# Method for Estimating the Optimal Coefficient of L1C/B1C Signal Correlator Joint Receiving
## 5 Discussion  
The amplitude ratio combination is the optimal coefficient of the correlators joint, which is estimated in this paper by the equivalence principle of correlator joint and discriminator joint and code tracking accuracy of the correlator joint. Next, we quantitatively compared the amplitude ratio joint with the equal weight joint in Ref. [29] and the power weighting in Ref. [30], which can further prove the superiority of the optimal coefficients proposed in this paper.  
Figure 16: The relative improvement in the positioning accuracy coordinate axis.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \(\Delta X\) **(m)** & \(\Delta Y\) **(m)** & \(\Delta Z\) **(m)** & \(\Delta E\) **(m)** & \(\Delta N\) **(m)** & \(\Delta U\) **(m)** & \(\Delta B^{\ast}\) & \(\Delta H^{\ast}\) & \(\Delta L\) **(m)** \\ \hline
1 & 15.21 & 14.23 & 3.55 & 14.18 & 7.90 & 13.52 & 7.07 \(\times\) 10\({}^{-5}\) & 1.54 \(\times\) 10\({}^{-5}\) & 13.52 \\ \hline
2 & 9.29 & 9.0 & 2.15 & 8.41 & 5.04 & 8.70 & 4.51 \(\times\) 10\({}^{-6}\) & 9.15 \(\times\) 10\({}^{-6}\) & 8.70 \\ \hline
3 & 8.18 & 7.91 & 1.87 & 7.30 & 4.51 & 7.70 & 4.04 \(\times\) 10\({}^{-6}\) & 7.94 \(\times\) 10\({}^{-6}\) & 7.70 \\ \hline
4 & 8.12 & 7.83 & 1.86 & 7.29 & 4.45 & 7.60 & 3.98 \(\times\) 10\({}^{-6}\) & 7.93 \(\times\) 10\({}^{-6}\) & 7.60 \\ \hline
5 & 7.95 & 7.71 & 1.81 & 7.08 & 4.40 & 7.51 & 3.94 \(\times\) 10\({}^{-6}\) & 7.70 \(\times\) 10\({}^{-6}\) & 7.51 \\ \hline \end{tabular}
\end{table}
Table 5: The std of the position error for L1C.  
We compared the positioning accuracy improvement ratio in the ECEF coordinate system. In order to quantify the value into a number, we synthesize the errors in the three coordinate axes according to Formula (42).  
\[E_{ECCF}=\sqrt{\left(X_{r}-x_{0}\right)^{2}+\left(Y_{r}-y_{0}\right)^{2}+\left(Z _{r}-z_{0}\right)^{2}} \tag{42}\]  
\(x_{0}\), \(y_{0}\) and \(z_{0}\) represent the actual positions of the receiver in different coordinate systems. \(X_{r}\), \(Y_{r}\), \(Z_{r}\) are the receiver positions estimated by the method in this paper.  
According to Table 6, the accuracy of the B1C signal \"amplitude ratio combined positioning\" relative to data component positioning and pilot component positioning improved by approximately 42.56% and 11.431%. The accuracy of the L1C signal \"amplitude ratio combined positioning\" relative to data component positioning and pilot component positioning improved by approximately 46.886% and 14.377%.  
It is well known that the pilot performance in B1C/L1C signal is better than the data, so users are most concerned about the improvement effect of combined positioning relative to pilot positioning.  
Table 6 shows that the positioning accuracy of the amplitude-ratio combination is improved by 2% compared to the 1:1 combination, and by 1.3% compared to the power-ratio combination for B1C signal. The positioning accuracy of the amplitude-ratio combination is improved by 2.37% compared to the 1:1 combination, and by 1.6% compared to the power-ratio combination for L1C signal. These numerical results show that the optimal coefficients estimated in this paper greatly improve the correlators joint positioning accuracy for the unequal power distribution signals.