2522 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 6, JUNE TDOA Based Positioning in the Presence of Unknown Clock Skew

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1 5 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. NO. JUNE 3 TDOA Based Postonng n the Presence of Unknown Clock Skew Mohammad Reza Gholam Student Member IEEE Snan Gezc Senor Member IEEE anderkg.ström Senor Member IEEE Abstract Ths paper studes the postonng problem of a sngle target node based on tme-dfference-of-arrval TDOA measurements n the presence of clock mperfectons. Employng an affne model for the behavour of a local clock t s observed that TDOA based approaches suffer from a parameter of the model called the clock skew. Modelng the clock skew as a nusance parameter ths paper nvestgates jont clock skew and poston estmaton. The maxmum lkelhood estmator MLE s derved for ths problem whch s hghly nonconvex and dffcult to solve. To avod the dffculty n solvng the MLE we employ sutable approxmatons and relaxatons and propose two suboptmal estmators based on semdefnte programmng and lnear estmaton. To further mprove the estmaton accuracy we also propose a refnng step. In addton the Cramér-Rao lower bound CRLB s derved for ths problem as a benchmark. Smulaton results show that the proposed suboptmal estmators can attan the CRLB for suffcently hgh sgnal-to-nose ratos. Index Terms Wreless sensor network tme-dfference-ofarrval TDOA clock skew semdefnte programmng lnear estmator maxmum lkelhood estmator MLE Cramér-Rao lower bound CRLB postonng clock synchronzaton. I. INTRODUCTION POSITIONING of sensor nodes based on tme-of-arrval TOA measurements s a popular technque for wreless sensor networks WSNs [] []. TOA-based postonng can potentally provde hghly accurate estmaton of target s poston n some stuatons e.g. n lne-of-sght condtons and for suffcently hgh sgnal-to-nose ratos SNRs [] [7]. Despte ts hgh performance TOA-based postonng s strongly affected by the clock offset mperfecton a fxed devaton from a reference clock at tme zero. To resolve ths problem tme-dfference-of-arrval TDOA based postonng has been proposed as an alternatve approach n the lterature [] [] [] whch has found varous applcatons n practce e.g. n the Global Postonng System. Manuscrpt receved June 5 ; revsed December 5 and January 5 3. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was E. Serpedn. M. R. Gholam and E. G. Ström are wth the Dvson of Communcaton Systems Informaton Theory and Antennas Department of Sgnals and Systems Chalmers Unversty of Technology SE- 9 Gothenburg Sweden e-mal: {moreza erk.strom}@chalmers.se. S. Gezc s wth the Department of Electrcal and Electroncs Engneerng Blkent Unversty Ankara Turkey e-mal: gezc@ee.blkent.edu.tr. Ths work was supported n part by the European Commsson n the framework of the FP7 Network of Excellence n Wreless COMmuncatons # contract no. 33 n part by the Swedsh Research Councl contract no and n part by Turk Telekom contract no Dgtal Object Identfer.9/TCOMM /3$3. c 3 IEEE The clock of an oscllator can be descrbed va an affne model whch nvolves the clock offset and clock skew parameters [9]. Whle the clock offset corresponds to a fxed tme offset due to clock mperfectons the clock skew parameter defnes the rate of varatons n the local clock compared to the real tme [] []. Whle the TDOA technque resolves the clock offset ambguty t can stll suffer from the clock skew. It means that the actual dfference between two TOAs whch forms a TDOA measurement n a target node mght be larger or smaller than the actual dfference even n the absence of the measurement nose. For an deal clock the clock skew s equal to one and t mght be larger or smaller than one for an unsynchronzed clock. Thus a poston estmate may be consderably affected by a non-deal clock skew for an unsynchronzed network n practcal scenaros dependng on how much the clock skew devates from one. Durng the last few years varous synchronzaton technques have been proposed n the lterature; e.g. see [] [3] and references theren. Whle tradtonally synchronzaton and postonng are separately studed n MAC and physcal layers respectvely the authors n [] formulate a jont synchronzaton and postonng problem n the MAC layer. If the major delay s the fxed delay due to propagaton through the rado channel the jont poston and tmng estmaton technque works well. The method developed n [] s based on a two-way message passng protocol that can be consdered as a counterpart to two-way TOA rangng n the physcal layer [5]. The authors n [7] nvestgate the postonng problem based on tme of flght measurements for asynchronous networks n the physcal layer and propose a technque based on the lnear least squares. Usng approxmatons the authors n [] [] propose dfferental TDOA to mtgate the effects of mperfect clock mparments. Ths method can cause nose enhancement and performance degradaton n some scenaros. Such an approach s effectve when only clock offsets exst n target and reference nodes and when there are more than one target node. In addton the proposed teratve method based on a nonlnear least squares crteron may converge to a local mnmum resultng n a large postonng error snce the objectve functon s nonconvex. In ths paper we study the sngle node postonng problem n the physcal layer for one way rangng an unsynchronzed target node tres to fnd ts poston by computng TDOA measurements self-postonng. We assume that a number of reference nodes are perfectly synchronzed wth a reference clock and transmt ther sgnals at a common

2 GHOLAMI et al.: TDOA BASED POSITIONING IN THE PRESENCE OF UNNOWN CLOC SEW 53 tme nstant. Then the target node measures the TOAs of the receved sgnals and forms a set of the TDOA measurements. By constructng a TDOA measurement the clock offset vanshes n the TDOA measurement but as mentoned prevously an unsynchronzed clock skew stll affects the TDOA measurements. Snce the clock skew s unknown n ths study we consder t as a nusance parameter and nvolve t n the poston estmaton. In fact we deal wth the jont estmaton of the clock skew and the poston of the target node. Note that we consder a fxed clock skew durng the TDOA measurements snce ts varatons durng a perod of tme s assumed to be neglgble. For Gaussan measurement errors the maxmum lkelhood estmator MLE for ths problem s hghly nonconvex and dffcult to solve. In order to derve a computatonally effcent algorthm we consder a number of approxmatons and relaxatons and propose two suboptmal estmators whch can be effcently solved to provde coarse poston estmates. The frst estmator s based on relaxng the nonconvex problem to a semdefnte programmng SDP. Usng a lnearzaton technque we derve a lnear model and consequently apply a lnear least squares LLS approach to fnd an estmate of the target poston. We then apply a correcton technque [9] to mprove the estmaton accuracy. In order to mprove the accuracy of the coarse estmate provded by the SDP or the LLS we lnearze the measurements usng the frst-order Taylor seres expanson around the estmate and obtan a lnear model n whch the estmaton error can be approxmated. Based on that model the coarse poston estmate can be further mproved. To compare dfferent approaches we derve the Cramér- Rao lower bound CRLB as a benchmark. We also study the CRLB