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Joint Node Localization and Time-Varying Clock Synchronization in Wireless Sensor Networks Aitzaz Ahmad, Erchin Serpedin, Hazem Nounou, and Mohamed Nounou
Abstract—The problems of node localization and clock synchronization in wireless sensor networks are naturally tied from a statistical signal processing perspective. In this work, we consider the joint estimation of an unknown node’s location and clock parameters by incorporating the effect of imperfections in node oscillators, which render a time varying nature to the clock parameters. The data exchange mechanism is based on a two-way message exchange with anchor nodes. In order to alleviate the computational complexity associated with the optimal maximum a-posteriori estimator, two iterative approaches are proposed as simpler alternatives. The first approach utilizes an ExpectationMaximization (EM) based algorithm which iteratively estimates the clock parameters and the location of the unknown node. The EM algorithm is further simplified by a non-linear processing of the data to obtain a closed form solution of the location estimation problem using least squares (LS). The performance of the estimation algorithms is benchmarked by deriving the Hybrid Cram´er-Rao lower bound (HCRB) on the mean square error (MSE) of the estimators. The theoretical findings are corroborated by simulation studies which reveal that the LS estimator closely matches the performance of the EM algorithm for small time of arrival measurement noise, and is well suited for implementation in low cost sensor networks. Index Terms—Clock synchronization, node localization, EM algorithm, least squares, wireless sensor networks.
I. I NTRODUCTION
W
IRELESS sensor networks (WSNs) comprise a large number of inexpensive devices that are deployed for observing and initial processing of physical or environmental changes taking place in their vicinity. An on-board sensing equipment enables the sensors to summarize the useful information to be transmitted to a distant fusion center (FC), resulting in reduced communication requirements. Recent technology breakthroughs in micro-electro-mechanical systems (MEMS) have enabled successful deployment of large scale
Manuscript received February 2, 2013; revised June 6, 2013; accepted August 6, 2013. The associate editor coordinating the review of this paper and approving it for publication was H. Lin. A. Ahmad was with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA. He is currently with Corporate Research & Development, Qualcomm Technologies Inc., San Diego, CA 92121 USA (e-mail:
[email protected]). E. Serpedin is with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, 77843 USA (e-mail:
[email protected]). H. Nounou is with the Department of Electrical and Computer Engineering, Texas A&M University at Qatar (TAMUQ), Doha, Qatar (e-mail:
[email protected]). M. Nounou is with the Department of Chemical Engineering, Texas A&M University at Qatar (TAMUQ), Doha, Qatar (e-mail:
[email protected]). This work was supported in part by QNRF-NPRP grants 4-1293-2-513 and 09-341-2-128. Digital Object Identifier 10.1109/TWC.2013.090413.130324
WSNs. Several applications such as health care, industrial process monitoring, battlefield surveillance, target localization and tracking, etc., have been identified for WSNs [1]. Node localization is an important aspect of several WSN applications that require location-awareness such as geographical routing, disaster rescue, etc., [2]–[4]. There is an extensive literature on location estimation algorithms in WSNs [5], [6]. In general, the range-based localization algorithms utilize the metrics of time of arrival (TOA) [7], time difference of arrival (TDOA) [8], [9] and received signal strength (RSS) [10] to determine the distance between the unknown node and the anchors. These distance-based measurements are then used for node localization. Distributed location estimation algorithms have also been studied for cooperative and passive sensors using the above mentioned metrics [11]. Using a hybrid TOA and TDOA approach, positioning of multiple target nodes in a cooperative wireless network has been proposed in [12]. Recently, a cooperative localization algorithm that is resistant to malicious anchors has been proposed in [13] by employing semi-definite programming techniques. Clock synchronization is a crucial requirement to ensure optimal data fusion and coordinated wake and sleep periods. Since TOA and TDOA are time-based techniques, synchronization is an important prerequisite in node localization as well [14], [15]. Several data exchange protocols have been proposed for clock synchronization in WSNs [16]. A common data exchange mechanism employed in synchronization protocols is the two-way data exchange process [17]. It involves a pair of nodes attempting to estimate their relative clock offset and skew by exchanging their timing information. Several clock offset estimation algorithms are proposed in [18] for the two-way exchange model. The maximum likelihood (ML) estimate of the clock offset is obtained in [19] using graphical maximization. Joint clock offset and skew estimation algorithms for different network delay models, and the theoretical performance limits are derived in [20]. Assuming unknown propagation delay, clock offset and skew estimation algorithms are proposed in [21]. Joint ML clock offset and skew estimation is proposed in [22] by using convex optimization techniques. An important extension of pairwise clock synchronization is the network-wide clock synchronization. A globally distributed synchronization scheme based on broadcasting synchronization pulses is presented in [23]. Factor graphs using belief propagation (BP) have also been utilized for distributed network synchronization [24]. An extension to a multi-hop sensor network synchronization is studied in [25] using heuristic algorithms. The close connection between the problems of localization
c 2013 IEEE 1536-1276/13$31.00
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and synchronization necessitates a joint estimation approach. Recently, several contributions have studied joint localization and synchronization from a statistical signal processing viewpoint. Optimal and sub-optimal algorithms for estimating an unknown node’s position and clock parameters have been derived in [26]. The performance of the estimation algorithms developed therein is also compared with the Cram´er-Rao lower bound (CRB). A weighted least squares approach for joint estimation is devised in [27]. Robust algorithms for joint estimation that are resistant to target node’s uncertainties are derived in [28]. Joint synchronization and localization based on an asymmetrical time-stamping and passive listening protocol is proposed in [29]. However, a common theme in these contributions is the assumption of fixed clock parameters. Sensor nodes are often deployed in harsh environmental conditions which can introduce degradations in the quartz crystals over time. Failure to cope with the temporal variations can result in frequent re-synchronization requests. Since power is primarily consumed in radio transmission delivering timing information [30], exchanging time-stamps for resynchronization can quickly drain a sensor’s already