joint estimation of clock skew and offset in pairwise broadcast synchronization mechanism

2508 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 6, JUNE 2013

Joint Estimation of Clock Skew and Offset inPairwise Broadcast Synchronization Mechanism

Xuanyu Cao, Feng Yang, Xiaoying Gan, Jing Liu, Liang Qian, Xiaohua Tian, and Xinbing Wang

Abstract—The problem of jointly estimating clock skew andoffset for wireless sensor networks (WSNs) in a pairwise broad-cast synchronization (PBS) protocol is considered. The randompart of the delay is supposed to be an exponential random vari-able. We consider two estimators, i.e., joint maximum-likelihoodestimator (JMLE) and generalized ML-like estimator (GMLLE)proposed by Leng and Wu [15]. For both estimators, thecorresponding algorithms are explicitly derived and presented.For the GMLLE, the corresponding performance bound based onthe reduced set of observations is derived and the optimal valueof a user-defined parameter is identified accordingly. At last,analytical results are corroborated by numerical experiments. Weobserve that: (i) JMLE usually outperforms GMLLE at the costof larger computational complexity; (ii) JMLE, while achievingthe same estimation accuracy as that of the LP method presentedin [23], enjoys significantly lower computational complexity thanthat of the latter.

Index Terms—Clock synchronization, pairwise broadcast syn-chronization (PBS), wireless sensor networks (WSNs).

I. INTRODUCTION

W IRELESS Sensor Networks (WSNs) have a largenumber of critical applications such as industrial pro-

duction, battlefield reconnoitring and health monitoring, etc.,[1], [2] and [3]. Many of these applications require clocksynchronization for transmission scheduling or data fusion.In fact, for a wireless sensor node, approximately 17% of itstotal energy consumption is spent on clock synchronizationaccording to [4]. Since WSNs generally consist of manysmall sensors whose energy and spectrum resource are quitelimited, energy efficient synchronization protocols are highlydesirable. The energy consumption for synchronization ismainly composed of two parts: (1) the communication be-tween those wireless nodes, (2) the computation needed forimplementing the synchronization algorithms. Thus, to saveenergy, we need both accurate (to reduce necessary number ofcommunications) and low complexity (to reduce computationrelated energy cost) synchronization protocols. However, thereusually exists a tradeoff between these two aspects, for betteraccuracy often means higher computational complexity. But,as is reported by Pottie et al. [5] that the energy requiredto transmit 1 bit over 100 meters is equivalent to that of

Manuscript received July 20, 2012; revised November 10, 2012; January26 and March 07, 2013. The associate editor coordinating the review of thispaper and approving it for publication was E. Serpedin.

The authors are with the Department of Electronic Engineering, Shang-hai Jiao Tong University, China (e-mail: {apogne, fengyang, ganxiaoying,jingliu lj, lqian, xtian, xwang8}@sjtu.edu.cn). X. Cao is also with the StateKey Laboratory of Integrated Services Networks, Xidian University, China. X.Wang is also with the National Mobile Communications Research Laboratory,Southeast University, China.

Digital Object Identifier 10.1109/TCOMM.2013.032713.120511

executing 3 million computer instructions, we usually inclineto use high-accuracy synchronization mechanisms at the costof larger computational complexity unless the computationalability of the wireless sensor is extremely limited.

The clock synchronization in WSNs has been extensivelystudied in recent years and many protocols and algorithmshave been proposed. Here, we just concentrate on the mostrelevant works, i.e., those which study the clock synchroniza-tion in a statistical signal processing point of view. Freriset al. [6] have pointed out that not all clock parameterscan be determined with absolute accuracy even in a perfectsituation and hence, to some extent, explain the difficultyin synchronization for WSNs. In real circumstances, thesituation is even worse since the link delay between twowireless nodes usually contains a nondeterministic part [7].Many distributions such as Gaussian, exponential ,Gamma andWeibull, have been proposed to model the random part of thelink delay [8]. Thanks to those models, a statistical signalprocessing framework of clock synchronization is formed andmany previous papers have followed this line. The recentsynchronization mechanisms can be roughly divided into twocategories: (1) point to point synchronization protocols suchas two-way message exchange mechanism and one-way mes-sage mechanism, (2) broadcast synchronization protocols suchas pairwise broadcast synchronization (PBS) and referencebroadcast synchronization (RBS) [9].

In a two-way message exchange mechanism, Jeske [10]derives the MLE for clock offset by assuming an exponentialrandom delay model. Chaudhari et al. [11] derive the minimumvariance unbiased estimator (MVUE) of clock offset underboth symmetric and asymmetric exponential delay model.Recently, Ahmad et al. [12] study the time-varying clock offsetestimation by using the factor graph theory and consider aclass of delay models. Since energy is the major concern fordesigning synchronization protocols, people always wish toachieve long term synchronization which can make the WSNsspend less energy on repeating transmitting synchronizationsignals. Besides, it turns out that the rate (speed) of the clockis not exactly the same for every sensor [13], i.e., a clockskew exists. Consequently, to achieve long term synchroniza-tion, estimation of clock skew is necessary and even moreimportant than the estimation of clock offset. For this, Nohet al. [14] study the estimation of clock skew and offset intwo-way message exchange mechanism. Then Leng and Wu[15] propose three estimators for clock skew and offset byassuming a Gaussian delay model and an unknown fixed delay.Maximum likelihood estimation (MLE) is known as a goodestimator in estimation theory [16] because it is asymptotically

0090-6778/13$31.00 c© 2013 IEEE

CAO et al.: JOINT ESTIMATION OF CLOCK SKEW AND OFFSET IN PAIRWISE BROADCAST SYNCHRONIZATION MECHANISM 2509

pm

q

Fig. 1. PBS: When the master node m and a node p are adopting the two-way message exchange synchronization mechanism, inactive nodes located inthe common transmission range of m and p (the green ones) like node q canalso overhear the synchronization messages and thus estimate its clock skewand offset.

unbiased and its covariance approaches to the Cramer-Raolower bound (CRLB) as the number of observations goes toinfinity. As a result, Chaudhari et al. [17] derive the joint MLE(JMLE) of clock skew and offset under the exponential delaymodel. Later on, Leng and Wu [18] design a low complexityJMLE algorithm based on novel geometric analysis of thesupport region formed by the exponential delay model. AMLE based on reduced set of parameters is proposed and thecorresponding approximate CRLB is analyzed in [19].

However, there is a big disadvantage for the two-way mes-sage exchange synchronization mechanism: the synchroniza-tion is just point-to-point and the protocol does not exploit thebroadcast nature of wireless media. This drawback essentiallyleads to large communication overhead. Specifically, whentwo sensors are adopting a two-way message exchange syn-chronization mechanism, the sensors located in the commontransmission range of these two nodes can also overhear themessages (Fig.1) and should be able to synchronize its ownclock to some extent. Hence, a large number of inactivesensors can be synchronized at zero communication energyusage and this will save lots of energy for WSNs. The aboveis just the basic idea of PBS which is first proposed by Noh etal. [20] under a Guassian delay model. Afterwards, Chaudhariet al. [21] apply PBS to an exponential delay model and derivethe MVUE and MLE of clock offset (actually MLE is shownto coincide with MVUE in this scenario). Later on, Chaudhariet al. [22] also take the asymmetric exponential delay intoconsideration.

As previous works have shown, PBS and joint estimationof clock skew and offset are two important ways of savingenergy for synchronization in WSNs. However, to the best ofour knowledge, few works have combined these two methodsunder an exponential delay model. The reason may be that theanalysis is challenging: PBS involves one more node in theanalysis procedure compared to two-way message exchangemechanism and the introduction of clock skew usually makesthe problem much more complicated for exponential delaymodel as previous works have shown [17]. Ahmad et al. [23]takes the first step to estimate clock skew in PBS mechanismand convert the related estimation problem into a linearprogramming (LP). However, [23] has its limitation: standardLP algorithm (e.g., Newton’s method) is carried out throughiterations and the LP therein is related with five parameters,

which results in high computational complexity. Hence, the LPmethod in [23] may not be suitable for practical cheap wirelesssensors, whose computational ability is quite limited. In thispaper, we consider joint estimation of clock skew and offset byinvoking PBS. We adopt the exponential delay model whichcan represent the M/M/1 queue link delay better than Gaussiandelay model [24]. Although the derivation for Gaussian delaymodel is usually much simpler than exponential delay model,the latter turns out to be more suitable for practical WSNs.The primary contributions of this paper are summarized asfollows.