when an estmate of the clock skew s avalable through smulatons and nvestgate the effectveness of the proposed approaches. In summary the man contrbutons of ths work are: the dea of jont clock skew and poston estmaton based on TDOA measurements; dervaton of the MLE and the CRLB for the problem consdered n ths study; 3 dervng two suboptmal estmators to provde coarse estmates of the target locaton based on lnearzaton and relaxaton technques; proposng a smple estmator based on the frst order Taylor-seres expanson around the coarse estmate to obtan a refned poston estmate. The remander of the paper s organzed as follows. Secton II explans the sgnal model consdered n ths paper. In Secton III the maxmum lkelhood estmator and a theoretcal lower bound are derved for the problem. Two suboptmal estmators are studed n Secton IV. Smulaton results are dscussed n Secton V. Fnally Secton VI makes come concludng remarks. Notaton: The followng notatons are used n ths paper. Another alternatve s to measure TOAs of the sgnal transmtted by a target node n the reference nodes and then to transfer the measurements to a central unt to compute the TDOAs from whch the poston of the target s estmated remote postonng. Although ths method can resolve the clock mperfecton of the target node t needs a central processng unt and requres that the fnal estmate should be sent back to the target node. Fg.. θ Local clock Local clock versus real clock. θ + wt θ > w > Ideal clock θ =w = Reference Lowercase and bold lowercase letters denote scalar values and vectors respectvely. Matrces are wrtten n bold uppercase letters. M and denote the vector of M ones and the vector matrx of all zeros respectvely. I M s an M by M dentty matrx. The operators tr and E{ } are used to denote the trace of a square matrx and the expectaton of a vector varable respectvely. The Eucldan norm of a vector s denoted by.theblkdagx...x N s a block dagonal matrx wth dagonal elements blocks X...X N. da b = a b s the dstance between a and b and denotes the ronecker product. Gven two matrces A and B A B means that A B s postve semdefnte. S m and R m + denote the set of all m m symmetrc matrces and the set of all m vectors wth postve elements respectvely. [A] j denotes the element of matrx A n the th row and the jth column. II. SYSTEM MODEL Consder a two-dmensonal -D network wth N + sensor nodes. Suppose that the frst N sensors are reference anchor nodes whch are located at known postons a = [a a ] T R =... N and the last sensor node s the target node whch s placed at unknown poston x = [x x ] T R. It s assumed that the reference nodes are synchronzed wth a reference clock whle the clock of the target node s left unsynchronzed. The followng affne model s employed for the local clock of the target node []: Ct =θ + wt θ and w denote respectvely the relatve clock offset and the clock skew between the target node and the reference tme t. To get some nsght nto ths model consder Fg. whch llustrates the relaton between a local clock and a real clock. For the example n the fgure the local tme vares faster than the deal tme.e. w>. The affne model for the clock s a common model and has been justfed n the lterature e.g. see [9] [] [] and references theren. Therefore ths model s employed throughout the paper. Assume that the target node s able to measure the TOAs of the receved sgnals from The generalzaton to a three-dmensonal scenaro s straghtforward but s not explored n ths paper.

3 5 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. NO. JUNE 3 T Δt j... T Δt j Target Reference node Reference node j Fg.. TDOA measurement at the target node for sgnals from two reference nodes and j. the reference nodes. Suppose that the synchronzed reference nodes send ther sgnals at the tme nstant T k see Fg.. The TOA measurement for the sgnal transmtted from reference node at the target node for the kth measurement can be wrtten 3 as [] [] t k = θ + w T k + da x +ñ k c =...N k =... c s the speed of propagaton da x s the Eucldan dstance between reference node and the pont x ñ k s the TOA estmaton error at the target node for the sgnal transmtted from the th reference node at tme T kand s the number of TOA measurements messages for every lnk between a reference node and the target node collected n the target node. The estmaton error s often modeled by a zero-mean Gaussan random varable wth varance σ /c ;.e. ñ k Nσ /c [] [5]. In addton t s assumed that E{ñ l ñm j } =for j or l m. Note that we assume that θ and w are fxed unknown parameters for k =... The precedng measurement model ndcates that n order to obtan an estmate of the dstance between the target node and a reference node parameters θ w andt k as nusance parameters should be estmated as well. For nstance the measurements n can be collected by the target node to derve an optmal estmator for estmatng the unknown parameters ncludng the nusance parameters whch makes the problem qute complex and challengng. One way to get rd of some of the unknown parameters s to subtract TOA measurements of the sgnals sent from reference nodes and j and form a TDOA measurement as follows: Δt k j = t k t k da x j = w da j x +ñ k ñ k j c c j =...N. 3 As observed from 3 the clock offset θ and T k have no effect on TDOA measurements snce they cancel out n the TDOA calculaton. The clock skew however stll affects the TDOA measurements and t should be consdered when 3 If tme stampng s performed n the MAC layer a model ncludng fxed and random delays wth no measurement nose can be consdered. Such a model has been extensvely studed n the synchronsaton lterature e.g. n [] and references theren. estmatng the target node poston. Throughout ths paper we assume that the TDOA measurements are computed by subtractng all the TOA measurements except the frst one from the frst TOA. Consequently the range-dfference-ofarrval RDOA measurements are obtaned as z k = cδt k = wd + n k n k =...N k =... n k = c ñk and d = da x da x. Defne the vector of measurements z as z = [ z T T ] zt R N 5 z k = [ z k... T N] zk R N. In order to fnd the poston of the target node based on the measurements n 5 one needs to estmate the clock skew w as well. III. MAXIMUM LIELIHOOD ESTIMATOR AND THEORETICAL LIMITS In ths secton we frst derve the MLE for the postonng problem based on the measurements n. In the sequel we obtan a theoretcal lower bound on the varance of any unbased estmator. Note that the estmator obtaned n ths secton s optmal for the new set of measurements n 5 and not necessarly optmal for the orgnal TOA measurements n. A. Maxmum Lkelhood Estmator MLE To fnd the MLE we need to solve the followng optmzaton problem [3 Ch. 7]: [ˆx T ŵ] = arg max p Z z; xw [x T w] R 3 7 p Z z; xw s the probablty densty functon of vector z whch s ndexed by parameters x and w. Snce the TOA errors are Gaussan random varables z n 5 s modeled as a Gaussan random vector.e. z N μ C wth mean μ k and covarance matrx C beng computed as follows: μ = μ R N C =blkdag C...C R N N tmes μ = w [d... d N ] T C = dagσ...σ N +σ N T N. 9 Therefore consderng the model n the MLE formulaton can be expressed as [ˆx T [ ] T [ ] ŵ]= argmn z μ C z μ. [x T w] R 3 Usng Woodbury s dentty [] whch s a specal case of the matrx nverson lemma one can wrte C = dag σ...σ N s dag σ...σ N N T N dag σ...σ N.

