A Statistically Modelling Method for Performance Limits in Sensor Localization

In this paper, we study performance limits of sensor localization from a novel perspective. Specifically, we consider the Cramer-Rao Lower Bound (CRLB) in single-hop sensor localization using measurements from received signal strength (RSS), time of arrival (TOA) and bearing, respectively, but differently from the existing work, we statistically analyze the trace of the associated CRLB matrix (i.e. as a scalar metric for performance limits of sensor localization) by assuming anchor locations are random. By the Central Limit Theorems for $U$-statistics, we show that as the number of the anchors increases, this scalar metric is asymptotically normal in the RSS/bearing case, and converges to a random variable which is an affine transformation of a chi-square random variable of degree 2 in the TOA case. Moreover, we provide formulas quantitatively describing the relationship among the mean and standard deviation of the scalar metric, the number of the anchors, the parameters of communication channels, the noise statistics in measurements and the spatial distribution of the anchors. These formulas, though asymptotic in the number of the anchors, in many cases turn out to be remarkably accurate in predicting performance limits, even if the number is small. Simulations are carried out to confirm our results

Error Propagation in Sensor Network Localization with Regular Topologies

Location information for sensors in wireless sensor networks (WSNs) is essential to many tasks. In the presence of noise, locations must be estimated and thus the errors are unavoidable. Moreover, the errors can propagate (i.e. increase) as sensors progressively more distant from anchors are localized. Understanding the rules governing error propagation is quite helpful to deploying WSNs and improving performances of localization systems. In this paper, we investigate error propagation measured by the Cramer-Rao Lower Bound (CRLB) in a type of regular 1-Dimensional WSNs whose Fisher Information Matrices are symmetric band Toeplitz matrices. Approximate analytic formulas for the CRLBs in the regular and almost regular WSNs are derived, and properties of error propagation are also obtained. In addition, we derive a magic number relating to the number of range measurements, which indicates a turning point as to system localization accuracies

Analyzing Error Propagation in Range-based Multihop Sensor Localization

Location information for sensors in wireless sensor networks (WSNs) is essential to many tasks. If sensors are mobile and are to be controlled to certain locations, localizability is indispensable. In a noisy environment, locations must be estimated, an

Understanding error propagation in multihop sensor network localization

In multihop localization procedures where not every node at unknown positions (i.e., sensors) can directly measure distances to nodes at known positions (i.e., anchors), sensor localization errors normally propagate (i.e., increase) as sensors progressiv

Analyzing localization errors in one-dimensional sensor networks

One-dimensional sensor networks can be found in many fields and demand node location information for various applications. Developing localization algorithms in one-dimensional sensor networks is trivial, due to the fact that existing localization algorithms developed for two- and three-dimensional sensor networks are applicable; nevertheless, analyzing the corresponding localization errors is non-trivial at all, because it is helpful to improving localization accuracy and designing sensor network applications. This paper deals with localization errors in distance-based multi-hop localization procedures of one-dimensional sensor networks through the CramérRao lower bound (CRLB). We analyze the fundamental behaviors of localization errors and show that the localization error for a sensor is locally determined by network elements within a certain range of this sensor. Moreover, we break down the analysis of localization errors in a large-scale sensor network into the analysis in small-scale sensor networks, termed unit networks, in which tight upper and lower bounds on the CRLB can be established. Finally, we investigate two practical issues: the applicability of the analysis based on the CRLB and the optimal anchor placement

Noisy localization on the sphere: Planar approximation

In real localization systems, noise from hardware and the environment makes it impossible for any algorithms to precisely localize objects, e.g. sensors and targets. Besides that, planar approximation is another source of error when dealing with localization over spherical surfaces, e.g. the surface of the earth, though it is neglected in many algorithms. This work deals with evaluating the error arising from planar approximation in localization problems over spherical surfaces. We characterize the error as arising for two different though related causes, and accordingly introduce concepts of radial error and angular error to account for these. A localization algorithm, based upon a Cayley-Menger Determinantal condition introduced recently for localization problems, is utilized for the analysis, and analytical results are confirmed through a number of simulations. As an end result of the study, we characterize the regions over which a planar approximation will be satisfactory, given an upper bound on the acceptable error it introduces in comparison with treating localization as a task in three-dimensional space

On the Performance Limit of Sensor Localization

In this paper, we analyze the performance limit of sensor localization from a novel perspective. We consider distance-based single-hop sensor localization with noisy distance measurements by Received Signal Strength (RSS). Differently from the existing studies, the anchors are assumed to be randomly deployed, with the result that the trace of the associated Cramér-Rao Lower Bound (CRLB) matrix becomes a random variable. We adopt this random variable as a scalar metric for the performance limit and then focus on its statistical attributes. By the Central Limit Theorems for U-statistics, we show that as the number of anchors goes to infinity, this scalar metric is asymptotically normal. In addition, we provide the quantitative relationship among the mean, the standard deviation, the number of anchors, parameters of communication channels and the distribution of the anchors. Extensive simulations are carried out to confirm the theoretical results. On the one hand, our study reveals some fundamental features of sensor localization; on the other hand, the conclusions we draw can in turn guide us in the design of wireless sensor networks

On the Performance Limit of Single-Hop TOA Localization

In this paper, we analyze the performance limit of sensor localization from a novel perspective. We consider distance-based single-hop sensor localization with noisy distance measurements by time of arrival (TOA). Differently from the existing studies, t