Location estimation using RSS measurements with unknown path loss exponents
© Zeytinci et al.; licensee Springer. 2013
Received: 13 October 2012
Accepted: 16 June 2013
Published: 26 June 2013
The location of a mobile station (MS) in a cellular network can be estimated using received signal strength (RSS) measurements that are available from control channels of nearby base stations. Most of the recent RSS-based location estimation methods that are available in the literature rely on the rather unrealistic assumption that signal propagation characteristics are known and independent of time variations and the environment. In this paper, we propose an RSS-based location estimation technique, so-called multiple path loss exponent algorithm (RSS-MPLE), which jointly estimates the propagation parameters and the MS position. The RSS-MPLE method incorporates antenna radiation pattern information into the signal model and determines the maximum likelihood estimate of unknown parameters by employing the Levenberg-Marquardt method. The accuracy of the proposed method is further examined by deriving the Cramer-Rao bound. The performance of the RSS-MPLE algorithm is evaluated for various scenarios via simulation results which confirm that the proposed scheme provides a practical position estimator that is not only accurate but also robust against the variations in the signal propagation characteristics.
KeywordsMobile positioning; Location estimation; Cramer-Rao bound; Received signal strength
Recently, location estimation has been among the most attractive research topics in the area of cellular communications. With accurate position estimation, a variety of applications and services such as emergency services, monitoring and tracking fraud protection, asset tracking, fleet management, mobile yellow pages, and even cellular system design and management can become feasible for cellular networks . These potential applications of wireless positioning have also been recognized by the IEEE, which set up a standardization group 802.15.4a for designing a new physical layer for low-data rate communications combined with positioning capabilities . Furthermore, the Federal Communications Commission (FCC) in the USA has required wireless providers to locate mobile users within tens of meters for emergency 911 calls .
The position of a mobile station (MS) can be determined using multiple radio signals transmitted or received by the MS. Some location estimation methods like assisted global positioning system (A-GPS) are based on signals transmitted from satellites, while others rely on measurements of signals between MS and base stations (BS), the so-called network-based methods. Currently, the best positioning accuracy in cellular systems is provided by A-GPS at the expense of a significant increase in network and handset complexity . For instance, modifications on the handset such as an embedded GPS receiver and deployment of location management units (LMU) into the network are needed in order to operate A-GPS systems . Compared to A-GPS, network-based methods are relatively less complex. Moreover, they can be used in many situations where the A-GPS method cannot be applied, i.e., indoor positioning, but generally with a degradation in accuracy. So far, a wide variety of network-based positioning techniques have been proposed which use measurements obtained within the cellular networks, such as received signal strength (RSS), time of arrival (TOA), time difference of arrival (TDOA), and angle of arrival (AOA) methods [6–15].
Positioning technology is often based on trilateration in time-based methods like TOA and TDOA, in which the MS position is obtained as the intersection point of three circles constituted by distance estimates . Assurance of proper operation of time-based methods requires the deployment of LMUs . These network elements perform timing measurements of all local transmitters, based on which the actual relative time difference for each BS can be estimated. In a similar manner, AOA-based positioning methods require the implementation of adaptive array antennas, by which the direction of signal arrival can be estimated. At the same time, the handset typically requires software modifications to enable positioning functionality with such a method. Consequently, these methods are not widespread in commercial systems due to their high deployment costs. Therefore, improving the accuracy of positioning systems based on the existing cellular network infrastructure is desired, which is the main motivation of this study.
The system model that is used for location estimation in this paper is based on RSS measurements. One fundamental MS function is to find the BSs with the strongest signal strength for cell selection purposes. Thus, RSS-based methods can be implemented without any hardware enhancement in either the MS or the BS. Distance information between MS and BS can be extracted from RSS measurements using accurate information about the propagation characteristics of the measurement channel. However, in most of the RSS-based location estimation techniques in the literature, it is assumed that the propagation parameters of the measurement channel are accurately known a priori, either through a training period or by assuming a perfect free-space channel condition [18–24]. Such an approach results in degraded positioning accuracy in many practical application scenarios, including mobile tracking techniques in [25, 26]. More specifically, both [25, 26] study Monte Carlo-based mobility tracking algorithms under the assumption that the mobile moves in a 2D environment with input as acceleration. In , RSS data are used to predict mobile position and velocity using particle filtering techniques, but the method relies on known propagation parameters. In , the so-called interacting multiple model method has been extended to predict the state of the mobile for indoor applications by considering multiple fixed environment parameters at the expense of an increase in the number of random processes, whereas the technique herein relies on continuous parameter adaptation.
