In this paper, the problem of clock synchronization for Wireless Sensor Network (WSN) under exponential delay is analyzed based on two-way message exchange mechanism. The Maximum Likelihood Estimator (MLE) for joint estimation of the clock offset and clock skew is derived, and an approximate Cramer-Rao Lower Bound (CRLB) is also developed. Simulation results verify that the proposed estimator gives improved performance compared to an existing algorithm. ©2010 IEEE.published_or_final_versionThe IEEE International Symposium on Circuits and Systems (ISCAS), Paris, France, 30 May-2 June 2010. In Proceedings of IEEE-ISCAS, 2010, p. 461-46

Leng, M

Wu, YC

HKU Scholars Hub

Title On joint synchronization of clock offset and skew for WirelessSensor Networks under exponential delayAuthor(s) Leng, M; Wu, YCCitationThe IEEE International Symposium on Circuits and Systems(ISCAS), Paris, France, 30 May-2 June 2010. In Proceedings ofIEEE-ISCAS, 2010, p. 461-464Issued Date 2010URL http://hdl.handle.net/10722/134515Rights IEEE International Symposium on Circuits and SystemsProceedings. Copyright © IEEE.On joint synchronization of clock offset and skewfor Wireless Sensor Networks under exponentialdelayMei Leng and Yik-Chung WuAbstract—In this paper, the problem of clock synchronizationfor Wireless Sensor Network (WSN) under exponential delay isanalyzed based on two-way message exchange mechanism. TheMaximum Likelihood Estimator (MLE) for joint estimation ofthe clock offset and clock skew is derived, and an approximateCramer-Rao Lower Bound (CRLB) is also developed. Simula-tion results verify that the proposed estimator gives improvedperformance compared to an existing algorithm.I. INTRODUCTIONWireless Sensor Network (WSN), emerged as an impor-tant research area in recent years, consists of many small-scale miniature devices (or sensor nodes) capable of onboardsensing, computing and communications. WSNs are used inindustrial and commercial applications to monitor data thatwould be difficult or inconvenient to monitor using wiredequipment, such as monitoring the health status of environ-ment, controlling industrial machines and home appliances,fire detection and object tracking, etc. [1] [2]. Most of theseapplications require collaborative execution of a distributedtask amongst a large set of synchronized sensor nodes. Fur-thermore, data fusion, power management and transmissionscheduling require all the nodes running on a common timeframe. However, every individual sensor in a WSN has its ownclock. Different clocks will drift from each other with time dueto many factors, such as imperfection of the oscillators andenvironmental changes. This makes clock synchronization be-tween different nodes an indispensable piece of infrastructure.Clock synchronization is not an easy task in practice dueto several unique properties of WSN. The first and mostimportant one is the limited power supply in low-end sensornodes. Due to harsh operating conditions, nodes in WSNsare mostly left unattended for their lifetimes without anymaintenance or battery replacement. To save power, eachsynchronization procedure should be simple and the frequencyof re-synchronization should be low. This makes simplicityand accuracy the primary concerns of clock synchronizationalgorithms for WSNs.The second challenge of clock synchronization in WSN isthe unknown message delays in physical and MAC layers.Kopetz and Ochsenreiter [4] for the first time analyzed theprocess of message delay and decompose the unknown delayThe authors are with the Department of Electrical and Electronic En-gineering, The University of Hong Kong, Hong Kong. Email: {meileng,ycwu}@eee.hku.hk.The work was supported in part by the HKU Seed Funding Programme,Project No. 200611159233into several components: send time, access time, transmissiontime, propagation time, reception time and receive time. Thesedelay components can be grouped into two portions: thefixed delay and the random delay. The fixed delay is usuallyunknown, and if it is not modeled explicitly, it will be treatedas a part of time offset, thus lowering the accuracy of timingparameter estimation. On the other hand, the random delay hasbeen modeled as random variables following different distribu-tions (such as Gaussian