# A Joint Dual-Frequency GNSS/SINS Deep-Coupled Navigation System for Polar Navigation
## 3 Deep-Coupled Navigation System Design
### Filtering Model Design of the Joint Coherent Prefilter
#### Measurement Model  
In coherent prefilter algorithm, each accumulated correlator in the tracking channel outputs \(I\)s and \(Q\)s are directly used as measurements of the Kalman filter. The 12-measurement model with in-phase and quadrature-phase prompt, early, and late (\(I_{\text{P}}\), \(Q_{\text{P}}\), \(I_{\text{E}}\), \(Q_{\text{E}}\), \(I_{\text{L}}\), and \(Q_{\text{L}}\) for L1 C/A and L2 CM) measurements is used in each prefilter. The measurement vector of the prefilter can be written as  
\[\begin{array}{lcl}\mathbf{Z}_{\text{pre}}&=&\left[Z_{\text{L1 CA}},Z_{\text {L2 CM}}\right]^{T}\\ &=&\left[I_{\text{E, L1 CA}},\,Q_{\text{E, L1 CA}},\,I_{\text{P, L1 CA}},\,Q_{\text{P}},\,\text{L1 CA},\,I_{\text{L, L1 CA}},\,Q_{\text{L, L1 CA}},\,Q_{\text{L, L1 CA}},\right.\\ &&I_{\text{E, L2 CM}},\,Q_{\text{E, L2 CM}},\,I_{\text{P, L2 CM}},\,Q_{\text{P}},\,\text{L2 CM},\,I_{\text{L, L2 CM}},\,Q_{\text{L, L2 CM}},\,Q_{\text{L, L2 CM}}\right]^{T}\end{array} \tag{18}\]  
According to the expressions of I/Q introduced in Equation (12), the measurement model of the joint prefilter for GPS L1 C/A and L2 CM can be described as  
\[\mathbf{Z}_{\text{pre, k}}=D\cdot h_{\text{ k}}\Big{(}\mathbf{X}_{\text{pre, k}}\Big{)}+\mathbf{n}_{\text{k}}=\left[\begin{array}{c}D_{\text{L1 CA}}\cdot h_{ \text{L1 CA, k}}(\mathbf{X}_{\text{L1 CA, k}})+\mathbf{n}_{\text{L1 CA,k}}\\ D_{\text{L2 CM}}\cdot h_{\text{L2 CM, k}}(\mathbf{X}_{\text{L2 CM, k}})+\mathbf{n}_{\text{L2 CM,k}}\end{array}\right] \tag{19}\]  
where \(D\) is the navigation data bits, \(\mathbf{n}_{\text{k}}\) is the measurement noise vector at time \(k\), and \(h_{\text{k}}(\cdot)\) is the nonlinear measurement matrix and can be expressed as  
\[h_{\text{k}}(\cdot)=\left[h_{\text{L1 CA, k}}(\cdot),\,h_{\text{L2 CM, k}}(\cdot)\right]^{T}=\left[\begin{array}{c}A_{\text{L1 CA}}\cdot\text{sinc}( \delta f_{\text{L1 CA}}\cdot T)\cdot R(\delta\eta_{\text{L1 CA}}+\Delta_{-1}) \cdot\cos(\delta\overline{\Phi}_{\text{L1 CA}})\\ A_{\text{L1 CA}}\cdot\text{sinc}(\delta f_{\text{L1 CA}}\cdot T)\cdot R(\delta \eta_{\text{L1 CA}}+\Delta_{-1})\cdot\sin(\delta\overline{\Phi}_{\text{L1 CA}}) \\ A_{\text{L1 CA}}\cdot\text{sinc}(\delta f_{\text{L1 CA}}\cdot T)\cdot R(\delta \eta_{\text{L1 CA}}+\Delta_{0})\cdot\cos(\delta\overline{\Phi}_{\text{L1 CA}}) \\ A_{\text{L1 CA}}\cdot\text{sinc}(\delta f_{\text{L1 CA}}\cdot T)\cdot R(\delta \eta_{\text{L1 CA}}+\Delta_{0})\cdot\sin(\delta\overline{\Phi}_{\text{L1 CA}}) \\ A_{\text{L1 CA}}\cdot\text{sinc}(\delta f_{\text{L1 CA}}\cdot T)\cdot R(\delta \eta_{\text{L1 CA}}+\Delta_{+1})\cdot\cos(\delta\overline{\Phi}_{\text{L1 CA}}) \\ A_{\text{L1 CA}}\cdot\text{sinc}(\delta f_{\text{L1 CA}}\cdot T)\cdot R(\delta \eta_{\text{L1 CA}}+\Delta_{+1})\cdot\sin(\delta\overline{\Phi}_{\text{L1 CA}}) \\ A_{\text{L2 CM}}\cdot\text{sinc}(\delta f_{\text{L2 CM}}\cdot T)\cdot R(\delta \eta_{\text{L2 CM}}+\Delta_{-1})\cdot\cos(\delta\overline{\Phi}_{\text{L2 CM}}) \\ A_{\text{L2 CM}}\cdot\text{sinc}(\delta f_{\text{L2 CM}}\cdot T)\cdot R(\delta \eta_{\text{L2 CM}}+\Delta_{0})\cdot\cos(\delta\overline{\Phi}_{\text{L2 CM}}) \\ A_{\text{L2 CM}}\cdot\text{sinc}(\delta f_{\text{L2 CM}}\cdot T)\cdot R(\delta \eta_{\text{L2 CM}}+\Delta_{0})\cdot\sin(\delta\overline{\Phi}_{\text{L2 CM}}) \\ A_{\text{L2 CM}}\cdot\text{sinc}(\delta f_{\text{L2 CM}}\cdot T)\cdot R(\delta \eta_{\text{L2 CM}}+\Delta_{+1})\cdot\sin(\delta\overline{\Phi}_{\text{L2 CM}}) \\ A_{\text{L2 CM}}\cdot\text{sinc}(\delta f_{\text{L2 CM}}\cdot T)\cdot R(\delta \eta_{\text{L2 CM}}+\Delta_{+1})\cdot\sin(\delta\overline{\Phi}_{\text{L2 CM}}) \end{array}\right] \tag{20}\]Finally, the fifth-degree Cubature Kalman filter (5 th-CKF) for the prefilter, which was proposed and described in [19] in detail, is used in this paper to solve the nonlinear problem of the prefilter and get a higher filter accuracy. Besides this, the noise variance in observations can be computed as a function of C/N\({}_{0}\) as reported in [22; 24].