In this paper, a method that aims to resolve major shortcomings of the existing RSS-based positioning techniques is proposed. In particular, an RSS-based location estimation technique that jointly estimates the propagation parameters and the MS position is explained in Section 2. The Cramer-Rao bound (CRB) derivation and accuracy evaluation are given in Section 3 as a benchmark for performance comparison. In Section 4, simulation results under various scenarios are presented to evaluate the performance of the proposed algorithm. Finally, some concluding remarks are given in Section 5.
2 RSS-based location estimation model
Using RSS together with a path loss (PL) and shadow fading model, a distance estimate between the BS and the MS can be obtained. The propagation model used throughout this paper is a modified version of the log normal model that is widely used in the literature [27–30]. The PL exponent (PLE) is the key parameter in the log normal model. An accurate value of the PLE is required in order to obtain an accurate estimate of the MS-BS distance from the corresponding RSS measurement.
In most of the existing studies of RSS-based location estimation techniques in the literature, the channel model is assumed to be known a priori, that is, the path loss characteristics of the coverage area are considered known, either by assuming that the environment is a perfect free space or by extensive measurement and modeling prior to the deployment of location estimation systems. However, the PLE parameter is environment dependent [31, 32]. Even in the same environment, propagation characteristics may change considerably over a long period of time, e.g., due to seasonal and/or weather changes . In , it is experimentally demonstrated in an omnidirectional antenna system that the PLE is strongly dependent on the base antenna height and the terrain category. Extension of the experimental study in  to directional antenna systems can be found in  where the authors show that antenna beamwidth has an additional gain reduction influence on the PLE. In other experimental studies that consider path loss modeling for directional antenna systems [35, 36], the authors compute different PLEs for different areas of the cell that consist of different terrains (but not different PLEs for different directional antennas of the same base station). There are also related studies that utilize single PLE for directional antenna systems [37–39].
The proposed algorithm in this paper estimates the MS location using the available RSS measurements without any need for a training period or any need for the knowledge of the PLE value in the path loss model. The PLE values are determined and calibrated in real time for every mobile using the RSS measurements. By incorporating the antenna radiation patterns of the sectors in a BS into the signal model, an additional improvement in position estimates is provided. In determining the position of a given mobile user, it is assumed that the PLE is the same for all sectors of a given base station, whereas the PLE might be different for another. This is a quite realistic assumption in light of the discussion of the related literature in the previous paragraph, since the mobile user is in the same location with respect to the base station and directional antennas are at the same height.
2.1 Antenna model
2.2 Path loss model
In this paper, it is assumed that the channels used by distant BSs may have different propagation characteristics. More specifically, the channels used by the cells that are served by the same BS are assumed to have the same PLE, and this PLE is not necessarily the same for other channels. Hence, the value of Rk,s and μk,s depends on α k in addition to the mobile position, (x,y).
2.3 Problem formulation and the positioning algorithm
where is the value of θ for which the given function is minimized over the given data set. Note that a distinctive advantage of ML estimate is that it can always be found for a given data set .
where and d k represents the distance between k th BS and the MS, that is, . Herein, k index represents the BS, and s represents the sector that belongs to BS k. Since each channel used by different BSs may have a distinct PLE, α parameters are indexed with BS ID, k.
In Algorithm 1, kmax is the maximum number of allowed iterations (kmax=400 is used in the simulations), and ε1 is used to detect how close the estimate is to the desired value (e.g., ε1=10−15). Both parameters are chosen by the user. The damping parameter of the LM algorithm, h, is positive, which guarantees that Δ is a descent direction. Note that for large values of h, we have Δ=J T (θ)f(θ)/h, which implies a short step in the descent direction, which in turn is good if the current iterate is far from the solution. On the other hand, if h is small, then Δ is approximately equal to what we have from the Gauss-Newton iteration. Since the damping parameter
influences both the direction and the size of the step, its update is controlled by the gain ratio ρ in the algorithm. A large positive value of ρ indicates a good approximation which allows us to decrease h so that the LM step is closer to the Gauss-Newton step, whereas a small or negative ρ is a poor approximation which requires an increase of damping by twofold in order to get closer to the steepest direction and hence increase chances of faster convergence. By this choice of parameters similar to , we have observed linear to superlinear convergence in our problem, although it is harder to make specific statements on the convergence rate for the problem in hand. However, it is well known that the Levenberg-Marquardt method has a quadratic rate of convergence when Jacobian is a nonsingular square matrix and if the parameter is chosen suitably at each step. The condition of the nonsingularity of Jacobian is too strong, and it is not valid in our problem either. Although the authors show in [42, 43] that the method has quadratic convergence under appropriate assumptions and the choice of the damping parameter, the results are valid only locally. In the next section, we derive the CRB bounds for the proposed method.