4 GHOLAMI et al.: TDOA BASED POSITIONING IN THE PRESENCE OF UNNOWN CLOC SEW 55 s / N = σ. Then the MLE can be obtaned as [ˆx T ŵ]=argmn [x T w] R 3 k= = s N z k wd σ N z k wd zj k wd j j= σ σ j. As observed from the MLE problem s hghly nonconvex and therefore s dffcult to solve. To obtan the soluton of ths problem a grd search approach or an teratve search e.g. gradent-based approach ntalzed close to the target poston and close to the clock skew can be used. A grd search method has some drawbacks such as complexty. Moreover fndng a good ntal pont n the postonng problem s often a challengng task []. In Secton IV we derve suboptmal estmators to fnd good ntal ponts. Before the detaled dscussons on these suboptmal estmators n Secton IV the CRLBs are obtaned n the followng subsecton n order to provde performance benchmarks. B. Cramér-Rao Lower Bound CRLB Consderng the measurement vector n 5 wth mean μ and covarance matrx C as n the elements of the Fsher nformaton matrx can be computed as [3 Ch. 3] [ μ J nm =[J] nm = ψ n ψ n = ] T C [ μ ψ m { x n f n = w f n =3. From 9 μ / ψ n can be obtaned as follows: [ ] μ ψ n μ ψ n = [ μ =... ψ n ] nm = 3 3 ] T μ N n = 3 ψ n 5 xn an da x f n = { xn a w +n da +x d + f n =3. After some calculatons the entres of the Fsher nformaton matrx can be computed as follows: J = w J = w = N = N = I si Ī I si Ī N d N J 33 = s σ J = J = w J 3 = J 3 = w J 3 = J 3 = w N = N = j= d d j σ σ j I I si Ī I d sσ Ī d N I d sσ Ī d = I n = xn a n σ da x x n a n da x N Ī n = σl σ l= xn a ln da l x x n a n da x 7. The CRLB whch s a lower bound on the varance of any unbased estmator s gven as Var ˆφ [J ]. 9 Then the lower bounds on the error varances for any unbased estmates of the poston and the clock skew can be computed as usng the nverse of a 3 3 square matrx [] E{ ˆx x } J 33 J + J J3 + J3 J 33 J J J +J 3J 3 J J J3 J J3 a E{ ŵ w } J J J J 33 J J J +J 3J 3 J J J3 J J3. b In the rest of ths secton we derve an alternatve CRLB for poston estmaton when an estmate of the clock skew s avalable. To that am we model the clock skew estmate as a Gaussan random varable ŵ = w + ξ w ξ w s the error n the clock skew estmaton that s modeled as a zero-mean Gaussan random varable ξ w Nσ w and rewrte as z k =ŵd ξ w d + n k nk =...N. We assume that ξ w and n k are ndependent. We collect all the measurements when an estmate of the clock skew s avalable as follows: z = [ z T z T ] T R N

5 5 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. NO. JUNE 3 z k = [ z k... zk N] T R N. 3 Consderng that the vector z s a Gaussan random vector z N μ C wth and μ = ŵ [d...d N ] T C =blkdag C... C tmes C =dagσ...σ N +σ N T N + σ w [d... d N ] T [d... d N ] 5 the entres of the Fsher nformaton matrx s obtaned as [3 Ch. 3] [ ] T [ ] μ J nm =[ J] nm = C μ x n x m + tr C C C C x m x n n = m =. Then the CRLB for the poston estmate s gven by E{ ˆx x } J + J J J J 7 Ths CRLB expresson wll be useful for provdng theoretcal lmts on the performance of poston estmators that are based on already avalable estmates of the clock skew parameter. In addton for σ w =.e. no estmaton errors the CRLB expresson covers the specal case n whch the clock skew parameter s perfectly known. IV. SUBOPTIMAL ESTIMATORS To solve the MLE formulated n usng an teratve algorthm we need a sutable ntal pont that s suffcently close to the optmal soluton. In ths secton we propose two suboptmal estmators that provde such ntal ponts. In partcular we consder a two step estmaton procedure: coarse and fne. For the coarse estmaton step we derve two suboptmal estmators based on semdefnte programmng SDP relaxaton and lnear least squares LLS. In the fne estmaton step we derve a lnear model and employ a technque based on the regularzed least squares crtreron. A. Coarse estmate We frst express the clock skew parameter as w =+δ δ s a small value. Dvdng both sdes of 3 by w and usng the approxmaton /+δ δ we can approxmate the RDOA measurement n as z k δ d + δn k nk =...N k =... Ths s a reasonable model snce the devaton of the clock skew parameter from the deal value of w =s not sgnfcant for most practcal clocks. whch can be further smplfed for the purpose of obtanng the approxmate MLE n Secton IV-A n whch the covarance matrx s ndependent of the unknown parameter δ as z k δ d +n k nk =...N k =... 9 It s noted that keepng δ n for the SDP formulaton n the next secton complcates the problem. In fact the covarance matrx of measurement nose wll be dependent on the unknown parameter δ and therefore t s dffcult to convert the correspondng MLE to an SDP problem. However for the LLS formulaton we apply a nonlnear processng on measurements n that makes the measurement nose be dependent on both δ and unknown dstance. Hence neglectng δ does not change the complexty of the problem consderably. As explaned later n the LLS approach we frst neglect the effect of the unknown parameters on the covarance matrx of the measurement nose and fnd a frst estmate of the unknown parameters. We then use the frst estmate to approxmate the covarance matrx. Semdefnte Programmng: In ths secton we frst apply the maxmum lkelhood crteron to the model n 9 and then change t to an SDP problem. The MLE for the model n 9 can be obtaned as [ˆx T ˆδ] = arg mn [x T δ] R 3 k= z k δ Pd T C z k δ Pd 3 z k s as n and matrx P and vector d are gven by P = a... d =[da x da x... da N x] T. 3b To solve 3 we use an alternatve projecton approach. That s we frst optmze the MLE objectve functon wth respect to the unknown parameter δ. Takng the dervatve of the objectve functon n 3 wth respect to δ and equatng to zero yeld the followng expresson for δ: δ = g T d 3 g = k= PT z k C k= zt k C z k. 33 In the next step of the alternatve projecton approach we nsert the expresson n 3 nto the MLE cost functon n 3. We also note that δ s small and then mpose a constrant an upper bound on ts absolute value.e δ δ max δ max s a reasonable upper bound on δ. Aftersome manpulaton we can express the MLE for the poston as mnmze d T Qd x R subject to g T d δ max 3

