# Method for Estimating the Optimal Coefficient of L1C/B1C Signal Correlator Joint Receiving
## 3 The Joint Receiving Scheme of Correlator
### Optimal Coefficient Estimation of Carrier Loop  
Combined punctual correlation \(P_{J}\) is used in the carrier loop. To derive the optimal coefficient of the carrier loop, we consider the coherence loss induced by Doppler \(Sa(\pi\delta_{f,d}T)\) and code phase loss \(R(\delta_{\tau,d})\) to be 0 dB, regardless of the message \(D\) and secondary code symbols \(C_{\text{sec}}\). According to Equations (13), (16), (19) and (22), the discriminator output of the data, pilot and correlator joint can be obtained as follows:  
\[\Delta\hat{\varphi}_{d}=\text{arctan}(\frac{Q_{P,d}}{I_{P,d}})=\text{arctan}( \frac{A_{d}\cdot\sin\delta_{\varphi,d}+n_{d,Q}}{A_{d}\cdot\cos\delta_{\varphi,d }+n_{d,l}}) \tag{23}\]\[\Delta\hat{\phi}_{p}=\arctan(\frac{Q_{p,p}}{I_{p,p}})=\arctan(\frac{A_{p}\cdot \sin\delta_{q,p}+n_{p,Q}}{A_{p}\cdot\cos\delta_{q,p}+n_{p,I}}) \tag{24}\]  
\[\Delta\hat{\phi}_{I}=\arctan(\frac{Q_{p,I}}{I_{p,p}})=\arctan(\frac{\alpha A_{ d}\cdot\sin\delta_{q,d}+\alpha n_{d,Q}+\beta A_{p}\cdot\sin\delta_{q,p}+ \beta n_{p,Q}}{\alpha A_{d}\cdot\cos\delta_{q,d}+\alpha n_{d,I}+\beta A_{p} \cdot\cos\delta_{q,p}+\beta n_{p,I}}) \tag{25}\]  
where \(Q_{p,x}\) and \(I_{p,x}\) are the imaginary and real parts of punctual correlation for the \(x\) component \(P_{x}\); \(n_{x,Q}\) and \(n_{x,I}\) represent the noise of the quadrature branch and the direction branch of the \(x\) component. The carrier phase error tends to 0 when the carrier loop tends to be stable, that is, \(\delta_{q,d}\to 0\), \(\delta_{q,p}\to 0\). According to the principle of equivalent infinitesimal, we can obtain \(\sin\delta_{q,d}\to 0,\cos\delta_{q,d}\to 1\). The discriminator output of the data, pilot and correlator joint can be approximately written as follows:  
\[\Delta\hat{\phi}_{d}=\arctan(\frac{n_{d,Q}}{A_{d}+n_{d,I}}),\ \Delta\hat{\phi}_{p}=\arctan(\frac{n_{p,Q}}{A_{p}+n_{p,I}}) \tag{26}\]  
\[\Delta\hat{\phi}_{I}\approx\arctan(\frac{\alpha n_{d,Q}+\beta n_{p,Q}}{\alpha A _{d}+\alpha n_{d,I}+\beta A_{p}+\beta n_{p,I}}) \tag{27}\]  
The noise term \(\alpha n_{d,I}\), \(\beta n_{p,I}\) of the denominator can be omitted, which is infinitesimal relative to the signal amplitude \(\alpha A_{d}+\beta A_{p}\). Equations (26) and (27) can be approximately written as follows:  
\[\Delta\hat{\phi}_{d}\approx\arctan(\frac{n_{d,Q}}{A_{d}}),\ \Delta\hat{\phi}_{p} \approx\arctan(\frac{n_{p,Q}}{A_{p}}) \tag{28}\]  
\[\Delta\hat{\phi}_{I}\approx\arctan(\frac{\alpha n_{d,Q}+\beta n_{p,Q}}{\alpha A _{d}+\beta A_{p}}) \tag{29}\]  
The purpose of the joint-tracking method for the data and pilot is to make full use of the energy of the two components. The correlator joint only needs one discriminator, whose hardware complexity is lower than that of the discriminator joint method. The ideal performance is equivalent for the discriminator-level joint and the correlator-level joint, which depend on the design of the satellite signal. Therefore, the discriminator functions of the correlator joint and the discriminator joint should be equivalent. According to the equivalence principle of joint method, the discriminator functions of the correlator joint method and the discriminator joint method are equivalent. Therefore, the discriminator the correlator joint can be written as a normalized linear combination of the data and pilot. \[\Delta\hat{\phi}_{I}=\lambda\Delta\hat{\phi}_{d}+\omega\Delta\hat{\phi}_{p} \tag{30}\]  
where \(\lambda\) and \(\omega\) are the weight coefficients of the discriminator joint. The coefficients of the discriminator joint are determined by the variance in the loop noise, which depends on the power ratio of the data/pilot.