# An Overview of a Class of Clock Synchronization Algorithms for Wireless Sensor Networks: A Statistical Signal Processing Perspective

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## Abstract

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## 1. Introduction

## 2. System Model for Pairwise Clock Synchronization

#### 2.1. Clock Offset and Skew

#### 2.2. Two-Way Message Exchange Mechanism

## 3. Pairwise Clock Synchronization under Gaussian Delays

- The central limit theorem (CLT) states that the PDF of the sum of a large number of independent and identically-distributed (i.i.d.) random variables is approximately normally distributed. Therefore, the Gaussian model is appropriate if the random delays are assumed to be the summation of multiple independent random variables.
- Experimental results based on two Texas Instruments ez430-RF2500 evaluation boards were recorded in [22] to demonstrate the fitness of the Gaussian distribution in modeling the random portion of delays in WSNs.

**Φ**and d is expressed as:

**Φ**is given by:

**Φ**is obtained by plugging the corresponding estimate of d back into Equation (6). Finally, the MLEs of θ and f are obtained from the MLE of

**Φ**using the invariance principle [23].

## 4. Pairwise Clock Synchronization under Exponential Delays

- For the point-to-point hypothetical reference connection (HRX) between two nodes, a single-server M/M/1queue can appropriately represent the aggregate link delay, where the random delays are modeled as independent exponential random variables [25].
- Among all distributions with a fixed mean in the support $[0,+\mathrm{\infty})$, the exponential distribution achieves the maximum differential entropy, and thus, it is the least informative.

#### 4.1. Clock Offset Estimation under Exponential Delays

#### 4.1.1. Maximum Likelihood Estimator

#### 4.1.2. Best Linear Unbiased Estimator

#### 4.1.3. Minimum Variance Unbiased Estimator

**Φ**denote the parameters to be estimated and z represent the set of the observations; then, the Neyman–Fisher factorization theorem states that if the likelihood function $p\left(\mathbf{z}\right|\mathbf{\Phi})$ can be factorized as:

**z**only through $T\left(\mathbf{z}\right)$ and h is a function depending on

**z**only, then $T\left(\mathbf{z}\right)$ is a sufficient statistic for

**Φ**. In a two-way message exchange mechanism under symmetric exponential delays, $\mathbf{\Phi}={[d,\theta ,\lambda ]}^{T}$ and $\mathbf{z}={[{U}_{1},{U}_{2},\cdots ,{U}_{N},{V}_{1},{V}_{2},\cdots ,{V}_{N}]}^{T}$. The MVUE of

**Φ**was derived in [31] by implementing the following procedure. Express first the likelihood function in terms of the unit step function $u[\xb7]$ as follows:

**Φ**and ${g}_{1},{g}_{2},{g}_{3}$ are functions depending on the data only through $\mathbf{T}=\{{\displaystyle \mathrm{\sum}_{i=1}^{n}}({U}_{i}+{V}_{i}),{U}_{\left(1\right)},{V}_{\left(1\right)}\}$. Using the Neyman–Fisher factorization theorem, it turns out that

**T**is a sufficient statistic. Moreover, the Rao–Blackwell–Lehmann–Scheffe theorem (p. 109 in [23]) claims that a sufficient statistic

**T**is complete if there is only one function $c(\xb7)$ of

**T**that is unbiased, and this function leads to the MVUE, i.e., ${\mathbf{\Phi}}_{\text{MVU}}=c\left(\mathbf{T}\right)$. Therefore, the remaining task is to prove that

**T**is complete or, equivalently, that only one function of

**T**is unbiased, and to find that function.

**T**, denoted as ${\mathbf{T}}^{\prime}=\{{\displaystyle \mathrm{\sum}_{i=1}^{N}}({U}_{i}+{V}_{i}-{U}_{\left(1\right)}-{V}_{\left(1\right)}),{U}_{\left(1\right)},{V}_{\left(1\right)}\}$. Then, it was shown that ${\mathbf{T}}^{\prime}$ is complete by assuming that there are two functions of ${\mathbf{T}}^{\prime}$ leading to unbiasedness, i.e., $\text{E}\left(c\left({\mathbf{T}}^{\prime}\right)\right)=\text{E}\left(h\left({\mathbf{T}}^{\prime}\right)\right)=\mathbf{\Phi}$, and then, proving that, actually, $c\left({\mathbf{T}}^{\prime}\right)=h\left({\mathbf{T}}^{\prime}\right)$. It turns out that

