distance terms equal to ,

then from the result in Appendix-C, the proposed MLLE can be expressed as

(24) In the sequel, using the set of distances, the proposed MLLE for exponential random delays (EMLLE) can be rewritten as

yields (25) (22)

Define the differences of the first and last time stamps as , , , and , respectively. Equations (21) and (22) can then be rewritten, respectively, as

where and set of the actual time stamps ( same results.

and

. Notice that using the ) yields exactly the

Now we are interested in the lower bound of the EMLLE to evaluate its asymptotic behavior. The derivative of the log likelihood function becomes

(26) The expected value of the square of (26) is then given by

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where is due to the fact that and are independent. Therefore, the lower bound of the EMLLE is given by

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Finally, the lower bound of the GMLLE is given by

(27)

(30)

In fact, we have followed the same steps used in CRLB derivation since the same reasoning and proof can also be applied to the lower bound derivation for the MLLE.

Note that the complexity of the MLLEs is far less than that of the GMLE. In fact, for the GMLE, the number of required muland , tiplications and additions are approximately respectively, while both MLLEs require only a few multiplications and additions (less than five), regardless of the number of beacons . Moreover, for the GMLE, the fixed portion of delays must also be estimated, which requires additional computations.

B. Gaussian Delay Model Similarly, assuming , , , and are i.i.d. normal distributed RVs with variance , and become zero mean , respectively. The normal distributed RVs with variance joint PDF of and is then given by

C. Combination of Clock Offset and Skew Estimation Since the proposed MLLEs are only for estimating the clock skew , we still need to estimate the clock offset for a complete clock synchronization. Considering the given clock skew model, the th observations of delays of timing message exand can be rewritten, change respectively, as

Hence, the likelihood function becomes

Differentiating the log-likelihood function with respect to yields

Since and are known values and can be estimated using the MLLE, the sets of delay observations between two nodes can be recomposed by (31) (32)

Thus, the proposed MLLE for the Gaussian delay model (GMLLE) is given by

(28) Again, similar procedures can be applied to derive a lower bound for the GMLLE. The second-order derivative of the log likelihood function becomes

(29) The expected value of (29) is given by

where , , and , respectively. Notice that it can be applied to the same clock offset estimator as in (1) and (7) for exponential and Gaussian delay models, respectively. Substituting the sets of delay observations yields the following clock offset estimators: exponential delays

(33)

Gaussian delays

(34)

Consequently, the proposed joint clock offset and skew estimators consist of the following steps. Step 1) Estimate clock skew using the proposed MLLE eior according to the type of ther random delays. Step 2) Recompose the sets of delay observations and , as shown in (31) and (32). Step 3) Estimate clock time offset using the estimator, either (33) or (34) corresponding to the given delay model.

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Fig. 6. MSE of the MLE of the Gaussian delay model (GMLE) and the Gaussian MLLE (GMLLE) for Gaussian random delays ( = 1).

Fig. 7. MSE of the GLME and the exponential MLLE (EMLLE) for exponential random delays ( = 1).

In fact, the proposed MLLEs require multiple message exto obtain the set of distances changes in a sync period . However, these estimators can be applied not only within the same sync period, but also throughout several consecutive sync periods. In other words, a new set of observations present in the next sync period can be substituted for the set of time stamps of the initial message exchange in the initial sync period. This substitution can be sequentially performed thereafter. Therefore, the proposed MLLEs can also like be applied to the single message exchange model TPSN without further modifications. The performance of the MLLEs is analyzed in Section VI. VI. SIMULATION RESULTS Fig. 6 compares the mean square error (MSE) of the GMLLE with the joint GMLE of clock skew and corresponding CRLB when is 1. It can be seen that the GMLLE performs close to (typical the GMLE for a reduced number of observations

Fig. 8. MSE of the GLME and the MLLEs for Gamma random delays ( = 2).

Fig. 9. MSE of the joint ML clock offset estimate and the proposed estimator for Gaussian random delays ( = 0:5).

values for energy efficient regimes), and its variance goes to zero as the number of observations increases (consistent and asymptotically efficient). Note that the GMLLE works well without knowing the fixed portion of delays , whereas the same is required by the joint GMLE. Fig. 7 illustrates the MSE of the EMLLE relative to the joint GMLE in exponential random delay channels when is 1. It can be seen that again the proposed MLLE is consistent and comparable to the GMLE. The consistency of the proposed MLLEs can also be checked from (27) and (30) since the corresponding MSE bounds approach 0 as increases. In order to evaluate the robustness of estimators, Fig. 8 compares the performance of the GMLE with the MLLEs in standard Gamma distributed (one of the most widely used models for capturing random queuing delays) random delay channels when is 2. Both MLLEs exhibit similar performance compared to the GMLE regardless of the type of random delays. This is due to the fact that the performance of the MLLE is dom, which do not vary inated by the set of distances much with respect to the type of random delays.