6 GHOLAMI et al.: TDOA BASED POSITIONING IN THE PRESENCE OF UNNOWN CLOC SEW 57 Q s gven by T k= Q = z zt k C P k P C k= zt k C z k k= k= z zt k C P k P. 35 k= zt k C z k The optmzaton problem n 3 s nonconvex and dffcult to solve. In order to obtan a convex problem we express d T Qd as d T Qd =trqv V = dd T andrelax the nonconvex constrant V = dd T as follows. Recallng that v j =[V] j = a x a j x we can represent the dagonal entres of V as [ ] v = a x I a = tr a T a Z 3 Z = [ x T ] T [ x T ].e. Z s a rank- postve semdfnte matrx. In addton usng Cauchy-Schwarz nequalty we can express v j j as [ ] v j = a x a j x tr I a Z a T j a T a. j 37 Hence the problem n 3 can be wrtten as: mnmze d T Qd z S 3 ;d R N + ;V SN [ ] I a subject to tr a T j a T a Z v j j j [ ] I a tr a T j a T a Z v j j j [ ] I a tr a T a Z = v g T d +δ max g T d δ max V = dd T Z rankz = [Z] 33 = j =...N. 3 The nonconvex problem n 3 can be changed to a convex problem by droppng the rank- constrant rankz =and relaxng the nonconvex constrant V = dd T to a convex one.e. V dd T. Then the convex optmzaton problem called SDP can be cast as mnmze d T Qd z S 3 ;d R N + ;V SN [ I a subject to tr a T j a T a j [ I a tr a T j a T a j [ I a tr a T a g T d +δ max ] Z v j j ] Z v j j ] Z = v g T d δ max [ ] V d d T Z [Z] 33 = j =...N. 39 Note that the constrant V dd T s expressed as a lnear matrx nequalty usng the Schur complement [5]. If the optmal soluton of 39.e. Ẑ has rank- property and V = dd T then the optmal soluton s at hand. Otherwse we can apply a rank- approxmaton technque to mprove the poston estmate []. Lnear least squares LLS: In ths secton we derve a lnear estmator to estmate the poston of the target node based on a nonlnear processng technque. We frst translate the network such that the frst reference node les at the orgn. In partcular we defne a = a a for =...N and t = x a. Then we move da x n rememberng that d = da x da x to the rght-hand-sde n the translated coordnates as z k δ+ t da t+ δnk nk =...N. Assume that the nose s small compared to the dstance da t. Then squarng both sdes of and droppng the small term we get z k δ +z k δ t + t a a T t + t +da t δnk nk =...N. Assumng small δ we can wrte δ δ. Hence we obtan a lnear model based on unknown vector θ =[t T t δ] T as z k =gk T θ + ξ k g k =[ a T z k zk ] T z k =zk a and ξ k =da t δnk nk +δ t z k. Followng the procedure explaned above for all measurements we obtan a lnear model n the matrx form as h = Gθ + ξ 3 matrx G and vectors h and ξ are computed as G = [ g... g... g... ] T g N R N 3 h =[ z... z N... z... z N ] T R N ξ =[ξ... ξn... ξ... ξn ] T R N. Note that the nose vector ξ s a random vector wth a nonzero mean. In fact Eξ =δ t μ μ s gven n. The covarance matrx of ξ can be computed as C ξ =blkdag D...D tmes C blkdag D...D tmes 5 D =dag da t δ+δ t...da N t δ + δ t. Usng the least squares crteron a soluton to 3 s obtaned as [3] ˆθ =G T C ξ G G T C ξ h Eξ. 7 Note that the mean vector Eξ and the nverse of the covarance matrx C ξ are unknown n advance snce they

7 5 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. NO. JUNE 3 are dependent on the unknown parameters. We frst assgn a zero vector and an dentty matrx to the mean vector and the covarance matrx respectvely and fnd an estmate of the unknown parameters. Then we approxmate the mean vector and the covarance matrx and recalculate the estmate gven n 7. The covarance matrx of the estmate n 7 s gven by [3] Cˆθ =G T C ξ G. Remark : In cases that the observaton matrx G s llcondtoned we can use a regularzaton technque n 7 to obtan a soluton to the lnear model n 3. When the regularzaton parameter apples to the last component of the unknown vector θ t has a nce nterpretaton. That s the devaton of the clock skew from the deal clock s extremely small. Remark : The procedure for approxmatng the covarance matrx and mean vector of ξ can be terated for several tmes. However n practce one round of updatng s enough to acheve good performance. We can further mprove the accuracy of the estmate n 7 by takng the relaton between the elements of the estmate vector ˆθ nto account. Each element of 7 can be wrtten as [ˆθ] = t + χ [ˆθ] = t + χ [ˆθ] 3 = t + χ 3 9 χ =[χ χ χ 3 ] T denotes the estmaton error.e. χ = ˆθ θ andt =[t t ] T. Suppose that the estmaton errors are consderably small. Therefore squarng both sdes of the elements n 9 yelds [ˆθ] t +t χ [ˆθ] t +t χ [ˆθ] 3 t + t χ 3. 5 Hence the relaton between the estmated elements n 7 can be obtaned usng 5 as u = Bφ + ν 5 ν =[t χ t χ t χ 3 ] T [ u = [ˆθ] [ˆθ] [ˆθ] T 3] φ = [ t T ] t B =. 