On the Accuracy of Passive Hyperbolic Localization in the Presence of Clock Drift

Transcription

1 On the Auray of Passive Hyperboli Loalization in the Presene of Clok Drift Saeed Shoaee, Johannes Shmitz, Rudolf Mathar RWTH Aahen University Aahen, Germany {shoaee, shmitz, Sivan Toledo Tel-Aviv University Tel-Aviv 69978, Israel Abstrat We disuss reeiver lok orretion and assoiated performane bounds for passive emitter loalization using TDOA measurements from asynhronous sensor networks. In the onsidered system, passive reeiving sensors are augmented with beaons at known loations that perform approximately periodial transmissions, used for alibration of the system synhronization. The preise transmission times as well as the transmission interval of the beaon messages are unknown. Similarly the transmissions of the target are irregular and do not, in general, our simultaneous with the beaon transmissions. The loks of eah sensor are desribed by a linear error model. Based on that, we ompare different snapshot based approahes for lok orretion and derive theoretial limits for the loalization of the target based on the modified Cramér- Rao lower bound MCRLB). Simulation results are presented to illustrate the theoretial findings and show that our proposed estimators perform lose to the MCRLB. Additional experimental results verify the analysis and the approah in a realisti large sale senario. Index Terms TDOA, Loalization, Clok drift, Beaon, Cramér-Rao bound I. INTRODUCTION Time measurement based loalization systems either rely on very aurate synhronization between the anhor points, or, in asynhronous systems, require aurate estimation of the lok offsets to ahieve optimum performane. Network infrastruture assisted passive loalization has been in the enter of various researh ativities for several deades. In omparison, results on the operation of suh systems with asynhronous, drifting loks and wireless synhronization are rather young. The most ommon approah to loalization is divided in two stages, where first a physial signal parameter is estimated, in our ase the time of arrival TOA), and seond the target loation is obtained using a seond algorithm that takes the measurements, TOAs or the resulting time differenes of arrival TDOAs) as an input, e.g., [] [4]. Due to hardware imperfetions and physial onstraints, impairments of the first stage are not unommon in pratie. The requirement of nanoseond auray for deimeter level loalization is hallenging even for the most aurate loks. For optimum loalization performane, it is therefore neessary to apply orretion methods to ope with the effets of asynhronous loks. Aordingly, in [5] the idea of differential TDOA DTDOA) has been introdued in order to orret for lok offsets. Further, in [6] the use of a alibration emitter, also known as beaon, has been proposed. However, the authors fous on inaurate knowledge of sensor loations rather than asynhronous loks. Later, [7] proposed a protool for passive loalization with asynhronous loks where one of the anhors atively transmits a referene signal. The problem has also been investigated in a downlink fashion [8], where the transmitting anhors loks are assumed to be perfet, but the reeiving target possesses a drifting lok. An extension of this senario to ooperative loalization is given in [9]. Most of the publiations in this area apply the Cramér-Rao lower bound CRLB) to derive theoretial performane limits. As desribed in [9], for problems with many interlinked parameters that all need to be estimated, the modified CRLB MCRLB) often is more pratial and provides more insight into the problem. It has been introdued in [0] and further thoroughly examined in [] and [2]. In the present paper, the methodology of the modified bound is followed, as it onstitutes a feasible way of deriving theoretial performane limits. In ontrast to the previous works, we onsider a different system model, where the investigated system onsists of distributed passive sensors, i.e., they an only reeive, and additional beaons, that an only transmit. The beaons are not olloated with the sensors, sensor and beaon loations are known and the sensor loks are running asynhronously and experiene drift over time. Reently, in [3] a similar system has been onsidered. However, this paper does not present any theoretial bounds and assumes a very preisely timed beaon signal, where the exat time differene between two beaon transmissions is known, whih we do not have in our ase. Another implementation of suh a system for wildlife traking has