meagre energy resources. Accurately tracking the drifts in clock parameters can boost the network lifetime by minimizing energy consumption. Recently, several synchronization-only approaches have considered time-variations in clock parameters in WSNs. A Bayesian approach for clock offset estimation has been presented in [31] and [32], while a factor graph approach is utilized in [33] to obtain a closed form solution of the clock offset estimator when the likelihood function of the network delays is Gaussian, exponential or log-normally distributed. Time-variations in clock skew have been incorporated in [34] for clock synchronization. A Kalman filter based approach is used for tracking clock skew variations in [35], [36]. In this work, we aim to introduce the notion of temporal variation in clock parameters in the realm of joint node localization and clock synchronization in WSNs. We develop two iterative estimation algorithms which have varying degrees of accuracy and simplicity. The performance of the estimators is also benchmarked by deriving the theoretical lower bounds on the MSE of the estimators. Our main contributions in this paper are summarized as follows. 1) An Expectation-Maximization (EM) based joint localization and time-varying synchronization algorithm is proposed that iteratively determines the time-varying clock parameters using a Kalman smoother followed by a likelihood maximization for the location estimation. 2) In order to alleviate the computational complexity that comes with the two-dimensional likelihood optimization required for localization, a linearization based least squares (LS) method is presented which yields a closed form solution and is therefore, a simpler alternative to the EM algorithm. Moreover, it is observed through numerical simulations that the performance of the LS based location estimator is fairly close to the EM algorithm for small to moderate measurement noise errors. 3) Theoretical lower bounds on the MSE of an estimator are obtained by deriving the Hybrid Cram´er-Rao bound (HCRB) in our estimation framework. This helps to
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compare the performance of the aforementioned estimators. This paper is organized as follows. The joint localization and time-varying clock synchronization system model is outlined in Section II. The iterative algorithms for joint localization and synchronization are presented in Section III. The HCRB is derived in Section IV and numerical simulation results are discussed in Section V. Conclusions are drawn in Section VI. The notation used in this paper is as follows. Upper (lower) case bold faced letters denote matrices (vectors). The matrices 0M×N and 1M×1 denote the M × N matrix of zeros and the M × 1 matrix of ones, respectively. The notations diag(x) and blkdiag (X, Y ) denote a diagonal matrix with the vector x on its main diagonal and a block diagonal matrix with matrices X and Y on its main diagonal, respectively. The operators Tr, ⊗ and denote the trace of a matrix, the Kronecker product and the Hadamard product between matrices, respectively. Ex,y [·] stands for the expectation with respect to the joint distribution f (x, y). The partial derivative of a function h (x) with respect T ∂h(x) ∂h(x) . to x, ∂h(x) ∂x , is the column vector ∂x1 , . . . , ∂xN II. S YSTEM M ODEL Consider a network composed of N anchor nodes and an unknown node, denoted as Node X, that needs to be synchronized and localized to the anchors. The locations j j of T th the j anchor and Node X are given by sj = sx , sy T and x = [x1 , x2 ] , respectively. It is assumed that the anchors are synchronized with the same reference time t and their locations are accurately known. The process of joint localization and synchronization proceeds by exchanging timestamps between Node X and the anchors using a two-way message exchange mechanism as shown in Fig. 1. At the k th message exchange, Node X transmits its current timing information to the j th anchor through time-stamp Sj,k . The anchor records the time Rj,k at which this message is received according to its own time scale. After some time has elapsed, the j th anchor replies at time S¯j,k and transmits a synchronization packet containing both the time-stamps Rj,k ¯ j,k and S¯j,k to Node X. This message is received at time R by Node X according to its own clock. Therefore, after K th exchanges X is equipped with time with the j anchor,Node ¯ j,k K which are to be used to stamps Sj,k , Rj,k , S¯j,k , R k=1 ascertain its location and clock parameters. In this work, it is assumed that the clock of Node X is related to the reference time t as follows: CX (t) = αt + β , where α and β denote the clock skew and clock offset with respect to the reference time, respectively. Hence, the aforementioned two-way timing exchange process can be expressed as [37] Sj,k = α (Rj,k − dj − wj,k ) + β ¯ j,k = α S¯j,k + dj + w R ¯j,k + β ,
(1)
where the measurement noise errors wj,k and w ¯j,k are as2 sumed i.i.d. Gaussian with zero mean and variance σw . The
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¯ j,K R
By collecting data for all k = 1, . . . , K, it follows that
α
Sj,K
(7) y − d (x) ⊗ 1K×1 = HΘ + w , T T T T T where y = y1 , . . . , yK , Θ = θ1T , . . . , θK , w = T T T w1 , . . . , wK and H = diag {H1 , . . . , HK }. The joint distribution of {y, Θ}, parameterized by x, can be expressed as
¯ j,1 R Sj,1 Node X
... β
j th Anchor Rj,1
Fig. 1.
S¯j,1
Rj,K
f (y, Θ; x) = f (Θ) f (y|Θ; x)
S¯j,K
= f (θ0 )
A two-way timing message exchange mechanism
K
f (θk |θk−1 )
k=1
fixed line-of-sight propagation delay, denoted by dj , is given by1 dj = x − sj . By defining 1 Δ β , θ2 = , α α we can equivalently express (1) as Δ
θ1 =
Rj,k − dj = Sj,k θ1 − θ2 + wj,k ¯ j,k θ1 + θ2 + w −S¯j,k − dj = −R ¯j,k .
(2)
(3)
By stacking data from all N anchors at the k th message exchange, the system model in (3) can be compactly expressed as (4) yk − d (x) = Hk θ + wk , T where yk = R1,k , −S¯1,k , . . . , RN,k , −S¯N,k , the parameter Δ ¯1,k , . . . , wN,k , w ¯N,k ], vector θ = [θ1 θ2 ]T , wk = [w1,k , w T d (x) = d1 1T2×1 , . . . , dN 1T2×1 , and the 2N × 2 matrix Hk is given by ⎤ ⎡ S1,k −1 ¯ 1,k ⎢ −R 1⎥ ⎥ ⎢ ⎢ .. .. ⎥ . Hk = ⎢ . ⎥ . ⎥ ⎢ ⎣ SN,k −1⎦ ¯ N,k 1 −R Since sensor nodes are usually deployed in harsh environmental conditions, degradations in quartz oscillators render a time-varying nature to the clock skew and offset of Node X. Several recent contributions have proposed clock synchronization schemes by considering temporal variations in the clock parameters [31], [33], [36]. In this work, it is assumed that the variations in the clock parameters induce a Gauss-Markov evolution model for θ at the k th message exchange, i.e., θk = θk−1 + nk ,
(5) T where nk is zero mean Gaussian noise such that E nk nk = P , where P is a diagonal matrix with entries σ12 and σ22 . A brief discussion on the assumed Gauss-Markov model is provided in Appendix A. This model helps to capture time variations and also lends mathematical simplicity to gain a theoretical insight into the problem of joint localization and time-varying clock synchronization. Using (5), the two-way message exchange model (4) at the k th round is now expressed as (6) yk − d (x) = Hk θk + wk . 1 The
speed of light constant c is omitted for brevity.
