Linear Least Squares Approach for Accurate Received Signal Strength Based Source Localization
TL;DR: It is proved that the performance of the improved LLS estimator achieves Cramer-Rao lower bound at sufficiently small noise conditions and the variances of the position estimates are derived and confirmed by computer simulations.
Abstract: A conventional approach for passive source localization is to utilize signal strength measurements of the emitted source received at an array of spatially separated sensors. The received signal strength (RSS) information can be converted to distance estimates for constructing a set of circular equations, from which the target position is determined. Nevertheless, a major challenge in this approach lies in the shadow fading effect which corresponds to multiplicative measurement errors. By utilizing the mean and variance of the squared distance estimates, we devise two linear least squares (LLS) estimators for RSS-based positioning in this paper. The first one is a best linear unbiased estimator while the second is its improved version by exploiting the known relation between the parameter estimates. The variances of the position estimates are derived and confirmed by computer simulations. In particular, it is proved that the performance of the improved LLS estimator achieves Cramer-Rao lower bound at sufficiently small noise conditions.
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TL;DR: A novel semidefinite programming (SDP) relaxation technique is derived by converting the ML minimization problem into a convex problem which can be solved efficiently and requires only an estimate of the path loss exponent (PLE).
Abstract: Cooperative localization (also known as sensor network localization) using received signal strength (RSS) measurements when the source transmit powers are different and unknown is investigated. Previous studies were based on the assumption that the transmit powers of source nodes are the same and perfectly known which is not practical. In this paper, the source transmit powers are considered as nuisance parameters and estimated along with the source locations. The corresponding Cramer-Rao lower bound (CRLB) of the problem is derived. To find the maximum likelihood (ML) estimator, it is necessary to solve a nonlinear and nonconvex optimization problem, which is computationally complex. To avoid the difficulty in solving the ML estimator, we derive a novel semidefinite programming (SDP) relaxation technique by converting the ML minimization problem into a convex problem which can be solved efficiently. The algorithm requires only an estimate of the path loss exponent (PLE). We initially assume that perfect knowledge of the PLE is available, but we then examine the effect of imperfect knowledge of the PLE on the proposed SDP algorithm. The complexity analyses of the proposed algorithms are also studied in detail. Computer simulations showing the remarkable performance of the proposed SDP algorithm are presented.
231 citations
Cites background or methods from "Linear Least Squares Approach for A..."
...Here, we derive a similar technique to the one used in [4] with considering unknown transmit powers....
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...These measurements may i nclude received signal strength (RSS) [2]–[4], time-of-arrival [5], [6], time-difference-of-a rrival [7]–[9], and angle-of-arrival [10], [11]....
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...Linear estimators for RSS localization have been previously considered in the literature [4]....
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...Most studies mentioned above on RSS local ization assume that the source transmit powers are the same and known [4], [13], [17], [23]....
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TL;DR: This letter derives a weighed least squares (WLS) formulation to jointly estimate the sensor node location and the transmit power, based on the unscented transformation (UT) for the case of unknown transmit power and unknown PLE.
Abstract: In this letter, we consider the received-signal-strength (RSS) based localization problem with unknown transmit power and unknown path loss exponent (PLE). For the case of unknown transmit power, we derive a weighed least squares (WLS) formulation to jointly estimate the sensor node location and the transmit power, based on the unscented transformation (UT). For the case of unknown PLE, we propose an alternating estimation procedure to alternatively estimate the sensor node location and the PLE. The estimation procedure can also be applied to the case when both the transmit power and the PLE are unknown. Simulation results confirm the effectiveness of the proposed method.
152 citations
Cites background or methods from "Linear Least Squares Approach for A..."
...It is seen from the figure that the proposed method performs better than the LLS method in [5], in the sense that it achieves much lower biases and RMSE’s....
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...Remark 1: The above procedure is different from that in [5]....
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...Our alternating estimation procedure is different from [5] in that our procedure guarantees the estimate of the PLE being in a reasonable interval to avoid divergence....
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...For comparison, the performance of the SDR method in [2], the LLS method in [5], and the NML method in [6] is also shown here....
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...120428 an alternating method to alternatively estimate the sensor node location and the PLE, where the sensor node location is estimated using the linear least squares (LLS) method [5]....
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01 Jan 2012
TL;DR: This chapter contains sections titled: Introduction Measurement Models and Principles for source Localization Algorithms for Source Localization Performance Analysis for LocalizationAlgorithms and Conclusion.
