# 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  
The real-world tests evaluate the probability function with realistic data. Therefore, two scenarios have been selected: (1) an open sky case on a rooftop antenna in order to see the probability function under ideal conditions and (2) an urban case, where multipath is present and a significant shift and deformation on the probability function should be present. The data was recorded in Graz, Austria and post-processed with the SX3 software receiver. The receiver was configured to dump the multi-correlator maps to files, which contain the correlation values \(|P|\), which are related to the latest PVT. The multi-correlator maps contain correlation values for a range of code phases \(\tau\) and Dopplers \(f_{d}\), as shown in (3) of Figure 1. For all real-world tests, the red crosses in the upper plots correspond to the correlation values along the Doppler bin, which contains the correlation maxima. In order to obtain a high-resolution probability function, the correlation values (red crosses) have been interpolated with sinc function (black line). It is of major importance that the input correlation values \(|P|\) are of correct amplitude because it basically drives the shape of the probability function and variance estimate. In order to obtain the same amplitude for \(|P|\) as in Equation (15), all amplitude scaling elements have been verified in the receiver. The GNSS Signal Samples \(\hat{s}_{\mu}\) in Equation (15) have been recorded with a quantization of 2 bits in the Analog to Digital Conversion (ADC) stage. The two bits refer to the value range [\(-3-11\) 3] and thus the RMS of the recorded samples differs from that in Equation (15) because the Automatic Gain Control (AGC) within the receiver front-end steers the amplitude of the GNSS signal in a way to optimally use the available quantization range. The RMS value of the samples \(\hat{s}_{\mu}\) can be measured by the receiver. In this experiment, the RMS for the GPS L1 band was measured with \(\beta_{s_{\mu},RMS}=1.71\), as also shown in Figure 12. Due to computational efficiency, the replica signal used for the correlation was generated with an amplitude of 8, thus it is simply \(\beta_{rep}=8\). Based on this, the correlation values can be obtained after applying the scaling factors as  
\[|P(\tau,\omega)|=|P_{MC,map}(\tau,\omega)|\ (\beta_{s_{\mu},RMS}\ \beta_{rep})^{-1}, \tag{51}\]  
Figure 11: An increase of the multipath amplitude \(\alpha_{MP}\) shifts the weighted mean \(\mu\) and increases the weighted standard deviation \(\sigma_{w}\). The probability function naturally covers the uncertainty also in the case of \(\alpha_{MP}=1\), when the multipath signal is as strong as the LOS signal. The dotted black line refers to the LOS signal. where \(|P_{MC,map}(\tau,\omega)|\) refers to the correlation value from the multi-correlator map of the software receiver. For both scenarios, open sky and urban, the code delay bias standard deviation was set to \(\sigma_{\Delta\tau}=3\) m and has been evaluated at three epochs. The open sky case is shown in Figure 13 and refers to good open sky conditions of GPS L1 C/A satellite PRN 12. The probability functions for all epochs show a slight bias of \(\mu\approx-1\) m with respect to the feed back PVT. The weighted standard deviation \(\sigma_{w}\) nearly approaches the theoretical lower limit of \(\sigma_{\Delta\tau}=3\) m. No other significant effects are visible. The urban scenario was recorded during a test drive through Graz. The data has been post-processed and analyzed with the software receiver. An urban environment with significant deformation on the correlation function was selected and analyzed with MATLAB. The analysis focuses on the GPS L1 C/A satellites PRN 27, PRN 21 and PRN 18, as respectively shown in Figure 14, Figures 15 and 16. A overview of the analyzed scenario is given in Figure 17, which shows the three analyzed positions (red dots) and the satellite constellation.