# Joint Delay-Doppler Estimation Performance in a Dual Source Context
## 5 Validation and Discussion  
### Methodology  
To validate the new CRBs derived in Section 3, the asymptotic properties of the CMLE introduced in Section 4.1 are exploited. It is know that for a high SNR the statistical mean square error (MSE) of the delay and Doppler frequency CMLE are accurately described by the corresponding CRBs. Two representative signals are considered: (i) a Global Positioning System (GPS) L1 C/A signal [45], and (ii) a Linear Frequency Modulated (LFM) chirp signal classically defined as following:\[s(t)=e^{\beta\pi f(t)},\;f(t)=\frac{B}{2T}\left(\frac{t}{T}-\frac{1}{2}\right)^{2},\;T=NT_{s}, \tag{28}\]  
with \(B\) the chirp bandwidth equal to 8 MHz (which is the receiver bandwidth in this case) and \(N\) the number of samples. The received signal is built with two delayed and attenuated replicas with arbitrary values of Doppler frequencies. The parameters for the three different scenarios considered are given in Table 1. Notice that the L1 C/A chips unit is 1/1023 ms. In this table \(\rho_{1}/\rho_{0}\) is the ratio between the reflected amplitude and the main one (also referred to as multipath-to-direct ratio in GNSS [42]) and \(F_{di}=b_{i}f_{c}\) refers to the Doppler frequency in Hertz corresponding to the Doppler stretch \(b_{i}\).  
As discussed in Section 4, depending on the time-delay difference between the two replicas, \(\Delta\tau\), one can apply a sub-optimal estimators such as the CRE [40], which may behave like a CMLE in the asymptotic region (high SNR). In this case it is possible to see the estimator threshold region in a reasonable computation time. However, when it comes to very small values of \(\Delta\tau\), meaning less than \(1/F_{s}\) chip delay, a sub-optimal algorithm like the CRE is in general not able to converge to the CRB. As a consequence, during the simulations, when the product \(\Delta\tau F_{s}\) is smaller than or equal to 1, the CRE is no longer efficient. In such limit case, a direct implementation of the CMLE is preferred. Computational cost soars but at very high-SNR it is possible to limit the search area at \(3\sigma\), where \(\sigma\) is the expected variance of the estimated values, directly provided by the derived CRBs. Therefore the CMLE simulation is valid only at very high SNR, because in this asymptotic region the CMLE is Gaussian distributed and its variance is equal to the CRB.  
Three scenarios are then considered: scenario (a) presents the simple case where signals are clearly p  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Estimator** & \(F_{s}\) **(MHz)** & \(\Delta\tau\) **(L1 C/A Chips)** & \(\Delta\tau\) **(m)** & \(\rho_{1}/\rho_{0}\) & \(\Delta\phi\) **(\({}^{\circ}\))** & \(Fd_{0}/Fd_{1}\) **(Hz)** \\ \hline (a) & CRE & 8 & 2 & 600 & 0.5 & 15 & 20/50 \\ (b) & CRE & 8 & 1/4 & 75 & 0.5 & 15 & 20/50 \\ (c1) & CRE & 8 & 1/8 & 37.5 & 0.5 & 15 & 20/50 \\ (c2) & CMLE & 8 & 1/8 & 37.5 & 0.5 & 15 & 20/50 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulations scenarios for the CRB validation: (a) two signals totally parted (\(\Delta\tau=2\) chips), (b) a secondary path very close to the main signal (\(\Delta\tau=1/4\) chip), and (c) considered to showcase the difference between the CMLE and CRE (\(\Delta\tau=1/8\) chip).  
Figure 2: _Cont._