Abstract: Time of arrival (TOA), time difference of arrival (TDOA), time sum of arrival (TSOA), received signal strength (RSS), and direction of arrival (DOA) of the emitted signal are commonly used measurements for source localization. This chapter introduces two categories of positioning algorithms based on TOA, TDOA, TSOA, RSS, and DOA measurements. The first category works on the nonlinear equations directly obtained from the nonlinear relationships between the source and measurements. Corresponding examples, namely, nonlinear least squares (NLS) and maximum likelihood (ML) estimators, are presented. The second category attempts to convert the equations to linear. The chapter discusses the linear least squares, weighted linear least squares (WLLS), and subspace approaches. It develops the mean and variance expressions for any positioning method which can be formulated as an unconstrained optimization problem. The Cramer‐Rao lower bound (CRLB), which is a lower bound on the variance attainable by any unbiased location estimator using the same data, is also discussed.
134 citations
TL;DR: This work considers simultaneous estimation of source-measurement associations and the source locations, in addition to finding the initial signal transmission time and proposes an efficient three-step algorithm that progressively simplifies the original problem through convex relaxation and sensible approximations.
Abstract: We investigate the localization of multiple signal sources based on sensors performing time-of-arrival (TOA) measurement in wireless sensor networks. Moving beyond the widely studied single source localization problem, concurrently active multiple sources substantially complicate the problem since anchored sensor nodes are unaware of associations between measured signals and source nodes. At the same time, as the total number of possible source-measurement associations grows exponentially with the number of sensor nodes, it is inefficient to attempt conventional single-source localization algorithm for each possible association in a brute-force manner. In this work, we address this difficult problem from a joint optimization perspective. Specifically, we consider simultaneous estimation of source-measurement associations and the source locations, in addition to finding the initial signal transmission time. This joint optimization problem includes both discrete and continuous variables. We propose an efficient three-step algorithm that progressively simplifies the original problem through convex relaxation and sensible approximations. Our proposed algorithm demonstrates results comparable to a genie-aided method that utilizes known source-measurement associations.
122 citations
TL;DR: Experimental results on distance estimation, location, and detection accuracy show that BLE beacon is a promising solution for an interactive smart museum.
Abstract: The Internet of Things (IoT) can enable smart infrastructures to provide advanced services to the users. New technological advancement can improve our everyday life, even simple tasks as a visit to the museum. In this article, an indoor localization system is presented, to enhance the user experience in a museum. In particular, the proposed system relies on Bluetooth Low Energy (BLE) beacons proximity and localization capabilities to automatically provide the users with cultural contents related to the observed artworks. At the same time, a received signal strength-based technique is used to estimate the location of the visitor in the museum. An Android application is developed to estimate the distance from the exhibits and collect useful analytics regarding each visit and provide a recommendation to the users. Moreover, the application implements a simple Kalman filter in the smartphone, without the need of the cloud, to improve localization, precision and accuracy. Experimental results on distance estimation, location, and detection accuracy show that BLE beacon is a promising solution for an interactive smart museum. The proposed system has been designed to be easily extensible to the IoT technologies and its effectiveness has been evaluated through experimentation.
121 citations
Cites methods from "Linear Least Squares Approach for A..."
...In [19], convex optimization is used to address the RSS-based noncooperative and cooperative localization problems, while a linear least squares estimator is proposed in [20]....
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References
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TL;DR: The Fundamentals of Statistical Signal Processing: Estimation Theory as mentioned in this paper is a seminal work in the field of statistical signal processing, and it has been used extensively in many applications.
Abstract: (1995). Fundamentals of Statistical Signal Processing: Estimation Theory. Technometrics: Vol. 37, No. 4, pp. 465-466.
14,342 citations
"Linear Least Squares Approach for A..." refers background or methods in this paper
...where is the variable for . The solution for (16) is [ 21 ]...
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...[ 21 ] Q. Guo and D. Huang, “A concise representation for the soft-in soft-out...
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...According to BLUE, the covariance matrix for , denoted by ,i s [ 21 ]...
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...The first one is a best linear unbiased estimator (BLUE) [10], [ 21 ], which is shown to be equivalent to [19] and [20] but with smaller computational requirement, while the second is its improved version by exploiting the known relation between the position and range estimates....
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TL;DR: Using the models, the authors have shown the calculation of a Cramer-Rao bound (CRB) on the location estimation precision possible for a given set of measurements in wireless sensor networks.
Abstract: Accurate and low-cost sensor localization is a critical requirement for the deployment of wireless sensor networks in a wide variety of applications. In cooperative localization, sensors work together in a peer-to-peer manner to make measurements and then forms a map of the network. Various application requirements influence the design of sensor localization systems. In this article, the authors describe the measurement-based statistical models useful to describe time-of-arrival (TOA), angle-of-arrival (AOA), and received-signal-strength (RSS) measurements in wireless sensor networks. Wideband and ultra-wideband (UWB) measurements, and RF and acoustic media are also discussed. Using the models, the authors have shown the calculation of a Cramer-Rao bound (CRB) on the location estimation precision possible for a given set of measurements. The article briefly surveys a large and growing body of sensor localization algorithms. This article is intended to emphasize the basic statistical signal processing background necessary to understand the state-of-the-art and to make progress in the new and largely open areas of sensor network localization research.
