# 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  
#### 3.3.1 State Model  
The state vector of the joint coherent prefilter is defined as  
\[\begin{array}{rcl}\mathbf{X}_{\text{pre}}&=&[\mathbf{X}_{\text{L1 CA}},\; \mathbf{X}_{\text{L2 CM}}]^{T}\\ &=&[A_{\text{L1 CA}},\;\delta\tau_{\text{L1 CA}},\;\delta\phi_{\text{0,L1 CA}},\; \delta f_{\text{0,L1 CA}},\;\delta a_{\text{0,L1 CA}},\\ &&A_{\text{L2 CM}},\;\delta\tau_{\text{L2 CM}},\;\delta\phi_{\text{0,L2 CM}},\; \delta f_{\text{0,L2 CM}},\;\delta a_{\text{0,L2 CM}}]^{T}\end{array} \tag{14}\]  
where \(A_{\text{L1 CA}}\) and \(A_{\text{L2 CM}}\) represent the accumulated amplitudes of L1 C/A and L2 CM signals, respectively; \(\delta\tau_{\text{L1 CA}}\) and \(\delta\tau_{\text{L2 CM}}\) are the code phase biases between the local replica code and the incoming signals; and the other states are similar to the variables in Equation (13) but for L1 C/A and L2 CM signals.  
Then, the system model of the joint coherent prefilter can be described as  
\[\mathbf{X}_{\text{pre,k}}=\textit{Ph}_{\text{pre}[\text{k,k-1}}\cdot\mathbf{X} _{\text{pre,k-1}}+\mathbf{w}_{\text{k-1}} \tag{15}\]  
where \(\mathbf{w}\) is the process noise vector and \(\textit{Ph}_{\text{pre}[\text{k,k-1}}\) is the state transition matrix from time \(k-1\) to \(k\). \(\textit{Ph}_{\text{pre}[\text{k,k-1}}\) can be expressed as  
Figure 2: Block diagram of joint L1 C/A and L2 CM vector-tracking-based global navigation satellite system (GNSS)/strapdown inertial navigation system (SINS) deep-coupled navigation system structure.  
\[\begin{split} Phi_{\text{pre}|k,k-1}=\left[\begin{array}{cccccccc} Phi_{\text{LICA}|k-1}&0_{5\text{s}5}\\ 0_{5\text{s}5}&Phi_{\text{L2 CM}|k-1}\end{array}\right]=\left[\begin{array}{ ccccccccc}1&0&0&0&0&0&0&0&0&0\\ 0&1&0&\beta_{1}T&\frac{\beta_{2}T^{2}}{2}&0&0&0&0&0\\ 0&0&1&T&\frac{T^{2}}{2}&0&0&0&0&0\\ 0&0&0&1&T&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&0&1&0&0&0&0\\ 0&0&0&0&0&1&0&\beta_{2}T&\frac{\beta_{2}T^{2}}{2}\\ 0&0&0&0&0&0&0&1&T\\ 0&0&0&0&0&0&0&1\end{array}\right]\end{split} \tag{16}\]  
where \(\beta_{1}\) and \(\beta_{2}\) are used to convert from units of chips to units of cycles (\(\beta_{1}=1/1540\) for L1 C/A and \(\beta_{2}=1/1200\) for L2 CM).  
In general, the process noise covariance is given by \(\mathbf{Q}=E[\mathbf{ww}^{\text{T}}]\), whose diagonal elements are given by the spectral densities of the process noises \(\left[S_{\Delta\varphi_{0}},S_{\Delta\delta_{0}},S_{\Delta\delta_{0}}\right]\) by taking in account the clock noise and the dynamics. However, due to ionospheric scintillation noise contributions, the spectral densities of the a priori fixed process noise related to the phase and the frequency errors are modified by [23]  
\[\left\{\begin{array}{l}S_{\Delta\varphi_{0}}=S_{\Delta\varphi,clk}+S_{ \Delta\varphi,scint}\\ S_{\Delta\delta_{0}}=S_{\Delta\delta,clk}+S_{\Delta\delta,\text{scint}}\end{array}\right. \tag{17}\]  
where \(S_{\Delta\varphi,clk}\) and \(S_{\Delta\delta,clk}\) depend on the type of the oscillator and can be modelled as in [12]. \(S_{\Delta\varphi,scint}\) and \(S_{\Delta\delta,\text{scint}}\) are determined by the scintillation phase noise power spectral density and can be modelled as in [23].