Upper bounds on position error of a single location estimate in wireless sensor networks
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Upper bounds on position error of a single location estimate in wireless sensor networks Mohammad Reza Gholami1* , Erik G Ström1 , Henk Wymeersch1 and Sinan Gezici2
Abstract This paper studies upper bounds on the position error for a single estimate of an unknown target node position based on distance estimates in wireless sensor networks. In this study, we investigate a number of approaches to confine the target node position to bounded sets for different scenarios. Firstly, if at least one distance estimate error is positive, we derive a simple, but potentially loose upper bound, which is always valid. In addition assuming that the probability density of measurement noise is nonzero for positive values and a sufficiently large number of distance estimates are available, we propose an upper bound, which is valid with high probability. Secondly, if a reasonable lower bound on negative measurement errors is known a priori, we manipulate the distance estimates to obtain a new set with positive measurement errors. In general, we formulate bounds as nonconvex optimization problems. To solve the problems, we employ a relaxation technique and obtain semidefinite programs. We also propose a simple approach to find the bounds in closed forms. Simulation results show reasonable tightness for different bounds in various situations. Keywords: Wireless sensor networks; Positioning problem; Projection onto convex set; Convex feasibility problem; Semidefinite relaxation; Position error; Worst-case position error
1 Introduction Position information is often one of the vital requirements for wireless sensor networks (WSNs), especially for location-aware services [1]. Position information can be extracted via GPS but also from the network [2]. During the last few years, a vast number of positioning algorithms have been proposed in the literature [1,3-6], just to cite a few. Such algorithms can be assessed in different ways, for example on the basis of complexity, accuracy, or coverage [6]. Accuracy is one of the performance measures that is commonly used to evaluate positioning algorithms. In the literature, the accuracy metric has been studied widely through the position error, defined as the norm of the difference between the estimated and the true position [1,6]. For instance, the Cramér-Rao lower bound, employed to evaluate position estimates, provides a lower bound on the variance of any unbiased estimator ([7], chap. 3).
*Correspondence: [email protected] 1 Division of Communication Systems, Information Theory, and Antennas, Department of Signals and Systems, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden Full list of author information is available at the end of the article
In addition to the lower bound assessment, for some applications, it may be useful to know the maximum position error contained in an estimate of the target node position [1,8,9]. For example, it can be imagined that a specific service can be offered to a user if its maximum location error is smaller than a
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