K
f (yk |θk ; x) . (8)
k=1
The conditional pdfs f (θk |θk−1 ) and f (yk |θk ; x) are given by (θk − θk−1 )T P −1 (θk − θk−1 ) f (θk |θk−1 ) = C1 exp − 2 (9) f (yk |θk ; x) = T (yk − d (x) − Hk θk ) (yk − d (x) − Hk θk ) C2 exp − 2 2σw (10) where C1 and C2 are constants. Our goal is to T using the time-stamps jointly estimate ξ = ΘT , xT ¯ j,k K as well as the known anchor loSj,k , Rj,k , S¯j,k , R k=1 cations sj , for j = 1, . . . , N . The joint estimates of Θ and x can be obtained as ˆ x ˆ = arg max ln f (y, Θ; x) . Θ, (11) Θ,x
This joint estimation problem is computationally demanding. The nodes in a WSN are generally inexpensive devices characterized by limited capabilities of computation and communication. This computational complexity necessitates the development of simpler alternative algorithms that lower the computational burden while maintaining a desired performance level. In the next section, two iterative methods are explored for joint localization and timing synchronization. III. I TERATIVE A PPROACHES In this section, two iterative estimation algorithms are proposed which differ mainly in their approach to determine the location of the unknown node. A. The EM Algorithm The EM algorithm is an iterative method used to determine the ML estimate of the parameters of a given distribution from incomplete data [38]. The EM algorithm offers a simpler alternative to an otherwise intractable ML estimation problem by assuming additional unobserved parameters in the underlying distribution. The ML estimates are then computed by iterating between the Expectation and Maximization steps. Due to its analytical tractability, the EM algorithm finds numerous applications in diverse fields [39]. Assuming that the data y is incomplete, the complete data T Δ vector is defined as z = y T , ΘT . The expectation and
AHMAD et al.: JOINT NODE LOCALIZATION AND TIME-VARYING CLOCK SYNCHRONIZATION IN WIRELESS SENSOR NETWORKS
maximization steps in the EM algorithm can be described as follows. E-Step: ˆ (i) of the unknown node’s location at Given an estimate x iteration i, and the observed data y, determine the likelihood function Δ ˆ (i) = EΘ|y,xˆ (i) [ln f (z; x)] . (12) Q x, x M-Step: Obtain an estimate of x at iteration index i + 1 by maximizing ˆ (i) , i.e., Q x, x ˆ (i+1) = arg max Q x, x ˆ (i) . x (13) x
The E-Step and M-Step are repeated until convergence. After each iteration, we are guaranteed to converge towards a local maximum [38]. Using (8), (9) and (10), it follows that
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ˆ (i) , it can be observed that (5) and Given an estimate x (6) represent a linear Gaussian model. The minimum mean (i) square error (MMSE) estimator θˆk|K can be obtained from a standard Kalman smoother. The forward recursion for (i) obtaining θˆk|k can be expressed as follows [40]. Forward Recursion Prediction: (i) (i) θˆk|k−1 = θˆk−1|k−1
ˆ (i) ˆ (i) Σ k|k−1 = Σk−1|k−1 + P
(18)
Correction:
ˆ (i) H T + σ 2 I ˆ (i) H T Hk Σ κk = Σ w k|k−1 k k|k−1 k (i) (i) (i) (i) ˆ − Hk θˆk|k−1 θˆk|k = θˆk|k−1 + κk y˜k x
ˆ (i) = (I − κk Hk ) Σ ˆ (i) Σ k|k k|k−1
(19)
ˆ (i) is the MMSE matrix. where Σ k|k The operation of a smoother is completed by employing a K 1 ˆ(i) T ln f (z; x)= C− 2 (y˜k (x) − Hk θk ) (y˜k (x) − Hk θk ) backward sweep that produces the smoothed estimates θk|K 2σw ˆ (i) . The recursions of the Rauch-Tung-Striebel (RTS) and Σ k=1 k|K (14) smoother are given as follows [41]. Δ where y˜k (x) = yk − d (x) and the terms that do not depend on x are collected in the constant C . The likelihood function Backward Recursion ˆ (i) , can be evaluated as at the ith iteration, Q x, x −1 ˆ (i) ˆ (i) Bk−1 = Σ k−1|k−1 Σk|k−1 ˆ (i) = Q x, x (i) (i) ˆ(i) ˆ ˆ(i) ˆ = θ + B − θ θ θ k−1 k−1|K k−1|k−1 k|K k|k−1 K 1 (i) (i) (i) T T ˆ ˆ ˆ ˆ (i) EΘ|y,xˆ (i) − 2 (y˜k (x) − Hk θk ) (y˜k (x) − Hk θk ) Bk−1 Σ =Σ + Bk−1 Σ −Σ . k−1|K k−1|k−1 k|K k|k−1 2σw k=1 (20) K 1 Remark 1: Intuitively, the backward step yields an im=− 2 Tr EΘ|y,xˆ (i) (y˜k (x) − Hk θk ) · 2σw provement in the forward step-only approach since it uses k=1 the entire data sequence to smooth out the estimates of Θ. T . (15) This improvement comes at the cost of some additional pro(y˜k (x) − Hk θk ) cessing. The extent of this improvement is quantified through simulations in Section V. By defining Therefore, the E-step of the EM algorithm (12) yields an (i) Δ (i) Δ T ˆ ˆ MMSE (equivalently, MAP) estimate θk for k = 1, . . . , K θk|K = EΘ|y,xˆ (i) [θk ] , Rk|K = EΘ|y,xˆ (i) θk θk , of (i) ˆ . The estimates of as a by-product while calculating Q x, x the likelihood function in (15) can be expressed as α and β can, in turn, be obtained by using the transformation in (2). The resulting estimates are sub-optimal since, in K 1 (i) (i) T T ˆ general, the MAP estimator does not commute over non-linear ˆ Q x, x =− 2 Tr y˜k (x) y˜k (x) + Hk Rk|K Hk 2σw transformations. However, the sub-optimal estimators show k=1 T good fidelity performance and closely match the theoretical (i) (i) −y˜k (x) θˆk|K HkT − Hk θˆk|K y˜kT (x) . (16) lower bounds derived in Section IV. The M-step can now be expressed using (13) as After some algebraic steps, (16) can be equivalently written as 1 ˆ (i) = − 2 Q x, x 2σw
K k=1
T ˆ Tr Hk Σ k|K Hk
(i) + y˜k (x) − Hk θˆk|K
(i)
(i) y˜k (x) − Hk θˆk|K
T
(17) where
K −1 ˆ (i) H T Tr Hk Σ k k|K 2 x 2σw k=1 T (i) (i) + y˜k (x) − Hk θˆk|K y˜k (x) − Hk θˆk|K .
ˆ (i+1) = arg max x
T
Δ ˆ (i) (i) (i) ˆ (i) = Σ Rk|K − θˆk|K θˆk|K . k|K
ˆ (i+1) is given as the After some simplifications, the estimate x solution of a 2-D norm minimization problem K 2 (i) ˆ (i+1) = arg min x (21) y˜k (x) − Hk θˆk|K . x
k=1
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A closed form solution of the optimization problem in (21) does not exist. The interior point methods can be used to ˆ (i+1) efficiently [42]. obtain the estimates x The EM algorithm, therefore, provides estimates of Θ and x by alternating between the E and M-steps, respectively. The ˆ (2) , x ˆ (3) , . . . ˆ (1) , x algorithm is terminated when the sequence x converges. The EM algorithm for joint localization and timevarying clock synchronization of Node X is summarized in Algorithm 1.
where
Algorithm 1 The EM Algorithm ¯j,k , R ¯ j,k K and known 1: Input time-stamps Sj,k , Rj,k , S k=1 anchor locations sj , for j = 1, . . . , N . ˆ (0) . 2: Initialize x 3: for k = 1, . . . , K do 4: Determine Q x, x(i) in (17) using the MMSE estima(i) tor θˆk|K from (19) and (20). 5: end for ˆ (i+1) by solving the optimiza6: Obtain the ML estimate x tion problem (21). 7: return
and η = [xT , x ]T . By augmenting data from all N anchor nodes, (29) can be written as
⎡ T 2sj ⎢2sTj ⎢ ⎢ Aj = ⎢ ... ⎢ ⎣2sTj 2sTj (i)
⎤ −1 −1⎥ ⎥ .. ⎥ , . ⎥ ⎥ −1⎦ −1 (i)
⎤ (i)2 sj 2 − tj,1 2⎥ ⎢ ⎥ ⎢sj 2 − t¯(i) j,1 ⎥ ⎢ ⎥ ⎢ . . .... =⎢ ⎥ , ⎥ ⎢ 2 ⎢ (i) ⎥ 2 ⎣sj − tj,K ⎦ (i)2 2 sj − t¯j,K ⎡
(i)
pj
(i)
(i)
(i)
uj = [uj,1 , u¯j,k , . . . , uj,K , u ¯j,K ]T ,
(29)
2
Aη = p(i) + u(i) ,
(30) (i)
T
(i)
T
where A = [AT1 , . . . , ATN ]T , p(i) = [p1 , . . . , pN ]T , (i)T (i)T and u(i) = [u1 , . . . , uN ]T . It can be observed that (30) represents a linear matrix equation for the estimation of x. 2 2 By neglecting the second order noise terms, wj,k and w ¯j,k , (i) in u and letting (i)
(i)
(i)
(i)
q (i) = [2t1,1 , 2t¯1,1 , . . . , 2t1,K , 2t¯1,K , (i) (i) (i) (i) . . . , 2tN,1, 2t¯N,1 , . . . , 2tN,K , 2t¯N,K ]T ,
B. The LS estimator The location estimator in (21) requires a costly 2-D norm minimization that may be computationally infeasible for a sensor node. Therefore, it becomes imperative to simplify the location estimation method in Section III-A. Similar to the above discussion, it can be noticed that with ˆ (i) available at iteration i, the parameters θk an estimate x evolve according to a linear state space Gaussian model and can be efficiently estimated by the Kalman smoother described in Section III-A. Using estimates θˆk , (3) can be expressed at iteration i as (i) (i) Rj,k − dj = Sj,k θˆ1,k − θˆ2,k + wj,k
(22)
¯ j,k θˆ(i) + θˆ(i) + w ¯j,k . −S¯j,k − dj = −R 1,k 2,k
(23)
(i) Δ (i) (i) tj,k = Rj,k − Sj,k θˆ1,k + θˆ2,k .