been desribed in [4]. Previously, the sensors have been synhronized with GPS and therefore the lok drift has been negleted in pratie, while for the present paper we onsider the same system running with asynhronous, drifting loks. For numerial evaluation we then obtain reorded TOAs from the fusion enter of the system under this deteriorated onditions. Due to stringent onstraints in battery size of the target and beaons, the rate of transmissions in the system is stritly limited. Further, target and beaon transmissions are not synhronized and therefore the time differene between the reeived messages an lead to large differenes in the loal loks, during one set of measurements. Hene, we

2 derive the neessary lok orretion algorithms and evaluate their performane using simulations and a omparison with the MCRLB. For validation, we then apply the orretion algorithms to the reorded TOAs. The remainder of the paper is strutured as follows. In Se. II the model for the loalization system and the erroneous loks is defined. Subsequently, in Se. III estimators for the orretion are disussed. Then in Se. IV the bounds for different lok orretion approahes are derived. Finally, Se. V provides simulation and experimental results to demonstrate and verify the performane of the disussed approahes. A short onlusion of the work is given in Se. VI. II. SYSTEM AND CLOCK MODEL We onsider a system of M distributed reeiving sensors that are loalizing a moving transmitter. The system is onsidered to be passive and has no ommuniation between the sensors and the transmitting target. Additionally, two transmitting beaons at known loations are available for synhronization. We assume that eah sensor i is independently able to estimate the TOA t i,k of the reeived signal from either the target k = 0 or the two beaons k {, 2}, based on a pilot sequene. The loation an then be determined using the TDOAs τ i,k = t i,k t,k between reeiver i and the referene reeiver. In the following we apply [3] to find the solution and exlusively fous on the problem of orretly determining the TOAs and resulting TDOAs respetively. For that, the time offset aused by the drifting loks at reeiver i is modeled as [5] t i = ɛ i t + φ i, ) where t is the real time, φ i a onstant initial offset at time instane t 0) = 0 and ɛ i the drift rate, whih is assumed to be onstant over the ourse of a measurement period and Gaussian distributed with respet to different loks. The mean of ɛ i is and its standard deviation σ ɛi depends on the speifi lok type, it an reah as low as 0 2 for GPS disiplined osillators GPSDOs) and up to 0 5 for a temperature ompensated rystal osillator TCXO). This espeially means that in a system with several sensors, their loks will often drift in different diretions with respet to real time. Hene, assuming that target and beaon transmissions our at a relatively low rate, e.g., Hz, onsiderable relative errors between sensor loks might be introdued. This an also lead to an assoiation problem of the reeived transmissions at the different sensors. We assume that a mehanism exist to resolve this problem, suh as unique message identifiers embedded in the signal. Another physial layer approah is to perform a plausibility hek by omparing known and estimated beaon loations. In order to perform lok synhronization between two sensors, two beaons at different loations are needed. However, a single beaon is suffiient to ahieve a reasonable approximate solution as desribed later. Figure shows the transmission sequene of the involved signal bursts at time instanes t q). A transmitted signal from beaon or target k at time index q Fig. : Transmission sequene of a beaon augmented passive TDOA loalization system using TOA measurements. Different ombinations of measurements an be used to estimate the lok parameters and subsequently loalize the target. reeives the sensor i at the TOA t q) i,k = ɛ i t q) + d i,k ) + φ i + n q) i, 2) where n q) i is a realization of zero-mean Gaussian noise with variane σn 2 i and d i,k is the distane between the beaon or target and the sensor d i,k = p k p i 2, 3) with the two dimensional oordinate vetor p k = [x k, y k ] T. The TDOA is then given as the differene τ i,k = t q) i,k tq),k = ɛ i ɛ )t q) d i,k d,k + ɛ i ɛ + φ i φ + n q) i n q). Loalization algorithms eventually require the distane differenes δ i,k = d i,k d,k, 5) in order to geometrially determine the target oordinates. Hene, orretion of the parameters ɛ i and φ i is neessary. A possible way to remove parameters that are onstant between two TDOA measurements is the differential time differene of arrival DTDOA) τ q,r) i,kl 4) = τ q) i,k τ r) i,l. 6) Different types of DTDOA measurements are possible. A DTDOA between reeptions of the same transmitter, e.g., the same beaon, at different time instane yields τ q,r) i,kk = ɛ i ɛ )t q) t r) )+n q) i n q) n r) i +n r). 7) Clearly, the onstant term φ i φ is anelled out. Another type of DTDOA is between different transmitters k and l, i.e.,