• We propose a low complexity JMLE algorithm. Thealgorithm is explicitly described and its computationalcomplexity is shown to be O(N3) (N is the number ofthe rounds involved in the synchronization) in the worstcase. Exploiting the special structure of the formulatedoptimization problem, we compress the number of pa-rameters to two in the final search problem. So, comparedwith the LP based method involved with five parametersin [23], the proposed algorithm possesses significantlylower complexity. Without the need for iteration, theproposed JMLE algorithm always finds out the exact MLpoint, which is a widely accepted good estimator in theliterature [16]. Hence, the performance of the proposedJMLE is well guaranteed.

• Given the limited computational capability of real sensornodes, we further endeavor to find estimators with evenlower complexity. A generalized ML-like estimator (GM-LLE) proposed by Leng and Wu [15] is invoked herein.Although the idea remains analogous, the algorithm andthe performance bound derivation are totally differentsince (i) we adopt PBS here while [15] adopts a two-way message exchange mechanism, (ii) more importantly,we assume an exponential delay model rather than aGaussian delay model used in [15] and this makes ouranalysis much more challenging. Note that the GMLLEhere is also different from the algorithm in [19]: (i) theway to reduce the parameter set is different and thepresented GMLLE depends on a user defined parameterk, (ii) the CRLB-like performance bound derived in[19] is only an approximation. In contrast, we derivethe exact CRLB-like performance bound1 and identifythe optimal value for k accordingly. In addition, weshow that the computational complexity of GMLLE isO((N − k)2 + I(N − k)), where I is the number ofiterations and I = 3 generally suffices to guaranteeconvergence. Therefore, we analytically justify that thecomputational complexity of GMLLE is much smallerthan that of JMLE.

• Last but not least, numerical experiments are imple-mented to corroborate the theoretical analysis. The op-timality of the theoretically chosen value of the user-defined parameter k for GMLLE is confirmed via sim-ulations. We observe that JMLE usually outperforms

1We notice that the derived CRLB-like performance bound can only serveas performance benchmark for GMLLE and does not apply to JMLE since itis based on the reduced set of observations. Hence, throughout this paper, soas to avoid misunderstanding, we call this bound the performance bound (orPB for short) for GMLLE rather than CRLB.


sjm rj

pm

rjmp sj

p

rjmq

rjpq

p

m

q

Fig. 2. PBS: Node m and node p are synchronizing by using a two-waymessage exchange mechanism while the inactive node q is overhearing tosynchronize.

GMLLE in terms of accuracy. However, the average CPUtime for implementing JMLE algorithm is about twicethat of GMLLE, which verifies that GMLLE has lowercomplexity than JMLE. The LP method in [23], thoughhaving the same accuracy as JMLE, consumes muchmore (more than 100 times of) CPU time than that ofJMLE and GMLLE.

The rest of this paper is organized as follows. In Section II,we describe the system model and formulate the problem. InSection III, we derive the JMLE algorithm. In Section IV, wepresent the GMLLE algorithm and derive the correspondingperformance bound, based on which we identify the optimalvalue of a user-defined parameter k. In Section V, we showthe simulation results. In Section VI, we conclude this paper.

II. STATEMENT OF THE PROBLEM

Fig.1 shows a WSN consisting of some sensor nodes.Node m is selected as the reference node whose clock isconsidered to be absolutely accurate. Node m and node p aresynchronizing via a two-way message exchange mechanism.The procedure for the j-th round exchange is shown in Fig.2and depicted as follows. Node m transmits a synchronizationsignal with time stamp smj . Then node p receives this signaland gives it a time stamp rmp

j according to its own clock.Afterwards, node p transmits a synchronization signal back tonode m at time spj of its own clock. Then node m receives thesignal and records the time as a time stamp of rpmj . Thus the j-th round is finished. As shown in Fig.1, there may exist severalsilent sensor nodes located in the common transmission rangeof node m and p. Suppose that node q is one of such nodes.Hence, when node m and p are exchanging messages, nodeq also overhears them and performs some synchronizationaccordingly. Since node q is inactive, i.e., node q does not needto pay any energy for communication, this synchronizationmechanism (PBS) turns out to be much more energy efficientthan other point-to-point mechanisms. Specifically, as Fig.1illustrates, node q will receive the message sent by m and pand give them time stamps rmq

j and rpqj respectively. Note thatq will also know the value of rmp

j since node p is required tosend this information back to node m in a two-way messageexchange mechanism. In sum, node q will get the value of{smj , spj , r

mpj , rmq

j , rpqj }.

As for the clock model, different from [21], we assumethat there exist clock skews for node p and node q in thispaper. Denote the clock rate for node p and q as ωp andωq respectively. The clock offsets certainly still exist and wedenote them as φp and φq for node p and q respectively.Hence, if the clock time of node m is t, then the clock timeof node p and q are ωpt + φp and ωqt + φq , respectively.Experiment results have shown that the link delay consists oftwo parts: deterministic delay and nondeterministic delay [7].In this paper, we assume that the deterministic part of the delayis d for all links. We further assume that the nondeterministicpart of delay is an exponential random variable with meanα. We exploit symmetric delay model, i.e., the mean of thenondeterministic part of delay for all links is the same. Here,in order to capture the reality of WSNs, we assume that d andα are fixed but unknown. A justification of this delay modelcan be found in many previous works such as [9], [15], [21]and [22].

Now, for node q, the model depicted above can be summa-rized by the following equations, where 1 ≤ j ≤ N [23]:

rmpj = ωp(s

mj + d+ zmp

j ) + φp, (1)

rmqj = ωq(s

mj + d+ zmq

j ) + φq, (2)

rpqj = ωq

(spj − φp

ωp+ d+ zpqj

)+ φq. (3)

In the above equations, zmpj , zmq

j , zpqj are nondeterministicpart of link delays, which are assumed to be i.i.d. exponentialrandom variables with common mean α. Compared with [21]and [22], due to the introduction of clock skew, the obser-vations as well as parameters can not be perfectly separatedand hence the MVUE is hard to find. Compared with [17],to derive JMLE, we now have two more parameters ωq andφq , which make the dimension of the involved parametersbigger and the situation more complicated. However, if wecan estimate the related parameters accurately, significantreduction of energy cost is expected as explained in Sec-tion I, which motivates us to search for good estimatorsin this scenario. The objective of this work is to estimateωq, ωp, φq, φp accurately with low computational complexity.In the following, we will present two estimation algorithmsand analyze the corresponding performance.

III. JOINT MAXIMUM LIKELIHOOD ESTIMATOR (JMLE)

In this section, we derive the algorithm for the JMLE.Rewriting Eqn. (1), (2) and (3), we obtain:

rmpj − φp

ωp− d− smj = zmp

j , (4)

rmqj − φq

ωq− d− smj = zmq

j , (5)

rpqj − φq

ωq− d− spj − φp

ωp= zpqj . (6)


Since zmpj , zmq

j , zpqj are i.i.d. exponential delay with mean α,the likelihood function can be written as follows [23]:

L(α, d, φp, φq, ωp, ωq)

= α−3N exp

[− 1

α

N∑j=1

(rmpj

ωp+

rmqj − φq

ωq+

rpqj − φq

ωq

− spjωp

− 3d− 2smj

)]×

N∏j=1

[u

(rmpj − φp

ωp− d− smj

)

u

(rmqj − φq

ωq− d− smj

)u

(rpqj − φq

ωq− d− spj − φp

ωp

)]

Here u(·) stands for the unit step function, i.e., u(x) = 1 ifx > 0, u(x) = 0 if x < 0. Denote the sum inside exp as ξ, i.e.,

ξ �∑N

j=1

(rmpj

ωp+

rmqj −φq

ωq+

rpqj −φq

ωq− spj

ωp− 3d− 2smj

).

Define λ � 1/α. When the support condition is satisfied,i.e., the quantities in the unit step function are all positive,we have L = λ3Ne−ξλ. Suppose parameters other than λhave already been given, then the λ̂ that maximizes L mustsatisfy ∂L/∂λ

∣∣∣λ=λ̂

= 0. So we let ∂L∂λ = λ3N−1e−ξλ(3N −

λξ) = 0. This leads to λ̂ = 3N/ξ which is equivalent to α̂ =ξ/(3N). Substituting this condition back into the expression oflikelihood function L, when the support condition is satisfied,we get L ∝ ξ−3N . Note that the sum

∑Nj=1 s

mj is a constant

in the expression of ξ. Therefore, in order to maximize L, theproblem becomes to minimize the following expression

ξ̃ � −3dN +1

ωp

N∑j=1

(rmpj − spj)+

1

ωq

N∑j=1

(rmqj + rpqj − 2φq),

(7)subject to the following constraints:

ωp > 0, ωq > 0, (8)

d ≤ rmpj − φp

ωp− smj 1 ≤ j ≤ N, (9)

d ≤ rmqj − φq

ωq− smj 1 ≤ j ≤ N, (10)

d ≤ rpqj − φq

ωq− spj − φp

ωp1 ≤ j ≤ N. (11)

The constraints (9) and (11) can be written as follows for1 ≤ j ≤ N, 1 ≤ l ≤ N :

rmpj − ωps

mj − ωpd ≥ φp ≥ ωpd+ spl −

ωp

ωq(rpql − φq). (12)

Note that φp only exists in constraints (9) and (11) and itsrange is from −∞ to +∞. Besides, the value of ξ̃ does notdepend on φp. Hence, (9) and (11) can be substituted by thefollowing constraints for 1 ≤ j ≤ N, 1 ≤ l ≤ N :

rmpj − ωps

mj − ωpd ≥ ωpd+ spl −

ωp

ωq(rpql − φq). (13)

The reason of this substitution is that as long as (13) is satis-fied, we can always find a φp which satisfies the constraints(9) and (11) and the value of this φp will not affect the value ofthe objective function ξ̃, which means that we can arbitrarilychoose a value for φ̂p as long as the constraint (13) is satisfied.

