# Optimal Particle Filter Weight for Bayesian Direct Position Estimation in a GNSS Receiver
## 1 Introduction  
The Position, Velocity and Time (PVT) estimates of a Global Navigation Satellite System (GNSS) receiver are sometimes not sufficiently accurate, in particular in difficult environments such as urban areas, forests or indoors. In such environments, the observations do not follow a Gaussian distribution. This fact adds a bias to the PVT solution, which can be generally described as close to Gaussian distributed [1; 2; 3; 4; 5]. Thus, the PVT estimation in difficult environments can benefit from Sequential Monte Carlo (SMC) methods because they consider all available statistical information in the estimation process. Particle filters [6; 7] belong to the group of non-parametric Bayesian filters and are an implementation of an SMC method [8]. Thus, they allow for nonlinear and non-Gaussian distributed system and observation models. The performance comparison between particle filters and the Extended Kalman Filter (EKF) within a nonlinear and non-Gaussian environment shows that improved position estimates can be expected using particle filters with the drawback of computational intensive processing [9]. The application of particle filters in the field of PVT estimation can mostly be found in research. Often, particle filters are applied in tracking multiple objects (see e.g., [10]). For this purpose, particlefilters are used to combine different sensors with the GNSS receiver PVT estimate or the receiver observables, pseudorange and Doppler. For instance, the shadow matching algorithm is based on a particle filter that aims to improve the positioning results in dense urban areas by fusing three-dimensional geometry information of buildings with the PVT estimates from GNSS signals [11]. In [12], the indoor positioning results are improved by a particle filter that fuses PVT estimation from GNSS signals with high power 4G Long Term Evolution (LTE) mobile network signals compared to GNSS signals. Furthermore, in [13], a particle filter is used to improve the position estimate to support visual impaired people by fusing non-Gaussian distributed probabilistic maps with GNSS. Apart from these applications that combine GNSS PVT estimates with other sensors, particle filters are also applied within the Bayesian Direct Position Estimation (BDPE) [14; 15] that aims to estimate the PVT of a GNSS receiver directly from the signal. This promising direct positioning and signal processing approach increases the sensitivity in PVT tracking by 10 \(log_{10}\)(N)-dB, where N is the number of used GNSS signals [16]. Thus, this method was investigated in different GNSS related topics like GNSS signal acquisition performance in [17; 18; 19], improved positioning accuracy when fusing GNSS with visual information in [20] or improving the robustness of position and time estimation during jamming and meaconing attacks in [21; 22]. The first real-time capable implementation of BDPE into a commercial software based GNSS receiver was shown in [23], even if suboptimal but computational efficient algorithms and limitations in number of tracked GNSS signals have been made as a tradeoff to achieve real-time performance. The previously mentioned work presents the concept of particle filters and their elements, performance and implementation [6; 7; 8; 9]. Current research topics of particle filters in GNSS are given in [10; 11; 12; 13]. The concept and first implementations of (B)DPE can be found in [14; 15; 16; 17; 18; 19; 20; 21; 22; 23], but the optimal weight update for BDPE is still an open topic. One major aspect of BDPE is to perform an ideal weight update during incorporation of a new set of measurements. Therefore, the goal of this work is to derive an optimal particle weight update function for multiple GNSS signals, to propose a solution for numerical stability during the update steps and to analyze the method using representative scenarios.