5 Then the least squares soluton to 5 s obtaned as ˆφ =B T C ν B C ν B T u 53 the covarance matrx of C ν can be computed as C ν =dagt t t [ ] Cˆθ dagt :3:3 t t. 5 To compute the covarance matrx C ν we use the estmate gven n 7 nstead of the true values of t t and t whch are unknown a-pror. Based on the precedng calculatons the target poston can be obtaned as follows: x j =sgn[θ] j [ˆφ]j + aj j = 55 the sgnum functon sgnx s defned as { f x sgnx = 5 f x<. Note that usng a smlar approach as employed n [9] [7] we can compute the covarance of the estmate n 55. B. Fne estmate The approaches consdered n the coarse estmaton step provde good ntal ponts for further refnng the poston estmates. One method s to mplement the MLE usng an teratve search approach ntalzed wth the estmate n the coarse estmaton step. In ths secton we propose another approach wth lower complexty. To that end we frst update the estmate of the clock skew. Assumng an estmate of the locaton x x = ˆx for ˆx gven by the SDP soluton n 39 or x = x for x provded by the LLS n 55 an estmate of the clock skew can be obtaned from usng the method of moments [3] as N k= = ŵ = zk N d 57 = d = d d and d = x a. Now consderng an estmate of the clock skew n 57 and applyng the frst order Taylor seres expanson about x to we get the followng expresson: z k ŵ d + ḡ T Δx + n k n k 5 ḡ =ŵ x a / d ŵ x a / d andδx = x x. Thus we arrve at the followng lnear model to estmate the estmaton error Δx: t = ḠΔx + ϑ 59 ϑ =[n n... n N n...n n...n N n ]T t =[z ŵ d... zn ŵ d N... z ŵ d... zn ŵ d N ] T G = I [ḡ T... ḡn T ] T. The assumpton n dervng the model n 59 requres that the estmaton error Δx be small enough. We take ths assumpton nto account and apply a regularzed least squares Tkhonov regularzaton technque to fnd an estmate of Δx as [9] [] [9] ˆΔx =ḠT C Ḡ + λi Ḡ T C t λ defnes a trade-off between Δx and ḠΔx t T C ḠΔx t. Fnally the updated estmate s obtaned as ˆ x = x + ˆΔx.

8 GHOLAMI et al.: TDOA BASED POSITIONING IN THE PRESENCE OF UNNOWN CLOC SEW 59 C. Complexty analyss In ths secton we evaluate the complexty of the estmators consdered n ths study based on the total number of the floatng-pont operatons or flops. We assume that an addton subtracton and multplcatonn operton n the real doman can be computed by one flop [] and that a dvson or square root operaton needs r flops usually to 3 flops [3]. We calculate the total number of flops for every method and express t as a polynomal of the free parameters. Then we compute the complexty as the order of growth for each approach. To smplfy the results we keep only the leadng terms of the complexty expressons. The maxmum lkelhood estmator: As prevously mentoned the MLE s nonlnear and nonconvex. Therefore the complexty of the MLE hghly depends on the soluton method. In addton the complexty of each method also depends on a number of parameters e.g. the number of teratons the ntal pont and the soluton accuracy. Here we compute the cost of evaluatng the objectve functon of the MLE n for a certan pont and also the cost of the Gauss-Newton GN approach to solve the MLE when a good ntal pont s avalable. We note that we need r +5 flops to compute a dstance. The number of flops requred to evaluate the objectve functon of the MLE s approxmately gven by N N ++r +N + r. Then consderng the leadng term the complexty of evaluatng the MLE objectve functon s expressed as ON. It can be shown that the complexty of every Newton step s on the order of N 3. Then the total cost of the GN approach for solvng the MLE s OI GN N 3 I GN s the number of teratons n the GN method to converge to the soluton. The semdefnte programmng: The worst-case complexty of the SDP n 39 s gven by OI SDP N c L s + Ls 3 +L3 log/ɛ [3] L s the number of equalty constrants s s the dmenson of the th semdefnte cone N c s the number of semdefnte constrants I SDP s the number of teratons and ɛ s the accuracy of the SDP soluton. Therefore the complexty of the SDP formulated n 39 s gven as SDP cost O I SDP N + N N N + +N + 3 log/ɛ. 3 Then the number of teratons I SDP I SDP ON / [3]. s approxmated as 3 The lnear least squares: To compute the complexty of the LLS we note that the matrx C ξ s a block dagonal matrx. In addton snce the matrx C n s fxed and dagonal the nverse of C can be computed once and used later. Then the complexty of the lnear estmator n 7 can be computed as Flops of LLS n 7 N 3 + r + + r +5 cost of computng h Eξ +N r +7+r +5++ N r + cost of computng D cost of computng C ξ +N cost of G T C ξ N cost of G T C ξ G cost of G T C ξ G G T C + 3N cost of C ξ Gh Eξ ξ h Eξ It can easly be verfed that the complexty of the correcton technque compared to the LLS n 7 s neglgble. Then the complexty of the the lnear estmator for large and N can be computed as ON + rn. Fne estmaton step: In a smlar way the complexty of the fne estmaton step can be computed as Fnestep N5 + r + N ++r cost of computng d cost of computng ŵ + +N +N +Nr + cost of computng t cost of computng ḠT C + N ++ + N cost of ḠT C Ḡ+λI cost of Ḡ T C t +. 5 cost of ḠT C Ḡ+λI ḠT C t Then the complexty of the fne estmaton step can be approxmated as O N. Table I summarzes the complexty of the dfferent approaches for large and N. Note that for small and N the cost for dfferent approaches can be dfferent from the ones n Table I. We have also measured the average runnng tme of dfferent algorthms for a network consstng of reference nodes as consdered n Secton V. The algorthms have been mplemented n Matlab on a MacBook Pro Processor.3 GHz Intel Core 7 Memory GB MHz DDR3. To mplement the MLE we use the Matlab functon named lsqnonln [3] ntalzed wth the true values of the locaton and the clock skew. To mplement the SDP we use the CVX toolbox [33]. We run the algorthms for realzatons of the network and compute the average runnng tme n ms as shown n Table II. It s noted that although the MLE has lower complexty than the SDP we need a good ntal pont for the GN algorthm whch n turn poses further complexty. From Table I and Table II t s observed that the proposed approaches have reasonable complexty especally the lnear estmator. V. SIMULATION RESULTS A. Smulaton Setup Computer smulatons are conducted n order to evaluate the performance of the proposed approaches. Fg. 3 llustrates the

