## 6 Analysis of the CRB(\(\boldsymbol{\
### Effects of the Signal Modulation on the \(CRB(\boldsymbol{\tau}_{long})\)  
We now present the results of computing the CRB of \(\boldsymbol{s}_{\Gamma}\) and \(h\), given \(\Delta\theta_{el}\) spans of different lengths and the GNSS signals considered. For each signal, we have considered a particular \(B_{fe}~{}=~{}B_{Tx}/2\). We have considered the Leica AR25 antenna model and a true receiver height \(h=3\) m. We also have used the Z-L model for \(\Gamma(\theta_{el})\) given a dry ground surface with a permittivity (real part) \(\varepsilon_{r}=30\) and a conductivity \(\sigma=0.2\) S/m [76]. The initial \(\theta_{el}\) is \(10^{\circ}\) for all of the elevation spans considered. Figure 6: \(\sqrt{\mathrm{CRB}(h)}\) for different \(\Delta\theta_{el}\)’s (vertical axis), starting from \(10^{\circ}\), and true receiver heights \(h\) (horizontal axis). Complete left-hand circular polarization (LHCP) reflection with constant \(|\Gamma|^{2}=-10\) dB is assumed. (**a**) The results when an ideal isotropic antenna is considered, while (**b**) shows the results when the Leica AR25 antenna pattern is considered. Black lines are used to represent different level curves for the values labeled on top of them. The results obtained are summarized in Figure 7. As expected, the longer the \(\Delta\theta_{el}\) covered during the observation time, the lower the CRB values obtained, for all three parameters. However, this improvement is not linear. We observe how the CRB decreases significantly for \(\Delta\theta_{el}\) covering up to \(\sim\)2\({}^{\circ}\), while it decreases at a much slower rate when extending \(\Delta\theta_{el}\) beyond that value. This observed behavior can be explained as follows. For our scenario and given the GNSS signals' carrier wavelength, i.e., from 0.19-0.25 m, a \(\Delta\theta_{el}\) of \(\sim\)2\({}^{\circ}\) is required to observe a 2\(\pi\) phase shift in the reflected signal phase, caused exclusively due to the change in \(\Delta\rho\). In the IPT context, this corresponds to observing a single complete oscillation of the SNR interference pattern. Longer \(\Delta\theta_{el}\), i.e., longer observation times still provide some improvement. It is important to remark about the existing trade-off between extending \(\Delta\theta_{el}\), i.e., extending the observation time, and the displacement of the specular reflection point over the reflection surface. The change in \(\theta_{el}\) also changes the size of the first Fresnel zone. For low elevation angles, such as the ones that we need to consider to ensure the smooth surface assumption, the change  
Figure 7: \(\sqrt{\mathrm{CRB}}\) for (**a**) \(h\), (**b**) \(\varepsilon_{r}\), and (**c**) \(\sigma\), for different \(\Delta\theta_{el}\) starting from an initial \(\theta_{el}=10^{\circ}\). Each data point in the plots represents the results for a \(\Delta\theta_{el}\), with its value on the \(x\)-axis representing the final \(\theta_{el}\) angle achieved in that span. The first data point (left of the plot) corresponds to a single \(T_{coh}\) observation. in the area covered by the first Fresnel zone and the displacement of the specular reflection point is significant, as shown in Figure 8. In practice, it is unlikely to find real surfaces that are that flat and homogeneous (and with no obstacles), except maybe for the case of calm water surfaces. On the other hand, the antennas used in practice (e.g., in geodetic sites) have lower cross-coupling at lower elevation angles, i.e., less attenuation on the LHCP signal for these elevations. This is why higher oscillation amplitudes of the SNR interference pattern are observed for \(\theta_{el}\) angles in standard GNSS geodetic receivers [7; 55]. We shall also notice that even for long observation times, where the signal's phase information can be used in the parameter estimation, the greater the signal's bandwidth, the lower the CRB for all of the parameters.