Now, except constraint (8), only constraints (10) and (13)exist. Rewriting (10) and (13), we get:

ωqd+ φq ≤ rmqj − ωqs

mj , (14)

2ωqd+ φq ≤ ωq

ωp(rmp

j − spl ) + rpql − smj ωq, (15)

where 1 ≤ j, l ≤ N . Define f(ωp, ωq) � min1≤j≤N{rmqj −

ωqsmj } and g(ωp, ωq) � min1≤j,l≤N

{(ωq/ωp)(rmpj − spl ) + rpql − smj ωq}. Then (14) and (15) are

equivalent to the following two constraints:

ωqd+ φq ≤ f(ωp, ωq), 2ωqd+ φq ≤ g(ωp, ωq). (16)

Note that the value of f and g does not depend on d and φq .Hence, the only constraints on d and φq is (16), which is alinear constraint with respect to d and φq . We further noticethat the objective function ξ̃ in Eqn. (7) is just a linear functionin terms of d and φq . Hence we rewrite Eqn. (7) as follows:

ξ̃ = −2N

ωq

(3ωq

2d+ φq

)+

1

ωp

N∑j=1

(rmpj − spj

)

+1

ωq

N∑j=1

(rmqj + rpqj

)(17)

≥ −N

ωq(f(ωp, ωq) + g(ωp, ωq)) +

1

ωp

N∑j=1

(rmpj − spj

)

+1

ωq

N∑j=1

(rmqj + rpqj

). (18)

The reason for (18) is that 3ωq

2 d + φq = 12 [(2ωqd + φq) +

(ωqd+ φq)]. Applying (16), we immediately obtain (18) andthe equality in (18) holds iff both conditions in (16) hold withequality, i.e.,

d =g(ωp, ωq)− f(ωp, ωq)

ωq, φq = 2f(ωp, ωq)− g(ωp, ωq).

(19)Denote the following expressions:

f̃(ωp, ωq) =f(ωp, ωq)

ωq= min

1≤j≤N

{rmqj

ωq− smj

}, (20)

g̃(ωp, ωq) =g(ωp, ωq)

ωq= min

1≤j,l≤N

{rmpj − splωp

+rpqlωq

− smj

}

= min1≤j≤N

{rmpj

ωp− smj

}+ min

1≤l≤N

{− splωp

+rpqlωq

}. (21)

Suppose the optimal value for d and φq have already beenchosen as in Eqn. (19). Then the inequality in (18) becomesequality and we can rewrite (18) as follows

ξ̃ = −N(f̃(ωp, ωq) + g̃(ωp, ωq)) +1

ωp

N∑j=1

(rmpj − spj

)

+1

ωq

N∑j=1

(rmqj + rpqj

). (22)

So far the estimation problem has been reduced to the problemof minimizing Eqn. (22). Note that this is a LP problem


y

xx x x x x1 2 3 4 5O

Fig. 3. An illustration for the curve of general expression y =min1≤j≤N{aj + bjx}. The curve is divided into several line segments bysome turning points xi. The skews of the line segments should decrease asx increases.

with respect to two positive parameters 1ωp

and 1ωq

. Hence,the optimal point of ( 1

ωp, 1ωq

) should lie in the set of theturning points identified by the minimization operator in theexpression of f̃(ωp, ωq) and g̃(ωp, ωq).

All the minimization operators in f̃ and g̃ take the formof min1≤j≤N{aj + bjx}. The general curve of this functionis depicted in Fig.3. The curve of y = min1≤j≤N{aj + bjx}is divided into several line segments by some turning points.The skews of the line segments should decrease as x increases.The first line segment should be the one with the minimum aj .Denote j1 = argminj{aj} and y = aj1 + bj1x should be thefirst line segment. Denote j2 the index for the second line seg-ment. Since the skew must decrease, we have j2 = argj{thesolution for aj + bjx = aj1 + bj1x is minimum

∣∣bj1 > bj}.This procedure continues until the index jl = argminj{bj} isselected. Then all the turning points are founded. This processis summarized as Algorithm 1.

Algorithm 1 JMLE: Finding the turning points formin1≤j≤N{aj + bjx}

Arrange bj in decreasing order {bjl} such that bjl ≥ bjl+1.

Find i1 = argminj{aj} and l1 such that jl1 = i1, m = 1.while lm �= N do

Find lm+1 = argminl

{ajl

−ajlm

bjlm−bjl

∣∣l > lm

}and turning

point xm =ajlm+1

−ajlm

bjlm−bjlm+1

.

m = m+ 1.end whileOutput the turning points {xi}1≤i≤m−1.

After applying Algorithm 1 to the three minimizationoperators in the expressions of f̃ and g̃, we get three setsof turning points for 1/ωp, 1/ωq and 1/ωq

1/ωprespectively.

Denote the three sets as {1/ωp = aj}1≤j≤m1 , {1/ωq =

bj}1≤j≤m2 and{

1/ωq

1/ωp= cj

}1≤j≤m3

. Each turning point of

the three sets is actually one line in the two dimensionalplane of (1/ωp, 1/ωq). These lines together tessellate the plane(1/ωp, 1/ωq) into several small regions. The optimal point of(1/ωp, 1/ωq) must lie in the intersections of these lines sinceξ̃ is a linear function (in terms of 1/ωp and 1/ωq) with fixedcoefficients inside each of these small regions. Therefore, wejust do a search among those intersections and find the onewith the minimum ξ̃. This will give the JMLE (ω̂p, ω̂q). Then,

invoking Eqn. (12) and (19), we can give the JMLE (φ̂p, φ̂q).The detailed process is summarized in Algorithm 2.

Algorithm 2 JMLE: Finding the estimates (ω̂p, ω̂q, φ̂p, φ̂q)

1: Exploit Algorithm 1 to get three sets of turningpoints: {1/ωp = aj}1≤j≤m1 , {1/ωq = bj}1≤j≤m2 and{

1/ωq

1/ωp= cj

}1≤j≤m3

.

2: Identify three sets of intersection points (1/ωp, 1/ωq):{(aj , bl)}1≤j≤m1,1≤l≤m2 , {(aj, ajcl)}1≤j≤m1,1≤l≤m3 and{(

bjcl, bj

)}1≤j≤m2,1≤l≤m3

.

3: Exploit the above intersections and Eqn. (22) to computeξ̃. Find the one which gives the minimum ξ̃. Accordingly,output the corresponding (ω̂p, ω̂q).4: Exploit Eqn. (12) and (19) to give the estimates (φ̂p, φ̂q).Note that when using (12), φ̂p can be chosen arbitrarily aslong as (12) is satisfied.

Note that the number of turning points is O(N), i.e., thenumber of lines in the (1/ωp, 1/ωq) is O(N). Hence, thenumber of intersection points needed to be checked duringthe search for JMLE is O(N2). In addition, the number ofinstructions needed to calculate ξ̃ is O(N) for each inter-section point. Therefore, the computational complexity forthe proposed JMLE algorithm is O(N3) in the worst case.One thing we should mention is that this is the worst casecomplexity. In Section V, simulation results show that theexperimental complexity can be significantly lower than thisupper bound and almost scales linearly with N .

Remark 1: The proposed JMLE possesses both low com-plexity and high accuracy. The low complexity results fromthe fact that the ultimate optimization problem (22) is a LPproblem only related to two parameters. Besides, since wegive the exact JMLE based on all the raw observations, theproposed JMLE algorithm shows good accuracy.

