## 2 Conventional MIMU/GNSS UTC Integration Architecture
### Channel Prefilter Model  
The channel prefilters are utilized to generate PR and PRR measurements as illustrated in Figure 1. In harsh environments, the carrier phase tracking is more vulnerable than the frequency tracking. For this reason, states of the prefilter include code phase error \(\tau_{p}\), carrier frequency error \(f_{e}\), and carrier frequency rate \(\delta a\), excluding carrier phase error.  
Figure 1: A typical MIMU/GNSS UTC navigation architecture with channel prefilters.  
The discrete-time system model of the prefilter is  
\[\mathbf{x}_{k}=\mathbf{\Phi}_{k/k-1}\mathbf{x}_{k-1}+\mathbf{\eta}_{k-1} \tag{2}\]  
where \(\mathbf{x}=\left[\begin{array}{cc}\tau_{p}&f_{\epsilon}&\delta a\end{array} \right]^{T}\) is the state vector, and \(\mathbf{\eta}\) is the system noise vector.  
The state one-step transition matrix is  
\[\mathbf{\Phi}_{k/k-1}=\left[\begin{array}{ccc}1&\beta T&\beta T^{2}/2\\ 0&1&T\\ 0&0&1\end{array}\right] \tag{3}\]  
where \(T\) is the update period, \(\beta=f_{\infty}/f_{\alpha}\) is a scale factor, \(f_{\alpha}\) is the nominal carrier frequency, and \(f_{\infty}\) is the pseudo-random noise (PRN) code chip rate. For the GPS L1 C/A code, \(\beta\) is equal to 1/1540.  
The prefilter measurement model is modeled as  
\[\mathbf{z}_{k}=\mathbf{H}\mathbf{x}_{k}+\mathbf{v}_{k} \tag{4}\]  
where \(\mathbf{z}_{k}=\left[\begin{array}{cc}z_{\tau}&z_{f}\end{array}\right]^{T}\) is the measurement vector, and \(\mathbf{v}_{k}\) is the measurement noise vector. The variables \(z_{\tau}\) and \(z_{f}\) denote the outputs of the code discriminator and carrier frequency discriminator, respectively.  
The measurement matrix is defined as  
\[\mathbf{H}=\left[\begin{array}{ccc}1&-\beta T/2&\beta T^{2}/6\\ 0&1&-T/2\end{array}\right]. \tag{5}\]  
A normalized non-coherent early-minus-late envelope code discriminator [20] is applied to compute code tracking errors.  
\[z_{\tau}=\frac{1}{2}\frac{E-L}{E+L} \tag{6}\]  
where \(E\) and \(L\) are the autocorrelation amplitudes of the early and late correlators, respectively, given by  
\[\begin{array}{l}E=\sqrt{l_{E}^{2}(n)+Q_{E}^{2}(n)}\\ L=\sqrt{l_{L}^{2}(n)+Q_{L}^{2}(n)}\end{array}. \tag{7}\]  
A four-quadrant arctangent frequency discriminator [20] is used in this paper, given by  
\[z_{f}=\frac{\text{arctan2}(P_{cross},P_{dot})}{\pi T_{coth}}. \tag{8}\]  
\(P_{cross}\) and \(P_{dot}\) in (8) can be expressed as follows:  
\[\begin{array}{l}P_{cross}=I_{p,1}Q_{p,2}-Q_{p,1}I_{p,2}\\ P_{dot}=I_{p,1}I_{p,2}+Q_{p,1}Q_{p,2}\end{array} \tag{9}\]  
where \(I_{p,1}\) and \(Q_{p,1}\) denote the coherent integration values of the first half of the coherent integration period, and \(I_{p,2}\) and \(Q_{p,2}\) denote the coherent integration values of the second half of the coherent integration period.  
The code discriminator variance \(\sigma_{\tau}^{2}\) and frequency discriminator variance \(\sigma_{f}^{2}\) can be approximated as follows [21; 22]:  
\[\begin{array}{l}\sigma_{\tau}^{2}\approx\frac{1}{(c/n_{0})T_{coth}}\Big{(} \frac{1}{2}+\frac{1}{2(c/n_{0})T_{coth}}\Big{)}\\ \sigma_{f}^{2}\approx\frac{1}{4\pi^{2}(c/n_{0})(T_{coth}/2)^{3}}\Big{(}1+\frac {1}{(c/n_{0})T_{coth}/2}\Big{)}\end{array} \tag{10}\]  
where \(c/n_{0}\) is the carrier-to-noise ratio in Hz.