3 The Cramer-Rao lower bound
where F=−E[Δ θ (Δ θ lnf(p;θ)) T ] is the Fisher information matrix (FIM) .
where , , δ k l =1 if k=l, and δ k l =0, otherwise.
In the next section, we further define quantitative performance measures for location estimators based on CRB.
3.1 Accuracy measures
where εrms is defined as the root-MSE (RMSE) of location estimators.
where |F| is the determinant of the Fisher information matrix and εrms is defined as the RMSE of location estimators. The closed form expression of the CRB bound is not explicitly given here due to its complexity; instead, only the numerical solutions are presented. On the other hand, each component of the FIM matrix is given in the Appendix.
4 Simulation results
The proposed algorithm computes the ML estimate of the MS position using the RSS measurements and concurrently calibrates the PLE parameters of the channels occupied by different BSs. Recall that this algorithm is referred to as RSS-MPLE algorithm. In order to demonstrate the improvement on the positioning accuracy provided by the RSS-MPLE algorithm, its performance is compared with those of other algorithms, such as RSS with single PLE algorithm (RSS-SPLE) , which finds and calibrates a single PLE for all channels, and RSS with known PLE algorithm (RSS-KPLE)  in which PLE values are known as a priori. Furthermore, the Cramer-Rao bound has been evaluated and compared with the RMSE results of the proposed algorithm.
4.1 Effect of truncated RSS measurements
RSS measurements below −110 dBm and above −48 dBm are truncated in GSM systems. In Figure 3, the effect of such truncation of RSS measurements in the performance of the proposed algorithm is investigated. The case in which all BSs have the same PLE value, i.e., α1=α2=α3=3, is considered in this simulation, during which RSS-MPLE and RSS-KPLE algorithms are operated both with truncated and original RSS measurements. The truncated measurements represent all RSS measurements below −110 dBm and above −48 dBm. In Figure 3 (and the subsequent figures in the paper), the following legend clarification is necessary to better interpret the results:
‘Truncated RSS are omitted’ means that RSS measurement are truncated and measurements below −110 dBm and above −48 dBm are not used in the simulations.
‘Truncated RSS are used’ means that RSS measurements being truncated and measurements below −110 dBm and above −48 dBm are used in the simulations.
‘RSS are not truncated’ means that RSS measurements are not truncated.
From Figure 3, we note that the positioning RMSE obtained with RSS-MPLE algorithm does not exceed 20 m even for σ v =10 if RSS measurements are not truncated. On the other hand, truncation of RSS measurements dramatically degrades the performance of both RSS-MPLE and RSS-KPLE algorithms due to the decrease in the number of RSS measurements, i.e., the performance of the proposed algorithm is expected to improve with an increase in the number of available measurements.
Since the truncated measurements represent all RSS measurements below −110 dBm and above −48 dBm, they introduce a large bias in the position estimate when incorporated in the RSS measurement set. Because of this, positioning accuracy of the RSS-MPLE algorithm severely degrades when the truncated RSS measurements are not omitted. Compared to the RSS-MPLE algorithm, the RSS-KPLE algorithm performs better for all σ v values when truncated RSS measurements are used. This is an expected result since RSS-MPLE algorithm estimates PLE values in addition to the coordinates with the same number of RSS measurements.
4.2 Effect of inaccurate knowledge of PLE values
4.3 Effect of distinct PLE values
This subsection focuses on the performance analysis of RSS-MPLE, RSS-SPLE, and RSS-KPLE algorithms under different channel conditions. In the first scenario, it is assumed that all channels have the same PLE value. In the second scenario, PLE values differ for channels occupied by different BSs.