distribution, exponential distribution,Gamma and Weibull distribution) in the literature based ondifferent justifications and applications, and the difficulty ofdesigning an optimal algorithm for joint estimation of clockoffset and clock skew largely depends on the modeling of thisrandom delay.When the random delay follows Gaussian distribution, theoptimal estimator has been given in [7]. However, as pointedout in [3], in many cases, (e.g., when the point-to-point HRX(Hypothetical Reference Connection) topology is of interest),the link delay between two nodes is appropriately representedas a regular M/M/1 queue, and the random delay should bemodeled as exponential random variables. In this case, it ismuch more difficult to design the optimal clock synchronizer.Jeske [5] derived the Maximum Likelihood Estimator (MLE)of clock offset with an unknown fixed delay. But unfortunatelythe clock skew is not considered, which may result in frequentre-synchronization. Therefore, Noh et al. [6] proposed analgorithm for joint estimation of clock offset and clock skewby treating the fixed delay as a nuisance paramter (denoted asEMLLE in this paper). Unfortunately, in EMLLE, not all theavailable data are used, thus the performance of the EMLLEis limited.In this paper, the MLE for joint estimation of the clockoffset and clock skew is derived when the fixed delay is un-known and the random delay follows exponential distribution.In general, the Cramer-Rao Lower Bound (CRLB) does notexist for exponential delay, but by using a mild approximation,an approximate CRLB is derived. Simulation results confirmthat the proposed estimator provides improved performanceover the EMLLE.II. SYSTEM MODELWe consider the synchronization between a parent node Pand its child node S based on a two-way timing messageexchange mechanism as shown in Fig. 1. In the ith round ofmessage exchange, node S sends a synchronization message to978-1-4244-5309-2/10/$26.00 ©2010 IEEE 461Fig. 1. Two-way time-stamps exchange between two nodes S and P .node P at T1,i. Node P records its time T2,i at the reception ofthat message, and replies node S at T3,i. The replied messagecontains both time-stamps T2,i and T3,i. Then node S recordsthe reception time of node P ’s reply as T4,i. Note that T1,iand T4,i are the time stamps recorded by the clock of nodeS, while T2,i and T3,i are recorded by that of node P . AfterN rounds of message exchange, node S obtains a set of timestamps {T1,i, T2,i, T3,i, T4,i}Ni=1. The above procedure can bemodeled as [6]T2,i = β1 × T1,i + β0 + β1 × (d+Xi), (1)T3,i = β1 × T4,i + β0 − β1 × (d+ Yi), (2)where β0 and β1 represents the clock offset and clock skew ofnode S with respect to node P , respectively; d stands for thefixed portion of message delay from one node to another; andXi and Yi are variable portions of the message delay. Basedon the reasons explained in Section I and the fact that thedown/up links between two nodes are usually symmetric, Xiand Yi are assumed to be independent and identical distributed(i.i.d.) and follow exponential distribution with common rateparameter. The goal is to estimate clock offset β0 and clockskew β1 based on the observation of a set of time-stamps{T1,i, T2,i, T3,i, T4,i}Ni=1.III. MAXIMUM LIKELIHOOD ESTIMATOR (MLE) FORCLOCK OFFSET AND SKEWFirst notice that the fixed delay d is unknown and it isa nuisance parameter in the case of clock synchronization.By observing that the uplink and downlink undergo the sameamount of fixed delay, we can rewrite the original model byadding (1) to (2), and we haveT2,i+T3,i = β1× (T1,i+T4,i)+2β0+β1× (Xi−Yi) . (3)Dividing the above equation by β1, defining θ1 = 1/β1 andθ0 = β0/β1, and stacking all the time-stamps in matrix form,the model becomes T1,1 + T4,1...T1,N + T4,N= T2,1 + T3,1 −2... ...T2,N + T3,N −2[θ1θ0]+ Y1 −X1...YN −XN .