Auto-regressive modeling of shadowing for RSS mobile tracking Hadi Noureddine Abstract—In this paper, we consider the tracking of mobile terminals based on the received signal strength (RSS) measured from several base stations. The spatial correlation of the random shadowing is exploited in order to improve the position tracking. We define an auto-regressive (AR) model of the temporal evolu- tion of the shadowing. This model allows for performing a joint tracking of the position and the shadowing by applying a Rao- Blackwellized (RB) particle filter approximating the posterior probability distributions numerically. The simulation results show that the tracking can be improved by considering sufficiently high auto-regressive orders. I. I NTRODUCTION Navigation tasks and fleet management are among the first applications of positioning and tracking. These applications have been diversified to include geo-localization of emergency calls, network control, and a variety of location-based services. Wireless networks have been considered as a supplement or an alternative to the GPS solution for localizing mobile terminals. Several position dependent properties of the radio signals can be exploited and several solutions have been developped [1]. The position tracking relies on the measurements obtained from the radio signals or from the outputs of an inertial navigation system (INS), and on a displacement model. It performs the estimation of the hiden state vector composed of the position and other parameters of interest. In this paper, we focus on mobile tracking using the received signal strength (RSS) measurements. The obstacles in the propagation path between a user equipment (UE) and the base station (BS) cause attenuations in the form of slow fading or shadowing. The shadowing is usually assumed to follow a spatially correlated log-normal distribution [2][3]. As the UE moves, the spatial correlation is transformed into a temporal one. Several RSS-based algorithms have been proposed in order to improve the position tracking in presence of random shad- owing. In [4], a predicition of the shadowing, modeled by a first-order auto-regressive Gaussian process, is used along with the RSS measurements for estimating position. Unfortunately, this solution is sub-optimal, and can be improved by using a probabilistic approach. Thus, our solution is based on Bayesian filtering, which efficiently exploits the incoming measurements by recursively updating the posterior probability distribution. The update is performed by taking the shadowing as a part of the state vector, whose stochastic process is no more Markovian. The transition equations of this process are obtained by an auto-regressive modelling of the shadowing evolution. As a remark, an alternative approach was used in a general context in [5], where the shadowing was considered as a mea- surement noise and the temporal correlation was accounted for in evaluating the likelihood function. Recursive Monte Carlo methods, also known as particle filters [6][7], allows for approximating the probability density functions which are analytically untractable because of the non-linearity of the AR model and the RSS measurements with respect to the position. More specifically, a Rao-Blackwellized particle filter will be implemented where the part of the state vector consisting of the position and its derivatives is represented by particles and the shadowing part is tracked by means of a Kalman filter. This solution has the advantage of reducing the needed amount of particles. This paper is organized as follows. In section II, the transi- tion model describing the evolution of the state vector and the observation model are presented. In section III, we introduce the auto-regressive modeling of the shadowing evolution. The RB particle filter for solving the tracking problem is presented in section IV. In section V, we show that for a particular case of a collinear trajectory and under an exponential correlation function, the shadowing has a first order AR(1) model. Simulation results and conclusions are presented in sections VI and VII. II. SYSTEM MODEL In this section, we describe the transition model of the hidden state vector and the observation model. They allow for computing the a priori and likelihood probabilities in the bayesian filtering processing, respectively. A. Transition model We define the kinematic vector c k at time kT , where k ∈ N and T is a time step, comprising the UE position x k = [x k ,y k ] T and other kinematic parameters (e.g. velocity). This vector is issued from a known Markov process of transition probability p(c k |c k−1 ) and initial distribution p(c 0 ). We denote Ω k =[ǫ 1 (x k ), ··· ,ǫ N BS (x k )] T , where ǫ i (x k ) is the shadowing vaule in decibel (dB) corresponding to the i-th base station out of N BS . The shadowing is assumed invariant in time for a fixed position. The state vector s k is defined by s k =[c T k , Ω T k ] T . (1) We can decompose the transistion equation as p(s k |s 0:k−1 )= p(c k |c k−1 )p(Ω k |x 0:k , Ω 0:k−1 ) (2)