4.3.1 Equal α 1 , α 2 , and α 3
To evaluate the RSS-MPLE and RSS-SPLE algorithms with respect to the FCC requirements, the positioning error cumulative distribution function (CDF) is shown in Figure 6 for σ v =6 dB, which is a realistic value in a microcellular environment . The RSS-SPLE and RSS-KPLE algorithms satisfy the FCC requirements, which mandate 67% CERP within 100 m and 95% CERP within 300 m. Although the RSS-MPLE algorithm does not satisfy the FCC requirements, this algorithm offers a solution for environments that possess distinct and variable PLEs.
4.3.2 Distinct α 1 , α 2 , and α 3
Figure 7 shows that the performance of the RSS-SPLE algorithm deteriorates when PLE values of the BSs are unequal. The scenario considered in this simulation can be experienced when a BS that is in the vicinity of the MS is in NLOS condition and other BSs are in LOS condition with the MS. Compared to the RSS-SPLE algorithm, positioning accuracy of the RSS-MPLE algorithm does not change significantly under these conditions. On the other hand, position estimates obtained with RSS-SPLE algorithm are erroneous due to the bias in the α estimate. Moreover, positioning accuracy of RSS-MPLE and RSS-KPLE algorithms are close even when PLE values vary. Thus, RSS-MPLE algorithm satisfies the requirements for a position estimate with low error variance, independent from unknown propagation parameters, i.e., σ v and α values. The positioning error CDF shown in Figure 8 indicates that the accuracy of mobile positioning can substantially be improved by employing the RSS-MPLE algorithm.
4.4 CRB performance evaluation
In this paper, a practical positioning method that can be implemented in mobile networks with simple modifications in the existing infrastructure is presented. The proposed method, the so-called RSS-MPLE, is based on RSS measurements and jointly estimates the MS position and the propagation parameters, namely the PLE value of the measurement channel. The RSS-MPLE method does not need a training period to estimate the PLE value of the channel. The most significant feature of the proposed RSS-MPLE algorithm is its ability to separately calibrate the PLE value of each channel occupied by different BSs. Moreover, the RSS-MPLE algorithm incorporates the antenna radiation pattern information that provides additional improvement in positioning accuracy. Via extensive simulations, the performance of the proposed method has been compared with those of the existing algorithms in terms of positioning RMSE, bias, availability, and CERP under different environmental conditions by changing PLE and SNR values. Simulation results indicate that the RSS-MPLE algorithm is robust against variations in the PLE values under different environment conditions.
6.1 Cramer-Rao bound for the three BS case
The authors would like to thank the referees for their constructive comments which improved the exposition of the paper.
The authors dedicate this article in memory of one of its co-authors, Frederic Kerem Harmanci, who passed away between the completion of his article and its publication.
- Caffery JJ: Wireless Location in CDMA Cellular Radio Systems. Springer, Heidelberg; 2000.Google Scholar
- IEEE P802.15.4a/D4: Part 15.4: Wireless Medium Access Control(MAC) and Physical Layer (PHY) Specifications for Low-Rate wireless personal area networks (LR-WPANs): Amendment of IEEE Std 802.15.4, July 2006. IEEE, Piscataway; 2006.Google Scholar
- Commission FC: Revision of the Commission’s Rules to Ensure Compatibility with Enhanced 911 Emergency Calling Systems, FCC 94-102. FC Commission, Washington DC; 1996.Google Scholar
- Borkowski J, Niemela J, Lempiäinen J: Cellular location techniques supporting A-GPS positioning. IEEE Vehicular Technol. Conf 2005, 1: 429-433.Google Scholar
- Gustafsson F, Gunnarsson F: Mobile positioning using wireless networks: Possibilities and fundamental limitations based on available wireless network measurements. IEEE Signal Process. Mag 2005, 22(4):41-53.View ArticleGoogle Scholar