(24)
Define
Squaring (22), yields (i)2
2
which can be re-written as 2
2
(i)2
(i)
where
(i) Δ uj,k =
(i) 2tj,k wj,k
(25)
2 − wj,k . Similarly, define
(i) Δ ¯ ˆ(i) ˆ(i) ¯ t¯j,k = R j,k θ1,k − θ2,k − Sj,k .
(26)
(i) Using t¯j,k and squaring (23), we have 2
2
(i)2
(i)
2sTj x − x = sj − t¯j,k + u ¯j,k ,
2 = σw diag(q (i) q (i) ) ,
(27)
(i) Δ (i) 2 where u¯j,k = 2t¯j,k w ¯j,k − w ¯j,k . By stacking all K observations th for the j anchor, (25) and (27) can be written in matrix form as (i) (i) Aj η = pj + uj , (28)
(31)
˜ = [w1,1 , w where w ¯1,1 , . . . , w1,K , w ¯1,K , , . . . , wN,K , w ¯N,K ]T . The LS solution for the estimation of η at iteration i + 1 can now be expressed as −1 −1 −1 ηˆ(i+1) = AT Σ(i) A AT Σ(i) p(i) . (32) u u The Kalman smoother stage employed at iteration i yielding (i) the estimates θˆk , k = 1, . . . , K, can be used to compute Σu (i) and p (cf. (29) and (31)). The estimate can be further refined by exploiting the relationship between elements of η [26]. We have ¯x ˆ (i+1) = ηˆ(i+1) + wLS , M where
(i)
2 x − sj = tj,k − 2tj,k wj,k + wj,k ,
2sTj x − x = sj − tj,k + uj,k ,
(i)
the noise covariance matrix Σu can be obtained as T (i) (i) (i)T (i) (i) ˜ q w ˜ Σu ≈ E u u =E q w
⎡
1 ¯ =⎣0 M xˆ1
(33)
⎤ 0 1⎦ x ˆ2
and wLS indicates the estimation error in estimating ηˆ(i+1) . The estimates {ˆ x1 , x ˆ2 } are available from (32). Finally the ˆ (i+1) at iteration i + 1 is given refined LS location estimator x by −1 (i+1) ¯ T AT Σ(i)−1 Aηˆ(i+1) ¯ T AT Σ(i)−1 AM ¯ ˆ LS = M x M u u −1 −1 ¯ T AT Σ(i) AM ¯ ¯ T AT Σ(i)−1 p(i) . = M M u u (34) Remark 2: The LS location estimator in (34) is derived by neglecting the second order measurement noise errors. Since the measurement noise errors are usually small, it is expected
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that the performance of the LS estimator will closely match that of the EM algorithm. The aforementioned LS-based approach presents a simpler closed form alternative to the potentially costly 2-D norm minimization problem for location estimation in Section III-A. The steps of the algorithm are summarized in Algorithm 2. Algorithm 2 The LS Algorithm ¯j,k , R ¯ j,k K and known 1: Input time-stamps Sj,k , Rj,k , S k=1 anchor locations sj , for j = 1, . . . , N . ˆ (0) . 2: Initialize x 3: for k = 1, . . . , K do (i) 4: Determine the MMSE estimator θˆk|K from the Kalman smoother in (19) and (20). 5: end for ˆ (i+1) using (34). 6: Obtain the LS estimate x 7: return
C. Complexity Analysis In this section, we present a complexity analysis of the EM and LS algorithms based on the total number of flops. It is assumed that multiplication, subtraction and addition operations can be computed using one flop, and a division and a square root operation can be computed using m flops [42], [43]. It is clear that both the EM and the LS algorithms employ the same Kalman smoother to estimate θk for k = 1, . . . , K. Thus, the main computation difference between the two algorithms is the evaluation of the 2-D norm minimization in (21) and the LS solution in (34) at a particular iteration. The complexity of the 2-D norm minimization will depend on the minimization algorithm employed, the initial point and accuracy. In order to evaluate complexity, we determine the number of operations required to compute the objective function in (21) for a certain point and the cost involved in the minimization algorithm when a initial point is available. We need r + 5 flops to evaluate ||x − sj ||. The total number of flops required to compute the objective function can be approximated as KN (19 + r) − K. The complexity of the minimization algorithm will depend on the nature of algorithm employed. Thus, the approximate computational complexity of the norm minimization will be O((rKN + Minimization Algorithm Complexity)) for large K and N . In order to compute the complexity of the LS algorithm, (i) note that the diagonal nature of Σu implies that approximately (4 + 2r)KN flops are required to determine its inverse. Similarly, we need approximately O(12KN ) and ¯ ¯ and Σ(i) O(4KN )operations to compute AM u AM , respectively. After the outer matrix inverse, and the subsequent −1 ¯ T AT Σ(i) p(i) , the total operations are multiplication with M u approximately given by O(44KN + 2rKN ), which for large K and N can be approximated as O(2rKN ). Hence, we can see that the minimization algorithm required to compute the 2D norm minimizer is the major source of increased complexity of the EM algorithm compared to the simpler LS algorithm at a particular iteration.
