# Optimal Particle Filter Weight for Bayesian Direct Position Estimation in a GNSS Receiver
## 9 Conclusions  
This paper discusses Bayesian Direct Position Estimation and describes a real-time capable implementation in a software based receiver. The Bayesian estimation framework is described in detail and a link to direct position estimation is established. The work shows the derivation of the optimal particle weight for BDPE and delivers a solution for performing the weight update in the logarithmic scale to overcome limited precision of digital computation. It further points out problems within Bayesian filters when performing the optimal particle weight update directly. Therefore, the paper proposes a method that introduces an additional nuisance parameter on the code measurement to cover the residual user range errors. The impact on several scenarios for a single GPS C/A BPSK(1) signal is given. These scenarios include a brief discussion on the impact of different signal strengths, different code delay bias variances, constructive and destructive multipath and different multipath amplitudes. Additionally, two real-world examples show the results of the optimal particle weighting applied on ideal open sky data and on multipath-prone urban environment data. The work extends the original particle filter from [23], which followed a heuristic approach for the particle filter weight update. The original filter used a suboptimal chosen exponential weight update function, which causes a dense concentration of the particle cloud. In order to avoid such a dense  
Figure 17: Environment for the urban scenario with three red measurement points. The driving direction was to the east, thus the red measurement points refer from left to the right column in Figure 14, Figures 15 and 16. The upper left plot shows the satellite constellation. The three analyzed satellites PRN 27, PRN 21 and PRN 18 are marked with a black circle. The measurement was taken in the Steyrergasse in Graz, Austria. The point at W/S 1901/317839.4 refers to LAT = 47.06430622 deg, LON = 15.45391867 deg. Map image © 2017 Google, Landsat/Copernicus  
concentration, the process noise was increased to cover also the residual user range errors. This is a working and viable approach to cover these errors, but it has caused rapid resampling steps of the particle filter. From the proposed method, it is expected that a Bayesian filter can perform an optimal particle weight update with numerical stability. Furthermore, it is expected that the particle filter positioning performance and robustness will improve. The positioning benefits from the optimal particle weight update and it is expected that the estimated position becomes more accurate and the estimated accuracy of the position becomes more reliable. The introduced nuisance parameter allows for overlapping GNSS signals in the PVT domain in the case of unmodeled residual user range errors, which is expected to make, on one hand, the filter more robust and, on the other hand, reduce the number of resampling steps. [PERSON] implemented the particle filter, derived the formulas, did the analysis and wrote the paper. [PERSON] supported the mathematical derivation of the equations. [PERSON] provided the concept for the introduced nuisance parameter and initiated the work. This research received no external funding. The authors would like to thank IFEN GmbH, Poing, Germany, for providing the GNSS Software Receiver. Moreover, we want to acknowledge Professor [PERSON] from the Graz University of Technology, Institute of Geodesy, Austria, for helpful discussions on BDPE. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: