## Appendix A GNSS Basics
Thus, both parameters, range and radial velocity, are estimated simultaneously, by a search in the range (delay)-velocity (Doppler frequency) plane. To perform this first search of Doppler frequency and of delay, the acquisition sub-system of the GNSS receiver uses a frequency and delay lookup table to find the correlation peak. If the incoming signal is affected by thermal noise or/and multipath propagation, the output of the coherent correlator will be affected as well. The best Maximum Likelihood (ML) estimation of the range and relative velocity of the satellite can be obtained as:  
\[\left((\hat{\tau},\hat{\
u})_{\text{ML}}=\underset{\tau,f_{d}}{\operatorname{ argmax}}\Big{(}|A_{s,c}(\tau,\
u)|^{2}\Big{)}\right) \tag{3}\]  
where \(\
u\) is the Doppler frequency and \(A_{s,c}(\tau,\
u)\) represents the cross-ambiguity function of the signals \(s(t)\) and \(c(t)\), defined as:  
\[A_{s,c}(\tau,\
u)=\int\limits_{-\infty}^{\infty}s\Big{(}t+\frac{\tau}{2}\Big{)} c^{*}\Big{(}t-\frac{\tau}{2}\Big{)}e^{-j2\pi\
u t}dt. \tag{4}\]The cross-ambiguity function is a time-frequency representation which has a wide utilization in target localization problems, such as in radar, sonar, GPS and so on. In simple scenarios, without multipath propagation, such as in the case of a car travelling in a plain rural region, the satellite is in Line-of-Sight (LoS) and the received signal has the expression in (2) with \(b=1\). In more complicated scenarios, such as in natural or urban canyons, reflections from (possibly moving) objects in the surroundings produce multipath propagation, whose effect is the multipath fading. The Cramer-Rao theorem specifies the conditions for an unbiased estimator of a single parameter, \(\theta\), of a data set \(\mathbf{x},f(\mathbf{x})\), to have minimum variance and to be efficient. The theorem specifies the value of this minimum variance as well. This value represents a lower bound for all the unbiased estimators of the considered parameter and is named the Cramer-Rao Lower Bound (CRLB). The unbiased estimator which has this variance, \(f(\mathbf{x})\), is named the optimal estimator. Its expression can be deduced by solving the equation:  
\[\frac{\partial\ln p(\mathbf{x};\theta)}{\partial\theta}=I(\theta)[f(\mathbf{x })-\theta], \tag{10}\]  
where \(p(\mathbf{x};\theta)\) represents the data probability density function dependent on the parameter \(\theta\) and \(I(\theta)\) represents the Fisher information. In the following, we will consider the delay as parameter, \(\theta=\tau\) and we will present the expressions of the CRLB for GNSS delay estimation in cases without and with Non-Line-of-Sight (NLoS) multipath propagation. In the first case, the signal model is:  
\[s(t)=c(t-\tau_{0})e^{j2\pi v_{0}t}+\eta(t). \tag{11}\]  
The corresponding Cramer-Rao Lower Bound has the expression [29]:  
\[CRLB_{\text{without multipath}}=\frac{1}{4\pi^{2}SNR\sigma_{f}^{2}}, \tag{12}\]  
where \(SNR\) represents the signal-to-noise ratio, defined as \(2E_{c}/N_{0}\), and \(\sigma_{f}^{2}\) represents the square of the effective bandwidth of the transmitted signal \(c(t)\), defined as:  
\[\sigma_{f}^{2}=\frac{\int_{-\infty}^{\infty}f^{2}|C(f)|^{2}df}{E_{c}}. \tag{13}\]  
Therefore, in the case when there is no multipath propagation, and the satellite is in LoS, the \(CRLB\) varies inverse proportionally with the square of effective bandwidth of the transmitted signal \(\sigma_{f}\) and with the signal-to-noise ratio of the received signal, \(SNR\). The range estimation precision increases with the decreasing of the \(CRLB_{\text{without multipath}}\). This explains why GNSS designers have chosen the CDMA modulation, which has a large effective bandwidth. The single impairment which affects the precision of range estimation in the case of propagation without multipath is the receiver's thermal noise, whose power spectral density decreases the \(SNR\).