**T**is also complete, since the sufficient statistics are unique within one-to-one transformations [23]. To this end, what remains to prove is to find an unbiased estimator for

**Φ**as a function of

**T**, which represents the MVUE according to the Rao–Blackwell–Lehmann–Scheffe theorem. It seems difficult to find three unbiased functions of

**T**for each of $d,\theta $ and λ just by inspection. However, it can be observed that the BLUE ${\mathbf{\Phi}}_{\text{BLUE-S}}$ in Equation (16) is a function of

**T**and is also unbiased. Therefore, it is concluded that the BLUE is also the MVUE:

#### 4.1.4. Comparison of Estimators

Clock Offset | Symmetric Delays | Asymmetric Delays | ||||||
---|---|---|---|---|---|---|---|---|

Formula | Bias | Variance | MSE | Formula | Bias | Variance | MSE | |

MLE [28,29] | $\frac{{U}_{\left(1\right)}-{V}_{\left(1\right)}}{2}$ | 0 | $\frac{{\lambda}^{2}}{2{N}^{2}}$ | $\frac{{\lambda}^{2}}{2{N}^{2}}$ | $\frac{{U}_{\left(1\right)}-{V}_{\left(1\right)}}{2}$ | $\frac{{\lambda}_{1}-{\lambda}_{2}}{2N}$ | $\frac{{\lambda}_{1}^{2}+{\lambda}_{2}^{2}}{4{N}^{2}}$ | $\frac{{\lambda}_{1}^{2}+{\lambda}_{2}^{2}-{\lambda}_{1}{\lambda}_{2}}{2{N}^{2}}$ |

BLUE [30,31] | $\frac{{U}_{\left(1\right)}-{V}_{\left(1\right)}}{2}$ | 0 | $\frac{{\lambda}^{2}}{2{N}^{2}}$ | $\frac{{\lambda}^{2}}{2{N}^{2}}$ | $\frac{N({U}_{\left(1\right)}-{V}_{\left(1\right)})-(\overline{U}-\overline{V})}{2(N-1)}$ | 0 | $\frac{{\lambda}_{1}^{2}+{\lambda}_{2}^{2}}{4N(N-1)}$ | $\frac{{\lambda}_{1}^{2}+{\lambda}_{2}^{2}}{4N(N-1)}$ |

MVUE [31] | $\frac{{U}_{\left(1\right)}-{V}_{\left(1\right)}}{2}$ | 0 | $\frac{{\lambda}^{2}}{2{N}^{2}}$ | $\frac{{\lambda}^{2}}{2{N}^{2}}$ | $\frac{N({U}_{\left(1\right)}-{V}_{\left(1\right)})-(\overline{U}-\overline{V})}{2(N-1)}$ | 0 | $\frac{{\lambda}_{1}^{2}+{\lambda}_{2}^{2}}{4N(N-1)}$ | $\frac{{\lambda}_{1}^{2}+{\lambda}_{2}^{2}}{4N(N-1)}$ |