IV. GENERALIZED ML-LIKE ESTIMATOR (GMLLE)

Due to the quite limited computational capability of thesensor nodes in WSN, we endeavor to find estimators witheven lower computational complexity than the proposed JMLEalgorithm in the previous section. Specifically, we invoke thegeneralized ML-like estimator (GMLLE) proposed in [15].Although the method of reducing the number of parameters issimilar, the overall algorithm and performance analysis hereare very different since (i) we adopt PBS here while [15]adopts a two-way message exchange mechanism; (ii) moreimportantly, we assume an exponential delay model rather thana Gaussian delay model used in [15]. Both differences havemade our work much more challenging.

A. Algorithm

The basic idea behind GMLLE in [15] is that by subtractionof the observation equations, the fixed part of delay and theclock offset can be removed and thus the number of unknownparameters is reduced. The idea still works for exponentialdelay model in a PBS protocol herein. Specifically, subtracting


the j-th and (j + k)-th equations in (4), (5) and (6), we get:

rmpj+k − rmp

j

ωp− smj+k + smj = zmp

j+k − zmpj , (23)

rmqj+k − rmq

j

ωq− smj+k + smj = zmq

j+k − zmqj , (24)

rpqj+k − rpqjωq

− spj+k − spjωp

= zpqj+k − zpqj , (25)

where 1 ≤ j ≤ N − k. Here �N/2� ≤ k ≤ N − 1 isa user-defined parameter. One thing we should mention isthat the GMLLE is different from the algorithm in [19]. Thealgorithm in [19] add together the two equations involved inthe two-way message exchange mechanism in every round.This may not work well in the PBS mechanism since bydoing so for Eqn. (4), (5) and (6), we still have to dealwith four parameters ωp, ωq, φp, φq and only the fixed delay ddisappears. In constrast, by invoking a user defined parameterk, GMLLE can eliminate three parameters ωp, ωq and d.

In GMLLE, we constrain k to be larger than N/2 in orderto avoid the statistical dependence of the R.H.S. of Eqn. (23),(24) and (25). In fact, the random variables in the R.H.S.of (23), (24) and (25) are i.i.d. Laplace variables with zeromean. Specifically, the distribution is p(x) = 1

2α exp(− |x|

α

).

Consequently, the likelihood function is as follows:

L(ωp, ωq, α)

=

(1

2α

)3(N−k)

× exp

[− 1

α

N−k∑j=1

(∣∣∣∣∣rmpj+k − rmp

j

ωp− smj+k + smj

∣∣∣∣∣+

∣∣∣∣∣rmqj+k − rmq

j

ωq− smj+k + smj

∣∣∣∣∣+

∣∣∣∣∣rpqj+k − rpqj

ωq− spj+k − spj

ωp

∣∣∣∣∣)]

. (26)

Now we apply MLE to the above likelihood function andthe situation becomes much simpler than the strict JMLEdiscussed in Section III. To maximize L(ωp, ωq, α), we onlyneed to minimize the following expression:

ξ �N−k∑j=1

( ∣∣∣∣∣rmpj+k − rmp

j

ωp− smj+k + smj

∣∣∣∣∣+

∣∣∣∣∣rmqj+k − rmq

j

ωq− smj+k + smj

∣∣∣∣∣+

∣∣∣∣∣rpqj+k − rpqj


ωp

∣∣∣∣∣). (27)

Again we are confronting with a LP problem. Instead ofconducting an exhaustive search, we now use a low complexitybut still accurate iteration method to minimize ξ as in [19]and [29]. Suppose we have already got an estimate ω̂q, thento minimize ξ in terms of ωp is equivalent to minimizing thefollows expression (the middle absolute value term in Eqn.

(27) is omitted since it is independent of ωp):

ξ1

(1

ωp

)=

N−k∑j=1

( ∣∣∣rmpj+k − rmp

j

∣∣∣∣∣∣∣∣ 1ωp

− smj+k − smjrmpj+k − rmp

j

∣∣∣∣∣+∣∣∣spj+k − spj

∣∣∣∣∣∣∣∣ 1ωp

− rpqj+k − rpqjspj+k − spj

1

ω̂q

∣∣∣∣∣). (28)

The minimization of ξ1 is a weighted median value prob-lem [29]. For this, we should rearrange the sequence{

smj+k−smjrmpj+k−rmp

j

}1≤j≤N−k

⋃{ rpqj+k−rpqj

spj+k−spj

1ω̂q

}1≤j≤N−k

in increas-

ing order. Note that regardless of the value of ω̂q, we can

rearrange{


j

}1≤j≤N−k

and{

rpqj+k−rpqj

spj+k−spj

1ω̂q

}1≤j≤N−k

in increasing order separately in advance. So, when a new ω̂q

is generated, we only need to merge two increasing sequencesinto one long combined sequence still in increasing order. Thiscan reduce the complexity significantly. After merging the twosequences in this way, we rewrite ξ1 into the following form

ξ1 =

2(N−k)∑j=1

aj

∣∣∣∣ 1ωp− bj

∣∣∣∣ . (29)

where {bj}1≤j≤2(N−k) is in increasing order. Then we findthe minimum M such that

∑Mj=1 aj ≥ 1

2

∑2(N−k)j=1 aj . Thus

optimal 1ω̂p

, i.e., the weighted median value, is bM .After updating the estimate ω̂p, we again update ω̂q in a

similar way and keep iterating until convergence. Numericalexperiments show that three rounds of iteration are usuallyenough for convergence (one round means updates for bothω̂p and ω̂q once). As for the initial estimate, we chooseω̂p = 1, ω̂q = 1 because the clock skew is generally verysmall in real WSNs [13]. The reason that we do not exploita LS initial estimate as [19] is that the pseudo-inverse ofthe matrix involved in the LS estimation may incur bigcomputation overhead. Actually, the complexity of computingthe pseudo-inverse turns out to be even larger than that of theiterations in GMLLE. It is meaningless to spend a majority ofcomputational capability on just finding a coarse initial point.

The iterations of GMLLE raise the convergence issue,which is analyzed as follows. Note that each iteration findsthe corresponding minimum points for either ωp or ωq , i.e.,the value of the target function ξ always decreases after eachiteration. Specifically, suppose that the estimates at the k-thiteration are ω

(k)p and ω

(k)q . We have:

ξ

(1

ω(k)p

,1

ω(k)q

)≥ ξ

(1

ω(k+1)p

,1

ω(k)q

)

≥ ξ

(1

ω(k+1)p

,1

ω(k+1)q

).

Furthermore, the objective function ξ in Eqn. (27) is convexwith respect to 1

ωpand 1

ωq. Therefore, the proposed GMLLE

will finally converge to the global optimum point.With the estimates ω̂p and ω̂q available, we can now easily

give good estimates for the rest of the parameters by exploitingthe results of [21]. Plugging the estimates ω̂p and ω̂q back into


Eqn. (4), (5) and (6), we obtain:

Uj �rmpj

ω̂p− smj = d+

φp

ω̂p+ zmp

j , (30)

Vj �rmqj

ω̂q− smj = d+

φq

ω̂q+ zmq

j , (31)

Wj �rpqjω̂q

− spjω̂p

= d+

(1

ω̂q− 1

ω̂p

)φp + zpqj . (32)

Viewing φp/ωp and φq/ωq as new clock offsets, we get a pureclock offset estimation problem. Thanks to [21], the MLE andMVUE are the same in this case and can be written as follows:

φ̂p = ω̂p(V(1) −W(1)), (33)

φ̂q = ω̂q(2V(1) − U(1) −W(1)), (34)

d̂ = U(1) +W(1) − V(1), (35)

α̂ =1

3N

N∑j=1

(Uj + Vj +Wj)− 1

3U(1) − 1

3V(1) +

2

3W(1).(36)

where U(1) � min1≤j≤N{Uj}, V(1) and W(1) are similarlydefined.

The computational complexity is analyzed as follows.

The ordering of the sequence{


j

},{

rpqj+k−rpqj

spj+k−spj

},{

smj+k−smjrmqj+k−rmq

j

}and

{spj+k−spjrpqj+k−rpqj

}can be accomplished with a

complexity of O ((N − k)2)

while the iteration part will incurO(I(N−k)) complexity, where I is the iteration rounds. Thusthe total complexity is at most O ((N − k)2 + I(N − k)

)and

we choose I = 3 in practice.Remark 2: The low complexity of the GMLLE algorithm

is justified by the simulation results in Section V. Note that ifwe use some advanced sorting algorithms (e.g., quick sort) todo the sequence ordering in GMLLE, the complexity can beeven lower. We do not consider this here.

