# Method for Estimating the Optimal Coefficient of L1C/B1C Signal Correlator Joint Receiving
## 3 The Joint Receiving Scheme of Correlator  
### Correlator Joint Receiving Algorithm  
A pilot channel undergoes similar path delays and shifts to a data channel, thus making data/pilot combined tracking an attractive option. The two components have the same frequency, and the carrier phase relationship is fixed. The traditional single-component tracking method wastes useful power. Based on the specific relationship between the data and the pilot, the receiver adopts correlator joint receiving, where the total signal power is increased to obtain a higher tracking accuracy. The combination receiving design framework is shown in Figure 3, which includes a tracking module and a positioning module. In the tracking module, \"Costas\" is adopted by the data-component carrier tracking loop due to the unknown message information that it modulates. However, a pure phase-locked loop with better performance is designed in the pilot-component carrier loop because it modulates the known secondary code information. After the digital intermediate frequency signal is stripped from the carrier, it is coherently integrated with the early, punctual, and late local copy code. Figure 2: The noncoherent code-tracking error. The receiver correlator outputs early branch E, punctual branch P and late branch L, which can be expressed as follows:  
\[E_{d}=A_{d}D\frac{\sin(\pi\delta_{f}T)}{\pi\delta_{f}T}[R(\delta_{\tau,d}+\frac{ d}{2})\cdot\cos(\delta_{\varphi,d})+R(\delta_{\tau,d}+\frac{d}{2})\cdot\sin(\delta_{ \varphi,d})]+n_{E_{d,l}}+n_{E_{d,Q}} \tag{12}\]  
\[P_{d}=A_{d}D\frac{\sin(\pi\delta_{f}T)}{\pi\delta_{f}T}[R(\delta_{\tau,d})\cdot \cos(\delta_{\varphi,d})+R(\delta_{\tau,d})\cdot\sin(\delta_{\varphi,d})]+n_{p_ {d,l}}+n_{p_{d,Q}} \tag{13}\]  
\[L_{d}=A_{d}D\frac{\sin(\pi\delta_{f}T)}{\pi\delta_{f}T}[R(\delta_{\tau,d}-\frac {d}{2})\cdot\cos(\delta_{\varphi,d})+R(\delta_{\tau,d}-\frac{d}{2})\cdot\sin( \delta_{\varphi,d})]+n_{L_{d,l}}+n_{L_{d,Q}} \tag{14}\]  
\[E_{p}=A_{p}C_{\text{sec}}\frac{\sin(\pi\delta_{f}T)}{\pi\delta_{f}T}[R(\delta _{\tau,p}+\frac{d}{2})\cdot\cos(\delta_{\varphi,p})+R(\delta_{\tau,p}+\frac{d} {2})\cdot\sin(\delta_{\varphi,p})]+n_{E_{p,l}}+n_{E_{p,Q}} \tag{15}\]  
\[P_{p}=A_{p}C_{\text{sec}}\frac{\sin(\pi\delta_{f}T)}{\pi\delta_{f}T}[R(\delta _{\tau,p})\cdot\cos(\delta_{\varphi,p})+R(\delta_{\tau,p})\cdot\sin(\delta_{ \varphi,p})]+n_{P_{p,l}}+n_{P_{p,Q}} \tag{16}\]  
\[L_{p}=A_{p}C_{\text{sec}}\frac{\sin(\pi\delta_{f}T)}{\pi\delta_{f}T}[R(\delta _{\tau,p}-\frac{d}{2})\cdot\cos(\delta_{\varphi,p})+R(\delta_{\tau,p}-\frac{d} {2})\cdot\sin(\delta_{\varphi,p})]+n_{L_{p,l}}+n_{T_{p,Q}} \tag{17}\]  
where \(A_{d}\) and \(A_{p}\) are the signal amplitude of the data component and the pilot component, respectively. \(D\) is the message symbol of the data component. \(C_{\text{sec}}\) is the secondary code symbol of the pilot. \(\delta_{f}\) is the frequency error (Hz) between the local carrier and the received signal carrier, and \(T\) is the coherent integration time. \(\delta_{\tau,d}\) and \(\delta_{\tau,p}\) are the code phase error between the local code and the received signal of the data and the pilot component. \(\delta_{\varphi,d}\)  
Figure 3: The framework of correlator joint receiving for BIC/L1C signals. and \(\delta_{\varphi,p}\) are the carrier tracking phase error of the data and the pilot component; \(n\) is the noise of the correlation integral. To make full use of all the energy of the two components, a combination is performed after the correlator. According to Table 1, the phase relationship between the pilot and data component is the same for the L1C (GPS, QZS2-4) signals. The phase relationship between the pilot and data component is orthogonal for B1C and L1C(QZS1). Consequently, the output of the correlator after combination can be expressed as follows:  
\[\left|E_{J}\right|=