(4)When Xi and Yi are i.i.d. and follow the exponentialdistribution with rate parameter λ, it is easy to see thatZi , Yi − Xi follows Lapacian distribution with locationparameter 0 and scale parameter 1/λ, which can be repre-sented as Zi ∼ Laplace(0, 1/λ). Denote TS,i = T1,i + T4,iand TP,i = T2,i + T3,i, the likelihood function can be writtenasln f({TS,i, TP,i}Ni=1; θ1, θ0)= N lnλ2− λ ·N∑i=1∣∣∣TS,i − θ1TP,i + 2θ0∣∣∣. (5)Therefore, the MLE becomes minimizing the second termon the RHS of (5)[θ∗1 , θ∗0 ] = argminθ1,θ0N∑i=1∣∣∣TS,i − θ1TP,i + 2θ0∣∣∣, (6)which is an unconstrained L1-norm minimization problem.Denote the objective function as F (θ1, θ0), we observe thatif θ1 is given, F (θ0) = 2∑Ni=1∣∣∣θ0 − 0.5(θ1TP,i − TS,i)∣∣∣,and the MLE of θ0 is the median value of the sequence{0.5(θ1TP,i−TS,i)}Ni=1 [8]. On the other hand, if θ0 is given,the MLE of θ1 is the optimal solution of the following problemargminθ1N∑i=1TP,i∣∣∣θ1 − TS,i + 2θ0TP,i∣∣∣. (7)Treating (TS,i + 2θ0)/TP,i as the location of the ith particlewith weight TP,i on a horizontal line, it can be proved thatthe MLE of θ1 is the so-called weighted median value of thedata set {[TP,i, (TS,i + 2θ0)/TP,i]}Ni=1 [9]. The procedure forfinding the weighted median value is given as follow.1) Sort the location sequence ρi , (TS,i + 2θ0)/TP,i inascending order, such that ρ[1] < · · · < ρ[i] < · · · < ρ[N ],where {ρ[i]}Ni=1 is the order statistics of {ρi}Ni=1 (thatis, [i] is the index of the ith smallest element among{ρi}Ni=1);2) Find the smallest value of K such that∑[K]i=[1] TP,i ≥0.5∑Ni=1 TP,i, and denote the solution as K∗;3) The weighted median of the data set {[TP,i, (TS,i +2θ0)/TP,i]}Ni=1 is ρ[K∗] = (TS,[K∗] + 2θ0)/TP,[K∗].Based on the above observations, an iterative algorithm isproposed to find the optimal solution. First, the Least Squares(LS) solution is calculated as[θ01, θ00] =TP,1 −2... ...TP,N −2†·TS,1...TS,N , (8)where [A]† indicates the pseudo-inverse operation of matrixA. Using the LS solution as the initial estimates of θ1 andθ0, the proposed iterative algorithm repeats the followingoperation until convergence.1) Estimate θk1 as the weighted median value of the sequence{[TP,i, (TS,i + 2θk−10 )/TP,i]}Ni=1;2) Estimate θk0 as the median value of the sequence0.5{θk1TP,i − TS,i}Ni=1.After obtaining the MLE of θ1 and θ0, we have β∗1 = 1/θ∗1and β∗0 = θ∗0/θ∗1 .462Since both the weighted median and the median operationsfind the MLE of the corresponding parameter, we haveF (θk−11 , θk−10 ) ≥ F (θk1 , θk−10 ) ≥ F (θk1 , θk0 ). (9)Moreover, the objective function F (θ1, θ0) in (5) is convex,and there exists no local minimums. Therefore, the proposedalgorithm will finally converge to the global optimal solution.And simulation results show that the algorithm usually con-verges within 10 iterations.IV. APPROXIMATE CRAMER-RAO LOWER BOUND (CRLB)To find CRLB, we need to take derivative of the likelihoodfunction in (5) with respect to the unknown parameters. Butthe likelihood function (5) is non-differentiable at TS,i −θ1TP,i+2θ0 = 0. Nevertheless, (5) can be approximated witha function that is differentiable. Here we use the followingapproximation|t| ≈ 1rln [cosh(rt)] , (10)where r is a user-defined parameter and can be used to controlthe accuracy of the approximation. For example, when r =200, the function (10) gives nearly perfect approximation tothe absolute value function, as shown in Fig 2.Since Zi = TS,i − 1β1TP,i +2β0β1and follows Laplaciandistribution, i.e., Zi ∼ Laplace(0, 1/λ), the approximatedlikelihood function of (5) is given byln f({TP,i, TS,i}Ni=1;β1, β0)≈ N ln λ2− λN∑i=11rln{cosh[r(TS,i − 1β1TP,i +2β0β1)]}.(11)Taking second derivative of the log-likelihood function (11)with respect to β1 and β0, we have∂2 ln f∂2β1= −λ∑(TP,i − 2β0)[−2β−31 tanh(rZi)+ rβ−41 (TP,i − β0)sech2(rZi)], (12)∂2 ln f∂2β0= −λ∑[4rβ−21 sech2(rZi)], (13)∂2 ln f∂β1∂β0= −λ∑[−2β−21 tanh(rZi)+ 2rβ−31 (TP,i − 2β0)sech2(rZi)]. (14)To calculate CRLB, we further need to take expectationof (12) – (14) with respect to Zi. Based on the fact that∫ +∞0tanh(x)e−µxdx = − ∫ 0−∞ tanh(x)eµxdx, we have∫ +∞−∞tanh(rZi)λ2e−λZidZi = 0. (15)Furthermore, we also have∫ +∞−∞sech2(rZi)λ2e−λZidZi=λr{λ2r[ψ(λ+ 2r4r)− ψ(λ4r)]− 1}︸ ︷︷ ︸,V, (16)Fig. 2. Approximation of the Laplacian random variable.because∫ +∞0sech2(x)e−µxdx =∫ 0−∞ sech2(x)eµxdx =µ2[ψ(µ+24)− ψ (µ4 )] − 1, where ψ(x) is a special functionnamed Euler psi function and its precise definition can befound in [10, pp.892-896].Using (15) and (16), it is easy to calculate the expectationof (12) – (14) with respect to Zi, and the Fisher informationmatrix isFIM(β1, β0) =[−E∂2 ln f∂β21−E ∂2 ln f∂β1∂β0−E ∂2 ln f∂β0∂β1 −E∂2 ln f∂β20]=λ2β41[A BB C], (17)where A 4= ∑Ni=1 V (TP,i − 2β0)2, B 4= 2β1∑Ni=1 V (TP,i −2β0) and C 