9 53 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. NO. JUNE 3 TABLE I COMPLEXITY OF DIFFERENT APPROACHES. Method Complexty Evaluaton of the MLE objectve functon at a pont ON MLE usng GN true ntalzaton OI GN N 3 SDP OI SDP N + 5 log/ɛ I SDP ON / LLS ON + rn Fne estmaton step O N TABLE II AVERAGE RUNNING TIME OF DIFFERENT APPROACHES. Method Tme ms MLE usng true ntalzaton 7 SDP 55 LLS Fne estmaton step.9 a 7 a a 5 y coordnate [m] a x 5 x x 3 a 5 3 x x a a a x coordnate [m] Perfect knowledge Jont estmate Target node Fg. 3. A -D network deployment used n the smulatons blue squares and red crcles show reference and target nodes respectvely. Fg. 5. Ellpss uncertanty regon for =3 for σ =[m]. postons of the reference and target nodes for a -D network. In the smulatons the clock skew s randomly drawn from [.995.5] and t s assumed that the standard devaton of the nose s the same for all nodes.e. σ = σ. In the smulatons we assume that a reference node sends ts k +th sgnal after other reference nodes complete transmttng the kth sgnal. For smplcty of mplementaton we consder the order of TDOA measurements accordng to 5 and. To evaluate the performance of dfferent approaches we consder the root-mean-squared error RMSE and the cumulatve dstrbuton functon CDF of the poston error. B. CRLB Analyss In ths secton we nvestgate CRLBs on poston estmaton n the presence and absence of clock skew nformaton. Fg. shows the CRLBs for varous target nodes and for dfferent values of. We compare the CRLBs n two scenaros: jont estmaton of the poston and the clock skew parameter.e. unknown clock skew and perfect knowledge of the clock skew parameter.e. perfectly synchronzed clocks. It s observed that for small values of σ the CRLBs are close to each other n both scenaros and the degradaton due to the unknown clock skew ncreases wth the standard devaton of the nose σ. Except for target node two located at the center of the reference nodes addng a new unknown varable as a nusance parameter.e. the clock skew parameter deterorates the accuracy of poston estmates. To vsualze the effects of an unknown clock skew on the poston estmaton accuracy we plot the CRLB as an ellpsod uncertanty n Fg. 5 for σ =m. We scale the coordnates of the target nodes so that the dfference between the two scenaros s clearly vsble. We observe from the fgure that dfferent locatons for the target nodes show dfferent behavors. For target node two two ellpsods concde whle for the other target nodes the volume of the ellpsod for the unknown clock skew parameter scenaro s larger than the one for the perfect synchronzaton scenaro. In the next smulatons we compare the performance of the jont poston and clock skew estmaton wth the poston estmaton when an estmate of the clock skew s avalable. As mentoned n Secton III-B we model the avalable clock skew estmate as a Gaussan random varable that s ŵ N w σ w. The CRLBs for the two cases are plotted n Fg. for target one when =3. It s observed from the fgure that when the standard devaton of the clock skew estmate ncreases the jont estmaton technque outperforms the one that s based on avalable estmates of the clock skew. It s also seen for low SNRs hgh σ s that the jont estmaton

10 GHOLAMI et al.: TDOA BASED POSITIONING IN THE PRESENCE OF UNNOWN CLOC SEW = =3 = Jont estmate 5 Perfect knowledge Jont estmate = =3 5 3 = 3 Perfect knowledge = =3 = Standard devaton of nose σ [m] a Jont estmate Perfect knowledge Standard devaton of nose σ [m] c = =3 = Standard devaton of nose σ [m] b Jont estmate Perfect knowledge Standard devaton of nose σ [m] d Fg.. CRLBs for varous target nodes n two scenaros jont estmate of the target poston and the clock skew and perfect knowledge of the clock skew for a target node one b target node two c target node three and d target node four Jont estmate Partal knowledge σ =[m] σ =9[m] σ =7[m] σ =5[m] σ =3[m] Standard devaton of nose σ w Fg.. CRLBs of target node node one poston estmate when =3 for two scenaros: jont clock skew and poston estmaton partal knowledge of clock skew n the form of a clock skew estmate s avalable. approach has better performance than the other one. From the fgure we can derve thresholds for σ w and σ to specfy when the jont estmaton technque outperforms the other technque. For example for target node one when σ w.5 and σ = 3 meters the jont estmaton approach s superor. C. Performance of Estmators In ths secton we evaluate the performance of the poston estmaton technques developed n Secton III and Secton IV. To solve the MLE n we use Matlab s functon named lsqnonln [3] ntalzed wth the true value of the poston and the clock skew or wth the estmates from the SDP or LLS. To solve the SDP n 39 we employ the CVX toolbox [33]. The upper bound δ max n the SDP and the regularzaton parameter λ are set to. and. respectvely. Fg. 7 shows the RMSEs of dfferent approaches versus the standard devaton of the measurement nose. It s observed that the proposed estmators n the coarse estmaton step can provde good ntal ponts such that the MLE ntalzed wth the coarse estmaton step estmate the SDP or LLS estmate attans the CRLB. The fgure also shows that the fne estmaton step sgnfcantly mproves the accuracy of the coarse estmate for both the SDP and LLS. In some scenaros the SDP approach outperforms the LLS and n other scenaros the LLS has better performance compared to the SDP. For nstance the LLS approach for target one n Fg. 7a and Fg. 7b sgnfcantly outperforms the SDP and the LLS followed by the fne step estmaton s very close to the CRLB. Note that the performance of the estmators can be mproved by ncreasng the number of messages. From the fgure t s observed that the behavor

