## 3 Derivation of the CRB for Short Observation Intervals
### Definition and General Case for \(M\) Propagation Paths  
Estimation of the error bounds plays a major role in parameter estimation in general, since they provide benchmarks to assess the performance of practical estimators. The workhorse in this respect is the well-known Cramer-Rao bound. The CRB defines a lower bound on the variance of any unbiased estimator for a set of deterministic parameters when the probability density function (pdf) of the measurements is known ([68], Chapter 3). The CRB quantifies the dependency of the measurements' pdf on the unknown parameters to estimate. The stronger the pdf's dependency on these parameters,the lower the CRB. The CRB for a vector of \(L\) unknown real-valued parameters \(\mathbf{\xi}\in\mathbb{R}^{L\times 1}\) states that the covariance matrix of the estimates, \(\mathbf{C}_{\mathbf{\xi}}\), must satisfy (under standard regularity conditions):  
\[\mathbf{C}_{\mathbf{\xi}}\geq\mathbf{J}^{-1}\left(\mathbf{\xi}\right), \tag{7}\]  
where \(\mathbf{J}\left(\mathbf{\xi}\right)\) is the Fisher information matrix (FIM), whose inverse is the CRB matrix. The elements in the FIM are defined as:  
\[\left[\mathbf{J}\left(\mathbf{\xi}\right)\right]_{ij}=-E\left\{\frac{\partial^{2} \ln\,p\left(\mathbf{x};\mathbf{\xi}\right)}{\partial\xi_{i}\partial\xi_{j}}\right\}, \tag{8}\]  
where \(\ln\,p(\mathbf{x};\mathbf{\xi})\) is the log-likelihood function of \(\mathbf{\xi}\), given the vector of random measurements \(\mathbf{x}\). Taking the inverse from Equation (8), we can express the CRB of the \(i\)-th parameter in \(\mathbf{\xi}\) as:  
\[\text{CRB}\left(\xi_{i}\right)=\left[\mathbf{J}^{-1}\left(\mathbf{\xi}\right) \right]_{ii}. \tag{9}\]  
In Section 2, we have introduced the vector \(\mathbf{x}\), modeling the discrete output of the receiver front-end during the time interval \(T_{obs}\). As described there previously, the vector \(\mathbf{x}\) is assumed to obey a circularly-symmetric complex Gaussian distribution. Its expected value, defined as \(\mathbf{\mu}\), will then be \(\mathbf{r}=\sum\mathbf{r}_{m}\), as defined in Equation (3). We can express each signal component \(\mathbf{r}_{m}\) as a function of the following unknown parameters, grouped into the vector \(\mathbf{\xi}_{m}\) as:  
\[\mathbf{\xi}_{m}\triangleq\left[u_{m},\phi_{m},\tau_{m},\alpha_{m}\right]^{T}, \tag{10}\]  
where, for the \(m\)-th signal component, \(u_{m}\) represents its amplitude, \(\phi_{m}\) its phase shift, \(\tau_{m}\) its time-delay or code phase and \(\alpha_{m}\) its Doppler shift coefficient. In this case, we want to jointly estimate the vector \(\mathbf{\xi}\in\mathbb{R}^{4M\times 1}\), which has been defined as the vector concatenating all of the \(\mathbf{\xi}_{m}\) vectors. Now, given the Gaussian distribution of the vector \(\mathbf{x}\) assumed in our case, it is possible to make use of the Slepian-Bangs formula ([68], Chapter 3), which provides a convenient way to compute Equation (8) as:  
\[\left[\mathbf{J}\left(\mathbf{\xi}\right)\right]_{ij}=\text{tr}\left\{\mathbf{C}^ {-1}\frac{\partial\mathbf{C}}{\partial\xi_{i}}\mathbf{C}^{-1}\frac{\partial \mathbf{C}}{\partial\xi_{j}}\right\}+2\Re\left\{\frac{\partial\mathbf{\mu}^{H}}{ \partial\xi_{i}}\mathbf{C}^{-1}\frac{\partial\mathbf{\mu}}{\partial\xi_{j}}\right\}, \tag{11}\]  
where \(\Re\{\cdot\}\) denotes the real part of a complex vector or scalar, \(\text{tr}\{\cdot\}\) denotes the trace operator, the superscript \({}^{H}\) is used to denote the conjugate transpose operator and the \(\mathbf{C}\) represents the covariance matrix of \(\mathbf{x}\). Given that the noise component is independent from any of the parameters in \(\mathbf{\xi}\), then the first term of the sum in Equation (11) will be equal to zero, due to \(\partial\mathbf{C}/\partial\xi_{i}=\mathbf{0}\).