B. Performance Bound

The performance bound for GMLLE in a two-way messageexchange protocol under the assumption of Gaussian delaymodel is derived as CRLB based on reduced set of observa-tions in Eqn. (23)-(25). This bound is critical in evaluatinghow much performance loss is incurred by the manipulationof the observations. Compared to [15], since we are studyinga PBS protocol under the assumption of exponential delayhere, we need to derive the performance bound for GMLLEagain. This is not a trivial job since the conditions are quitedifferent and much more complex. Note that the derivation ofthe performance bound also differs from that of [19]: (i) thealgorithm and synchronization mechanism are different andour GMLLE algorithm involves a user defined parameter k;(ii) the CRLB-like performance bound derived in [19] is justan approximation while we derive the exact one for GMLLEin this section. Note that the derived performance bound canonly serve as performance benchmark for GMLLE and doesnot apply to JMLE for it is based on the reduced set ofobservations.

Denote ympj,k � rmp

j+k − rmpj , ymq

j,k � rmqj+k − rmq

j and ypqj,k �rpqj+k−rpqj , where 1 ≤ j ≤ N−k. Denote L(a, b) the Laplace

distribution: p(x) = 12b exp

(− |x−a|

b

). Then, from Eqn.

(23)-(25), we have ympj,k ∼ L(ωp(s

mj+k − smj ), αωp), y

mqj,k ∼

L(ωq(smj+k − smj ), αωq) and ypqj,k ∼ L(ωq

ωp(spj+k − spj ), αωq).

Then the joint probability density function (PDF) of ympj,k , y

mqj,k

and ypqj,k, where 1 ≤ j ≤ N − k, is as follows:

f(ymp, ymq, ypq) =

(1

2α

)3(N−k)(1

ωp

)N−k (1

ωq

)2(N−k)

× exp

[− 1

α

N−k∑j=1

(∣∣∣∣∣ympj,k

ωp− smj+k + smj

∣∣∣∣∣+

∣∣∣∣∣ymqj,k

ωq− smj+k + smj

∣∣∣∣∣+∣∣∣∣∣y

pqj,k


ωp

∣∣∣∣∣)]

. (37)

Remark 3: Note that we regard ymp, ymq, ypq as the onlyrandom outputs of the system, i.e., the observations. Andsp, sm are just the previously known system inputs and are notobservations. Strictly speaking, Eqn. (26) in Subsection IV-Ais just an approximation of the likelihood function when ωp

and ωq are near to 1 while Eqn. (37) is the standard likelihoodfunction. However, maximizing (37) is too complicated sinceit cannot be reduced to a LP problem. So we use the approx-imate likelihood function (26) when designing algorithm inSubsection A. Now, to derive the performance bound, we willturn to the standard likelihood function in (37).

Generally speaking, CRLB does not exist for exponentialdistribution since the support region is related to the unknownparameters and the regularity condition for CRLB is notsatisfied. However, when they are manipulated according tothe algorithm of GMLLE, the observations are changed toLaplace random variables, which are suitable for derivingCRLB. We first perform some parameter transformations:θ1 � 1/ωp, θ2 � 1/ωq, λ � 1/α. Then the joint PDF becomesas follows:

f(ymp, ymq, ypq) =

(λ

2

)3(N−k)

θN−k1 θ

2(N−k)2

× exp

[− λ

N−k∑j=1

( ∣∣∣ympj,k θ1 − smj+k + smj

∣∣∣+∣∣∣ymq

j,k θ2 − smj+k + smj

∣∣∣+ ∣∣∣ypqj,kθ2 − (spj+k − spj )θ1

∣∣∣ )].(38)

Now we derive the CRLB with respect to (θ1, θ2, λ) and thisCRLB based on the reduced set of observations is the desiredperformance bound for GMLLE. The derivation of CRLB isjust the derivation of the Fisher information matrix (FIM).This is accomplished by Lemma 1.

Lemma 1: The FIM based on the reduced set of observa-tions in Eqn. (38) is as follows, with respect to (θ1, θ2, λ).

FIM =

⎛⎝ F11 F12 F13

F12 F22 F23

F13 F23 F33

⎞⎠


where

F11 =3(N − k)

λ2, F12 =

N − k

λθ1, F13 =

2(N − k)

θ2,

F22 =N − k

θ21+

λ2

θ21

N−k∑j=1

(smj+k − smj

)2

+ λ2N−k∑j=1

(spj+k − spj

)2,

F23 = −θ1λ2

θ2

N−k∑j=1

(spj+k − spj

)2,

F33 = − (N − k)2

θ22+

2(N − k)

θ22+

λ2

θ22

N−k∑j=1

(smj+k − smj

)2

+λ2θ21θ22

N−k∑j=1

(spj+k − spj

)2.

Proof: The derivation is somewhat complicated and isavailable in Appendix A.Thanks to Lemma 1, the performance bound for (λ, θ1, θ2) isjust the diagonal elements of the inverse matrix of the FIM,i.e., for any unbiased estimator (λ̂, θ̂1, θ̂2):

Var(λ̂)≥ [FIM]−1

11 , (39)

Var(θ̂1

)≥ [FIM]−1

22 , (40)

Var(θ̂2

)≥ [FIM]−1

33 . (41)

Closed form of the performance bound can be got directlyfrom (39)-(41). However the expression is too complicatedand hence not given here. According to the parameter trans-formation rule of CRLB [16], the performance bound for theoriginal paramter vector (α, ωp, ωq) is as follows:

Var (α̂) ≥ α4[FIM]−111 , (42)

Var (ω̂p) ≥ ω4p[FIM]−1

22 , (43)

Var (ω̂q) ≥ ω4q [FIM]−1

33 . (44)

C. Identifying Optimal User-defined Parameter k

Obviously, the performance of the GMLLE depends on thechoice of a user-defined parameter k. In this subsection, weendeavor to select the optimal k which can give us the bestestimation performance, i.e., the minimum mean square error(MMSE). In this subsection, we make efforts to find suchan optimal k analytically. In Section V, computer simulationswill give us the same result which corroborates our theoreticalanalysis.

When the observations are reduced to Eqn. (23)-(25), wehave derived the corresponding performance bound in Sub-section IV-B. Note that the GMLLE is indeed a MLE basedon the reduced observations (it is not true MLE since thetrue MLE should be based on the original observations, i.e.,GMLLE is only suboptimal). From classical estimation theory,we know that MLE is asymptotically unbiased and approachesto the CRLB when the number of observations is large enough[16]. Consequently, the performance of GMLLE can be welldescribed and predicted by the CRLB-like performance bound

derived in Subsection IV-B, i.e., smaller performance boundmeans better performance. Based on the above analysis, theproblem turns out to be finding the k which can minimizethe performance bound. In the following, when referring toperformance bound, we mean the performance bound for θ2since: (i) the estimation of the clock skew of the inactivenode is the most important one when requiring long termsynchronization; (ii) the accuracy of the estimation of theclock offset φq is related with the accuracy of the estimationof skew through Eqn. (34).

Without loss of generality, we assume that smj+1 − smj =H > 0, spj+1 − spj = G > 0, since the interval time oftransmission can be chosen in advance, where G and H aretwo constants. The rationality of such an assumption is wellargued in [15]. Under this assumption, the optimal value of kcan be selected by the following theorem.

Theorem 1: The optimal choice of the user-defined param-eter k is the integer nearest to (2/3)N .

Proof: Under the previously mentioned assumption, theFIM can be further reduced to the form of [FIM] = (N −k) · [A] (the detailed expression of A is complex and omittedhere). Since2 PB(θ2) = [FIM]

−133 = 1

N−k [A]−133 , we need to

compute [A]−133 which can be written as follows:

[A]−133 =

det

(A11 A12

A12 A22

)det([A])

.

After some algebraic calculations, the performance bound forθ2 can be expressed as:

1

PB(θ2)= (N − k)×(

−4− 4λ2H2k2 − 8k2G2λ2θ21 − 3θ41k4λ4G4

2θ22 + 3λ2θ22H2k2 + 3k2G2λ2θ21θ

22

− N − k

θ22+

2

θ22+

λH2k2

θ22+

λ2θ21k2G2

θ22

).

Not surprisingly, the value of PB(θ2) depends on theunknown parameters and hence the optimal k cannot bedetermined strictly when we do not know the value of thoseparameters. However, for practical concern, we can invokesome approximations to make the problem solvable. First,the values of θ1 and θ2 are generally very close to 1 (notethat they are just the reciprocal of the clock skews) andthus can be approximated as 1 here. Second, the number ofobservations N is usually much larger than H,G and λ. Notethat N/2 ≤ k < N , we only need to consider those terms withthe highest sum degree of N and k and simply omit the otherterms. Third, we regard the variable k as a real number ratherthan an integer to make the optimization feasible analytically.Thanks to the above three approximation assumptions, we canwrite the derivative of the quantity 1/PB(θ2) with respect to

2By PB, we mean performance bound.