3,080 citations
"Linear Least Squares Approach for A..." refers background or methods in this paper
...[ 5 ] G. K. Kaleh and R. Vallet, “Joint parameter estimation and symbol...
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...The mean squares position errors (MSPEs) of the two proposed methods are also produced and it is proved that the performance of the improved LLS estimator achieves Cramer-Rao lower bound (CRLB) [ 5 ], [15] at sufficiently small noise conditions....
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...Time-of-arrival (TOA), timedifference-of-arrival (TDOA), received signal strength (RSS), acoustic energy and angle-of-arrival (AOA) of the emitted signal are commonly used measurements [ 5 ] for location determination....
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...which is identical to the CRLB for RSS-based positioning [ 5 ], indicating the optimality of ....
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...It is assumed that , and , are known a priori through a testing and calibration campaign [ 5 ], [19]....
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TL;DR: An effective technique in locating a source based on intersections of hyperbolic curves defined by the time differences of arrival of a signal received at a number of sensors is proposed and is shown to attain the Cramer-Rao lower bound near the small error region.
Abstract: An effective technique in locating a source based on intersections of hyperbolic curves defined by the time differences of arrival of a signal received at a number of sensors is proposed. The approach is noniterative and gives an explicit solution. It is an approximate realization of the maximum-likelihood estimator and is shown to attain the Cramer-Rao lower bound near the small error region. Comparisons of performance with existing techniques of beamformer, spherical-interpolation, divide and conquer, and iterative Taylor-series methods are made. The proposed technique performs significantly better than spherical-interpolation, and has a higher noise threshold than divide and conquer before performance breaks away from the Cramer-Rao lower bound. It provides an explicit solution form that is not available in the beamforming and Taylor-series methods. Computational complexity is comparable to spherical-interpolation but substantially less than the Taylor-series method. >
2,202 citations
"Linear Least Squares Approach for A..." refers background or methods in this paper
...Based on [ 11 ], we improve the LLS algorithm by including a second...
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...[ 11 ] J. Dauwels, S. Korl, and H.-A....
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...The RSS is different from the TOA and TDOA which are proportional to range [8]–[10] and range difference [ 11 ], [12], respectively....
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TL;DR: This work derives CRBs and maximum-likelihood estimators (MLEs) under Gaussian and log-normal models for the TOA and RSS measurements, respectively for sensor location estimation when sensors measure received signal strength or time-of-arrival between themselves and neighboring sensors.
Abstract: Self-configuration in wireless sensor networks is a general class of estimation problems that we study via the Cramer-Rao bound (CRB). Specifically, we consider sensor location estimation when sensors measure received signal strength (RSS) or time-of-arrival (TOA) between themselves and neighboring sensors. A small fraction of sensors in the network have a known location, whereas the remaining locations must be estimated. We derive CRBs and maximum-likelihood estimators (MLEs) under Gaussian and log-normal models for the TOA and RSS measurements, respectively. An extensive TOA and RSS measurement campaign in an indoor office area illustrates MLE performance. Finally, relative location estimation algorithms are implemented in a wireless sensor network testbed and deployed in indoor and outdoor environments. The measurements and testbed experiments demonstrate 1-m RMS location errors using TOA, and 1- to 2-m RMS location errors using RSS.
1,881 citations
TL;DR: It is shown that the CWLS estimator yields better performance than the LS method and achieves both the Crame/spl acute/r-Rao lower bound and the optimal circular error probability at sufficiently high signal-to-noise ratio conditions.
Abstract: Localization of mobile phones is of considerable interest in wireless communications. In this correspondence, two algorithms are developed for accurate mobile location using the time-of-arrival measurements of the signal from the mobile station received at three or more base stations. The first algorithm is an unconstrained least squares (LS) estimator that has implementation simplicity. The second algorithm solves a nonconvex constrained weighted least squares (CWLS) problem for improving estimation accuracy. It is shown that the CWLS estimator yields better performance than the LS method and achieves both the Crame/spl acute/r-Rao lower bound and the optimal circular error probability at sufficiently high signal-to-noise ratio conditions.
531 citations
"Linear Least Squares Approach for A..." refers background in this paper
...(7) and square both sides of (1) to yield [ 8 ]...
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...The RSS is different from the TOA and TDOA which are proportional to range [ 8 ]–[10] and range difference [11], [12], respectively....
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...[ 8 ] A. W. Eckford, “Channel estimation in block fading channels using the...
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