# Optimal Particle Filter Weight for Bayesian Direct Position Estimation in a GNSS Receiver
## 8 Simulations and Real-World Results
### Real-World Open Sky and Urban Scenario
Satellite PRN 27 in Figure 14 and satellite PRN 18 in Figure 15 show a significant variation in amplitude of the correlation values \(|P|\), basically due to shadowing and multipath effects. Satellite PRN 21 in Figure 15 is less affected by the environment, because the signal amplitude remains nearly constant and no other significant effects are visible. Nearly all of the probability functions face a significant bias in \(\mu\) and increased weighted standard deviation \(\sigma_{w}\) compared to a standard open sky signal. Especially the weighted standard deviation of PRN 18 at third column in Figure 16 shows a significant increase due to the lowered correlation amplitude. Furthermore, the correlation function as well as the probability function of satellite PRN 27 in Figure 14, position at time W/S 1901/317840.4 (middle row) seem to be significantly affected by multipath, which is assumed due to the flattened peak in the correlation function and the resulting deformation of the probability function. Figure 12: Frequency spectrum of the GPS L1 band with a root-mean-square (RMS) of the samples evaluating to \(\beta_{s_{\mu},RMS}=1.71\). At frequency offset of approximately 2.5 MHz a continuous wave (CW) interference peak is present. The dataset was recorded with a measurement van during a measurement campaign. Based on the CW interference being present during the complete measurement run, it is assumed that it was caused by one of the active on-board measurement instruments or radio connections. Additionally, it is assumed that the present CW interference does not influence the measurement because it is significantly outside the main lobe of the analysed GPS L1 C/A (Coarse/Acquisition) signal. Figure 14: Real-world urban scenario of satellite GPS L1 C/A PRN 27. The plots show from left to right three epochs at different places in a urban environment, referring to the red dots from left to right in Figure 17. It is expected that the correlation values \(|P|\) vary significantly due to shadowing and multipath. The weighted mean is shifted for all positions and both the correlation and probability function of the second point (middle row) seem to be significantly affected by multipath. The red crosses in the upper plots show the correlation values at code offset \(\tau\). The black line in the upper plot shows the sinc interpolated correlation values, which are used to obtain the weights \(w_{k}^{i}\) shown in the lower plots. Figure 13: Real-world open sky scenario of satellite GPS L1 C/A Pseudo-Random-Noise (PRN) number 12. The plots show from left to right three epochs referring to GPS Week/Second W/S. The weighted standard deviations \(\sigma_{\rm w}\) approach the theoretical lower limit of \(\sigma_{\Delta\tau}=3\) m for this ideal case. The dataset was recorded on the roof at LAT = 47.06446263 deg, LON = 15.40777110 deg at the Reininghausstrasse 13a, Graz, Austria. The red crosses in the upper plots show the correlation values at code offset \(\tau\). The black line in the upper plot shows the sinc interpolated correlation values, which are used to obtain the weights \(w_{k}^{i}\) shown in the lower plots. For clarification and as noted briefly above, all presented real-world scenarios show the resulting probability function with respect to the latest PVT estimate, not to an high accurate absolute reference. Figure 16: Real-world urban scenario of satellite GPS L1 C/A PRN 18. The plot content is analogical to Figure 14. This satellite signal is significantly affected by the environment. From the azimuth of satellite PRN 18 and the location of the buildings as shown in Figure 17, it can be assumed that the GNSS signal is blocked at the first and last positions, which fits to the amplitude of the correlation values. Interestingly, in the case of the third position, the significantly small correlation value leads to an increase in variance estimate. Figure 12.0 Real-world urban scenario of satellite GPS L1 C/A PRN 21.