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IV. H YBRID C RAM E´ R -R AO B OUND It is of significant theoretical interest to establish the best achievable estimation performance by placing lower bounds on an estimator. The HCRB, proposed in [44], is useful in cases where the complete parameter vector comprises deterministic as well as random parameters. This fits our estimation scenario since the location x is deterministic while Θ is a random process. The HCRB states that the covariance matrix of an estimator ξˆ is lower bounded as T −1 ˆ ˆ , (35) [H(Θ, x)] E ξ−ξ ξ−ξ where the matrix inequality is to be interpreted in the positive semi-definite sense. The 2(K + 1) × 2(K + 1) hybrid information matrix (HIM) is given by H(Θ, x) = EΘ [F (Θ, x)] + ! ∂ ln f (Θ|x) EΘ|x ∂ξ
∂ ln f (Θ|x) ∂ξ
!T , (36)
where the Fisher information matrix (FIM) F (Θ, x) is given by
F (Θ, x) = Ey|Θ,x
∂ ln f (y|Θ, x) ∂ξ
∂ ln f (y|Θ, x) ∂ξ
T
(37)
The HIM can be calculated as shown in the following theorem. Theorem 1: The sub-matrices in the HIM corresponding to Θ and x can be expressed respectively as ! T H1T H1 HK HK , . . . , H11 = blkdiag +Υ (38) 2 2 σw σw N 2K (x − sj ) (x − sj )T , (39) H22 = 2 2 σw j=1 x − sj where the tri-diagonal matrix Υ is ⎡ 2P −1 −P −1 ⎢−P −1 2P −1 −P −1 ⎢ ⎢ .. .. . . −P −1 Υ=⎢ ⎢ ⎢ . .. ⎣ 0 −P −1 Similarly, the cross term is given by !T H1T d (x) T H12 = H21 = ,..., 2 σw
0
⎤
⎥ ⎥ ⎥ ⎥ . (40) ⎥ ⎥ −1 ⎦ −P P −1
T HK d (x) 2 σw
!T T ,
(41) where d (x) is defined in (58). Proof: See Appendix B. Using the sub-matrices derived above, the following result is immediate. ˆ and x ˆ Lemma 1: The covariance matrices of estimators Θ can be lower bounded as T ˆ ˆ E Θ−Θ Θ−Θ (42) HCRBΘ ˆ − x) (x ˆ − x)T HCRBx (43) E (x
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−1 where HCRBΘ = H11 − H12 H−1 and HCRBx = 22 H21 −1 H22 − H21 H−1 H 12 11 Proof: The proof simply follows from Theorem 1 and the inversion properties of a block matrix . Lemma 1 can be used to benchmark the estimation performance of the EM and the LS algorithms discussed in Section III. Two special cases when x or Θ may be known are described below. Corollary 1: In case when the location x may be known, ˆ can be lower bounded as the covariance matrix of Θ T ˆ −Θ Θ ˆ −Θ (44) H−1 E Θ 11 . Similarly, if the parameter Θ is known, we have ˆ − x) (x ˆ − x)T H−1 E (x 22
(45)
ˆ in (44), an expression for the Using the lower bound for Θ T is given by corresponding lower bound on Ψ αT , β T [40] T ˆ ˆ E Ψ−Ψ Ψ−Ψ ∂E [g (Θ)] ∂Θ
!T HCRBΘ
∂E [g (Θ)] ∂Θ
! (46)
where g (Θ) is the transformation defined in (2) and HCRBΘ is given by Lemma 1. Due to the non-linear function g, an exact expression for (46) is mathematically intractable. Hence, ˆ we use numerical methods to evaluate the lower bound on Ψ. V. S IMULATION R ESULTS A. Simulation Setup In this section, we present simulation results to corroborate our findings in the earlier sections. In particular, we compare the relative performance of the estimators proposed above against the theoretical lower bounds. The anchors are located at (14, 21), (6, -8) and (24, 4). The clock skew is randomly drawn from [0.998, 1.002] and the offset is drawn randomly from [1, 10]. Unless stated otherwise, the location of Node X is generated by drawing x1 and x2 randomly from [1, 10]. The Gauss-Markov process noise variances σ12 and σ22 are set to 10−4 and 10−2 , respectively. The measure of accuracy used to benchmark different estimators isthe MSE. The MSE of the 2 2 x2 − x2 ) . location estimator is given by E (ˆ x1 − x1 ) + (ˆ ˆ and βˆ are given by The average MSEs of α 1 ˆ − α)T (α ˆ − α) MSEα = E (α K T 1 ˆ ˆ MSEβ = E β − β β−β . K
function of x as the number of iterations It can be i increases. ˆ (i) is unimodal noticed that at each iteration, exp Q x, x and hence, does not present any local maxima. This allows the algorithm to at the solution uniquely. Initially, converge ˆ (i) is located for i = 1, exp Q x, x away from the actual ˆ (i) starts to move location x. As i increases, exp Q x, x towards the left, finally settling at coordinates (2, 4) at about i = 12. Hence, the plot illustrates the improvement obtained ˆ (i) with each iteration of the EM algorithm. in estimates of x ˆ (i) Fig. 3 shows the improvement in the recursive estimate x as the number of iterations increases for different values of K. It is observed that the EM algorithm quickly settles at the true location coordinates in around 12 iterations for K = 2. Moreover, the convergence is faster with K = 4 compared to K = 2. Hence, the proposed iterative location estimation algorithm performs efficiently even with few message exchanges. The convergence of the EM algorithm with increasing iterations for the two cases of forward step-only EM and the backward step EM is also illustrated in Fig. 4. It is evident that the backward step converges at the true locations coordinates in fewer iterations compared to a forward steponly approach. This is a direct consequence of the smoothing of the random parameters using a backward sweep. However, this improvement comes at the cost of additional processing required to carry out the time-series smoothing. C. Backward Step versus Forward Step It is commonly known that smoothing the time series yields improvements in the estimator of the random parameters. Fig. 5 plots a comparison of the MSE performance of the estimators of clock skew and offset for the forward step-only 2 = and backward step EM with measurement noise variance σw −1 10 . It is indeed observed that the backward step approach consistently outperforms the forward-only step. Moreover, the performance gap increases as the number of message exchanges K increase. This is intuitively satisfying since a higher value of K ensures that the smoother has a greater room for improving the estimates by using the complete data sequence. In addition, it can be observed that the performance improvement of the backward step is more pronounced for the clock skew α as compared to the offset β and the MSE incurred in estimating the clock skew is lower than the corresponding MSE for the offset. The improvement using the backward step suggests that it would be useful to smooth the estimates of the clock parameters, since it yields higher accuracy and can result in a reduction of energy spent on exchanging timing information for re-synchronization. Similar MSE curves are also obtained for the iterative LS algorithm which are not shown here for brevity. D. Estimator Performance Analysis
B. Convergence Analysis of the EM Algorithm An important issue in the formulation of the EM algorithm is the impact of the increasing iterations on the recursive estimate of the location of Node X. In order to show the updates in the EM algorithm, the simulation results are plotted 2 = 10−1 and K = for x = (2, 4) with σw 2. Fig.(i)2 shows ˆ as a the updates in the likelihood function exp Q x, x
In this section, we present simulation results for comparing the MSE performance with the theoretical HCRB derived in Section IV. The MSE plots are provided for the estimates of location as well as the clock parameters of the unknown node to observe the impact of measurement noise and the number of message exchanges. The plots are averaged over 1000 independent simulation runs.
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Fig. 6 illustrates the MSE incurred in estimating the location 2 decreases. It can be observed that the MSE as the variance σw 2 decays with a decrease in σw . Moreover, the MSE decay also
increases as K increases from K = 4 to K = 8. The EM and the LS algorithm have the same performance for low to moderate noise variances. In addition, the MSE remains fairly close to the theoretical HCRB but does not attain it. This could potentially be due to the reason that HCRB is known to be less tight for the non-random part of the parameters [45]. The MSE performance of the skew estimates provided by the EM and LS algorithms is illustrated in Fig. 7 as the measurement noise decreases. It is observed that the proposed estimator has high fidelity and matches closely with the theoretical lower bound provided by HCRB. Moreover, the EM and LS approaches have similar performance. As expected, the performance improves as more messages are exchanged so that K increases. Similarly, Fig. 8 depicts the MSE curves for the estimation of clock offset versus measurement noise for K = 4 and K = 8. In this case, the MSE incurred in estimation is higher than the corresponding values for skew. However, the MSE performance is still close to the lower bounds and the gap diminishes as more messages are exchanged. Similar to the aforementioned observations, the EM and LS algorithms exhibit similar performance.
