## 4 Analysis of the CRB for Short Observation Intervals
### CRB when \(\Delta\rho>\rho_{chip}\)
The slight decrease of the CRB of \(h\) for all GNSS signals observed in Figure 3 for \(B_{fe}>B_{Tx}/2\) is explained due to the use of a non-ideal low-pass filter, like the one explained in Section 2, to model the limited transmission bandwidth. Also from Figure 3, we see how the \(\sqrt{\text{CRB}(h)}\) stays in the meter order, except for the Galileo E5 signal, which reaches decimeter order precision when \(B_{fe}>40\) MHz, due to its highest bandwidth. The CRB for the rest of parameters in \(\boldsymbol{\
abla}\) is jointly obtained when computing \(\mathbf{J}^{-1}(\boldsymbol{\
abla})\). In Equation (32), we have expressed the phase difference between the two signal components as the sum of two contributions: \(\phi_{\mathrm{T}}\) and \((2\pi/\lambda)\Delta\rho\), with the latter capturing the shift due to the extra propagation path corresponding to the specular reflection. For short observation intervals, where both of these contributions to \(\Delta\phi\) remain constant, it is not possible to individually estimate them precisely. This issue is easily spotted by examining the CRB (\(\phi_{\mathrm{T}}\)) expression, i.e., the fifth element in \([\mathbf{J}^{-1}(\boldsymbol{\
abla})]_{ii}\), derived in Appendix A, that can be approximated as:  
\[\mathrm{CRB}\left(\phi_{\mathrm{T}}\right)\approx-\frac{c^{2}k^{2}R\left(0 \right)}{2\,\gamma\,\mathrm{SNR}_{0}\,R^{\prime\prime}\left(0\right)}, \tag{39}\]  
where \(k=2\pi/\lambda\) is the wavenumber for the received signal carrier. Since the numerator in Equation (39) will usually be many orders of magnitude higher than its denominator, resulting in \(\sqrt{\mathrm{CRB}\left(\phi_{\mathrm{T}}\right)}\gg 2\pi\) rad, this implies that we cannot estimate \(\phi_{\mathrm{T}}\) properly. Given this ambiguity in individually estimating the two contributions to \(\Delta\phi\), performing precise phase altimetry is only possible if \(\phi_{\mathrm{T}}\) is assumed to be known or small enough that it can be neglected. Otherwise, a bias in the altimetry estimates is introduced. Thus, even if we considered the phase coherence in our derivation of CRB(\(h\)), Expression (35) is equivalent to the CRB for pure GNSS-R code altimetry only, i.e., as if no phase information were used. The CRB for code altimetry can be also derived following the same proposed approach, described in the previous section, but just assuming no phase coherence between the LOS and the reflected signal, i.e., considering \(\Delta\phi\) as the difference between two independent phases when computing the \(\mathbf{G}\) transformation matrix defined in Equation (34), which leads to the same results as in Equation (35). In addition, in [72; 73], in an effort made by the authors to characterize the code-tracking accuracy of GPS receivers, they computed the minimum code thermal noise jitter when using noncoherent delay-lock loop (DLL) discriminators, i.e., \(\sigma_{\mathrm{DLL}}^{2}\). The expressions provided in their results are very frequently used to characterize the tracking performance of different receiver  
Figure 3: \(\sqrt{\mathrm{CRB}(h)}\) as a function of the front-end’s bandwidth for the different GNSS signals considered. The vertical dashed lines are used to represent the \(B_{Tx}\) (one-sided) considered for each of the signals. architectures [62] (Chapter 5). Additionally, we can use these same expressions to compute the minimum variance on the \(h\) estimate, i.e., \(\sigma_{h}^{2}\), in a straightforward way, as:  
\[\sigma_{h}^{2}\geq\frac{c^{2}\left(\sigma_{\text{DLL}-0}^{2}+\sigma_{\text{DLL} -1}^{2}\right)}{4\sin^{2}\left(\theta_{el}\right)}, \tag{40}\]  
where \(\sigma_{\text{DLL}-0}^{2}\) and \(\sigma_{\text{DLL}-1}^{2}\) are the minimum variances of the LOS and the reflected signal code delays, respectively. In Equation (40), two independent tracking channels are assumed: one tracking the LOS signal and the other tracking the reflected signal.