3 two different beaons or one beaon and the target, at different time instanes τ q,r) i,kl = ɛ i ɛ )t q) t r) ) +ɛ i di,k +n q) i n q) d i,l ) d,k ɛ d,l n r) i + n r), again this eliminates the offset φ i φ. However the simple formation of DTDOAs is not able to remove ɛ i and hene, for larger time spans between beaon and target transmissions, ignoring these erroneous lok rates leads to a drasti degradation of the system auray. III. CLOCK ERROR CORRECTION Next, four different estimators are defined that are useful in different situations, whih depend on the amount of noise, the quality of the loks and the time span between transmissions. ) TDOA estimation without orretion: Assuming that the loks are of very high auray and synhronization is lose to perfet, i.e, ɛ i = and φ i = 0, simple TDOA estimation without further lok synhronization may be used based on 4). 2) TDOA estimation with offset orretion: If very aurate loks ɛ i = ) are available, that are not absolutely synhronized, i.e., φ i 0, a DTDOA with a single beaon may be used based on 8). Equivalently, the differene φ i φ an also be estimated expliitly using a single TDOA 4) from a beaon if the lok rates ɛ i are known ) 8) ˆφ = τ,k δ,k, 9) where without loss of generality, one may set the time of transmission t 0) = 0. The distane differenes needed by the loalization algorithm are then ˆδ,l = τ,l τ,k ) + δ,k. 0) 3) TDOA estimation with approximate offset and rate orretion: When lower quality, unsynhronized loks are used, i.e, ɛ i, φ i 0 it beomes neessary to orret for both parameters. This an be approximately ahieved using a single beaon, however, two transmissions from that beaon are now required. The estimation of φ i depends on ɛ i, hene we begin with the estimation ɛ i then subsequently estimate φ i and orret the initially measured TOAs. Let i = be the referene sensor. The time t 0) t ) between two beaon transmissions of a single beaon k an be estimated using a single sensor s lok. Compared to the true time differene between transmission this time differene is saled t 0) t ) = t 0),k ɛ t),k n0) + n ) ). ) Considering this saling, the DTDOA for the same beaon an be written as τ 0,),kk = ɛ ) t 0),k ɛ t),k ) + ɛ n 0) n ) ɛ ) n0) + n ). 2) From this, obviously not all lok rates an be estimated but it is possible to estimate them relative to the referene sensor, as β = ɛ ɛ. The estimator of β an be written as 0,) τ,kk ˆβ = + t0),k t),k ) t 0),k t),k ), 2,..., M. 3) Further, ˆɛ = is a reasonable hoie based on the the mean value of the distribution. Therefore, the estimate of lok rates will beome ˆɛ = ˆβ. Using those estimated values for the rates, next the absolute lok offsets of all sensors is estimated d,k ˆφ = t 0),k ˆɛ,,..., M. 4) With all of this the TOAs of the target an be orreted as 3) t,l ˆδ,l = ˆφ t 3),l ˆφ ), 5) ˆɛ ˆɛ and then used by the loalization algorithm to determine the loation p 0. Negleting estimation error, the lok orretion essentially enfores equal lok rates in all sensors, whih are idential to the true value of ɛ. A look at 4) reveals that this approah is reasonable as the term ɛ i ɛ )t q) beomes zero d and the saling in the term ɛ i,k d i ɛ,k is less ritial due to the errors in the rates being relatively small. 4) TDOA estimation with offset and rate orretion: Finally, if the system dimensions beome very large, e.g., satellite based systems, or the speed of the wave is low, e.g., ultrasound based systems, the error in the approximation of ɛ an no longer be negleted any more. Fortunately, using a seond beaon at a different loation, an estimate for ɛ an be obtained. Considering the time differene of two transmission from the two different beaons k and k t 0) t 2) = t 0),k ɛ d,k t2),k ) d ),k 6) n 0) n 2) ɛ ), this further results in τ 0,2),kk = β )t 0),k t2) + ɛ β d,k,k ) d,k ) d,k ɛ β d ),k + β n 0) n 2) ) n0) + n 2), 7) and leads to an estimator for ɛ ˆɛ = 0,2) τ,kk ˆβ )t 0),k t2),k ). 8) ˆβ δ,k δ,k ) where ˆβ is estimated using 3). Analogously to the last ase, using 4) and 5), the neessary input values for the loalization algorithm an be obtained.