#### 4.2. Joint Estimation of Clock Offset and Skew under Exponential Delays

#### 4.2.1. Removing Nuisance Parameters

#### 4.2.2. Direct Joint Estimation of Clock Offset and Skew

#### 4.3. Confidence Interval for Clock Offset

## 5. Pairwise Clock Synchronization under Unknown Random Delays

#### 5.1. Bootstrap Bias Correction

#### 5.2. Composite Particle Filtering

#### 5.3. Least Squares Estimators

## 6. Fully-Distributed Clock Synchronization Algorithms

#### 6.1. System Model

#### 6.2. Fully-Distributed Clock Synchronization Algorithms under Gaussian Delays

#### 6.2.1. Belief-Based Synchronization Algorithms

- (1)
- Each variable node ${\theta}_{i}$ transmits its current belief ${b}_{i}^{\left(l\right)}\left({\theta}_{i}\right)$ to all of its neighboring factornodes $\{{\gamma}_{ij},j\in {\beta}_{i}\}$.
- (2)
- Acting like an intermediate node, the message from a factor node ${\gamma}_{ij}$ to a variable node ${\theta}_{i}$ is calculated based on the belief received from ${\theta}_{j}$:$${m}_{{\gamma}_{i,j}\to {\theta}_{i}}^{\left(l\right)}\left({\theta}_{i}\right)=\int p\left({\mathbf{T}}_{i,j}\right|{\theta}_{i},{\theta}_{j}){b}_{j}^{\left(l\right)}\left({\theta}_{j}\right)d{\theta}_{j}\phantom{\rule{0.277778em}{0ex}}$$
- (3)
- After variable node ${\theta}_{i}$ receives all of the messages from its neighboring factor nodes, i.e., ${\{{m}_{{\gamma}_{i,j}\to {\theta}_{i}}^{\left(l\right)}\left({\theta}_{i}\right)\}}_{j\in {\beta}_{i}}$, it updates its belief ${b}_{i}^{(l+1)}\left({\theta}_{i}\right)$ as follows:$${b}_{i}^{(l+1)}\left({\theta}_{i}\right)=p\left({\theta}_{i}\right)\prod _{j\in {\beta}_{i}}{m}_{{\gamma}_{i,j}\to {\theta}_{i}}^{\left(l\right)}\left({\theta}_{i}\right)$$

