1089-7798 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LCOMM.2018.2849963, IEEE Communications Letters 1 RSS Localization Using Unknown Statistical Path Loss Exponent Model Rouhollah Sari, Hadi Zayyani, Member, IEEE, Abstract—Due to low complexity, Received Signal Strength (RSS) source localization in Wireless Sensor Network (WSN) has gained upsurge of interests in recent years. In this letter, we consider the Path Loss Exponent (PLE) as a random variable in the lognormal shadowing model. Assuming a general distribution function for the PLE, the nonlinear equation for finding the Maximum Likelihood (ML) distance estimator has been derived. Then, an approximated closed- form formula for the ML distance estimator is derived for uniform distribution of the PLE. Moreover, a Bayesian Minimum Mean Square Error (MMSE) estimator has been calculated for the PLE. Finally, localization is performed by a classical linear least square approach. Simulation results show the efficacy of the proposed algorithm in comparison to other methods. Index Terms—RSS localization, Unknown path loss exponent, Wire- less sensor network, Maximum likelihood estimator I. I NTRODUCTION W Ireless sensor networks have a broad spectrum of appli- cations in navigation, surveillance, habitat monitoring and other tactical applications [1]. In most applications, we must know, first, not only the measurement data but also the location where the data are obtained from [1]. There are several representative localization algorithms based on features of received signal or, equivalently, information available for localization [2], [3]. Time of arrival and time difference of arrival are two most common algorithms [1], [2], [4]. In these two methods, accurate localization can be obtained but accuracy is dependent on radio frequency bandwidth and time synchronization [1]. In contrast, the localization based on received signal strength is simple to implement and not sensitive to timing and radio frequency bandwidth [1]. In free space, signal power is inversely proportional to the square of the distance between the transmitter and receiver. However, in complex environment, due to multipath fading, this is not the case [5]. Because of the complexity, it is not straightforward to obtain a single RSS model for all environments [6]. Lognormal shadowing model is a simple model that shows the essence of signal propagation. The lognormal shadowing model considers the fact that the surrounding environmental clutter may be vastly different at two different location having the same transmitter receiver separation[8]. Also, we assume that the environment is uniform, so that, we can consider that the PLE in the lognormal shadowing model is the same for all sensors. This choice is a tradeoff between complexity and real signal propagation models [6]. RSS localization is inherently statistical because the PLE has statistical properties. Since the communication channel is dynamic, it varies over different scenarios and different locations [7]. Based on empirical results, the PLE takes a range of values (normally between 1 and 6) [8]. In practice, measurements for the PLE estimation carried out in almost all indoor and outdoor environments [9]. Recently, a considerable effort has been made by researchers on RSS R. Sari and H. Zayyani are with the Department of Electrical and Com- puter Engineering, Qom University of Technology (QUT), Qom, Iran (e-mails: rohol.sari@gmail.com, zayyani@qut.ac.ir). localization. Some researchers consider PLE as a known parameter while some others try to estimate it. Authors in [10] considered the path loss exponent as a known parameter and a maximum likelihood estimator for distance has been derived. In [11], assuming a known PLE, a Linear Least Square (LLS) approach is proposed for localization. This algorithm works effectively because it has the knowledge of the PLE a priori. The PLE is a key parameter in the LDPL model which will be intro- duced in section II. The model will deviate if we do not know PLE priori and this leads to higher localization error. Hence, it should be estimated from the RSS measurements. Assuming an unknown PLE, authors in [12] proposed a search method for the PLE. First, an approach is used to eliminate the uncertainty of transmitting power. Then, combined with a search method for the PLE under certain constraints, linear least square is utilized to determine the location of the source node [12]. In [13], a cost function is introduced for the localization error. Incremental values of PLE, between 2-5 are inserted in the cost function. The optimum PLE is the one which minimizes the cost function [13]. The authors in [14] proposed a weighed least squares formulation to jointly estimate the source location and the PLE. In [15], the authors consider the PLE as a random variable with Gaussian distribution. Then, a maximum a posteriori estimator is formulated that iteratively estimates the location coordinates. Leus in [7], proposed a closed-form total least squares method to estimate the PLE in which a singular value decomposition is computed for the PLE estimation. In this letter, lognormal shadowing model with a statistical path loss exponent is considered. Also, the transmitter power has been eliminated by taking a node as reference and then all measurements have been subtracted from reference node measurement. With these assumptions, a general nonlinear equation is derived for ML distance estimator, when the PLE has a general distribution. As a special case, we obtain an approximated closed form formula for ML distance estimator when the PLE has a uniform distribution. Furthermore, we derive a Bayesian MMSE estimator for the PLE. In short, the new algorithm decreases the complexity because it uses a closed form formula for distance estimator instead of search method. Due to closed form formula, the number of calculations and the simulation run time are considerably decreased. Also, the new algorithm does not need to know PLE and transmitter power priori and it uses uniform distribution for PLE. The rest of the paper is as follows: In section II, lognormal shad- owing model is introduced. In Section III we develop the closed- form nonlinear equation for ML estimator followed by uniform distribution as a special case. Also, a Bayesian estimator for the PLE is obtained in this section. In Section IV, an LLS approach for localization is reviewed. Simulation results are presented in Section V. Then, some conclusions are drawn in section VI. II. PROBLEM FORMULATION In the proposed method, the WSN consists of N sensors that are located at [x n ,y n ], (1 ≤ n ≤ N ). Also there is a source at