11 53 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. NO. JUNE 3 SDP SDP SDP+Fne SDP+Fne LLS LLS+Fne LLS LLS+Fne LLS+MLE LLS+MLE SDP+MLE MLEtrue ntalzaton CRLB SDP+MLE MLEtrue ntalzaton CRLB LLS LLS+Fne SDP SDP+Fne LLS+MLE SDP+MLE 7. Standard devaton of nose σ [m] MLEtrue ntalzaton CRLB a LLS SDP LLS+Fne SDP+Fne LLS+MLE SDP+MLE 5.9 Standard devaton of nose σ [m] MLEtrue ntalzaton CRLB b Standard devaton of nose σ [m] c Standard devaton of nose σ [m] d Fg. 7. RMSEs of varous approaches for a target node one and = b target node one and =3 c target node three and = and d target node three and =3. Note the dfferent scales on the vertcal axs of mprovement can vary for dfferent estmators. In fact snce we have derved the suboptmal estmators from the measurements usng two dfferent approaches the relaton between parameter and the performance of the estmators can be dfferent. It s also observed that the performance of the estmaton depends on the geometry of the network. The effect of the geometry can be studed through the so-called geometrc dluton of precson e.g. see [] whch relates the poston estmaton error to the geometry of the network and the measurement errors. Fnally n Fg. we plot the CDF of the poston errors defned as ˆx x ˆx s an estmate of the target poston. For ths fgure we set = 3 and σ =[m]. From the fgure we observe that the fne estmate consderably mproves the coarse estmate most of the tme and ts performance s close to the MLE. VI. CONCLUDING REMARS In ths paper we have studed the problem of selfpostonng a sngle target node based on TDOA measurements when the local clock of the target node s unsynchronzed. Although TDOA-based postonng s not senstve to a clock offset t suffers from another clock mperfecton parameter namely the clock skew. To address ths problem we have consdered a jont poston and clock skew estmaton technque and derved the MLE for ths problem. Snce the MLE s hghly nonconvex we have studed two suboptmal estmators that can be effcently solved. Usng relaxaton and approxmaton technques we have derved two estmators based on semdefnte programmng SDP and lnear least squares LLS approxmaton. To further refne the estmates we have lnearzed the measurements usng the frst order Taylor seres around the SDP or LLS estmate to derve a lnear model n whch the estmaton error can be approxmated. To compare dfferent approaches we have derved the CRLBs for the problem when ether no knowledge or partal knowledge of the clock skew s avalable. The smulaton results show that the proposed technques can attan the CRLB for suffcently hgh sgnal-to-nose ratos. REFERENCES [] G. Mao and B. Fdan Localzaton Algorthms and Strateges for Wreless Sensor Networks. Informaton Scence Reference 9. [] Z. Sahnoglu S. Gezc and I. Guvenc Ultra-Wdeband Postonng Systems: Theoretcal Lmts Rangng Algorthms and Protocols. Cambrdge Unversty Press. [3] N. Patwar A. O. Hero M. Perkns N. S. Correal and R. J. O Dea Relatve locaton estmaton n wreless sensor networks IEEE Trans. Sgnal Process. vol. 5 no. pp. 37 Aug. 3. [] N. Patwar J. Ash S. yperountas A. O. Hero and N. C. Correal Locatng the nodes: cooperatve localzaton n wreless sensor network IEEE Sgnal Process. Mag. vol. no. pp. 5 9 July 5. [5] S. Gezc A survey on wreless poston estmaton Wreless Personal Commun. vol. no. 3 pp. 3 Feb..

12 GHOLAMI et al.: TDOA BASED POSITIONING IN THE PRESENCE OF UNNOWN CLOC SEW 533 LLS.9..7 SDP LLS SDP+Fne LLS+Fne MLEtrue ntalzaton CRLB.9..7 SDP LLS+Fne SDP+Fne MLEtrue ntalzaton CRLB.. CDF.5 CDF Poston error ˆx x [m] a Poston error ˆx x [m] b Fg.. CDFs of the poston error for σ =mand =3for a target node one and b target node three. [] M. R. Gholam E. G. Ström and M. Rydström Indoor sensor node postonng usng UWB range measurments n Proc. 9 European Sgnal Process. Conf. pp [7] Y. Wang X. Ma and G. Leus Robust tme-based localzaton for asynchronous networks IEEE Trans. Sgnal Process. vol. 59 no. 9 pp. 397 Sept.. [] M. R. Gholam S. Gezc and E. G. Ström A concaveconvex procedure for TDOA based postonng IEEE Commun. Lett. 3. Avalable: fulltext/99/local 99.pdf [9] N. M. Frers S. R. Graham and P. R. umar Fundamental lmts on synchronzng clocks over networks IEEE Trans. Autom. Control vol. 5 no. pp June. [] Y.