## 3 PR Calculation in a GNSS Receiver
### Preliminary Results  
Since the purpose of the simulation experiments is to compare the effect of MP on the measured PR, two versions of the simulated signal are created:  
1. the signal \(s_{c}(t)\), which represents the clean scenario, containing only noise;
2. the signal \(s_{m}(t)\) obtained by adding MP to \(s_{c}(t)\) in the time windows given in Table 1.  
Figure 1 shows the time evolution of the code frequency corrections \(\Delta f_{c}[n]=f_{c}[n+1]-f_{n}\) where \(f_{n}\) is the nominal value of the code frequency (1.023 MHz in GPS L1 C/A), and the relative value in time calculated as \(\Delta T_{chip}[n]=T_{chip}-1/f_{c}[n+1]\) in the DLL. The red line is the average value for the corrections. Let's start by analysing this figure in the time windows specified in Table 1. What we want to observe is the behaviour of the PR computation just before and during a MP occurrence. The MP effect is clearly observed in the first time window shown in Figure 2, where a change of the code frequency appears during the disturbed time window. Then after a transient, the estimate of the code frequency returns to be stable. Figure 3 shows similar results in time windows affected by MPs with different durations.  
To make some comparisons with simulated data, we calculated PR \(\rho_{clean}\) in a scenario with only noise and PR \(\rho_{MP}\) in a scenario with noise and MP in the time windows given in Table 1. Then, we computed the difference \(\rho_{clean}-\rho_{MP}\) in order to see the impact of tracking errors at PR level. These operations are repeated for different values of the output rate \(T_{\rho}\) of the PR measurements. The persistent distortions could introduce errors on the estimate of \(f_{c}\) and consequently in PR computation based on samples counting (as we explained in Section 3.1).  
We observe that, if we use short \(T_{\rho}\), affected by MPs limited in time, the probability that next PR calculated is wrong is higher than if we use longer \(T_{\rho}\). It is possible to observe this effect in Figures 4-6,  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Multipath Instants** & **Duration (s)** \\ \hline
10.900–11.300 & 0.400 \\
27.900–28.100 & 0.200 \\
42.350–42.650 & 0.300 \\
50.750–50.900 & 0.150 \\
53.475–53.525 & 0.050 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Time windows when MP is present.  
In Figure 7 a zoom of the Figure 4 is depicted, where the circle points indicate that a MP is present in a time instant of PR computation. As expected the quantity \(\rho_{clean}-\rho_{MP}\) is different from zero where circle points are present.  
Figure 4: Behaviour of PR for PRN 31, with output every \(\Delta T_{\rho}=500\) ms. PR error (\(\rho_{clean}-\rho_{MP}\)) (**a**); \(\rho_{MP}\) trend in time (**b**); MP presence during a PR computation (**c**) and MP profile (**d**) are shown.  
Figure 3: Zoom of the \(\Delta T_{chip}[n]\) (**a**) and \(\Delta f_{c}[n]\) (**b**) for PRN 31 in the last time window of Table 1.  
Figure 5: Behaviour of PR for PRN 31, with output every \(\Delta T_{\rho}=1000\) ms. PR error \((\rho_{clean}-\rho_{MP})\) (**a**); \(\rho_{MP}\) trend in time (**b**); MP presence during a PR computation (**c**) and MP profile (**d**) are shown.  
Figure 6: Behaviour of PR for PRN 31, with output every \(\Delta T_{\rho}=2000\) ms. PR error \((\rho_{clean}-\rho_{MP})\) (**a**); \(\rho_{MP}\) trend in time (**b**); MP presence during a PR computation (**c**) and MP profile (**d**) are shown.