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TABLE I CONTRIBUTIONS ON CLOCK SYNC PROTOCOLS USING TWO-WAY MESSAGE EXCHANGES FOR WSNs

Fig. 9 compares the performance of the proposed clock offset estimator (34) with the joint Gaussian MLE of clock offset de. It can rived in (15) in the Gaussian delay model when be seen that the joint MLE outperforms the proposed estimator due to the help of the prior knowledge of and the complete set of time stamps.

Using the result in [14, p. 9], the PDFs of the order statistics and are given by

VII. CONCLUSIONS AND FUTURE STUDY In this paper, we have first derived the CRLB for the wellknown MLE of clock offset in TPSN assuming no clock skew and normally and exponentially distributed delays, respectively. Using a more realistic clock model, the joint MLE of clock offset and skew has then been proposed for Gaussian delays assuming the fixed portion of delays is known. Furthermore, we proposed novel MLLEs, requiring no prior knowledge of , for both Gaussian and exponential random delays, respectively. The proposed MLLEs can be implemented using simple modifications and present remarkably low complexity, which is an attractive feature for WSNs. These estimators and the derived performance bounds are targeting practical applications, and significant steps are conducted towards assessing the performance of different protocols currently popular for synchronization in WSNs. The proposed joint GMLE and MLLEs can be applied without further modifications to any clock synchronization protocols based on two-way timing message exchanges. The contributions of this paper are summarized in Table I. Future studies include assessing the performance of other popular synchronization protocols to achieve a global performance analysis of existing protocols.

Since the Jacobian of this transformation is 1, a joint distribution of RVs and is given by

(35) Integrating (35) with respect to

yields

which is equivalent to (2). B. Derivation of the Joint MLE of Clock Offset and Skew Plugging the expression of (14)] gives

[see (12)] into that of

[see

APPENDIX A. Derivation of Since and the order statistics and are independent, can be found by transforming a . From joint distribution using the dummy variable the assumptions, the PDF of the uplink and downlink delays and are given by

(36) Expanding (36) and after some manipulations, the MLE of clock offset can be expressed as

where . Plugging (15) into (12) yields the MLE of clock skew, shown in the equation at the top of the page.

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MLE of clock skew by

which is equivalent to (24).

f g

Fig. 10. Regions of the order statistics 

.

REFERENCES

C. Derivation of the Proposed Clock Skew Estimate The proposed clock skew can be derived by minimizing (23), which is given by

where order statistics in Fig. 10, the function

. Now divide the region of into three different regions, as shown in the first region then becomes region 1

Since is always positive, the corresponding estimate given by

is

region 1 Similarly, in the second region, the function

becomes

region 2 Hence, the estimate

is given by

region 2 any value Finally, in the third region, the function

takes the form region 3

Thus, the estimate

for the exponential delay model is given

in this region is region 3

Consequently, the estimate can be determined by choosing . The an adequate value between the order statistics maximizes the likelihood function and minmedian of imizes the mean square error of the estimate. Therefore, the

[1] I. Akyildiz et al., “Wireless sensor networks: A survey,” Comput. Netw., vol. 38, no. 4, pp. 393–422, Mar. 2002. [2] N. Bulusu and S. Jha, Wireless Sensor Networks: A Systems Perspective. Norwood, MA: Artech House, 2005. [3] B. Sundararaman et al., “Clock synchronization for wireless sensor networks: A survey,” Ad-Hoc Netw., vol. 3, no. 3, pp. 281–323, Mar. 2005. [4] S. Ganeriwal, R. Kumar, and M. B. Srivastava, “Timing synch protocol for sensor networks,” in Proceedings of 1st International Conference on Embedded Network Sensor Systems. New York: ACM Press, 2003, pp. 138–149. [5] J. Elson, L. Girod, and D. Estrin, “Fine-grained network time synchronization using reference broadcasts,” in Proc. 5th Operating Syst. Design and Implementation Symp., Dec. 2002, pp. 147–163. [6] M. Maroti, B. Kusy, G. Simon, and A. Ledeczi, “The flooding time synchronization protocol,” in Proceedings of the 2nd International Conference on Embedded Networked Sensor Systems. New York: ACM Press, 2004, pp. 39–49. [7] A. Syed and J. Heidemann, “Time synchronization for high latency acoustic networks,” Inf. Sci. Inst., Univ. Southern California, Marina del Rey, CA, Tech. Rep. ISI-TR-2005-602, 2005. [8] 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. [9] D. R. Jeske, “On the maximum likelihood estimation of clock offset,” IEEE Trans. Commun., vol. 53, no. 1, pp. 53–54, Jan. 2005. [10] V. Paxson, “On calibrating measurements of packet transit times,” in Proc. 7th ACM Sigmetr. Conf., Jun. 1998, vol. 26, pp. 11–21. [11] A. Papoulis, Probability, Random Variables and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [12] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 2nd ed. Reading, MA: Addison-Wesley, 1993. [13] S. M. Kay, Fundamentals of Statistical Signal Processing, Vol. I. Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1993. [14] H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. New York: Wiley, 2003.