15 20 25 3010-5

10-4

10-3

10-2

k

MS

E o

f the

clo

ck s

kew

H=1,G=0.7H=2,G=1H=5,G=4

(a) N = 30

8 9 10 11 12 13 1410-5

10-4

10-3

10-2

10-1

k

MS

E o

f the

clo

ck s

kew

H=1,G=0.7H=2,G=1H=5,G=4

(b) N = 15

Fig. 4. GMLLE: MSE of the clock skew estimate ω̂q with different G,Hand k. The number of observations are fixed to be 30 (left) or 15 (right)respectively. For all values of G,H , the optimal value of k is approximately20 (left) or 10 (right) as predicted by Theorem 1.

k as:

d

dk

(1

PB(θ2)

)≈ k5(3k − 2N)

(9λ5H6 + 9λ6H4G2 + 9λ6H2G4 +

18λ5H4G2 + 9λ5H2G4)/ (

2 + 3λ2H2k2 + 3k2G2λ2)2

.

Note that the factor which can decide the sign of the derivativeis just (3k − 2N). Thus, for all the values in the range ofN/2 ≤ k < N , there is only one point that can make the abovederivative be zero and this is just the optimal point we want.Letting the derivative be zero, we always have k = (2/3)Nregardless of the specific values of N,G,H and λ. Thus, wefinish the proof.

In Theorem 1, the choice coincides with that of [15]although the delay model, synchronization mechanism andderivation are totally different. In Section V, simulationssupport the theoretical analysis in this subsection and furtherindicate that even when N is not that large (e.g., N = 15),the optimality for the chosen k still holds. This validates therobustness of Theorem 1.

V. NUMERICAL EXPERIMENTS AND DISCUSSIONS

In this section, MATLAB simulations are implemented toverify the effects of the proposed JMLE and GMLLE. All thereported data points are obtained through an average of theresults of ten thousand simulation runs. According to Theorem1, the optimal value of the user-defined parameter k shouldbe (2/3)N . In order to justify this property in the simulation,we set the involved parameters as follows: ωp = 1.005, ωq =0.995, φp = −4, φq = 5, d = 3, α = 1, N = 30, which areall practical values frequently used in the literature. Fig.4(a)shows the MSE of the estimates of the two clock rates ωp, ωq

with different G,H and k under the fixed number of observa-tions, i.e., 30. In this case, the optimal value of k should be20 according to the theoretical result in Theorem 1. Fig.4(a)clearly corroborates this conclusion. Though sometimes thestrictly optimal value is 19, the difference between the MSEwhen k equals to 19 and 20 is very slight. In practice, wetend to use large k if the MSE does not increase much, sincethe computation overhead decreases with k increasing (thecomplexity of GMLLE is an increasing function of (N − k)).Fig.4(a) also tells us that generally speaking, the larger thevalues of G and H , the better the performance of GMLLE.

5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

N

MS

E o

f the

clo

ck s

kew

of n

ode

q

JMLEGMLLE, k=2N/3GMLLE, k=N-1PB for GMLLE,k=2N/3PB for GMLLE,k=N-1

(a) ωp = 1.005, ωq = 0.995, clockskew estimate ω̂q

5 10 15 20 25 3010-2

10-1

100

101

N

MS

E o

f the

clo

ck o

ffset

of n

ode

q

JMLEGMLLE, k=2N/3GMLLE, k=N-1

(b) ωp = 1.005, ωq = 0.995, clockoffset estimate φ̂q

5 10 15 20 25 3010-7

10-6

10-5

10-4

10-3

10-2

N

MS

E o

f the

clo

ck s

kew

of n

ode

q

JMLEGMLLE, k=2N/3GMLLE, k=N-1PB for GMLLE,k=2N/3PB for GMLLE,k=N-1

(c) ωp = 0.9996, ωq = 1.0005,clock skew estimate ω̂q

5 10 15 20 25 3010-2

10-1

100

101

N

MS

E o

f the

clo

ck o

ffset

of n

ode

q

JMLEGMLLE, k=2N/3GMLLE, k=N-1

(d) ωp = 0.9996, ωq = 1.0005,clock offset estimate φ̂q

Fig. 5. Comparison of the JMLE and GMLLE: (i) JMLE, (ii) GMLLE withk = (2/3)N , (iii) GMLLE with k = N − 1. The red lines are for (i). Theblue lines are for (ii). The green lines are for (iii). For clock skew estimationby GMLLE, the corresponding performance bound (PB) is also plotted. Thefigure shows that JMLE outperforms GMLLE and an optimal value of k (i.e.,k = (2/3)N ) is important for accuracy.

This is reasonable since GMLLE depends on the gap betweenobservations and if G and H become larger, this gap isalso amplified which leads to more accurate estimates. Thisproperty also holds for JMLE. Generally speaking, as G andH become larger or α becomes smaller (note that α is themean of the random delay, which acts like the noise), theperformance of both JMLE and GMLLE will become better.So, in practice, we need to ensure that (G +H)/α is abovea certain level, e.g., 20. Since the derivation of the optimalvalue of k in Subsection IV-C depends on the assumption thatN is large, one may be curious about the situation when Nis not very large. For this, we set N = 15 and do the samesimulation. The results are reported in Fig.4(b). We observethat the optimality of the analytically selected k still holds. Theabove property also holds for clock offset estimation since theaccuracy of the offset estimation depends heavily on that of theskew estimation. The simulation figures for offset estimationare omitted due to the space limitation.

Now we endeavor to compare the performance of JMLEand GMLLE. We fix G = H = 10. We test two sets of clockrates ωp = 1.005, ωq = 0.995 and ωp = 0.9996, ωq = 1.0005.In the latter case, the clock skews are much smaller, whichenables us to see if the values of clock skews influencethe estimation performance. Note that the values of theseparameters are reasonable since the order of the quantity|ω − 1| is roughly 10−3 times those of d, α and the clockoffset φ in real world WSNs [13]. In Fig.5(a) and Fig.5(b),for ωp = 1.005, ωq = 0.995, we compare three estimators:(i) JMLE, (ii) GMLLE with k = (2/3)N , (iii) GMLLE withk = N − 1. The computational complexity decreases from(i) to (iii) with the performance degrading, which can be


5 10 15 20 25 3010-5

10-4

10-3

10-2

10-1

N

Ave

rage

CP

U T

ime

(sec

ond)

JMLEGMLLE,k=2N/3LP in [23]

Fig. 6. A comparison of the average CPU time for running JMLE ,GMLLEand the LP based algorithm proposed in [23]. We observe that (1) the runningtime of JMLE is about twice as much as that of GMLLE (note that the y-axisis in logarithmic scale), (2) the running time of the LP based algorithm in[23] is much larger than that of JMLE and GMLLE.

seen from Fig.5(a) and Fig.5(b) obviously. The performancebound for the clock skew estimation in GMLLE is alsodepicted in Fig.5(a). When k = (2/3)N , or more generallyspeaking, when k is around the optimal user-defined value,the performance bound is very near to the MSE in GMLLE.This justifies that our performance bound based analysis ofchoosing optimal k in Section IV-C is reasonable. One thingwe should mention is that, for estimation of clock offsets,GMLLE with k = N − 1 actually fails to provide good esti-mate as shown in Fig.5(b). The MSE does not decrease as thenumber of observations increases. For this concern, if we wantto use GMLLE, we’d better choose k = (2/3)N to get goodestimates of the offsets. This essentially explains the necessityof selecting the optimal k. In Fig.5(c) and Fig.5(d), we conductthe same simulation when ωp = 0.9996, ωq = 1.0005. It isshown that MSE for both skew and offset estimates remainroughly the same, which means that the performance of JMLEand GMLLE are independent of the value of clock skews.