- Cheung K, So H, Ma W-K, Chan Y: Least squares algorithms for time-of-arrival-based mobile location. IEEE Trans. Signal Process 2004, 52(4):1121-1130. 10.1109/TSP.2004.823465MathSciNetView ArticleGoogle Scholar
- Dil BJ, Havinga PJM: Australasian Telecommunication Networks and Applications Conference, ATNAC 2010. Auckland; 31 October –3 November 2010.Google Scholar
- Lakmali B, Dias D: Database correlation for GSM location in outdoor & indoor environments. The 4th International Conference on Information and Automation for Sustainability, Colombo, 12-14 December 2008. IEEE, Piscataway; 2008.Google Scholar
- Ahonen S, Laitinen H: Database correlation method for UMTD location. IEEE Vehicular Technol. Conf 2003, 4: 2696-2700.Google Scholar
- Spirito M, Mattioli A: Preliminary experimental results of a GSM mobile phones positioning system based on timing advance. IEEE Vehicular Technol. Conf 1999, 4: 2072-2076.Google Scholar
- Spirito M: On the accuracy of cellular mobile station location estimation. IEEE Trans. Vehicular Technol 2001, 50(3):674-685. 10.1109/25.933304View ArticleGoogle Scholar
- Khalaf-Allah M: Nonparametric Bayesian filtering for location estimation, position tracking and global localization of mobile terminals in outdoor wireless environments. EURASIP J Adv. Signal Process 2008, 2008: 1-14.View ArticleGoogle Scholar
- Borkowski J, Lempiäinen J: Pilot correlation positioning method for urban UMTS networks. Proceedings of the 11th European Wireless Conference, Nicosia, 10–13 April 2005. vol. 2. IEEE, Piscataway; 2005.Google Scholar
- Pan W, Wu J, Jiang Z, Wang Y, You X: Mobile position tracking by TDOA-Doppler hybrid estimation in mobile cellular system, IEEE International Conference on Communications, Glasgow, 24–28 June 2007. IEEE, Piscataway; 2007.Google Scholar
- Türkyilmaz O, Alagöz F, Gür G, Tugcu T: Environment-aware location estimation in cellular networks. EURASIP J. Adv. Signal Process 2008, 2008: 276456. 10.1155/2008/276456View ArticleGoogle Scholar
- Sayed A, Tarighat A, Khajehnouri N: Network-based wireless location: challenges faced in developing techniques for accurate wireless location information. IEEE Signal Process. Mag 2005, 22(4):24-40.View ArticleGoogle Scholar
- Borkowski J, Lempiäinen J: Practical network-based techniques for mobile positioning in UMTS. EURASIP J. Appl. Signal Process 2006, 2006: 012930.Google Scholar
- Budka K, Calin D, Chen B, Jeske D: A Bayesian method to improve mobile geolocation accuracy. IEEE Vehicular Technol. Conf 2002, 2: 1021-1025.Google Scholar
- Chu KMK, Leung KRPH, JK-Y Ng, Li CH: Locating mobile stations with statistical directional propagation model in AINA ’04. Proceedings of the 18th International Conference on Advanced Information Networking and Applications, Washington, 29 - 31 March 2004. IEEE, Piscataway; 2004.Google Scholar
- Qi Y, Kobayashi H: Cramer-Rao Lower bound for geolocation in non-line-of-sight environment. IEEE Int. Conf. Acoustics, Speech, Signal Process. (ICASSP) 2002, 3: 2473-2476.Google Scholar
- Yamamoto R, Matsutani H, Matsuki H, Oono T, Ohtsuka H: Position location technologies using signal strength in cellular systems. IEEE Vehicular Technol. Conf 2001, 4: 2570-2574.Google Scholar
- Aso M, Saikawa T, Hattori T: Maximum likelihood location estimation using signal strength and the mobile station velocity in cellular systems. Vehicular Technol. Conf 2003, 2: 742-746.Google Scholar
- Zhou J, Ng J-Y: A data fusion approach to mobile location estimation based on ellipse propagation model within a cellular radio network, The 21st International Conference on Advanced Information Networking and Applications Niagara Falls, 21 - 23 May 2007. IEEE, Piscataway; 2007.Google Scholar
- Akar M, Mitra U: Variations on optimal and suboptimal handoff control for wireless communication systems. IEEE J. Selected Areas Commun 2001, 19(6):1173-1185. 10.1109/49.926373View ArticleGoogle Scholar
