## 2 Signal Analysis
### The Weighting Function
Figure 4 shows the variation in the improvement \(P_{m}\) in range with the coefficients \(a_{\phi}\) for the phase processing and \(a_{p}\) for the code processing. The dashed lines represent the case using only the elevation-dependent weighting functions, i.e., setting \(b_{\phi,j}\) and \(b_{p,j}\) to zero, and the solid lines represent the case when setting \(b_{\phi,j}\) and \(b_{p,j}\) to the C/N\({}_{0}\) coefficients leading to the largest \(P_{m}\) for the corresponding \(a_{\phi}\) and \(a_{p}\), respectively. The differences between the solid and the dashed lines for the same scenario, i.e., the same colour, denote thus the further contribution of the C/N\({}_{0}\) to the elevation-dependent weighting models. It can be observed that for both the phase and the code processing, incorporating the C/N\({}_{0}\) in the weighting model could generally lead to larger contributions in \(P_{m}\) in the suburban scenario (see the red lines) than in the open-sky scenario. From Figure 4, the highest value of the elevation-related weighting coefficient is observed for the phase processing in the suburban scenario. This is consistent with the information shown in Figure 2 (the blue and red lines in the top panel of Figure 2b). Using the weighting coefficients giving the largest positional improvements (Table 2), the between-receiver phase (\(\sigma_{\phi,j}\)) and code (\(\sigma_{p,j}\)) standard deviations in the zenith direction and at the \(S_{j}^{s}\) of \(\infty\) can be obtained with the least-squares variance component estimation (LS-VCE) procedure [38], and are given in Table 3 for both frequencies and both measurement environments. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Scenario**} & \multicolumn{4}{c}{**Phase**} & \multicolumn{4}{c}{**Code**} \\ \hline  & \(\hat{a}_{\phi}\) & \(\hat{b}_{\phi,1}\) [Hz] & \(\hat{b}_{\phi,2}\) [Hz] & \(P\) & \(P_{m}\) [m] & \(\hat{a}_{p}\) & \(\hat{b}_{p,1}\) [Hz] & \(\hat{b}_{p,2}\) [Hz] & \(P\) & \(P_{m}\) [m] \\ \hline Open-sky scenario & 10 & 0 & \(10^{4}\) & 16.40\% & 0.003 & 2 & \(10^{4}\) & 0 & 12.67\% & 0.096 \\ \hline Suburban scenario & 16 & \(10^{4}\) & 4000 & 34.11\% & 0.018 & 0 & \(10^{4}\) & 7000 & 18.81\% & 0.255 \\ \hline  & \multicolumn{4}{c}{\(b_{\phi,j}=0\)} & \multicolumn{4}{c}{\(b_{p,j}=0\)} \\ \hline Open-sky scenario & 16 & – & – & 8.44\% & 0.001 & 6 & – & – & 7.55\% & 0.057 \\ \hline Suburban scenario & 16 & – & – & 24.60\% & 0.013 & 4 & – & – & 8.41\% & 0.114 \\ \hline  & \multicolumn{4}{c}{\(a_{\phi}=0\)} & \multicolumn{4}{c}{\(a_{p}=0\)} & \multicolumn{4}{c}{\(a_{p}=0\)} \\ \hline Open-sky scenario & – & \(10^{4}\) & \(10^{4}\) & 13.50\% & 0.002 & – & \(10^{4}\) & 0 & 9.96\% & 0.076 \\ \hline Suburban scenario & – & \(10^{4}\) & 4000 & 7.30\% & 0.004 & – & \(10^{4}\) & 7000 & 18.81\% & 0.255 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The weighting coefficients (Equations (16) and (17)) leading to the smallest mean value in the positional results and the improvement ratios \(P\) of the mean of 3D positioning errors compared to the case that all the weighting coefficients are set to zero (i.e., no weighting model used). \(P_{m}\) refers to the actual improvement in the mean of the 3D positional errors. Figure 4: Improvement \(P_{m}\) in (**a**) phase-based and (**b**) code-based positional results with respect to the coefficients \(a_{\phi}\) and \(a_{p}\).