The Problem of Localization in Networks of Randomly Deployed Nodes: Asymptotic and Finite Analysis, and Thresholds

This paper deals with a network composed of two sets of nodes randomly distributed over a two dimensional domain S ⊆ ℜ 2 following two statistically independent Poisson point processes with intensities ρ L and ρ N L . The first process is associated with the nodes that have a-priori knowledge about their position (these are the so called L-nodes), while the other point process is associated with the nodes that are trying to localize themselves (these are the so called non-localized or NL-nodes). In particular, the paper focuses on the connection between some system level parameters and the node localization probability in a Poisson distributed configuration of Fred Daneshgaran is with ECE Dept., CSU, Los Angeles, USA. Massimiliano Laddomada and Marina Mondin are with DELEN, Politecnico di Torino, Italy. nodes, which are at the basis of topological network control. We do not propose any new or modified localization method. As it will become clear later, the primary assumptions in our analysis are: a) nodes are Poisson distributed over a bounded circular domain contained in ℜ 2 and b) each node has an average typically circular footprint representing its radio coverage. Hence, while we focus on a particular example involving range measurements using Received Signal Strength (RSS), the analysis can be applied to other range measurement methods as well. Notice that Poisson point processes are useful for modelling scenarios in which the deployment area, the number of deployed nodes, or both, are not a-priori known. The Poisson model is in fact a good approximation of a binomial random variable when the number of deployed nodes over a bounded domain is high while the node density is constant across the whole region of interest [1]. Nevertheless, the Poisson approximation leads in many cases of interest to a mathematically tractable problem.

This general framework can be recognized in many practical scenarios. A possible example is a Distributed Sensor Network (DSN), in which one may be interested in distributed power efficient algorithms to derive localization information in a randomly distributed collection of severely energy and computation power limited nodes. A second example may be that of a wireless network, in which the various network elements may communicate between themselves (in the case of wireless networks allowing peer-to-peer communication) or with a subset of nodes whose positions are known (this is the case of classic cellular networks and WLANs, whereby every node must communicate with at least one base-station or access point). With this scenario in mind, let us provide a brief overview of the localization methods that have been proposed in the literature.

Given the great difference between the communication and computation capability of the nodes, as exemplified by the DSN and WLANs, algorithms developed for localization should be tailored to the particular scenario at hand [2], [3].

Practical localization algorithms can be classified in at least two ways: centralized or distributed [2] and rangefree or based on ranging techniques [4]. The most common techniques are based on measured range, whereby the location of nodes are estimated through some standard methods such as triangulation. Cramer-Rao Bounds (CRBs) on the variance of any unbiased estimate based on the above ranging techniques are readily available and provide a benchmark for assessing the performance of any given algorithm [5], although we should note that the derivation of the CRB itself relies on a probabilistic model (often assumed to be Gaussian), that describes the connection between the parameter to be estimated and the raw observations.

In range-free localization, connectivity between nodes is a binary event: either two nodes are within communication range of each other or they are not [6]. For simplicity, we may view this event as obtained from hard quantization of, for instance, a RSS random variable. If RSS is above a certain detection threshold, the nodes can communicate, otherwise they cannot. Of course, the nature of path loss and the terrain characteristics influence both the coverage radius and the deviation of the coverage zone from the ideal circular geometry. In a typical scenario there may be multipath, Multiple Access Interference (MAI) and Non Line Of Sight (NLOS) propagation conditions [2]. Various range free algorithms have been proposed in the literature including the centroid algorithm [7], the DV-HOP algorithm [8], the Amorphous positioning algorithm [9], APIT [10], and ROCRSSI [4].

A review of various localization techniques proposed in the literature may be found in [11]. In [12], the authors propose an approach based on connectivity information for deriving the locations of nodes in a network. In [13], the authors present some work in the field of source localization in sensor networks.

A topic somewhat related to the problem dealt with in this paper is network connectivity. This topic has receive

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