## 4 Analysis of the CRB for Short Observation Intervals
### CRB when \(\Delta\rho>\rho_{chip}\)  
In this particular case, not only the diagonal terms for the resulting \(\mathbf{J}^{-1}(\mathbf{\Psi})\) matrix, i.e., \([\mathbf{J}^{-1}(\mathbf{\Psi})]_{ii}\), are much simpler, but this assumption is particularly relevant because it represents the best estimation case, i.e., when its CRB lower bounds all other cases with shorter \(\Delta\rho\). As shown in Appendix A, for \(h\) and \(|\Gamma|\) in \(\mathbf{\Psi}\), after some simple algebra, we obtain the following CRB expressions:  
\[\text{CRB}\left(h\right)=-\frac{1}{8\gamma\,\text{SNR}_{0}\,\sin^{2}\left( \theta_{el}\right)\left(\frac{R^{\prime\prime}\left(0\right)}{c^{2}}\right)}, \tag{35}\]\[\text{CRB}\left(\left|\Gamma\right|\right)=\frac{\left|\Gamma\right|^{2}+1}{2\, \text{SNR}_{0}\,R\left(0\right)}, \tag{36}\]  
where the coefficient \(\gamma\) introduced has been defined as:  
\[\gamma=\frac{\left|\Gamma\right|^{2}}{\left|\Gamma\right|^{2}+1}, \tag{37}\]  
and \(\text{SNR}_{0}\) represents the equivalent SNR as if only the LOS signal was present and, as such, can be expressed as:  
\[\text{SNR}_{0}=\frac{u_{0}^{2}}{\sigma_{w}^{2}}N=2\frac{C}{N_{0}}\,T_{obs}. \tag{38}\]  
Notice that the amplitude \(u_{0}\) takes into account the effects of the receiver's antenna gain pattern for the received LOS signal, which is dependent on the signal incident angle, i.e., a function of \(\theta_{el}\). From examining Equations (35) and (36), we can directly deduce the following:  
1. With little surprise, both CRB expressions are inversely proportional to \(\text{SNR}_{0}\). However, \(\text{CRB}(h)\) is also inversely proportional to \(\sin^{2}(\theta_{el})\). This poses a trade-off: we require low elevations for the smooth surface assumption to hold, but at the same time, the lower the elevation, the higher the \(\text{CRB}(h)\). Moreover, as shown by Equation (2), low elevation angles imply larger first Fresnel zone areas, which decrease the spatial resolution of our estimates. 2. The values that \(\text{CRB}(\left|\Gamma\right|)\) can take for different \(\left|\Gamma\right|\) values are bounded, as a consequence of having defined \(\Gamma\) as a ratio, with \(0<\left|\Gamma\right|\leq 1\). 3. In both CRB expressions, the effects of the assumed received signal bandwidth and the front-end filtering (possible losses and distortions) are modeled within the \(R^{\prime\prime}(0)\) and the ACF's peak, i.e., \(R(0)\). \(R^{\prime\prime}(0)\) can be understood as the curvature or the sharpness of the \(R(0)\). The higher the front-end bandwidth, the sharper the resulting ACF peak and, thus, the higher \(R^{\prime\prime}(0)\). This holds as long as the front-end bandwidth is narrower than the received signal bandwidth, otherwise there will be no sharpening on the resulting \(R(0)\). Additionally, for a higher front-end bandwidth, the \(\text{SNR}_{0}\) will also increase if the Nyquist sampling rate assumption is maintained, as a consequence of having more samples for the same \(T_{obs}\). In Figure 3, we show the results obtained when computing the CRB for \(h\) using Equation (35), as a function of the front-end's bandwidth \(B_{fe}\) for the different GNSS signals considered. Note that in Figure 3, the \(y\)-axis represents (the convention of representing the \(\sqrt{\text{CRB}}\) in the figures is followed consistently along the entire paper) \(\sigma_{\Psi_{i}}=\sqrt{\text{CRB}(\Psi_{i})}\). An isotropic antenna pattern was assumed, and \(\left|\Gamma\right|\) was set to \(\sqrt{0.1}\), which corresponds to an attenuation of 10 dB in the reflected signal power with respect to the LOS. The \(T_{obs}\) considered was equal to 1 s. The \(\Delta\rho\) was fixed to 600 m to ensure isolation between the LOS and the reflected signal components provided by the code correlation properties. We observe how the CRB of \(h\) consistently decreases with the increase of \(B_{fe}\) for all GNSS signals studied, as long as \(B_{fe}\) remains lower than the (one-sided) signal bandwidth \(B_{Tx}\).