## 3 Derivation of the CRB for Short Observation Intervals
### Two Propagation Path Case for a Ground-Based Static Receiver  
For the scenario described in Section 2, we assume \(M=2\), with the \(\mathbf{r}_{0}\) component used to represent the line-of-sight (LOS) signal and \(\mathbf{r}_{1}\) used for the specular reflection. In this scenario, we consider a stationary receiver, where subsequently, the LOS and the specular reflection will experience the same Doppler shift; thus, \(\alpha_{m}=\alpha_{l}:=\alpha\). This simplifies the FIM \(\mathbf{J}(\xi)\) defined in Equation (16). Its elements related to \(\alpha_{\left\{m,l\right\}}\) and a different parameter in \(\xi\), as defined in Equation (19), will now become zero. This implies that the estimates of \(\alpha\) will be independent from the estimates of the rest of parameters, i.e., they will decouple. The same conclusion was reached in [69] (Chapter 4) when computing the CRB for two propagation paths. Having decoupled parameters allows us to disregard the estimation of the \(\alpha\) for the rest of our analysis. The reason is that we are only interested in estimating the geometry of the scenario and the characteristics of the reflecting surface. In a stationary environment, \(\alpha\) cannot be related with any useful information about the scenario that we want to retrieve. Thus, we are now redefining the vector of unknown parameters \(\xi\), by redefining each \(\xi_{m}\) to just \([u_{m},\phi_{m},\tau_{m}]^{T}\). For the scenario described in Section 2, we have only two propagation paths; using Equation (16), the FIM can be expressed as the \(6\times 6\) block matrix:  
\[\mathbf{J}\left(\xi\right)=\left[ \begin{array}{cc}\mathbf{J}_{\xi 0}\xi_{0}&\mathbf{J}_{\xi 0}\xi_{1}\\ \mathbf{J}_{\xi 0}^{T}&\mathbf{J}_{\xi 1}\xi_{1}\end{array} \right], \tag{20}\]where each submatrix, computed using Expression (19), can be expressed as:  
\[\mathbf{J}_{\pi_{n}\xi_{l}}=\frac{2}{\sigma_{w}^{2}}\Re\left\{\left\{ \begin{array}[ ]{ccc}\mathbf{d}_{m}^{H}\mathbf{d}_{l}&ju_{l}\mathbf{d}_{m}^{H}\mathbf{d}_{l}&u _{l}\mathbf{d}_{m}^{H}\frac{\partial\mathbf{d}_{l}}{\partial\tau_{m}}\\ -ju_{m}\mathbf{d}_{m}^{H}\mathbf{d}_{l}&u_{m}u_{l}\mathbf{d}_{m}^{H}\mathbf{d} _{l}&-ju_{m}u_{l}\mathbf{d}_{m}^{H}\frac{\partial\mathbf{d}_{l}}{\partial\tau_ {l}}\end{array} \right\}e^{j\Delta\phi}\right\}, \tag{21}\]  
with \(m_{l},l\in\{0,1\}\) and \(\Delta\phi\triangleq\phi_{l}-\phi_{m}\). Let us now introduce the discrete autocorrelation function (ACF) of \(s(t)\), defined as:  
\[R\left(\tau\right)\triangleq\frac{1}{N}\sum_{n=0}^{N-1}s^{*}\left(\alpha t_{n} \right)s(\alpha t_{n}-\tau). \tag{22}\]  
The superscript \({}^{*}\) is used to denote the complex conjugate. Now, we introduce the following identities:  
\[R\left(\Delta\tau\right) =\frac{1}{N}\mathbf{d}_{m}^{H}\mathbf{d}_{l}, \tag{23}\] \[R^{\prime}\left(\Delta\tau\right) =-\frac{1}{N}\frac{\partial\mathbf{d}_{m}^{H}}{\partial\tau_{m}} \mathbf{d}_{l}=\frac{1}{N}\mathbf{d}_{m}^{H}\frac{\partial\mathbf{d}_{l}}{ \partial\tau_{l}},\] (24) \[R^{\prime\prime}\left(\Delta\tau\right) =-\frac{1}{N}\frac{\partial\mathbf{d}_{m}^{H}}{\partial\tau_{m}} \frac{\partial\mathbf{d}_{l}}{\partial\tau_{l}}. \tag{25}\]  
Notice that in order to keep our notation as simple as possible, we use \(R^{\left(n\right)}(\Delta\tau)\) to indicate the \(n\)-th order partial derivative of \(R\) with respect to \(\tau\) evaluated at the delay difference \(\Delta\tau\triangleq\eta_{l}-\tau_{m}\). The Identities (23)-(25) are obtained straightforwardly by applying the chain rule, and they allow us to express Equation (21) as:  
\[\mathbf{J}_{\pi_{n}\xi_{l}}=\frac{2N}{\sigma_{w}^{2}}\left[