Next, we test the CPU time for running the JMLE, GM-LLE and the LP method in [23] so as to compare theircomputational complexity. The average CPU time is reportedin Fig.6. First, we observe that the CPU time of JMLE isroughly twice as much as that of GMLLE, which indicatesthat GMLLE algorithm has a lower complexity. In previoussections, the worst case complexity of JMLE and GMLLEhas been analyzed theoretically. However, as shown in Fig.6,the experimental complexity of both algorithms are muchsmaller than the analytical upper bounds. In the simulation,the CPU time of both JMLE and GMLLE grow roughlylinearly with N (though it is not quite obvious from Fig.6since the y-axis is in logarithmic scale). Note that, whenchoosing estimators based on the tradeoff between complexityand accuracy, unless the computational ability of the wirelesssensor is extremely limited, we tend to apply computationallycomplicated algorithm to achieve better accuracy since theenergy cost of computation is much smaller than that ofcommunications in WSNs generally [5]. Hence, JMLE, thougha little complicated, is still a practical choice. Second, it canbe seen from Fig.6 that the CPU time of the LP method in[23] is much larger than (about 100 times of) that of eitherJMLE or GMLLE. In the simulation of the LP method of[23], we invoke the MATLAB function linprog and set the

5 10 15 20 25 3010-6

10-5

10-4

10-3

10-2

N

MS

E o

f the

clo

ck s

kew

of n

ode

q

JMLE, GammaMLLE, GammaJMLE, WeibullMLLE, Weibull

(a) clock skew estimate ω̂q

5 10 15 20 25 30

100

N

MS

E o

f the

clo

ck o

ffset

of n

ode

q

JMLE, GammaMLLE, GammaJMLE, WeibullMLLE, Weibull

(b) clock offset estimate φ̂q

Fig. 7. (Robustness) The random delay is either Gamma distribution orWeibull distribution: (i) JMLE, (ii) GMLLE with k = (2/3)N . The red linesand green lines represent JMLE with a Gamma(2,1) delay and a Weibull(2,2)delay, respectively. The blue lines and black lines represent GMLLE with aGamma(2,1) delay and a Weibull(2,2) delay, respectively.

maximum number of iterations to be �N3 � since the minimum

number of iterations necessary for obtaining the same accuracyas that of JMLE is about �N

3 � in our numerical setting, i.e., anyfurther reduction of the maximum number of iterations willmake the accuracy of the LP method degrade significantly.We observe that JMLE, while achieving the same accuracy asLP method in [23], enjoys much lower complexity. Hence, theproposed algorithms are much more practical for real worldlarge scale WSNs which consist of many cheap sensors withlow computational capability.

Finally, we examine the robustness of JMLE and GMLLEwhen the delay model is mismatched because the randompart of the delay may not be an exponential random variableexactly in real environment. For this, we apply JMLE andGMLLE to Gamma distributed random delay and Weibulldistributed random delay, respectively. For Gamma delay, theshape parameter is 2 and the scale parameter is 1. For Weibulldistribution, both shape parameter and scale parameter are2. For GMLLE, we choose the optimal k as (2/3)N . Thesimulation results for skew estimation and offset estimationare shown in Fig.7(a) and Fig.7(b), respectively. JMLE stilloutperforms GMLLE in a Gamma(2,1) delay while the per-formance gap between JMLE and GMLLE in a Weibull(2,2)delay is small. For both estimators, the MSE, although largerthan that of exact exponential delay model, still decreasesquickly as N becomes large. Therefore, the robustness of thetwo estimators is verified.

VI. CONCLUSION AND FUTURE WORK

In this paper, we study joint estimation of clock skewand offset in a PBS mechanism. Specifically, we investigatetwo estimators: JMLE and GMLLE. For both estimators,corresponding algorithms are derived and presented explicitly.For GMLLE, we derive its performance bound based on thereduced observations and identify the optimal user-definedparameter in both theoretical and experimental approach.Simulations are carried out to corroborate our algorithms andanalysis. This paper combines PBS and joint estimation ofskew and offset, which are two effective ways to save energyin clock synchronization in WSNs. The algorithms presentedwill provide fundamental schemes to designing better WSNs[26], [27].


There are several interesting directions for future work.First, the assumption that the fixed delay d is the same forall links is imprecise for almost all real WSNs. Actually,many localization algorithms have utilized the difference ind to determine the position of sensors. Hence, taking thedifference in d into account is a very practical concern.Second, in real WSNs, the clock parameters may not be aconstant. Rather, the clock skew and offset may vary slowlywith the time. Some pioneering works, e.g., [12] and [25],have already taken the time varying offset into consideration.But there still lack works on time varying clock skews inPBS mechanism. If one can solve the synchronization problemfor time varying clock parameters in PBS mechanism, thensignificant energy reduction is expected since we do not needto resynchronize the clocks that frequently. Finally, furtherwork on synchronizing the clocks over the entire network ina statistical signal processing perspective is expected [28].

ACKNOWLEDGMENT

This paper is supported by National Fundamental Re-search Grant (No. 2011CB302701); NSF China (No.61271219,61202373); Shanghai Basic Research Key Project(No. 11JC1405100); China Ministry of Education Fok YingTung Fund (No. 122002); China Ministry of EducationNew Century Excellent Talent (No. NCET-10-0580); ChinaPostdoctoral Science Foundation Grant 2011M500774 and2012T50417, STCSM Grant 12JC1405200 and SEU SKLproject #2012D13.

APPENDIX APROOF OF LEMMA1: DERIVATION OF THE FISHER

INFORMATION MATRIX

Proof: Taking logarithm of Eqn. (38) yields the follow-ing:

ln f(ymp, ymq, ypq)

= 3(N − k) ln

(λ

2

)+ (N − k) ln θ1 + 2(N − k) ln θ2

− λ

(N−k∑j=1

∣∣∣θ1ympj,k − smj+k + smj

∣∣∣+

N−k∑j=1

∣∣∣θ2ymqj,k − smj+k + smj

∣∣∣+

N−k∑j=1

∣∣∣θ2ypqj,k − θ1

(spj+k − spj

)∣∣∣).

Denote ξ the quantity inside the parentheses. Taking derivativewith respect to λ, θ1, θ2 respectively, we obtain:

∂ ln f

∂λ=

3(N − k)

λ− ξ,

∂ ln f

∂θ1=

N − k

θ1− λ

(N−k∑j=1

ympj,k sgn

(θ1y

mpj,k − smj+k + smj

)

+N−k∑j=1

(spj+k − spj

)sgn

(θ1

(spj+k − spj

)− θ2y

pqj,k

)),

∂ ln f

∂θ2=

2(N − k)

θ2− λ

(N−k∑j=1

ymqj,k sgn

(θ2y

mqj,k − smj+k

+smj

)+

N−k∑j=1

ypqj,ksgn(θ2y

pqj,k − θ1

(spj+k − spj

))),

where sgn(·) denotes the sign function, i.e., sgn(x) = 1 ifx > 0, sgn(x) = −1 if x < 0. Recall the definition ofymp, ymq, ypq , we can rewrite the above equations as:

∂ ln f

∂θ1=

N − k

θ1− λ

(N−k∑j=1

ympj,k sgn

(zmpj+k − zmp

j

)

+

N−k∑j=1

(spj+k − spj

)sgn

(zpqj − zpqj+k

)),

∂ ln f

∂θ2=

2(N − k)

θ2− λ

(N−k∑j=1

ymqj,k sgn

(zmqj+k − zmq

j

)

+

N−k∑j=1

ypqj,ksgn(zpqj+k − zpqj )

)),

ξ =N−k∑j=1

∣∣∣zmpj+k − zmp

j

∣∣∣+ N−k∑j=1

∣∣∣zmqj+k − zmq

j

∣∣∣+

N−k∑j=1

∣∣∣zpqj+k − zpqj

∣∣∣ .Before we proceed to derive FIM, we first summarize someuseful equations, which will be used frequently in the laterderivation, in Proposition 1. The proof of these equations canbe accomplished in a step-by-step approach and is omittedhere because of the limited space.

Proposition 1: Some frequently used formulas are listed asfollows.

E

∣∣∣zmpj+k − zmp

j

∣∣∣ = E

∣∣∣zmqj+k − zmq

j

∣∣∣ = E

∣∣∣zpqj+k − zpqj

∣∣∣ = 1

λ;

E

[(zmpj+k − zmp

j

)2]= E

[(zmqj+k − zmq

j

)2]

= E

[(zpqj+k − zpqj

)2]=

2

λ2;

E(ξ) =3(N − k)

λ,E(ξ2)=

3(N − k)(N − k + 1)

λ2;

E

[ympj,k sgn

(zmpj+k − zmp

j

)]=

ωp

λ;

E

[ymqj,k sgn

(zmqj+k − zmq

j

)]= E

[ypqj,ksgn

(zpqj+k − zpqj

)]=

ωq

λ;

E

[ympj,k

(zmpj+k − zmp

j

)]=

2ωp

λ2;

E

[ymqj,k

(zmqj+k − zmq

j

)]= E

[ypqj,k

(zpqj+k − zpqj

)]=

2ωq

λ2;

E

[(ympj,k

)2]= ω2

p

((smj+k − smj

)2+

2

λ2

);

E

[(ymqj,k

)2]= ω2

q

((smj+k − smj

)2+

2

λ2

);

E

[(ypqj,k

)2]=

ω2q

ω2p

(spj+k − spj

)2+ ω2

q ·2

λ2.


In what follows, we proceed to derive the elements of FIMone by one.

F11 = E

[(∂ ln f

∂λ

)2]

=9(N − k)2

λ2− 6(N − k)

λE(ξ) + E

(ξ2)

=3(N − k)

λ2.

