# Optimal Particle Filter Weight for Bayesian Direct Position Estimation in a GNSS Receiver
## 7 Delay Bias as Gaussian Nuisance Parameter  
This section describes an approach to handle user range uncertainties (due to e.g., multipath, orbit, clock and iono) within a DPE framework, which are much larger than the code noise. The idea is to introduce an unknown normally distributed bias in the signal model which reflects uncertainties in the user range. This artificially degrades the accuracy of the raw signal but better reflects the true measurement situation and avoids situations of non-overlapping probability functions. One may argue that this delay bias degrades the overall positioning accuracy, but this argument is only true for an ideal situation of vanishing range uncertainties. For real situations, Equation (27) does not reflect the range uncertainties and thus is therefore only an approximate model of the signal samples. Overall, we think that the parameter can be justified and its introduction produces more meaningful probability density function compared to the ideal case. Basically, this bias can be applied at code phase \(\tau\) and Doppler \(f_{d}\), referred to as code delay bias \(\Delta\tau\) and Doppler bias \(\Delta f_{d}\). This work discusses and analyses the introduction of a code delay bias \(\Delta\tau\) to the code delay \(\tau\rightarrow\tau+\Delta\tau\), but the procedure for the Doppler bias is basically the same. The code delay bias \(\Delta\tau\) should express orbit errors, ionospheric errors, clock errors and other modeling errors. It is assumed that these errors follow a normal distribution such that  
Figure 4: The plot is based on a MATLAB MATLAB (2015b) simulation (Release 2015b, The MathWorks Inc., Natic, MA, USA). The upper plot shows the correlation function from Equation (15) for different coherent integration times at a C/N0 = 45 dB-Hz. The simulated signal was generated without noise (i.e., the C/N0 merely defines the correlation amplitude). The lower plot shows the corresponding probability function after one weight update Equation (36), for one signal \(N=1\) and assuming a uniform distribution from the previous epoch, i.e., \(\widetilde{w}_{k-1}^{i}=0\) is considered in Equation (34). The statistics in the lower plot refer to the weighted mean \(\mu\) and weighted standard deviation \(\sigma_{w}\). \[\Delta\tau \sim N(0,\sigma_{\Delta\tau}^{2}), \tag{38}\] \[p(\Delta\tau) = \frac{1}{\sqrt{2\pi}\sigma_{\Delta\tau}}\exp\left\{-\frac{1}{2} \frac{\Delta\tau^{2}}{\sigma_{\Delta\tau}^{2}}\right\}. \tag{39}\]  
Considering for now only one signal and applying the delay bias from Equation (39) to Equation (26) leads to  
\[p(\mathbf{s}|\tau,\omega) = \int_{\Delta\tau=-\infty}^{\infty}p(\Delta\tau)p(\mathbf{s}|\tau+ \Delta\tau,\omega)d\Delta\tau. \tag{40}\]  
As this integral cannot be further simplified analytically, we target an approximate evaluation at selected grid points \(\Delta\tau_{k}\). Neglecting a normalization factor Equation (40) can be expressed as  
\[p(\mathbf{s}|\tau,\omega) \approx \sum_{k=1}^{K}p(\Delta\tau_{k})p(\mathbf{s}|\tau+\Delta\tau_{k}, \omega). \tag{41}\]  
The grid points \(\Delta\tau_{k}\) should be chosen in a way to cover a significant part of \(N(0,\sigma_{\Delta\tau}^{2})\) from Equation (38) with a reasonable resolution.