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E. LS Estimator Degrades Only at Large Measurement Noise The LS estimator is obtained by a non-linear processing of the data and ignoring the second order terms in measurement noise. Hence, we expect to see a performance degradation in 2 becomes large. This observation is the LS estimator when σw illustrated in Fig. 9 where the MSE in location estimation 2 and K = 4. Perforis plotted for various values of σw 2 mance starts to degrade as σw increases. Since in practical sensor networks, measurement noise is usually small, the LS approach is a viable alternative to the potentially costly EM algorithm which requires a 2-D norm minimization for location estimation in each iteration. VI. C ONCLUSIONS The problems of node localization and clock synchronization in a wireless sensor network are closely tied. In this work, the joint estimation of an unknown node’s location and its time-varying clock parameters is considered. Accurately tracking the temporal variation in the clock parameters
Fig. 9. MSE comparison of EM and LS algorithms versus measurement 2 for K = 4. noise variance σw
can lower the re-synchronization requirements and result in significant communication savings. The data exchange with the known anchors follows a two-way message exchange mechanism. In order to alleviate the computational complexity associated with the MAP estimator, two iterative approaches are proposed for joint localization and timing synchronization. The first approach utilizes the EM algorithm which iteratively estimates the unknown node’s location by considering the clock parameters as hidden variables and estimates the location as the solution to a 2-D norm minimization. In order to further simplify the EM algorithm, an LS based location estimation algorithm is presented. This algorithm results in closed form expressions for the joint estimation problem and is particularly suited in scenarios where the computational resources come at a high premium. The MSE of the various estimators is lower bounded by deriving the HCRB. Simulation results corroborate our theoretical findings and demonstrate the high accuracy of the iterative algorithms. It is observed that the
AHMAD et al.: JOINT NODE LOCALIZATION AND TIME-VARYING CLOCK SYNCHRONIZATION IN WIRELESS SENSOR NETWORKS
performance of the EM and LS algorithms are comparable and hence, LS can be used in low-cost sensor networks. An interesting future research direction is to extend the linear pre-processing approach introduced by [46] to the present estimation context. The estimation approach introduced by [46] cancels the effect of the unknown clock-offsets, leaving the Fisher information on node coordinates unchanged. As a consequence, there is no need for a two-way exchange message mechanism anymore with clear advantages on the signaling and estimation methods.
write EΘ|x
! !T ∂ ln f (Θ|x) ∂ ln f (Θ|x) = ∂ξ ∂ξ ⎡ T ∂ ln f (Θ) ∂ ln f (Θ) ∂Θ ∂Θ ⎣EΘ 02×2K
02×2
.
Using (8) and (9), K
ln f (θk |θk−1 )
and it follows that
(47)
where ck represents the small variation at the k th instant. Expanding 1/αk (first element of the transformation in (2)) in a Taylor Series, we have 1/αk = 1/(1 − ck ) = 1 + ck − c2k − ....
02K×2 ⎦
k=1
In real sensor networks, the variation of the clock skew random process α is usually very small around 1. Therefore, we can write the random process αk as αk = 1 − ck
⎤
(50)
ln f (Θ) = ln f (θ0 ) + A PPENDIX A G AUSS -M ARKOV S YSTEM M ODEL
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∂ ln f (Θ) = P −1 (nk+1 − nk ) , k = 1, . . . , K − 1 ∂θk ∂ ln f (Θ) = P −1 (−nK ) . ∂θK After some algebraic calculations, we have ! !T ∂ ln f (Θ) ∂ ln f (Θ) EΘ =Υ, ∂Θ ∂Θ
(51)
where Υ is defined in (40). (48)
Due to the small nature of the variation ck , the second order terms can be neglected. The process ck can be modeled as an AR(P) process [35], [36]. However, in this work, for simplicity, we model ck as an AR(1) process. Therefore, 1/αk can be sufficiently approximated by an AR(1) process. Similarly, for the second element of the transformation, we have βk /αk = βk /(1 − ck ) = βk 1 + ck − c2k − ... ≈ βk + βk ck (49) Since the variations in ck are small and the processes βk and ck are assumed independent, the variations in βk /αk will be largely due to the variations in the clock offset βk as var(βk ) >> var(βk ck ) = var(βk )var(ck ) Therefore, βk /αk ≈ βk . The process βk has been modeled as AR(1) process in several prior contributions e.g., [31], [33]. Hence, both the elements of θk in (2) can be modeled as independent AR(1) processes. Due to the independence of 1/αk ≈ 1−ck and βk /αk ≈ βk , we have a diagonal structure of the covariance matrix of noise σ12 and σ22 for ck and βk , respectively, nk . Assuming variances T we have E nk nk ≈ P = diag σ12 , σ22 . A PPENDIX B P ROOF OF T HEOREM 1 T (Θ|x) ∂ ln f (Θ|x) A. Computation of EΘ|x ∂ ln f∂ξ . ∂ξ We first compute the second term in (36). Since the parameters Θ and x are statistically independent, it follows that (Θ) = 0, we can f (Θ|x) = f (Θ). By using the fact that ∂f∂x
B. Computation of EΘ [F (Θ, x)] The FIM can be partitioned as F F (Θ, x) = 11 F21 where
F11 = Ey|Θ,x F12 = Ey|Θ,x F21 = Ey|Θ,x F22 = Ey|Θ,x
∂ ln f (y|Θ, x) ∂Θ ∂ ln f (y|Θ, x) ∂Θ ∂ ln f (y|Θ, x) ∂x ∂ ln f (y|Θ, x) ∂x
! ! ! !
