# Method for Estimating the Optimal Coefficient of L1C/B1C Signal Correlator Joint Receiving
## 3 The Joint Receiving Scheme of Correlator
### Optimal Coefficient Estimation of Code Loop  
In combination tracking mode, the autocorrelation power density of the local reference signal, and the cross-correlation power density of the local reference signal and the received signal can be determined from Equations (33) and (34), respectively. \[G_{L}(f)=\alpha^{2}*G_{data}(f)+\beta^{2}*G_{pilot}(f) \tag{35}\]  
\[G_{L,R}(f)=\alpha\sqrt{\frac{1}{4}}G_{data}(f)+\beta\sqrt{\frac{3}{4}}G_{pilot }(f) \tag{36}\]  
where \(G_{data}(f)\), \(G_{pilot}(f)\) are the normalized power spectral densities of the data and pilot, which can be calculated using Section 2. Combined with Equations (11), (35) and (36), the mathematical model of the joint code tracking accuracy for the BIC and the LIC signals is derived as follows:  
\[ \begin{split}\sigma_{tDLL}(\alpha,\beta)=C\times\sqrt{\frac{{{B _{L}}(1-0.5{B_{L}}{T_{p}})\int_{-{B_{f}^{\prime}}/2}^{{B_{f}}/2}{{{\left[{{{ \alpha^{2}}*G_{data}}(f)+\beta^{2}*G_{pilot}(f)}\right]}\sin^{2}(\pi/{dT_{c}})df }}}}{{{{\left({{{2\pi}}\right)}^{2}}\left({{\cdot}C/{\rm{N0}}\right)}\cdot{{{ \left[{{{B_{f}^{\prime}}/2}}\right]}\left[{{{\alpha^{2}}*G_{data}(f)+\beta \sqrt{\frac{3}{4}}G_{pilot}(f)}\right]}\sin(\pi{dT_{c}})df}\right]}^{2}}}}\\ \cdot\sqrt{1+\frac{1}{{{T_{p}}}}\frac{{{{\int_{-{B_{f}^{\prime}}/2} ^{{B_{f}^{\prime}}/2}{{{\left[{{{4^{2}}*G_{data}}(f)+\beta^{2}*G_{pilot}(f)} \right]}\cos^{2}(\pi{dT_{c}})df}}}}}}{{{{\left({{{\rm{C}}\rm{N0}}}\right)} \cdot{{{\left[{{{B_{f}^{\prime}}/2}}\left[{{{\alpha^{2}}*G_{data}(f)+\beta ^{2}*G_{pilot}(f)}\right]}\cos^{2}(\pi{dT_{c}})df}\right]}^{2}}}}}}.\end{split}  \tag{37}\]  
When the joint code tracking error function \(\sigma_{tDLL}(\alpha,\beta)\) is the smallest, the corresponding \(\sigma_{tDLL}(\alpha,\beta)\) is the optimal weight of the data, where \(\beta=1-\alpha\). \[\alpha=\arg[\frac{\partial\sigma_{tDLL}(\alpha,\beta)}{\partial\alpha}] \tag{38}\]  
It can be seen from the model that the choice of different weighting coefficients leads to a completely different joint tracking accuracy, and the joint tracking accuracy is related to parameters such as C/N0 and filter bandwidth \(B_{f}\). Therefore, it is difficult to obtain the optimal coefficient through theoretical derivation. Therefore, a numerical analysis method is used to search for the optimal weighting coefficient. The variation in the correlator joint code tracking error with the combination coefficient of the data \(\alpha\) is shown in Figure 4. The value range of \(\alpha\) is \([0,1]\). Curves with different colors from bottom to top in Figure 4 represent different C/N0 conditions; the C/N0 is 25-50 dB, which covers all possible signal strengths received by ordinary commercial antennas on land where the single-side bandwidth \(B_{f}\) is 40 MHz. For L1C and B1C, the correlator interval \(d\) generally selects (0, 0.15) chips, and, at this time, selects 0.08 chips. Figure 4 shows that the tracking error of the code loop decreases monotonically and then increases monotonically, which achieves the minimum value at the optimal coefficient \(\alpha\). Figure 5 is a traversal of the minimum error std  
The above conclusion is further verified in the Galileo-EIOS signal with equal proportions of data and pilot. For example, Figure 5 shows that the optimal coefficients of the correlator combination are \(\alpha=0.5\) and \(\beta=0.5\). \[\frac{\alpha}{\beta}=1=\frac{A_{d}}{A_{p}}\]  
To further verify that the amplitude ratio combination is the optimal coefficient of the correlator combination, the following is a more comprehensive and detailed comparison of the traditional equal weight 1:1 combination method [29], the power ratio combination method [30] and the amplitude ratio combination method estimated in this paper. Table 2 lists three combinations of coefficient allocation schemes. For the B1C and the L1C, the three schemes produce different effects.