Localization Algorithms under Correlated Shadowing in Wireless Sensor Networks

International Journal of Advanced Research in Engineering Applications, Volume 1, Issue 1, 6-8, 2014

Localization Algorithms under Correlated Shadowing in Wireless Sensor Networks Sameera V Mohd Sagheer Department of Electronics and Communication Engineering KMCT College of Engineering for Women Calicut, Kerala, India

Abstract- The existing models for radio channel shadowing was considered to be independent. However the shadowing on real world links are dependent. In this paper we review different wireless sensor network localization algorithms which considers the the shadowing between different links to be correlated. For this we have used the NeSH model.The shadowing between the links is modelled as log normal shadowing path loss model. The algorithms in the different works are analysed and their results are discussed.

Keywords— Wireless networks, adhoc networks, networking, localization, correlated shadowing ——————————  ——————————

1 INTRODUCTION

2 PROBLEM FORMULATION

Wireless sensor network has emerged as a major research area.Determining the location of sthe sensor nodes in the network has become an area of crucial research. Many solutions have been put forth for this problem. An obvious solution to this is to place GPS satellite receivers on the sensors. However this method is expensive and hence infeasible. A more realistic method is to assume that a small have knowledge of their positions. These nodes are known as anchor nodes. Then using the information that is communicated between these known sensor nodes the position of these unknown sensors are estimated. Currently time of arrival (TOA) and received signal strength(RSS) are used to determine the pairwise distances. Wireless sensor localization problem deals with the determination of the unknown position of sensor nodes using the position of anchor nodes and the distance between the sensor nodes (determined by TOA or RSS). The similar procedure has been applied in molecular conformation and distance geometry. Many localization algorithms have been proposed. They vary in their approaches.

Consider a wireless sensor network in with nodes. Here the anchors whose positions are known are m in number and is given by for . The sensor whose positions are known are in number and is denoted by for . The Euclidean distance between any two sensor nodes and is represented by . Let the distance between any sensor and anchor node which is unknown be represented by . Let ̂ be the noisy estimate of which is known where are the coordinates of sensor pairs. Let the set of all anchor sensor pairs be denoted by for which ̂ is known. The sensor localization problem can be stated as under: Estimate the position of unknown sensor nodes for with overall localization error minimized provided that we are given the position of the anchor nodes for . The distance estimate between the sensor nodes ̂ and between the anchor and sensor ̂ for .

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International Journal of Advanced Research in Engineering Applications, Volume 1, Issue 1, 6-8, 2014

3 SHADOWING MODEL

The NeSH model formulates the shadowing on a link as a normalized integral of the over the link endpoints .

3.1 Log normal shadowing A number of algorithms have been proposed and many of them used the log normal shadowing path loss model [6].

In this model, the distance estimates are given as the actual distance corrupted by additive Gaussian noise with zero mean and variance proportional to the distance. Consider a multi link sensor network. The links present in the network can be of three types: anchor to anchor, anchor to sensor or sensor to sensor. We assume the link to be symmetric, i.e. received power from node 1 to 2 is same as the received power from node 2 to 1. Since all the links are symmetric, we can assign a single received power (dBm) for link . The log normal shadowing path loss model states that the power at the receiver is inversely proportional to the transmitter receiver separation distance raised to some exponent (known as path loss exponent) and attenuated by a log normally distributed random variable. That is, for link , the received power in linear units Watts is given by

)

(equations 4) where is the transmit power,

∫

Single link properties: The NeSH model agrees with two important empirically observed link shadowing properties.

Property 1: The variance of link shadowing is approximately constant with path length.

Property 2: Shadowing losses are Gaussian.

Joint link property.

Given two links , belong to are functions of the same shadowing field . Thus NeSH model introduces correlation between them. The covariance between and is given by

√

∫ ∫

The received power between (1)two nodes represented by consists of losses due to both shadow fading and non shadow fading . NeSH model assumes that is independent and is modeled as Gaussian (2) with zero mean and variance .

( )

(

√

is the path

loss at close in reference distance , is the unitless path loss exponent, is the length of link and is the normally distributed random shadowing with zero mean and as variance.

In short we can say that the total fading on a link connecting two nodes is given as where is shadow fading loss and is non-shadow fading loss. It is assumed that and are independent and Gaussian.

Hence is also Gaussian with variance

3.2 NeSh Link Correlation Model

4 ALGORITHMS It is assumed that the shadowing on the links are correlated. Gudmundson [2] modeled shadow fading correlation for a mobile link with common base station. But this model does not provide correlation for links that [4] does not have common end points. Hence this model cannot be used in adhoc sensor networks. The NeSH model considers shadowing losses to be a function of that environment. Hence the losses in geographically proximate links are correlated. The shadowing is caused by an environment in an underlying spatial loss field . The NeSH model assumes that is an isotropic wide sense stationary Gaussian random field with zero mean and exponentially decaying spatial correlation.

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Dhalaan [1] approaches this as a non convex optimization problem. The localization problem can be written as convex SDP.

where be the vector of unknown distance square .

Vaghefi et al [10] defines a Cramer Rao Lower Bound which is a lower bound on the performance of any unbiased estimator. They formulated the maximum likelihood estimator as a Semi Definite Program.

Yang et al [12] used correlation among shadowing (3) losses to detect the location of the anchor nodes. An ML

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International Journal of Advanced Research in Engineering Applications, Volume 1, Issue 1, 6-8, 2014 Estimator and Kalman filter is combined to track mobile location and shadowing losses. The ML location estimator determines the target location. The Kalman filter determines the shadowing losses.

In [7]the Stochastic Proximity Algorithm was used for improved performance. The same correlated shadowing model discussed earlier was used.

6 CONCLUSION This paper reviewed localization algorithms that consider correlated shadowing for localization in wireless sensor networks. Most papers available in literature consider the errors in ranges used for localization as independent. This results in poor performance when the algorithms are deployed in real conditions. The performance improves when the correlation is considered.

5 SIMULATIONS, RESULT AND DISCUSSION Dhalaan et al [1] placed the nodes in a 5 5 area. The covariance matrix was formulated using the NeSH model.

Two nodes were assumed to be connected. The received power between them was greater than the threshold power.

Two situations were consideredd. First in which the covariance matrix was known, second in which the covariance matrix was unknown. The root mean squared distance was used as the measurement matrix.

Vaghefi et al [10] in his work fix the position of anchor nodes and place the sensor nodes in an area of . Uncorrelated and correlated shadowing were considered. The cumulative distribution function of the localization error was computed. The comparison of the proposed algorithm was dodne with other existing algorithms. The Cramer Rao Lower Bound was also calculated. For uncorrelated case, and and for correlated case, and . In both cases, the proposed SDP was found to be close to that of the Maximum Likelihood Estimator. It also performs better than other SDP algorithms [5, 8, 3, 9, 11].

The average running time of estimators were also compared in the same network having and . It is seen that among all the SDP algorithms compared in the paper, the proposed SDP algorithm has highest accuracy and lowest running time.

Sameera et al [7] solve the localization problem using the Stochastic Proximity Embedding. Different factors were considered, which included the standard deviation of the error model, the number of anchor nodes and various SPE parameters such as number of cycles, steps, learning parameter etc. The proposed algorithm was then compared with [1] and the result was found to be superior. Here the performance metric used was localization error.

Yang et al [12] considers a moving target and tracking is done on it. Although the tracking algorithm often does not give the exact path, it gives the correct direction.

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