#### 6.2.2. Consensus-Based Synchronization Algorithms

#### 6.3. Fully-Distributed Clock Synchronization Algorithms under Exponential Delays

## 7. Conclusions and Open Problems

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Durisic, M.P.; Tafa, Z.; Dimic, G.; Milutinovic, V. A survey of military applications of wireless sensor networks. In Proceedings of the IEEE 2012 Mediterranean Conference on Embedded Computing (MECO), Bar, Montenegro, 19–21 June 2012; pp. 196–199.
- Alemdar, H.; Ersoy, C. Wireless sensor networks for healthcare: A survey. Comput. Netw.
**2010**, 54, 2688–2710. [Google Scholar] [CrossRef] - Mainwaring, A.; Culler, D.; Polastre, J.; Szewczyk, R.; Anderson, J. Wireless sensor networks for habitat monitoring. In Proceedings of the 1st ACM International Workshop on Wireless Sensor Networks and Applications, Atlanta, GA, USA, 28 September 2002; pp. 88–97.
- Corke, P.; Wark, T.; Jurdak, R.; Hu, W.; Valencia, P.; Moore, D. Environmental wireless sensor networks. IEEE Proc.
**2010**, 98, 1903–1917. [Google Scholar] [CrossRef] - Ye, D.; Gong, D.; Wang, W. Application of wireless sensor networks in environmental monitoring. In Proceedings of the IEEE 2nd International Conference on Power Electronics and Intelligent Transportation System (PEITS), Shenzhen, China, 19–20 December 2009; Volume 1, pp. 205–208.
- Herring, C.; Kaplan, S. Component-based software systems for smart environments. IEEE Pers. Commun.
**2000**, 7, 60–61. [Google Scholar] [CrossRef] - Gokbayrak, A.B.; Divarci, S.; Urhan, O. Wireless sensor network gateway design for home automation applications. In Proceedings of the IEEE 22nd Signal Processing and Communications Applications Conference (SIU), Trabzon, Turkey, 23–25 April 2014; pp. 1770–1773.
- Flammini, A.; Ferrari, P.; Marioli, D.; Sisinni, E.; Taroni, A. Wired and wireless sensor networks for industrial applications. Microelectron. J.
**2009**, 40, 1322–1336. [Google Scholar] [CrossRef] - Wu, Y.C.; Chaudhari, Q.; Serpedin, E. Clock synchronization of wireless sensor networks. IEEE Signal Process. Mag.
**2011**, 28, 124–138. [Google Scholar] [CrossRef] - Rhee, I.K.; Lee, J.; Kim, J.; Serpedin, E.; Wu, Y.C. Clock synchronization in wireless sensor networks: An overview. Sensors
**2009**, 9, 56–85. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mills, D.L. Internet time synchronization: The network time protocol. IEEE Trans. Commun.
**1991**, 39, 1482–1493. [Google Scholar] [CrossRef] - Ganeriwal, S.; Kumar, R.; Srivastava, M.B. Timing-sync protocol for sensor networks. In Proceedings of the 1st International Conference on Embedded Networked Sensor Systems, Los Angeles, CA, USA, 5–7 November 2003; pp. 138–149.
- Sichitiu, M.L.; Veerarittiphan, C. Simple, accurate time synchronization for wireless sensor networks. In Proceedings of the IEEE Wireless Communications and Networking (WCNC), New Orleans, LA, USA, 16–20 March 2003; Volume 2, pp. 1266–1273.
- Van Greunen, J.; Rabaey, J. Lightweight time synchronization for sensor networks. In Proceedings of the 2nd ACM International Conference on Wireless Sensor Networks and Applications, San Diego, CA, USA, 19 September 2003; pp. 11–19.
- Maróti, M.; Kusy, B.; Simon, G.; Maróti, M.; Kusy, B.; Simon, G.; Lédeczi, Á. The flooding time synchronization protocol. In Proceedings of the 2nd International Conference on Embedded Networked Sensor Systems, Baltimore, MD, USA, 3–5 November 2004; pp. 39–49.
- Youn, S. A comparison of clock synchronization in wireless sensor networks. Int. J. Distrib. Sens. Netw.
**2013**, 2013. [Google Scholar] [CrossRef] - Simeone, O.; Spagnolini, U.; Bar-Ness, Y.; Strogatz, S.H. Distributed synchronization in wireless networks. IEEE Signal Process. Mag.
**2008**, 25, 81–97. [Google Scholar] [CrossRef] - Noh, K.L.; Chaudhari, Q.M.; Serpedin, E.; Suter, B.W. Novel clock phase offset and skew estimation using two-way timing message exchanges for wireless sensor networks. IEEE Trans. Commun.
**2007**, 55, 766–777. [Google Scholar] [CrossRef] - Ahmad, A.; Noor, A.; Serpedin, E.; Nounou, H.; Nounou, M. On clock offset estimation in wireless sensor networks with Weibull distributed network delays. In Proceedings of the IEEE 20th International Conference on Pattern Recognition (ICPR), Istanbul, Turkey, 23–26 August 2010; pp. 2322–2325.
- Papoulis, A. Random Variable and Stochastic Processes; McGraw-Hill: New York, NY, USA, 1991. [Google Scholar]
- Bovy, C.; Mertodimedjo, H.; Hooghiemstra, G.; Uijterwaal, H.; Van Mieghem, P. Analysis of end-to-end delay measurements in Internet. In Proceedings of the ACM Conference on Passive and Active Leasurements (PAM), Fort Collins, CO, USA, 25–26 March 2002.
- Etzlinger, B.; Wymeersch, H.; Springer, A. Cooperative synchronization in wireless networks. IEEE Trans. Signal Process.