Kyoung-Lae Noh received the B.S. and M.S. degrees in electronic communication engineering from Hanyang University, Seoul, Korea, in 2000 and 2002, respectively, and is currently working toward the Ph.D. degree in electrical engineering at Texas A&M University, College Station. From 2002 to 2003, he was with the Platform Research and Development Center, SK Telecom, Seoul, Korea, where he managed the service quality of cdma2000 1 and 1 evolution data optimized (EV-DO) networks. In Summer 2006, he was a Research Intern with the Digital Solution Center, Samsung Electronics, Seoul, Korea, where he was involved with an IEEE1394 over coax project. He is currently a Research Assistant with the Wireless Communications Laboratory, Department of Electrical Engineering, Texas A&M University. His research interests include wireless communications and ad hoc sensor networks.

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Qasim Mahmood Chaudhari received the B.E. degree in electrical engineering from the National University of Sciences and Technology, Rawalpindi, Pakistan, in 2001, the M.S. degree in electrical engineering from the University of Southern California, Los Angeles, in 2004, and is currently working toward the Ph.D. degree in electrical engineering at Texas A&M University, College Station. From 2001 to 2002, he was with the System-onChip (SoC) Tools Group, Communications Enabling Technologies. He recently completed an internship with the High Speed Downlink Packet Access (HSDPA) Performance Test Team, Qualcomm Inc. His research interests include wireless communications in general and synchronization in WSNs in particular.

Erchin Serpedin received the Diploma of electrical engineering degree (with highest distinction) from the Polytechnic Institute of Bucharest, Bucharest, Romania, in 1991, the Specialization degree in signal processing and transmission of information from the Ecole Superiéure D’Electricité, Paris, France, in 1992, the M.Sc. degree from the Georgia Institute of Technology, Atlanta, in 1992, and the Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, in 1999. In July 1999, he became an Assistant Professor with the Wireless Communications Laboratory, Texas A&M University, College Station, where he is currently an Associate Professor. His research interests are the areas of statistical signal processing and wireless communications. Dr. Serpedin served as a technical co-chair for the Communications Theory Symposium, Globecom 2006 Conference, and VTC Fall 2006: Wireless Access Track. He is currently an associate editor for the IEEE

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COMMUNICATIONS LETTERS, IEEE TRANSACTIONS ON SIGNAL PROCESSING, IEEE TRANSACTIONS ON COMMUNICATIONS, and IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, and EURASIP Journal on Applied Signal Processing. He was the recipient of the 2001 National Science Foundation (NSF) CAREER Award, the 2004 International Conference on Computing, Communication, and Control Technologies (CCCT) Best Conference Award, the 2004 Outstanding Faculty Award, the 2005 National Research Council (NRC) Fellow Award, and the 2006 American Society for Engineering Education (ASEE) Fellow Award.

Bruce W. Suter (M’85–SM’92) received the B.S. and M.S. degrees in electrical engineering and Ph.D. degree in computer science from the University of South Florida, Tampa, in 1972, 1972, and 1988, respectively. His past experience include academic positions with the University of Alabama at Birmingham, the Air Force Institute of Technology, and Harvard University, as well as industrial positions with Honeywell Incorporated and Litton Industries. He is currently a Director of the Center for Integrated Transmission and Exploitation (CITE), Air Force Research Laboratory (AFRL), Rome, NY. His current research interests are multiscale signal and image processing, network cross-layer optimization, unmanned aerial vehicles (UAVs) networking, and wireless communications. Dr. Suter is a member of Tau Beta Pi and Eta Kappa Nu. He was an associate editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING. He has served as organizer and chair of numerous workshops and conferences. He was the recipient of several honors including the title of Fellow of the Air Force Research Laboratory (2005) and the Arthur S. Fleming Award: Science Category (2002).