## 3 Derivation of the CRB for Short Observation Intervals
### Definition and General Case for \(M\) Propagation Paths
In order to compute the remaining term in Equation (11), we can express the elements of the gradient of \(\mathbf{\mu}\) as:  
\[\frac{\partial\mathbf{\mu}}{\partial u_{m}} =\mathbf{d}_{m}e^{i\phi_{m}}, \tag{12}\] \[\frac{\partial\mathbf{\mu}}{\partial\phi_{m}} =ju_{m}\mathbf{d}_{m}e^{i\phi_{m}},\] (13) \[\frac{\partial\mathbf{\mu}}{\partial\tau_{m}} =u_{m}\frac{\partial\mathbf{d}_{m}}{\partial\tau_{m}}e^{i\phi_{m}},\] (14) \[\frac{\partial\mathbf{\mu}}{\partial\alpha_{m}} =u_{m}\frac{\partial\mathbf{d}_{m}}{\partial\alpha_{m}}e^{i\phi_{ m}}, \tag{15}\]  
where the vector \(\mathbf{d}_{m}\) is defined as in Equation (5). It is worth noticing that \(\mathbf{J}(\mathbf{\xi})\) can be conveniently expressed as the following symmetric block matrix:\[\mathbf{J}\left(\xi\right)=\left[ \begin{array}{ccc}\mathbf{J}_{\xi 0}\xi_{0}&\mathbf{J}_{\xi 0}\xi_{1}&\cdots\\ \mathbf{J}_{\xi 0}^{T}&\ddots&\\ \vdots&\mathbf{J}_{\xi M-1}\xi M^{-1}\end{array} \right], \tag{16}\]  
where every element of each sub-matrix \(\mathbf{J}_{\xi m\xi_{l}}\) is computed as:  
\[\left[\mathbf{J}_{\xi m}\xi_{l}\right]_{ij}=2\Re\left\{\frac{\partial\left( \mathbf{r}_{m}^{H}\left(\xi_{m}\right)\right)}{\partial\xi_{i}}\mathbf{C}^{-1 }\frac{\partial\left(\mathbf{r}_{l}\left(\xi_{l}\right)\right)}{\partial\xi_{ j}}\right\}, \tag{17}\]  
with \(\xi_{i}\in\left\{\xi_{m}\right\}\) and \(\xi_{j}\in\left\{\xi_{l}\right\}\). Equation (17) can be used to compute the elements of \(\mathbf{J}(\xi)\), and the CRB of \(\xi\) is straightforwardly obtained as in Equation (9). This case corresponds to the CRB under the assumption of Gaussian noise, independent of the received GNSS signal. Let us now examine the case where only thermal noise is present. Since we have considered the sampling rate to be a multiple of the Nyquist sampling frequency, the samples in \(\mathbf{x}\) can be considered uncorrelated, which implies a diagonal covariance matrix, i.e., \(\mathbf{C}=\sigma_{w}^{2}\mathbf{I}\). The noise variance \(\sigma_{w}\) can be computed as:  
\[\sigma_{w}^{2}=\frac{N_{0}}{2}\int_{-\infty}^{\infty}\left|H_{Rx}\left(f\right) \right|^{2}df=N_{0}B_{n} \tag{18}\]  
where \(N_{0}/2\) is the assumed white noise spectral density in \(\mathrm{W}/\mathrm{Hz}\), \(H_{Rx}(f)\) is the front-end's normalized baseband filter response in the frequency domain and \(B_{n}\) is defined as the equivalent noise bandwidth (one-sided). Thus, for complex white Gaussian noise (CWGN), we can express Equation (17) as:  
\[\left[\mathbf{J}_{\xi m}\xi_{l}\right]_{ij}=\frac{2}{\sigma_{w}^{2}}\Re\left\{ \frac{\partial\left(\mathbf{r}_{m}^{H}\left(\xi_{m}\right)\right)}{\partial \xi_{i}}\frac{\partial\left(\mathbf{r}_{l}\left(\xi_{l}\right)\right)}{ \partial\xi_{j}}\right\}. \tag{19}\]  
If the narrowband approximation for the received signal is taken, i.e., \(\alpha_{\left\{m,l\right\}}\approx 1\), then the CRB obtained as a result of inverting the matrix defined in Equation (16) is analogous to the results described in [69] (Chapter 4), where the CRB was derived for two propagation paths using a similar approach. In addition, [PERSON] et al. used in [70] an approach similar to the one described in this paper, but for the case of up to four signal propagation paths.