4 IV. MODIFIED CRLB In this setion, theoretial limits of the estimation of target loation p 0 using TDOAs from 4) are derived. For any unbiased estimator, a lower bound on the variane of the error is provided by the Cramér-Rao lower bound CRLB) [6]. It is based on the inverse of the Fisher information matrix FIM) whih is in our ase given as { 2 } ln pτ ; u) [Iu)] n,m = E, 9) u n u m where τ = [τ 2,k,..., τ M,k ] T is the vetor of TDOAs 4) of the target when sensor i = is the referene sensor, and the vetor u of parameters to be estimated. For the problem at hand u onsists of the target oordinates as well as the lok parameters. The relationship between these parameters makes an analytial derivation of the CRLB very hallenging. An approah that provides a less tight bound but simplifies the omputation is the MCRLB for vetor parameter estimation []. To derive the MCRLB expression, the estimation parameters are split into u, the vetor of estimation parameters, and v, the vetor of unwanted parameters, whih are defined differently for eah ase onsidered below. We assume u to be deterministi, and v to be random with known probability distribution funtion. Then, the onditional FIM Iv; u) is alulated using the expetation operator as [Iu; v)] n,m = E τ v { 2 ln pτ v; u) u n u m with the probability density funtion pτ u; v) }, 20) pτ v; u)= 2π) M 2 detc ) 2 exp ) 2 τ µ)t C τ µ), 2) The onditional FIM an be alulated as [6] [ ] T [ µ [Iu; v)] n,m = C µ u n u m ]+ 2 tr C C u n C C u m ). 22) Aording to [], taking the expetation over v, one an then obtain the modified Fisher information matrix MFIM) I mod u) with [I mod u)] n,m = E v {[Iu; v)] n,m }. 23) This leads to the modified lower bound on the estimation variane varû m ) [ I mod u) ] m,m. 24) Next, this an be used to determine the bounds for the loalization with the different lok orretion approahes. ) TDOA estimation without orretion: In this ase there is only a single transmission from the target k = 0 available. We onsider u = [x 0, y 0 ] T as estimation parameters and v = [ɛ,..., ɛ M, φ,..., φ M ] T as the unwanted parameters. To alulate the lower bound on the estimation error, we diretly use the MCRLB from 24) with the mean µ = [µ 2,..., µ M ] T and the ovariane matrix C of τ µ i = ɛ ɛ i )t q) d,k d i,k + ɛ ɛ i + φ φ i, 25) C = diagσn 2 2,..., σn 2 M ) + σn 2 M T M. 26) For the ases that beaon transmissions are available, this simple MCRLB is unaeptably loose and we resort to a twostep approah, where we first obtain bounds for the lok parameters and then, in the seond step, derive the bound for the target loation. 2) TDOA estimation with offset orretion: In this ase a single beaon k = is available and the lok offsets φ i an be estimated. Hene, the estimation parameter vetor for the first step beomes u = [φ,..., φ M ] T and the vetor of unwanted parameters v = [ɛ,..., ɛ M ] T. Then, the MCRLB on the estimation of the lok offsets φ i is alulated using the mean and ovariane matrix from 25) and 26). In the seond step, we have u = [x 0, y 0 ] T, and v = [ɛ,..., ɛ M ] T. We model the lok offsets as ˆφ i = φ i + ω i, where ˆφ i is an estimate with the error ω i modeled as a Gaussian random variable, ω i N 0, σφ 2 i ), where σφ 2 i is given by the bound from the first step. The seond step mean and the ovariane matrix then beome µ i = ɛ ɛ i )t q) d,0 d i,0 + ɛ ɛ i + i ˆφ, 27) C = diags 2,φ,..., s 2 M,φ) + s 2,φ M T M, 28) where s 2 i,φ = σ2 n i + σφ 2 i. This leads to the MCRLB of the target loation. 3) TDOA estimation with approximate offset and rate orretion: When two transmissions of the beaon k = are available, it is possible to estimate the lok offsets φ i as well as the lok rates ɛ i, exept for one, e.g., ɛ. Thus, similar to the last ase, in the first step, we define the vetors as u = [ɛ 2,..., ɛ M, φ,..., φ M ] T and v = [ɛ ]. In the seond step, we model both the lok offset and rate as Gaussian random variables ˆφ i = φ i + ω i, i =,..., M and ˆɛ i = ɛ i +ξ i, i = 2,..., M, where ω i and ξ i are the errors in the lok offset and rate estimation, modeled as Gaussian random variables ω i N 0, σφ 2 i ) and ξ i N 0, σɛ 2 i ). Varianes σφ 2 i and σɛ 2 i are derived using the the first step MCRLB. To alulate the MCRLB on the estimation of the target position we inorporate the lok offset and lok rate models into 4). The seond step mean and ovariane matrix for this ase are µ i = ɛ ˆɛ i )t q) d,0 d i,0 + ɛ ˆɛ i + i ˆφ, 29) C = diags 2,ɛ,..., s 2 M,ɛ) + s 2,φ M T M, 30) where s 2 i,ɛ = σ2 ɛ i t q) + di,0 )2 + σ 2 n i + σ 2 φ i and s,φ is the same as in the last ase. Similarly, the seond step parameter vetors are u = [x 0, y 0 ] T and v = [ɛ ]. 4) TDOA estimation with offset and rate orretion: Finally, with the aid of two beaons k {, 2}, it is possible to estimate the lok rates of all the sensors. Hene, in this ase there are no unwanted parameters, neither in the first nor in the seond step. The first step estimation parameter vetor