F12 = E

(∂ ln f

∂λ

∂ ln f

∂θ1

)

=6(N − k)2

λθ1− N − k

θ1E(ξ)

− 3(N − k)

N−k∑j=1

E

(ympj,k sgn

(zmpj+k − zmp

j

))

+ λE

[ξ

(N−k∑j=1

ympj,k sgn

(zmpj+k − zmp

j

)

+

N−k∑j=1

(spj+k − spj

)sgn

(zpqj − zpqj+k

))]

=6(N − k)2

λθ1− 3(N − k)2

ωp

λ

+ωp

λ(N − k)(3N − 3k + 1)

=ωp(N − k)

λ.

F13 = E

(∂ ln f

∂λ

∂ ln f

∂θ2

)

=6(N − k)2

λθ2− 2(N − k)

θ2E(ξ)

− 3(N − k)

(N−k∑j=1

E

(ymqj,k sgn

(zmqj+k − zmq

j

))

+N−k∑j=1

E

(ypqj,ksgn

(zpqj+k − zpqj

)))

+ λE

[ξ

(N−k∑j=1

ymqj,k sgn

(zmqj+k − zmq

j

)

+

N−k∑j=1

ypqj,ksgn(zpqj+k − zpqj

))]

=6(N − k)2

λθ2− 2(N − k)

θ2· 3(N − k)

λ

− 6(N − k)2 · ωq

λ+

ωq

λ· (N − k)(6N − 6k + 2)

=2ωq(N − k)

λ.

F22 = E

[(∂ ln f

∂θ1

)2]

=

(N − k

θ1

)2

+ λ2E

[(N−k∑j=1

ympj,k sgn

(zmpj+k − zmp

j

)

+

N−k∑j=1

(spj+k − spj

)sgn

(zpqj − zpqj+k

))2]

− 2λ(N − k)

θ1E

(N−k∑j=1

ympj,k sgn

(zmpj+k − zmp

j

)

+

N−k∑j=1

sgn(zpqj − zpqj+k

))

=

(N − k

θ1

)2

+ λ2

(N−k∑j=1

ω2p

((smj+k − smj

)2+

2

λ2

)

+(ωp

λ

)2(N − k)(N − k − 1)

+

N−k∑j=1

(spj+k − spj

)2)− 2λ(N − k)

θ1· (N − k) · ωp

λ

=N − k

θ21+

λ2

θ21

N−k∑j=1

(smj+k − smj

)2

+ λ2N−k∑j=1

(spj+k − spj

)2.

F23 = E

(∂ ln f

∂θ1

∂ ln f

∂θ2

)

=2(N − k)2

θ1θ2− λ(N − k)

θ1E

((N−k∑j=1

ymqj,k ×

sgn(zmqj+k − zmq

j

))+

N−k∑j=1


))

− 2(N − k)λ

θ2E

(N−k∑j=1

ympj,k sgn

(zmpj+k − zmp

j

)

+

N−k∑j=1

(spj+k − spj

)sgn

(zpqj − zpqj+k

))

+ λ2E

[(N−k∑j=1

ympj,k sgn

(zmpj+k − zmp

j

)

+

N−k∑j=1

(spj+k − spj

)sgn

(zpqj − zpqj+k

))

×(

N−k∑j=1

ymqj,k sgn

(zmqj+k − zmq

j

)

+

N−k∑j=1


))]

=2(N − k)2

θ1θ2− 2λ(N − k)

θ1· ωq

λ· (N − k)

− 2(N − k)λ

θ2· (N − k) · ωp

λ+

2

θ1θ2· (N − k)2

− θ1λ2

θ2

N−k∑j=1

(spj+k − spj

)2

= −θ1λ2

θ2

N−k∑j=1

(spj+k − spj

)2.


F33 = E

(∂ ln f

∂θ2

)2

=4(N − k)2

θ22+ λ2

E

[(N−k∑j=1

ymqj,k sgn

(zmqj+k − zmq

j

)

+

N−k∑j=1


))2]

− 4(N − k)λ

θ2E

(N−k∑j=1

ymqj,k sgn

(zmqj+k − zmq

j

)

+

N−k∑j=1


))

=4(N − k)2

θ22+

λ2

θ22

N−k∑j=1

(smj+k − smj

)2+ 2ω2

q(N − k)

+2ω2q(N − k)(N − k − 1) +

λ2θ21θ22

N−k∑j=1

(spj+k − spj

)2

+2(N − k)

θ22+ (N − k)2ω2

q −8(N − k)2

θ22

= − (N − k)2

θ22+

2(N − k)

θ22+

λ2

θ22

N−k∑j=1

(smj+k − smj

)2

+λ2θ21θ22

N−k∑j=1

(spj+k − spj

)2.

Thereby, we finish the proof.

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Xuanyu Cao is currently pursuing his B.E. degreein electronic engineering in Shanghai Jiao Tong Uni-versity, China. His research interests are in the areaof statistical signal processing as well as asymptoticanalysis in wireless networks.

Feng Yang received the Ph.D. degree from ShanghaiJiao Tong University, Shanghai,China, in 2007.Since2008, he has worked on ShangHai Jiao Tong Uiv-ersity. His research interests include multimediacommunication and OFDM system design.


Xiaoying Gan received the Ph.D. degree in Depart-ment of Electronic Engineering from Shanghai JiaoTong University (SJTU), Shanghai, China, in 2006.From 2009 to 2010, she was a visiting researcherwith the California Institute for Telecommunicationsand Information Technology, University of Califor-nia, San Diego. She is currently with the Institute ofWireless Communication Technology, Departmentof Electronic Engineering, SJTU, where she is anAssociate Professor. Her current research interestsconcern cognitive network, dynamic spectrum allo-

cation and cellular system optimization.

Jing Liu received her Ph.D. degree in Communi-cations and Information Systems from Xidian Uni-versity, Xian, China, in 2005. Since July 2005, shejoined the Department of Electronic Engineering,Shanghai Jiao Tong University, China, where sheis currently an Associate Professor. From 2012 to2013, she worked as a visiting scholar at Universityof Waterloo, ON, Canada. Her research interestsfocus on wireless body area network, wireless sensornetwork and cognitive access networks.

Liang Qian joined the Shanghai Jiaotong University(SJTU) in 1989, where he is involved in teaching andresearch in signal and image processing and wirelesscommunications systems. He received his Ph.D. inCommunications and Information Processing fromSJTU in 2005, and had been a visiting scholarin Institute of information processing, Universityof Kalsruhe, in 2003. Dr. Qians research interestsinclude digital signal processing for wireless cellularsystem, satellite signal processing for navigationsystem, emergency wireless access for public se-

curity, as well as applications, performance and implementation of abovealgorithms. Among his current research projects, He is working on system-level integration and advanced algorithms for next generation wireless com-munications (LTE-A), and performance analysis of adaptive signal and imageprocessing algorithms.

Xiaohua Tian received his B.E. and M.E. degreesin communication engineering from NorthwesternPolytechnical University, Xi’an, China, in 2003 and2006, respectively. He received the Ph.D. degree inthe Department of Electrical and Computer Engi-neering (ECE), Illinois Institute of Technology (IIT),Chicago, in Dec. 2010. He is currently a postdoc inDepartment of Electronic Engineering of ShanghaiJiaotong University, China. He won the HighestStandards of Academic Achievement 2011 of IIT,Fieldhouse Research Fellowship 2009 of IIT, which

is awarded to only one student over IIT each year, and the IEEE EFSOI2010 Student Travel Grant Award. His research interests include application-oriented networking, Internet of Things and wireless networks. He serves asthe guest editor of International Journal of Sensor Networks, publicity co-chairof WASA 2012. He also serves as the Technical Program Committee memberfor Communications QoS, Reliability, and Modeling Symposium (CQRM) ofGLOBECOM 2011, Wireless Networking of GLOBECOM 2013 and WASA2011.

Xinbing Wang received the B.S. degree (with hons.)from the Department of Automation, Shanghai Jiao-tong University, Shanghai, China, in 1998, and theM.S. degree from the Department of Computer Sci-ence and Technology, Tsinghua University, Beijing,China, in 2001. He received the Ph.D. degree, majorin the Department of electrical and Computer Engi-neering, minor in the Department of Mathematics,North Carolina State University, Raleigh, in 2006.Currently, he is a faculty member in the Departmentof Electronic Engineering, Shanghai Jiaotong Uni-

versity, Shanghai, China. His research interests include scaling law of wirelessnetworks and cognitive radio. Dr. Wang has been an associate editor forIEEE/ACM TRANSACTIONS ON NETWORKING, IEEE TRANSACTIONS ON

MOBILE COMPUTING, and a member of the Technical Program Committeesof several conferences including ACM MobiCom 2012, ACM MobiHoc 2012,IEEE INFOCOM 2009-2014.

joint estimation of clock skew and offset in pairwise broadcast synchronization mechanism

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