F12 F22
∂ ln f (y|Θ, x) ∂Θ ∂ ln f (y|Θ, x) ∂x ∂ ln f (y|Θ, x) ∂Θ ∂ ln f (y|Θ, x) ∂x
!T !T !T !T . (52)
1) Sub-matrix EΘ (F11 ): The sub-matrix F11 can be computed more readily by using the alternative expression 2 ∂ ln f (y|Θ, x) . (53) F11 = Ey|Θ,x − ∂Θ∂ΘT It follows from the i.i.d nature of the network delays that K
∂ 2 ln f (y|Θ, x) ∂ 2 ln f (yk |θk , x) = ∂Θ∂ΘT ∂Θ∂ΘT
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k=1
where using (10) ln f (yk |θk , x)∝ −
(yk − d (x)−Hk θk )T(yk − d (x)−Hk θk ) 2 2σw
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[4] M. Nicoli, S. Gezici, Z. Sahinoglu, and W. Wemeersch, “Localization in mobile wireless and sensor networks,” EURASIP J. Wireless Commun. Netw., Dec. 2011. [5] N. Patwari, J. N. Ash, S. Kyperountas, A. O. I. Hero, R. L. Moses, and N. S. Correal, “Locating the nodes: Cooperative localization in wireless sensor networks,” IEEE Signal Processing Mag., vol. 22, no. 4, pp. 54–69, July 2005. [6] A. H. Sayed, A. Tarighat, and N. Khajehnouri, “Network-based wireless location: Challenges faced in developing techniques for accurate wireless location information,” IEEE Signal Process. Mag., vol. 22, no. 4, pp. 24–40, July 2005. . [7] Z. Ma and K. C. Ho, “TOA localization in the presence of random sensor position errors,” in IEEE International Conf. Acoustics, Speech Signal Process. (ICASSP 2011), May 2011, pp. 2468–2471. [8] K. C. Ho, “Bias reduction for an explicit solution of source localization using TDOA,” IEEE Trans. Signal Process., vol. 60, no. 5, pp. 2101– 2114, May 2012. [9] M. Sun and K. C. Ho, “An asymptotically efficient estimator for TDOA and FDOA positioning of multiple disjoint sources in the presence of sensor location uncertainties,” IEEE Trans. Signal Process., vol. 59, no. 7, pp. 3434–3440, July 2011. [10] B. R. Hamilton, X. Ma, J. Baxley, and B. Walkenhorst, “Node localization and tracking using distance and acceleration measurements,” in International Workshop Cognitive Inf. Process., June 2010, pp. 399–404. [11] M. R. Gholami, S. Gezici, M. Rydstr¨om, and E. G. Str¨om, “A distributed positioning algorithm for cooperative active and passive sensors,” in Personal Indoor Mobile Radio Commun. (PIMRC), Sept. 2010, pp. 1713–1718. [12] M. R. Gholami, S. Gezici, and E. G. Str¨om, “Improved position estimation using hybrid TW-TOA and TDOA in cooperative networks,” IEEE Trans. Signal Process., vol. 60, no. 7, pp. 3770–3785, July 2012. [13] D. Wang and L. Yang, “Cooperative robust localization against malicious anchors based on semi-definite programming,” in IEEE Military Commun. Conf. (MILCOM), Oct. 2012, pp. 1–6. [14] X. Tan and J. Li, “Cooperative positioning in underwater sensor networks,” IEEE Trans. Signal Process., vol. 58, no. 11, pp. 5860–5871, Nov. 2010. [15] Y. Wang and G. Leus, “Reference-free time-based localization for an asynchronous target,” EURASIP J. Advances Signal Process., Jan. 2012. [16] B. Sadler and A. Swami, “Synchronization in sensor networks: An overview,” in IEEE Military Commun. Conf. (MILCOM), Oct. 2006, pp. 1–6. [17] Q. Chaudhari and E. Serpedin, Synchronization in Wireless Sensor Networks: Parameter Estimation, Performance Benchmarks, and Protocols. Cambridge University Press, 2009. [18] H. Abdel-Ghaffar, “Analysis of synchronization algorithms with timeout control over networks with exponentially symmetric delays,” IEEE Trans. Commun., vol. 50, no. 10, pp. 1652–1661, Oct. 2002. [19] D. Jeske, “On maximum-likelihood estimation of clock offset,” IEEE Trans. Commun., vol. 53, no. 1, pp. 53–54, Jan. 2005. [20] K.-L. Noh, Q. Chaudhari, E. Serpedin, and B. Suter, “Novel clock phase offset and skew estimation using two-way timing message exchanges for wireless sensor networks,” IEEE Trans. Commun., vol. 55, no. 4, pp. 766–777, Apr. 2007. [21] M. Leng and Y.-C. Wu, “On clock synchronization algorithms for wireless sensor networks under unknown delay,” IEEE Trans. Veh. Technol., vol. 59, no. 1, pp. 182–190, Jan. 2010. [22] ——, “Low complexity maximum likelihood estimators for clock synchronization of wireless sensor nodes under exponential delays,” IEEE Trans. Signal Process., vol. 59, no. 10, pp. 4860–4870, Oct. 2011. [23] N. Khajehnouri and A. H. Sayed, “A distributed broadcasting timesynchronization scheme for wireless sensor networks,” in IEEE International Conf. Acoustics, Speech, Signal Process. (ICASSP), Mar. 2005. [24] M. Leng and Y.-C. Wu, “Distributed clock synchronization for wireless sensor networks using belief propagation,” IEEE Trans. Signal Process., vol. 59, no. 11, pp. 5404–5414, Nov. 2011. [25] K.-Y. Cheng, K.-S. Lui, Y.-C. Wu, and V. Tam, “A distributed multihop time synchronization protocol for wireless sensor networks using pairwise broadcast synchronization,” IEEE Trans. Wireless Commun., vol. 8, no. 4, pp. 1764–1772, Apr. 2009. [26] J. Zheng and Y.-C. Wu, “Joint time synchronization and localization of an unknown node in wireless sensor networks,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1309–1320, Mar. 2010. [27] S. Zhu and Z. Ding, “Joint synchronization and localization using TOAs: A linearization based approach,” IEEE J. Sel. Areas Commun., vol. 28, no. 7, pp. 1017–1025, Sept. 2010.
H T (yk − d (x)) − HkT Hθk Δ ln f (yk |θk , x) fk = = k . 2 ∂θk σw (55) This implies that ln f (yk |θk , x) = [0T2×1 , . . . , fkT , . . . , 0T2×1 ]T ∂Θ ! ∂ 2 ln f (yk |θk , x) HkT Hk = blkdiag 0 , . . . , , . . . , 0 2×2 2×2 2 ∂Θ∂ΘT σw Collecting all terms from (54) followed by the expectation in (53), it follows that ! T H1T H1 HK HK EΘ [F11 ] = blkdiag ,..., . (56) 2 2 σw σw 2) Sub-matrix EΘ (F22 ): Using the fact that ∂ x − sj x − sj = ∂x x − sj
(57)
we can write Δ ∂d (x) d (x) = = ∂x T x − s1 x − s1 x − sN x − sN , ,..., , x − s1 x − s1 x − sN x − sN (58) dT (x) (yk − d (x) − Hk θk ) ln f (yk |θk , x) = . 2 ∂x σw (59) The matrix F22 in (52) is given by T K d d F22 = Ey|Θ,x k 4 k σw Δ
dk =
k=1
=
N 2K T d (x) d (x) 2 σw j=1
(60)
which after some simplification yields (39). 3) The sub-matrix EΘ [F12 ]: The sub-matrix F12 in (52) can be computed by using (55) and (59). After matrix multiplication and taking expectation with respect to Ey,Θ , EΘ [F12 ] can be expressed as !T !T T T H1T d (x) HK d (x) EΘ [F12 ] = ,..., (61) 2 2 σw σw T = EΘ F21 . (62) The mathematical details in computing (61) are omitted for brevity. The proof of the theorem follows from the submatrices calculated above. R EFERENCES [1] I. F. Akyildiz, W. Su, Y. Sankarasubramanium, and E. Cayirci, “A survey on sensor networks,” IEEE Commun. Mag., vol. 40, no. 8, pp. 102–114, Aug. 2002. [2] S. Gezici, “A survey on wireless position estimation,” Wireless Personal Commun.(Special Issue Towards Global Seamless Personal Navigation), vol. 44, no. 3, pp. 263–282, Feb. 2008. [3] S. Gezici, Z. Tian, G. Giannakis, H. Kobayashi, A. Molisch, H. Poor, and Z. Sahinoglu, “Localization via ultra-wideband radios: A look at positioning aspects for future sensor networks,” Signal Process. Mag., IEEE, vol. 22, no. 4, pp. 70–84, July 2005.