-C. Wu Q. Chaudhar and E. Serpedn Clock synchronzaton of wreless sensor networks IEEE Sgnal Process. Mag. vol. no. pp. 3 Jan.. [] Q. M. Chaudhar E. Serpedn and. Qaraqe Some mproved and generalzed estmaton schemes for clock synchronzaton of lstenng nodes n wreless sensor networks IEEE Trans. Commun. vol. 5 no. pp. 3 7 Jan.. [] E. Serpedn and Q. M. Chaudhar Synchronzaton n Wreless Sensor Networks: Parameter Estmaton Performance Benchmarks and Protocols. Cambrdge Unversty Press 9. [3] Q. Chaudhar E. Serpedn and J. m Energy-effcent estmaton of clock offset for nactve nodes n wreless sensor network IEEE Trans. Inf. Theory vol. 5 no. pp Jan.. [] J. Zheng and Y. Wu Jont tme synchronzaton and localzaton of an unknown node n wreless sensor networks IEEE Trans. Sgnal Process. vol. 5 no. 3 pp Mar.. [5] Z. Sahnoglu and S. Gezc Rangng n the IEEE.5.a standard n IEEE Wreless Mcrowave Technol. Conf. [] H. Fan and C. Yan Asynchronous dfferental TDOA for sensor selflocalzaton n Proc. 7 IEEE Internatonal Conf. Acoustcs Speech Sgnal Process. vol. pp. 9. [7] C. Yan and H. Fan Asynchronous dfferental TDOA for non-gps navgaton usng sgnals of opportunty n Proc. IEEE Internatonal Conf. Acoustcs Speech Sgnal Process. pp [] Asynchronous self-localzaton of sensor networks wth large clock drft n Proc. 7 Internatonal Conf. Moble Ubqutous Syst.: Netw. Servces pp.. [9] M. R. Gholam S. Gezc and E. G. Ström Improved poston estmaton usng hybrd TW-TOA and TDOA n cooperatve networks IEEE Trans. Sgnal Process. vol. no. 7 pp July. [] I. Sar E. Serpedn. Noh Q. Chaudhar and B. Suter On the jont synchronzaton of clock offset and skew n RBS-protocol IEEE Trans. Commun. vol. 5 no. 5 pp May. [] M. R. Gholam Postonng algorthms for wreless sensor networks Lcentate Thess Chalmers Unversty of Technology Mar.. Avalable: [] Z. Sahnoglu Improvng range accuracy of IEEE.5.a rados n the presence of clock frequency offsets IEEE Commun. Lett. vol. 5 no. pp. Feb.. [3] S. M. ay Fundamentals of Statstcal Sgnal Processng: Estmaton Theory. Prentce-Hall 993. [] D. S. Bernsten Matrx Mathematcs: Theory Facts and Formulas nd edton. Prnceton Unversty Press 9. [5] L. Vandenberghe and S. Boyd Semdefnte programmng SIAM Rev. vol. 3 no. pp Mar. 99. [] Z.-Q. Luo W.-. Ma A. So Y. Ye and S. Zhang Semdefnte relaxaton of quadratc optmzaton problems IEEE Sgnal Process. Mag. vol. 7 no. 3 pp. 3 May. [7] M. R. Gholam S. Gezc E. G. Ström and M. Rydström Postonng algorthms for cooperatve networks n the presence of an unknown turn-around tme n Proc. IEEE Internatonal Workshop Sgnal Process. Advances Wreless Commun. pp. 7. [] S. Boyd and L. Vandenberghe Convex Optmzaton. Cambrdge Unversty Press. [9] M. R. Gholam S. Gezc E. G. Ström and M. Rydström Hybrd TW- TOA/TDOA postonng algorthms for cooperatve wreless networks n Proc. IEEE Internatonal Conf. Commun. [3] C. W. Ueberhuber Numercal Computaton. Sprnger-Verlag 997. [3] G. Wang and. Yang A new approach to sensor node localzaton usng RSS measurements n wreless sensor networks IEEE Trans. Wreless Commun. vol. no. 5 pp May. [3] The Mathworks Inc.. Avalable: [33] M. Grant and S. Boyd CVX: Matlab software for dscplned convex programmng verson. Feb.. Avalable: Mohammad Reza Gholam S 9 receved the M.S. degree n communcaton engneerng from Unversty of Tehran Iran n. From to he worked as a communcaton systems engneer n ndustry on desgnng recevers for dfferent dgtal transmsson systems. Currently he s workng toward the Ph.D. degree at Chalmers Unversty of Technology. Hs man nterests are statstcal nference convex optmzaton technques for sgnal processng and dgtal communcatons cooperatve and noncooperatve postonng approaches and network wde synchronzaton technques for wreless sensor networks. Snan Gezc S 3 M SM receved the B.S. degree from Blkent Unversty Turkey n and the Ph.D. degree n Electrcal Engneerng from Prnceton Unversty n. From to 7 he worked at Mtsubsh Electrc Research Laboratores Cambrdge MA. Snce 7 he has been wth the Department of Electrcal and Electroncs Engneerng at Blkent Unversty he s currently an Assocate Professor. Dr. Gezc s research nterests are n the areas of detecton and estmaton theory wreless communcatons and localzaton systems. Among hs publcatons n these areas s the book Ultra-wdeband Postonng Systems: Theoretcal Lmts Rangng Algorthms and Protocols Cambrdge Unversty Press. Dr. Gezc s an assocate edtor for IEEE Transactons on Communcatons IEEE Wreless Communcatons Letters and Journal of Communcatons and Networks..

13 53 IEEE TRANSACTIONS ON COMMUNICATIONS VOL. NO. JUNE 3 Erk G. Ström S 93 M 95 SM receved the M.S. degree from the Royal Insttute of Technology TH Stockholm Sweden n 99 and the Ph.D. degree from the Unversty of Florda Ganesvlle n 99 both n electrcal engneerng. He accepted a postdoctoral poston at the Department of Sgnals Sensors and Systems at TH n 995. In February 99 he was apponted Assstant Professor at TH and n June 99 he joned Chalmers Unversty of Technology Göteborg Sweden he s now a Professor n Communcaton Systems snce June 3. Dr. Ström currently heads the Dvson for Communcatons Systems Informaton Theory and Antennas at the Department of Sgnals and Systems at Chalmers and leads the competence area Sensors and Communcatons at the traffc safety center SAFER whch s hosted by Chalmers. Hs research nterests nclude sgnal processng and communcaton theory n general and constellaton labelngs channel estmaton synchronzaton multple access medum acccess multuser detecton wreless postonng and vehcular communcatons n partcular. Snce 99 he has acted as a consultant for the Educatonal Group for Indvdual Development Stockholm Sweden. He s a contrbutng author and assocate edtor for Roy. Admralty Publshers FesGas-seres and was a co-guest edtor for the PROCEEDINGS OF THE IEEE specal ssue on Vehcular Communcatons and the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS specal ssues on Sgnal Synchronzaton n Dgtal Transmsson Systems and on Multuser Detecton for Advanced Communcaton Systems and Networks. Dr. Ström was a member of the board of the IEEE VT/COM Swedsh Chapter. He receved the Chalmers Pedagogcal Prze n 99 and the Chalmers Ph.D. Supervsor of the Year award n 9.