5 ) DTDOA φ i, ɛ i 3) DTDOA φ i, ɛ i, ˆɛ = 2) DTDOA φ i ) TDOA 4) MCRLB, DTDOA φ i, ɛ i 3) MCRLB, DTDOA φ i, ɛ i, ˆɛ = 2) MCRLB, DTDOA φ i ) CRLB, TDOA Transmit freq. = 0.Hz Transmit freq. = 0.2Hz Transmit freq. = 0.5Hz Transmit freq. = Hz Transmit freq. = 2Hz Transmit freq. = 0Hz RMSE [m] 0 2 Geometry indued error RMSE [m] a) σ ɛ Fig. 2: a) The MCRLB ompared with simulations for different availability of beaon messages with φ i = 0. ) Baseline, no beaons used. 2) Single beaon message, estimation of φ i. 3) Two messages from the same beaon, estimation of φ i and ɛ i while ˆɛ =. 4) Three messages from two beaons, full estimation of φ i and ɛ i. b) Impat of time dependent lok rate modeled as a random walk, using method 3). b) σ η is v = [ɛ,..., ɛ M, φ,..., φ M ] T. We alulate the variane of the lok parameters estimation error using the standard CRLB. Analogous to the last ase in the seond step, we model all the lok rates and offsets with Gaussian random variables. Thus, the bound for the target loation is derived with the estimation parameter vetor u = [x 0, y 0 ] T, mean µ i = ˆɛ ˆɛ i )t q) d,0 + ˆɛ and ovariane matrix ˆɛ i d i,0 + ˆφ i ˆφ, 3) C = diags 2,ɛ,..., s 2 M,ɛ) + s 2,ɛ M T M. 32) V. RESULTS Simulative as well as experimental results have been obtained for verifiation. In the simulation, 5 sensors have been plaed using the same geometry as given in the experimental system, depited in Fig. 3 a). First, reeiver loks have been modeled aording to ) and the time between transmissions is 2 s. The standard deviation of the error in eah TOA measurement is σ ni = 0 8. It is then possible to ompare the analytial expressions for the bounds with Monte Carlo simulation result. This is shown in Fig. 2 a). From the plot it is found that if very good loks are available it is reommendable to not apply any orretion. That is due to the additional noise terms introdued into the solution due to additional measurements of the beaon signals. Note that this assumes that the offsets φ i are lose to zero. When the error in the TOA beomes large due to unompensated differing lok rates, it leads to large outliers of the estimated loation, aused by the nonlinear nature of the hyperboli geometry. This an be observed in the upper right orner of Fig. 2 a). Further, due to bad onditioning of the problem, as seen in 8), for the given system dimensions, full orretion of all ɛ i provides muh worse performane than the approximation ˆɛ =, whih itself should be used only for loks with σ ɛ However, this threshold obviously depends on the time between the beaon transmissions and also the measurement noise. Therefore, in a seond simulation, the time dependene of the lok rates ɛ i is modeled using a Brownian random walk approah where ση 2 is the variane of the additional Gaussian noise, that is umulatively added. We then vary the time between the transmissions and observe the degradation of the loalization in terms of root mean square error RMSE). The resulting plot is presented in Fig. 2 b). This an be used to selet the rate of the beaon transmission based on the speifiation of the used sensor loks. Experimental measurements have been obtained from the ATLAS system [4] with a subset of 5 sensors as shown in Fig. 3 a). Target tags, whih are usually attahed to birds for traking, transmit a binary frequeny shift keying signal every s with a bandwidth of 2 MHz. Beaons transmit an idential signal every 2 s. Transmitted data is a 892 long random ode sequene, that yields a large orrelation gain at the reeiver and enables simultaneous hannel aess for a large number of tags. The sensors are based on software defined radio with Ettus USRP N200 frontends that ontain a TCXO frequeny referene speified with 2.5 ppm. However, aording to the estimation for the reorded data set, all 5 sensor lok rates are within less than ppm. Fig. 3 b) shows the drift in the loation of a stati target when ust the offset orretion from Se. III-2 or the offset and rate orretion from Se. III-3 is

