## 4 GPS L1/L5 Receiver: L5 Pattern Assumption
### Ranging Error  
When the line of sight vector between the transmitter and the receiver crosses the ionosphere layer, the observation is available if both signals (L1 and L5) are tracked. In this case we will have two measurements \(\rho_{L1}\) and \(\rho_{L5}\), with errors the sum of different errors affecting the measurement as explained in Table 1. The ionospheric delay effect can be mitigated using an ionosphere-free combination, according to [18]:  
\[\sigma_{if}^{2}=\frac{1}{(\omega_{L1,L5}-1)^{2}}\Big{[}\omega_{L1,L5}^{2}\, \sigma_{L1}^{2}+\sigma_{L5}^{2}\Big{]}, \tag{3}\]  
where \(\omega_{L1,L5}\,=\,{f_{L1}}^{2}/{f_{L5}}^{2}\), and \(\sigma_{L1}^{2}\) and \(\sigma_{L5}^{2}\) are the variance at frequencies \(f_{L1}\) and \(f_{L5}\), respectively. We can see that the variance of the ionosphere-free pseudorange error is larger than the individual variances at frequency \(f_{L1}\) and \(f_{L5}\) (without considering the ionospheric variance). Thus, if, on the one hand, the ranging error due to the ionosphere delay is reduced using an ionosphere-free combination, the other error sources are amplified on the other hand.  
As was done in Section 3.3, Figures 27 and 28 show, respectively, the mean and max \(\sigma_{\text{tDLL}}\) for the three considered L5 patterns, assuming 10 dBi receiver antenna gain. We can see that the \(\sigma_{\text{tDLL}}\) in case we assume L5 pattern 1 is generally smaller than the other two cases. Indeed, the mean over all trajectory of the mean \(\sigma_{\text{tDLL}}\) is 0.57, 0.68, and 0.66 m when assuming L5 pattern 1, L5 pattern 2, and L5 pattern 3, respectively. Moreover, we can see that at the end of the trajectory the \(\sigma_{\text{tDLL}}\) in case we assume L5 pattern 3 has a bigger standard deviation in terms of mean and maximal value. As shown, at the end of the trajectory (from 330,000 km until the end), the third L5 pattern results in higher and smaller values of mean \(\sigma_{\text{tDLL}}\).  
Figure 27: Mean code tracking thermal noise range error for the assumed MTO. The thermal noise range error is computed for each available PRN and the mean is plotted.  
It is interesting to observe that the maximal \(\sigma_{DLL}\) up to 325,000 km does not exceed a certain value over time (we can observe a flat area at the top of the plots, which has an approximate value of 1.5 m). This value corresponds to the \(\sigma_{DLL}\) that the receiver experiences for the minimum \(C/N_{0}\) that can be tracked (12 dB-Hz for the L5 signal). However, after 325,000 km the maximal \(\sigma_{DLL}\) reaches higher values. This is the result of the iono-free combination, for which, as shown in Equation (3), the variance is larger than the variance that corresponds to the process of only one frequency (L1 or L5) individually. In our tracking strategy we use a free iono-combination only when the signal is crossing the ionosphere. Signals crossing the ionosphere are also processed in an iono-free combination below 325,000 km, but below this altitude the receiver is also able to process weaker signals from the side lobes, for which the individual \(\sigma_{DLL}\) is larger.