**2013**, 62, 2837–2849. [Google Scholar] - Kay, S. Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory; Prentice Hall: Englewood Cliffs, NJ, USA, 1993. [Google Scholar]
- Leng, M.; Wu, Y.C. On clock synchronization algorithms for wireless sensor networks under unknown delay. IEEE Trans. Veh. Technol.
**2010**, 59, 182–190. [Google Scholar] [CrossRef][Green Version] - Abdel-Ghaffar, H.S. Analysis of synchronization algorithms with time-out control over networks with exponentially symmetric delays. IEEE Trans. Commun.
**2002**, 50, 1652–1661. [Google Scholar] [CrossRef] - Moon, S.B.; Skelly, P.; Towsley, D. Estimation and removal of clock skew from network delay measurements. In Proceedings of the IEEE 18th Annual Joint Conference of the IEEE Computer and Communications Societies, New York, NY, USA, 21–25 March 1999; Volume 1, pp. 227–234.
- Paxson, V. On calibrating measurements of packet transit times. ACM SIGMETRICS Perform. Eval. Rev.
**1998**, 26, 11–21. [Google Scholar] [CrossRef] - Jeske, D.R. On maximum-likelihood estimation of clock offset. IEEE Trans. Commun.
**2005**, 53, 53–54. [Google Scholar] [CrossRef] - Ahmad, A.; Zennaro, D.; Serpedin, E.; Vangelista, L. A factor graph approach to clock offset estimation in wireless sensor networks. IEEE Trans. Inf. Theory
**2012**, 58, 4244–4260. [Google Scholar] [CrossRef] - Jeske, D.R.; Sampath, A. Estimation of clock offset using bootstrap bias-correction techniques. Technometrics
**2003**, 45, 256–261. [Google Scholar] [CrossRef] - Chaudhari, Q.M.; Serpedin, E.; Qaraqe, K. On minimum variance unbiased estimation of clock offset in a two-way message exchange mechanism. IEEE Trans. Inf. Theory
**2010**, 56, 2893–2904. [Google Scholar] [CrossRef] - Leng, M.; Wu, Y.C. On joint synchronization of clock offset and skew for wireless sensor networks under exponential delay. In Proceedings of the IEEE International Symposium on Circuits and Systems, Paris, France, 30 May–2 June 2010; pp. 461–464.
- Chaudhari, Q.M.; Serpedin, E.; Qaraqe, K. On maximum likelihood estimation of clock offset and skew in networks with exponential delays. IEEE Trans. Signal Process.
**2008**, 56, 1685–1697. [Google Scholar] [CrossRef] - Li, J.; Jeske, D.R. Maximum likelihood estimators of clock offset and skew under exponential delays. Appl. Stoch. Model. Bus. Ind.
**2009**, 25, 445–459. [Google Scholar] [CrossRef] - Li, J.; Jeske, D.R.; Pettyjohn, J. Approximate and generalized pivotal quantities for deriving confidence intervals for the offset between two clocks. Stat. Methodol.
**2009**, 6, 97–107. [Google Scholar] [CrossRef] - Li, J.; Jeske, D.R. Sequential Fixed Width Confidence Intervals for the Offset Between Two Network Clocks. Seq. Anal.
**2009**, 28, 475–487. [Google Scholar] [CrossRef] - Pettyjohn, J.; Jeske, D.R.; Li, J. Estimation and Confidence Intervals for Clock Offset in Networks with Bivariate Exponential Delays. Commun. Stat.-Theory Methods
**2013**, 42, 1024–1041. [Google Scholar] [CrossRef] - Kim, J.S.; Lee, J.; Serpedin, E.; Qaraqe, K. A robust approach for clock offset estimation in wireless sensor networks. EURASIP J. Adv. Signal Process.
**2010**, 2010, 19. [Google Scholar] [CrossRef] - Pettyjohn, J.S.; Jeske, D.R.; Li, J. Least squares-based estimation of relative clock offset and frequency in sensor networks with high latency. IEEE Trans. Commun.
**2010**, 58, 3613–3620. [Google Scholar] [CrossRef] - Efron, B.; Tibshirani, R.J. An Introduction to the Bootstrap; CRC Press: Boca Raton, FL, USA, 1994. [Google Scholar]
- Jeske, D.R.; Chakravartty, A. Effectiveness of bootstrap bias correction in the context of clock offset estimators. Technometrics
**2006**, 48, 530–538. [Google Scholar] [CrossRef] - Fujimoto, K.; Ata, S.; Murata, M. Playout control for streaming applications by statistical delay analysis. In Proceedings of the IEEE International Conference on Communications (ICC 2001), Helsinki, Finland, 11–14 June 2001; Volume 8, pp. 2337–2342.
- Loguinov, D.; Radha, H. Effects of channel delays on underflow events of compressed video over the Internet. In Proceedings of the IEEE 2002 International Conference on Image Processing, New York, NY, USA, 22–25 September 2002; Volume 2, pp. II–205–II–208.
- Ding, L.; Goubran, R.A. Speech quality prediction in VoIP using the extended E-model. In Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM'03), Francisco, CA, USA, 1–5 December 2003; Volume 7, pp. 3974–3978.
- Jeske, D.R. Jackknife Bias Correction of a Clock Offset Estimator. In Advances in Mathematical and Statistical Modeling; Springer: New York, NY, USA, 2008; pp. 245–254. [Google Scholar]
- Leng, M.; Wu, Y.C. Distributed clock synchronization for wireless sensor networks using belief propagation. IEEE Trans. Signal Process.