AHMAD et al.: JOINT NODE LOCALIZATION AND TIME-VARYING CLOCK SYNCHRONIZATION IN WIRELESS SENSOR NETWORKS
[28] Y. Wang, X. Ma, and G. Leus, “Robust time-based localization for asynchronous networks.” IEEE Trans. Signal Process., vol. 59, no. 9, pp. 4397–4410, Sept. 2011. [29] S. P. Chepuri, R. T. Rajan, G. Leus, and A.-J. van der Veen, “Joint clock synchronization and ranging: Asymmetrical time-stamping and passive listening,” IEEE Signal Process. Lett., vol. 20, no. 1, pp. 51–54, Jan. 2013. [30] G. J. Pottie, W. J. Kaiser, M. Giona, and S. Barbarossa, “Wireless integrated network sensors,” Commun., ACM, vol. 43, no. 5, pp. 51– 58, May 2000. [31] Q. Chaudhari, E. Serpedin, and J.-S. Kim, “Energy-efficient estimation of clock offset for inactive nodes in wireless sensor networks,” IEEE Trans. Inf. Theory, vol. 56, no. 1, pp. 582–596, Jan. 2010. [32] J.-S. Kim, J. Lee, E. Serpedin, and K. Qaraqe, “Robust clock synchronization in wireless sensor networks through noise density estimation,” IEEE Trans. Signal Process., vol. 59, no. 7, pp. 3035–3047, July 2011. [33] A. Ahmad, D. Zennaro, E. Serpedin, and L. Vangelista, “A factor graph approach to clock offset estimation in wireless sensor networks,” IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4244–4260, July 2012. [34] N. Freris, V. Borkar, and P. Kumar, “A model-based approach to clock synchronization,” in Proc. 48th IEEE Conf. Decision Control held jointly with the 2009 28th Chinese Control Conf. (CDC/CCC), Dec. 2009, pp. 5744–5749. [35] B. R. Hamilton, X. Ma, Q. Zhao, and J. Xu, “ACES: Adaptive clock estimation and synchronization using Kalman filtering,” in ACM International Conf. Mobile Comput. Netw., Sept. 2008, pp. 152–162. [36] H. Kim, X. Ma, and B. R. Hamilton, “Tracking low-precision clocks with time-varying drifts using Kalman filtering,” IEEE/ACM Trans. Netw., vol. 20, no. 1, pp. 257–270, Feb. 2012. [37] Q. Chaudhari, E. Serpedin, and K. Qaraqe, “On maximum likelihood estimation of clock offset and skew in networks with exponential delays,” IEEE Trans. Signal Process., vol. 56, no. 4, pp. 1685–1697, Apr. 2008. [38] A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Royal Statistical Society, vol. 39, Series B, pp. 1–38, 1977. [39] G. McLachlan and T. Krishnan, The EM Algorithm and Extensions. John Wiley & Sons, New York, NY, 1997. [40] S. M. Kay, Fundamentals of Statistical Signal Processing. Estimation Theory. Prentice-Hall, 1993. [41] H. E. Rauch, F. Tung, and C. T. Striebel, “Maximum likelihood estimates of linear dynamic systems,” AIAA J., vol. 3, no. 8, pp. 1445–1450, Aug. 1965. [42] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2003. [43] C. W. Ueberhuber, Numerical Computation. Springer, 1995. [44] Y. Rockah and P. M. Schultheiss, “Array shape calibration with sources in unknown locations, part I: Far field sources,” IEEE Trans. Acoustics, Speech, Signal Process., vol. 35, no. 3, pp. 286–299, Mar. 1987. [45] H. Messer, “The Hybrid Cram´er-Rao bound - From practice to theory,” in IEEE Workshop Sensors, Array Multichannel Signal Process., 2006. [46] M. Rydstrom, E. G. Strom, and A. Svensson, “Clock-offset cancelation methods for positioning in asynchronous networks,” in International Conf. Wireless Netw., Commun. Mobile Comput., June 2005, pp. 981– 986. Aitzaz Ahmad (S’12) received the B.E. and M.S. degrees in electrical engineering from National University of Sciences and Technology (NUST), Islamabad, Pakistan in 2006 and 2008, respectively, and the Ph.D. degree in electrical engineering from Texas A&M University, College Station, in 2012. He was awarded the President Gold Medal for academic distinction in his Masters degree. During Fall 2011, he was a research intern with Intel Corp, Santa Clara, CA. He is currently a Senior Engineer in Corporate R&D, Qualcomm Technologies Inc., San Diego, CA. His research interests include information theory, statistical signal processing in wireless sensor networks and bioinformatics, and message passing algorithms in graphical models.
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Erchin Serpedin (F’13) received the specialization degree in signal processing and transmission of information from Ecole Superieure DElectricite (SUPELEC), Paris, France, in 1992, the M.Sc. degree from the Georgia Institute of Technology, Atlanta, in 1992, and the Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, in January 1999. He is currently a professor in the Department of Electrical and Computer Engineering at Texas A&M University, College Station. He is the author of 2 research monographs, 1 textbook, 90 journal papers and 150 conference papers, and serves currently as associate editor for IEEE T RANSACTIONS ON C OMMUNICATIONS, Signal Processing (Elsevier), EURASIP Journal on Advances in Signal Processing, Physical Communication (Elsevier), and EURASIP Journal on Bioinformatics and Systems Biology. His research interests include statistical signal processing, information theory, bioinformatics, and genomics. He is an IEEE Fellow. Hazem N. Nounou (SM’08) received the B.S. degree (magna cum laude) from Texas A&M University, College Station, in 1995, and the M.S. and Ph.D. degrees from Ohio State University, Columbus, in 1997 and 2000, respectively, all in electrical engineering. In 2001, he was a Development Engineer for PDF Solutions, a consulting firm for the semiconductor industry, in San Jose, CA. Then, in 2001, he joined the Department of Electrical Engineering at King Fahd University of Petroleum and Minerals in Dhahran, Saudi Arabia, as an Assistant Professor. In 2002, he moved to the Department of Electrical Engineering, United Arab Emirates University, Al-Ain, UAE. In 2007, he joined the Electrical and Computer Engineering Program at Texas A&M University at Qatar, Doha, Qatar, where he is currently an Associate Professor. He published more than 80 refereed journal and conference papers. He served as an Associate Editor and in technical committees of several international journals and conferences. His research interests include intelligent and adaptive control, control of time-delay systems, system biology, and system identification and estimation. Dr. Nounou is a senior member of IEEE. Mohamed Nounou (SM’08) received the BS degree (magna cum laude) from Texas A&M University, College Station, in 1995, and the MS and PhD degrees from the Ohio State University, Columbus, in 1997 and 2000, respectively, all in chemical engineering. From 2000 to 2002, he was with PDF Solutions, a consulting company for the semiconductor industry, in San Jose, CA. In 2002, he joined the Department of Chemical and Petroleum Engineering at the United Arab Emirates University as an assistant professor. In 2006, he joined the Chemical Engineering Program at Texas A&M University at Qatar, Doha, Qatar, where he is currently an Associate Professor. He has published more than 80 refereed journal and conference papers and book chapters. He also served as an associate editor and in technical committees of several international journals and conferences. His research interests include process modeling and estimation, system biology, and intelligent control. He is a member of the American Institute of Chemical Engineers (AIChE) and a senior member of the IEEE.