- Mihaylova L, Angelova D, Honary S, Bull D, Canagarajah N, Ristic B: Mobility tracking in cellular networks using particle filtering. IEEE Trans. Wireless Commun 2007, 6(10):3589-3599.View ArticleGoogle Scholar
- Achutegui K, Rodas J, Escudero CJ, Miguez J: A model-switching sequential Monte Carlo algorithm for indoor tracking with experimental RSS data. The International Conference on Indoor Positioning and Indoor Navigation (IPIN), Zurich, 15–17 September 2010. IEEE, Piscataway; 2010.Google Scholar
- Turkka J, Renfors M: Path loss measurements for a non-line-of-sight mobile-to-mobile environment The 8th International Conference on ITS Telecommunications, Phuket, 24–24 October 2008. IEEE, Piscataway; 2008.Google Scholar
- Srinivasa S, Haenggi M: Path loss exponent estimation in large wireless networks. CERN Document Server, Meyrin; 2008.Google Scholar
- Shirahama J, Ohtsuki T: RSS-based localization in environments with different path loss exponent for each link. IEEE Vehicular Technology Conference, Singapore, 11–14 May 2008. IEEE, Piscataway; 2008.Google Scholar
- Popescu I, Nikitopoulos D, Constantinou P, Nafornita I: ANN prediction models for outdoor environment. IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications, Helsinki, 11–14 September 2006. IEEE, Piscataway; 2006.Google Scholar
- Rappaport T: Wireless Communications. Prentice Hall PTR, New Jersey; 2002.Google Scholar
- Erceg V, Greenstein LJ, Tjandra SY, Parkoff SR, Gupta A, Kulic B, Julius AA: An empirically based path loss model for wireless channels in suburban environments. IEEE J. Selected Areas Commun 1999, 17(7):1999-1205.View ArticleGoogle Scholar
- Martinez K, Hart JK, Ong R: Environmental sensor networks. Computer 2004, 37: 50-56.View ArticleGoogle Scholar
- Greenstein LJ, Erceg V: Gain reductions due to scatter on wireless paths with directional antennas. IEEE Commun. Lett 1999, 3(6):169-171.View ArticleGoogle Scholar
- Wilson PF, Papazian PB, Cotton MG, Lo Y: Advanced antenna test bed characterization for wideband wireless communications, NTIA Technical report. US Department of Commerce, Washington DC; 1999.Google Scholar
- Rahman TA, Ser GB: Propagation measurement in IMT-2000 band in Malaysia environment. TENCON 2000 2000, 1: 77-81.Google Scholar
- Wang S-W, Wang I: Effects of soft handoff, frequency reuse and non-ideal antenna sectorization on CDMA system capacity. 43rd IEEE Vehicular Technology Conference, Secaucus, 18–20 May 1993. IEEE, Piscataway; 1993.Google Scholar
- Vavoulas A, Vaiopoulos N, Varoutas DA, Chipouras A, Stefano G: Performance improvement of fixed wireless access networks by conjunction of dual polarization and time domain radio resource allocation technique. Int. J Commun. Syst 2010, 24: 483-491.View ArticleGoogle Scholar
- Park J, Jang JW, Park S-G, Sung W: Capacity of sectorized distributed networks employing adaptive collaboration from remote antennas. IEICE Trans 2010, E93-B: 3534-3537. 10.1587/transcom.E93.B.3534View ArticleGoogle Scholar
- Runyon D: Optimum directivity coverage of fan-beam antennas. IEEE Antennas Propagation Mag 2002, 44(2):66-70. 10.1109/MAP.2002.1003636View ArticleGoogle Scholar
- Kay SM: Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall PTR, New Jersey; 1993.Google Scholar
- Yamashita N, Fukushima M: On the rate of convergence of the Levenberg-Marquardt methods. Comput Suppl 2001, 15: 237-249.MathSciNetGoogle Scholar
- Fan J-Y, Yuan Y-X: On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption. Computing 2005, 74: 23-39. 10.1007/s00607-004-0083-1MathSciNetView ArticleGoogle Scholar
- Madsen K, Nielsen HB, Tingleff O: Methods for Non-linear Least Squares Problems. Informatics and Mathematical Modelling Technical University of Denmark, Lyngby; 2004.Google Scholar
- Pahlavan K, Levesque A: Wireless Information Networks. John Wiley and Sons, Inc., New York; 1995.Google Scholar
- Li X: RSS-based location estimation with unknown pathloss model. IEEE Trans. Wireless Commun 2006, 5(12):3626-3633.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.