# A Joint Dual-Frequency GNSS/SINS Deep-Coupled Navigation System for Polar Navigation
## 3 Deep-Coupled Navigation System Design
### GPS L1 C/A and L2 CM Signal Model  
The mathematical expression of the GPS L1 C/A and L2 CM intermediate frequency (IF) signal can be described as follows [8]:  
\[\left\{\begin{array}{l}S^{i}_{\text{L1C/A}}(t)=\sqrt{P_{\text{L1C/A}}}\cdot D ^{i}_{\text{L1 CA}}(t)\cdot C^{i}_{\text{L1C/A}}(t)\cdot\cos(2\pi f_{0,\text{ L1C/A}}t+\phi^{i}_{\text{L1C/A}})\\ S^{i}_{\text{L2 CM}}(t)=\sqrt{P_{\text{L2 CM}}}\cdot D^{i}_{\text{L2 CM}}(t)\cdot C ^{i}_{\text{L2 CM}}(t)\cdot\cos(2\pi f_{0,\text{ L2 CM}}t+\phi^{i}_{\text{L2 CM}})\end{array}\right. \tag{11}\]  
where \(i\) denotes the satellite number; \(P_{\text{L1C/A}}\) and \(P_{\text{L2 CM}}\) are the power of the L1 C/A and L2 CM signals, respectively; \(D(t)_{\text{L1 CA}}\) and \(D(t)_{\text{L2 CM}}\) are the navigation data modulated with the C/A code and CM code, respectively; \(C_{\text{L1C/A}}(t)\) and \(C_{\text{L2 CM}}(t)\) represent the 1.023 Mbps L1 C/A code and 1.023 Mbps L2 CM return-zero (RZ) code, respectively. \(f_{0,\text{ L1C/A}}\) and \(f_{0,\text{ L2 CM}}\) are the carrier frequencies of the L1 C/A and L2 CM signals, respectively; and \(\phi_{\text{L1C/A}}\) and \(\phi_{\text{L2 CM}}\) are the initial carrier phases of the L1 C/A and L2 CM signals, respectively.  
In the GNSS signal tracking loop, the incoming signal is multiplied by the local carrier signal generated by the carrier numerically controlled oscillator (NCO), removing the carrier frequency. Then, the result is multiplied by the local code signal generated by the code NCO. The outputs pass through the integrator and dump blocks to generate the in-phase/quadrature-phase (I/Q) correlator outputs as  
\[\left\{\begin{array}{l}I=A\cdot\frac{\sin(\pi\cdot\delta f\cdot T)}{\pi\cdot \delta f\cdot T}\cdot D\cdot R(\delta\tau+\Delta_{\text{k}})\cdot\cos(\delta \overline{\Phi})+n_{\text{I}}\\ Q=A\cdot\frac{\sin(\pi\cdot\delta f\cdot T)}{\pi\cdot\delta f\cdot T}\cdot D \cdot R(\delta\tau+\Delta_{\text{k}})\cdot\sin(\delta\overline{\Phi})+n_{ \text{Q}}\end{array}\right. \tag{12}\]  
where \(A\) represents the accumulated amplitude, \(T\) is the integration period, \(D\) is the navigation data bits of GNSS, \(R(\cdot)\) is the autocorrelation function of the ranging code, \(\delta\tau\) is the code phase bias between the local replica code and the incoming signals, and \(\delta f\) is the frequency error between the local replica frequency and the incoming signals. \(\Delta_{\text{k}}\) is the correlator spacing for early, prompt, and delay code, where \(k=-1,0,1\). \(n_{\text{I}}\) and \(n_{\text{Q}}\) are the noise of \(I\) and \(Q\). \(\delta\overline{\Phi}\) is the average phase error over the integration interval, which can be written as  
\[\delta\overline{\Phi}=\delta\phi_{0}+\frac{1}{2}\delta f_{0}T+\frac{1}{6} \delta a_{0}T^{2} \tag{13}\]  
where \(\delta\phi_{0}\) and \(\delta f_{0}\) indicate the initial phase error and the initial carrier frequency error at the start of the integration interval, respectively, and \(\delta a_{0}\) is the carrier phase acceleration error. GPS L1 C/A and L2 CM signals have correlator outputs similar to Equation (12) in the tracking loop.