## 3 Global Positioning System (GPS) L1 Receiver: 2D Pattern vs. 3D Pattern
### Ranging Error  
This section focuses on the impact of the two different assumed antenna patterns on the pseudorange error. The pseudorange error is the sum of different errors affecting the measurement, as we can see from Table 1. In this study, all the pseudorange errors were modeled as in our previous study [9]. However, it is relevant to mention here that, assuming the use of code based observations (as for the SANAG receiver), the code tracking thermal noise error is the main error affecting the pseudorange whose standard deviation can change over time due to a different signal power level at the receiver position, and then due to a different transmitter's antenna pattern model. Hence, for the two considered antenna patterns, for simplicity, we only report the simulated code tracking thermal noise.  
According to [17], for binary phase shift keying (BPSK) modulations, the thermal noise code tracking jitter is:  
\[\sigma_{\mathrm{DLL}}\cong\sqrt{\frac{B_{n}}{2\frac{C}{N_{0}}}D\Bigg{[}1+\frac {2}{T\frac{C}{N_{0}}(2-D)}\Bigg{]}},\ D\ \geq\ \frac{\pi R_{c}}{B_{fe}}, \tag{2}\]  
with \(B_{n}\) the code loop noise bandwidth (in Hz), \(D\) the early-to-late correlator spacing (in chips), \(T\) the coherent integration time (in s), \(B_{fe}\) the double-sided front-end bandwidth (in Hz), \(R_{c}=1/T_{c}\) the chipping rate (in chips/s) and \(C/N_{0}\) the carrier to noise ratio (in dB-Hz). In this work we assumed: \(B_{n}\) = 0.1 Hz and \(T\) = 20 ms for both signals; \(D\) = 0.25 chips for L1 and \(D\) = 0.8 chips for L5; \(B_{fe}\) = 13 MHz for L1 and \(B_{fe}\) = 40 MHz for L5.  
Figure 13: Percentage of continuous time intervals, a point of each curve identifies a time interval duration T (on the \(x\)-axis) and the corresponding percentage (on the \(y\)-axis) of continuous time intervals equal or longer than T.  
Figures 14 and 15 show respectively the mean and max \(\sigma_{\text{tDLL}}\), assuming 10 dBi antenna gain for the receiver. The mean and max code tracking thermal noise error are computed among the signals available at the same instant. For instance, the \(\sigma_{\text{tDLL}}\) for each available signal is displayed in Figure 16 in case of the 3D L1 antenna pattern assumption, as a function of the altitude. We can see from Figure 14 that it is difficult to state which antenna pattern (3D or 2D) would result in higher thermal noise, and thus ranging error. For example, at the beginning of the trajectory, up to 60,000 km, using the 3D pattern would bring a higher thermal noise, however from 75,000 km to 160,000 km using the 2D pattern would result in higher thermal noise. Finally, based on these results, using a 2D or 3D pattern would result in different \(\sigma_{\text{tDLL}}\) for each pseudorange; however, when computing the mean \(\sigma_{\text{tDLL}}\), the difference becomes small. From Figure 15 we can notice that at the beginning of the trajectory, up to 50,000 km the maximal \(\sigma_{\text{tDLL}}\) would be higher when assuming a 3D L1 pattern. It is important to note that the \(\sigma_{\text{tDLL}}\) for both 3D and 2D patterns does not exceed a certain value (approximately 10 m), which corresponds to the weakest signals that the receiver can process due to its limited sensitivity.  
Figure 14: Mean code tracking thermal noise range error, for the assumed MTO. The thermal noise range error is computed for each available pseudo random noise (PRN) and the mean is plotted.  
Figure 15: Max code tracking thermal noise range error, for the assumed MTO. The thermal noise range error is computed for each available PRN and the max is plotted.