## 3 Derivation of the CRB for Short Observation Intervals
### Two Propagation Path Case for a Ground-Based Static Receiver
\begin{array}{cc} R\left(\Delta\tau\right)\cos\left(\Delta\phi\right)&u_{l}R\left(\Delta\tau \right)\sin\left(\Delta\phi\right)&-u_{l}R^{\prime}\left(\Delta\tau\right)\cos \left(\Delta\phi\right)\\ -u_{m}R\left(\Delta\tau\right)\sin\left(\Delta\phi\right)&u_{m}u_{l}R\left( \Delta\tau\right)\cos\left(\Delta\phi\right)&u_{m}u_{l}R^{\prime}\left(\Delta \tau\right)\sin\left(\Delta\phi\right)\\ u_{m}R^{\prime}\left(\Delta\tau\right)\cos\left(\Delta\phi\right)&u_{m}u_{l}R^{ \prime}\left(\Delta\tau\right)\sin\left(\Delta\phi\right)&-u_{m}u_{l}R^{\prime }\left(\Delta\tau\right)\cos\left(\Delta\phi\right)\end{array} \right]. \tag{26}\]  
Now, since we can express \(\mathbf{J}_{\xi_{n}\xi_{l}}\) as in Equation (26), therefore \(\mathbf{J}(\xi)\) will be a function of the ACF of \(s(t)\) and first and second order derivatives with \(\tau\) evaluated at \(\Delta\tau\). It is helpful to notice that ACF and its derivatives can be expressed using their relationship with the signal's power spectral density (PSD). By making use of the well-known Wiener-Khinchin theorem [68] (pp. 576-577) and the properties of the Fourier transform, we have that:  
\[R^{\left(n\right)}\left(\Delta\tau\right)=\int_{-\infty}^{\infty}\left(j2\pi f \right)^{n}S_{s}\left(f\right)e^{j2\pi f/\Delta\tau}df, \tag{27}\]  
where \(S_{s}(f)\) represents the signal \(s(\alpha t)\)'s PSD. Notice that \(s(\alpha t)\) represents the filtered signal at the output of the receiver's front-end, scaled in the time domain due to the Doppler effect. It already takes into account any distortion effect on the signal due to the front-end's filtering. Thus, in our derivation, these effects are taken into account within the \(R()\) and its derivatives. Nonetheless, if we are interested in evaluating the influence of the front-end's filtering on the resulting CRB, it might be more convenient to express \(S_{s}(f)\) as:  
\[S_{s}(f)=|H_{Rx}(f)|^{2}S(f), \tag{28}\]  
where \(S(f)\) is the PSD of the signal received at the antenna. In the context of GNSS-R, in [54], the authors followed this approach to study the effects of using different front-end bandwidths and filter types in the estimation performance of the interferometric GNSS-R technique. Moreover, they considered also the band limitations of the signal transmitted in practice. In contrast, in our approach, we generate a model of \(s(t)\) simulating the samples at the output of the front-end, and then, we compute its ACF and its derivatives. The selected front-end bandwidth, \(B_{fe}\), will define the sharpness of the normalized autocorrelation peaks, which will have a great impact on its derivatives, \(R^{(n)}\). In addition, the total number of samples considered within \(T_{obs}\) will also depend on \(B_{fe}\), as \(N=T_{obs}/(2B_{fe})\). If we consider the support of \(R(\Delta\tau)\) to be approximately limited to the interval \([-T_{c},T_{c}]\), where \(T_{c}\) is the chip duration, then, according to Equation (26), we have \(\mathbf{J}_{\xi_{n}\xi_{j}}\approx 0\) for propagation path differences greater than \(\rho_{chip}=T_{c}c\), where \(c\) is the signal propagation speed. This is a consequence of the autocorrelation properties of PRNcodes [62]. Moreover, \(\mathbf{J}\left(\xi\right)\) becomes a diagonal matrix, which implies that all of the parameters in \(\xi\) are now uncoupled. From the FIM calculation point of view, this is equivalent to the scenario of having a receiver with two synchronized RF front-end channels, with uncorrelated noise components, and two different antennas: an up-looking one, exclusively receiving the LOS signal, and a second down-looking antenna, exclusively receiving the reflected component.