4= 4β21NV . By inverting the matrix (17), it can beshown that the CRLB for each parameter isCRLB(β1) =β41Cλ2(AC − B2) , (18)CRLB(β0) =β41Aλ2(AC − B2) . (19)V. SIMULATION RESULTS AND DISCUSSIONSAs mentioned in Section I, Noh et al. [6] proposed analgorithm EMLLE for joint estimation of the clock offset andclock skew. Here, simulation results are presented to comparethe performances of the EMLLE and the proposed estimator.The parameters used in the simulation are λ = 1, d = 2,β1 = 1.003 and β0 = −10. Each point in the figures is anaverage of 10000 simulation runs.Fig. 3 shows the mean squared error (MSE) for estimationof the clock skew β1 as a function of the number of round ofmessage exchange N . As shown in the figure, the proposed es-timator performs much better than EMLLE. The performanceof the proposed estimator deviates slightly from the CRLB,which is due to the approximation, and similar behavior isalso found in [11]. However, the approximate CRLB can stillpredict the trend of the performance of the proposed estimator463Fig. 3. MSE of estimated clock skew βˆ1 with respect to the number ofrounds of message exchange N .Fig. 4. MSE of estimated clock offset βˆ0 with respect to the number ofrounds of message exchange N .and serve as the lowest limit. Fig. 4 shows the correspondingresults for the clock offset β0. It can be seen from the figure,the same conclusions as in Fig. 3 can be drawn.VI. CONCLUSIONSClock synchronization for WSN in the presence of expo-nential delay was discussed based on the two-way messageexchange mechanism. The MLE for joint estimation of clockskew and clock offset was derived by treating the unknownfixed delay as a nuisance parameter. The CRLB was alsoderived by approximating the original non-differentiable prob-ability density function with a highly accurate differentiablefunction. Although the performance of the proposed estimatordeviates slightly from CRLB, which seems to be a commonresult due to the mild approximations [11], the CRLB pre-dicts the trend of the performance very well and gives thelowest limit. The proposed estimator was also compared toan existing algorithm, and simulation results indicate that theproposed estimator gives improved performance compared tothe EMLLE.REFERENCES[1] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, E. Cayirci, “Wirelesssensor networks: a survey”, Cumputer Networks, vol. 38, no. 4, pp.393-422, Mar. 2002.[2] N. Bulusu and S. Jha, Wireless Sensor Networks: A Systems Perspective.Artech House, 2005.[3] H. S. Abedel-Ghaffar, “Analysis of synchronization algorithms withtime-out control over networks with experimentally symmetric delays”,IEEE Trans. On Communications. vol. 50, no. 10, pp. 1652-1661. Oct.2002.[4] H. Kopetz and W. Ochsenreiter, “Clock synchronization in distributedreal-time systems”, IEEE Trans. on Computers., vol. 36, no. 8, pp. 933-940, Aug. 1987.[5] D. R. Jeske, “On the maximum likelihood estimation of clock offset”,IEEE Trans. on Communication., vol.53 pp. 53-54, Jan. 2005.[6] K. L. Noh, Q. M. Chaudhari, E. Serpedin and B. W. Suter, “Novelclock phase offset and skew estimation using two-way timing messageexchanges for wireless sensor networks”, IEEE Trans. on Communica-tions. vol. 55, no. 4, pp. 766-777, April 2007.[7] M. Leng and Y. C. Wu, “On clock synchronization algorithms forWireless Sensor Networks with unknown delay”, to appear in IEEETrans. on Vehicular Technology.[8] Junpei Sekino, “n-Ellipse and the minimum distance sum problem”, TheAmerican Mathematical Monthly, vol. 106, No. 3, pp. 193-202, Mar.,1999.[9] Chaya Gurwitz, “Weighted median algorithms for L1 approximation”,BIT Numerical Mathematics, 301-310. June 1990.[10] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, andproducts. The sixth edition, Academic press. July 2000.[11] Esa Ollila et al., “Compact Cramer-Rao Bound expression for indepen-dent component analysis”, IEEE Trans. on Signal Processing, vol.56,No.4, pp. 1421-1428, April 2008.464

On joint synchronization of clock offset and skew for Wireless Sensor Networks under exponential delay

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On joint synchronization of clock offset and skew for Wireless Sensor Networks under exponential delay

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