6 3.8 2 Sensors Referene sensor DTDOA-φ DTDOA-φ, ɛ Target Beaon CRLB, DTDOA-φ, ɛ 3.2 y [km] 6 CRLB, DTDOA-φ y [km] x [km] a) x [km] b) Fig. 3: a) Geometry of the onsidered loalization sensor network [4], b) drift in the loation of a stationary target with orretion of φ i ompared to the ase with orretion of φ i and ɛ i while ˆɛ =. used. The RMSE for the two ases is m and 8.40 m respetively. VI. CONCLUSION In this paper we have analyzed the sensitivity of a partiular type of passive loalization system with respet to different qualities of the sensor loks. Four different ases have been identified and orretion methods have been derived aordingly, together with the orresponding modified Cramér- Rao lower bounds. Results from simulation and a deployed experimental wildlife traking system learly illustrate that the onsidered system is able to perform well with drifting sensor loks. VII. ACKNOWLEDGMENTS This work was partially supported by the Deutshe Forshungsgemeinshaft DFG) proet CLASS grant MA 84/23-), by the Miverva Center for Movement Eology, and by grants 965/5 and 863/5 from the Israel Siene Foundation funded by the Israel Aademy of Sienes and Humanities). REFERENCES [] W. H. Foy, Position-loation solutions by Taylor-series estimation, IEEE Transations on Aerospae and Eletroni Systems, no. 2, pp , 976. [2] J. Smith and J. Abel, Closed-form least-squares soure loation estimation from range-differene measurements, IEEE Transations on Aoustis, Speeh, and Signal Proessing, vol. 35, no. 2, pp , 987. [3] Y. T. Chan and K. Ho, A simple and effiient estimator for hyperboli loation, IEEE Transations on signal proessing, vol. 42, no. 8, pp , 994. [4] Y. Huang, J. Benesty, G. W. Elko, and R. M. Mersereau, Real-time passive soure loalization: A pratial linear-orretion least-squares approah, IEEE transations on Speeh and Audio Proessing, vol. 9, no. 8, pp , 200. [5] C. Yan and H. H. Fan, Asynhronous self-loalization of sensor networks with large lok drift, in Mobile and Ubiquitous Systems: Networking & Servies, MobiQuitous Fourth Annual International Conferene on. IEEE, 2007, pp. 8. [6] K. Ho and L. Yang, On the use of a alibration emitter for soure loalization in the presene of sensor position unertainty, IEEE Transations on Signal Proessing, vol. 56, no. 2, pp , [7] Y. Wang, X. Ma, and G. Leus, Robust time-based loalization for asynhronous networks, IEEE Transations on Signal Proessing, vol. 59, no. 9, pp , 20. [8] M. R. Gholami, S. Gezii, and E. G. Strom, TDOA based positioning in the presene of unknown lok skew, IEEE Transations on Communiations, vol. 6, no. 6, pp , 203. [9] M. Leng, W. P. Tay, C. M. S. See, S. G. Razul, and M. Z. Win, Modified CRLB for ooperative geoloation of two devies using signals of opportunity, IEEE Transations on Wireless Communiations, vol. 3, no. 7, pp , 204. [0] A. N. D Andrea, U. Mengali, and R. Reggiannini, The modified Cramer-Rao bound and its appliation to synhronization problems, IEEE Transations on Communiations, vol. 42, no. 234, pp , 994. [] F. Gini, R. Reggiannini, and U. Mengali, The modified Cramer- Rao bound in vetor parameter estimation, IEEE Transations on Communiations, vol. 46, no., pp , 998. [2] M. Moenelaey, On the true and the modified Cramer-Rao bounds for the estimation of a salar parameter in the presene of nuisane parameters, IEEE Transations on Communiations, vol. 46, no., pp , 998. [3] J. Tiemann, F. Ekermann, and C. Wietfeld, Multi-user interferene and wireless lok synhronization in tdoa-based uwb loalization, in Indoor Positioning and Indoor Navigation IPIN), 206 International Conferene on. IEEE, 206, pp. 6. [4] A. Weller-Weiser, Y. Orhan, R. Nathan, M. Charter, A. J. Weiss, and S. Toledo, Charaterizing the auray of a self-synhronized reverse-gps wildlife loalization system, in 206 5th ACM/IEEE International Conferene on Information Proessing in Sensor Networks IPSN). IEEE, 206, pp. 2. [5] Y.-C. Wu, Q. Chaudhari, and E. Serpedin, Clok synhronization of wireless sensor networks, IEEE Signal Proessing Magazine, vol. 28, no., pp , 20. [6] S. M. Kay, Fundamentals of statistial signal proessing: estimation theory, 993.