**2011**, 59, 5404–5414. [Google Scholar] [CrossRef][Green Version] - Du, J.; Wu, Y.C. Distributed clock skew and offset estimation in wireless sensor networks: Asynchronous algorithm and convergence analysis. IEEE Trans. Wirel. Commun.
**2013**, 12, 5908–5917. [Google Scholar] [CrossRef][Green Version] - Yedidia, J.S.; Freeman, W.T.; Weiss, Y. Understanding belief propagation and its generalizations. Explor. Artif. Intell. New Millenn.
**2003**, 8, 236–239. [Google Scholar] - Li, Q.; Rus, D. Global clock synchronization in sensor networks. IEEE Trans. Comput.
**2006**, 55, 214–226. [Google Scholar] - Giridhar, A.; Kumar, P. Distributed clock synchronization over wireless networks: Algorithms and analysis. In Proceedings of the 2006 45th IEEE Conference on Decision and Control, San Diego, CA, USA, 13–15 December 2006; pp. 4915–4920.
- Schenato, L.; Gamba, G. A distributed consensus protocol for clock synchronization in wireless sensor network. In Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, 12–14 December 2007; pp. 2289–2294.
- Zennaro, D.; Dall'Anese, E.; Erseghe, T.; Vangelista, L. Fast clock synchronization in wireless sensor networks via ADMM-based consensus. In Proceedings of the IEEE 2011 International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks (WiOpt), Princeton, NJ, USA, 9–13 May 2011; pp. 148–153.
- Maggs, M.K.; O’Keefe, S.G.; Thiel, D.V. Consensus clock synchronization for wireless sensor networks. IEEE Sens. J.
**2012**, 12, 2269–2277. [Google Scholar] [CrossRef] - Berger, A.; Pichler, M.; Klinglmayr, J.; Potsch, A.; Springer, A. Low-Complex Synchronization Algorithms for Embedded Wireless Sensor Networks. IEEE Trans. Instrum. Meas.
**2015**, 64, 1032–1042. [Google Scholar] [CrossRef] - Schenato, L.; Fiorentin, F. Average TimeSynch: A consensus-based protocol for clock synchronization in wireless sensor networks. Automatica
**2011**, 47, 1878–1886. [Google Scholar] [CrossRef] - Gang, X.; Shalinee, K. Analysis of distributed consensus time synchronization with Gaussian delay over wireless sensor networks. EURASIP J. Wirel. Commun. Netw.
**2009**, 2009. [Google Scholar] [CrossRef] - Zennaro, D.; Ahmad, A.; Vangelista, L.; Serpedin, E.; Nounou, H.; Nounou, M. Network-wide clock synchronization via message passing with exponentially distributed link delays. IEEE Trans. Commun.
**2013**, 61, 2012–2024. [Google Scholar] [CrossRef] - Luo, B.; Cheng, L.; Wu, Y.C. Fully-distributed joint clock synchronization and ranging in wireless sensor networks under exponential delays. In Proceedings of the 2014 IEEE International Conference on Communication Systems (ICCS), Macau, China, 19–21 November 2014; pp. 152–156.
- Graham, S.; Kumar, P. Time in general-purpose control systems: The control time protocol and an experimental evaluation. In Proceedings of the 43rd IEEE Conference on Decision and Control (2004 CDC), Paadise Island, Bahamas, 14–17 December 2004; Volume 4, pp. 4004–4009.
- Freris, N.M.; Graham, S.R.; Kumar, P. Fundamental limits on synchronizing clocks over networks. IEEE Trans. Autom. Control
**2011**, 56, 1352–1364. [Google Scholar] [CrossRef] - Brown, D.R.; Klein, A.G. Timestamp-free network synchronization with random pairwise message exchanges. In Proceedings of the 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, Italy, 4–9 May 2014; pp. 5754–5758.
- Dwivedi, S.; de Angelis, A.; Zachariah, D.; Handel, P. Joint Ranging and Clock Parameter Estimation by Wireless Round Trip Time Measurements. IEEE Trans. Sel. Areas Commun.
**2015**, 99. [Google Scholar] [CrossRef] - Solis, R.; Borkar, V.S.; Kumar, P. A new distributed time synchronization protocol for multihop wireless networks. In Proceedings of the 2006 45th IEEE Conference on Decision and Control, San Diego, CA, USA, 13–15 December 2006; pp. 2734–2739.
- Jeske, D.R. CUSUM algorithm for detecting translations in gamma distributions. Qual. Reliab. Eng. Int.
**2015**, 31. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Wang, X.; Jeske, D.; Serpedin, E. An Overview of a Class of Clock Synchronization Algorithms for Wireless Sensor Networks: A Statistical Signal Processing Perspective. *Algorithms* **2015**, *8*, 590-620.
https://doi.org/10.3390/a8030590

**AMA Style**

Wang X, Jeske D, Serpedin E. An Overview of a Class of Clock Synchronization Algorithms for Wireless Sensor Networks: A Statistical Signal Processing Perspective. *Algorithms*. 2015; 8(3):590-620.
https://doi.org/10.3390/a8030590

**Chicago/Turabian Style**

Wang, Xu, Daniel Jeske, and Erchin Serpedin. 2015. "An Overview of a Class of Clock Synchronization Algorithms for Wireless Sensor Networks: A Statistical Signal Processing Perspective" *Algorithms* 8, no. 3: 590-620.
https://doi.org/10.3390/a8030590