## 4 Proposed MIMU/GNSS UTC Integration Architecture
### Adaptive Prefilter with Multiple Fading Factors  
The conventional channel prefilter depicted in Section 2.1 usually uses a CKF. A fatal drawback of the CKF is its poor performance in estimating mutation states. For the CKF in the steady state, the state covariance matrix of the filter tends to a fixed value, causing the gain matrix of the Kalman filter to tend to zero. In other words, for a steady-state CKF, the measurement correction is weakened. If a large frequency tracking error suddenly occurs, the estimated frequency error of a conventional prefilter will deviate from its true value. In this case, the CKF exhibits a serious model mismatch. An adaptive Kalman filter is an effective method to counteract the problem [30]. An adaptive Kalman filter with multiple fading factors is proposed to construct an adaptive channel prefilter. According to the basic theory of Kalman filtering, if there is no model mismatch, the innovation sequence of a CKF is a white noise sequence. In the process of filter updating, the variance matrix of the innovation sequence can be estimated by the following recursive expression [31]:  
\[\hat{V}_{k}=(1-\beta_{k})\hat{V}_{k-1}+\beta_{k}\epsilon_{k}\epsilon_{k}^{T} \tag{26}\]  
where \(\hat{V}_{k}\) is the estimate of the variance matrix at time \(k\), \(\hat{V}_{k-1}\) is the variance matrix estimate of the previous epoch, \(\beta_{k}\) is the fading weighting factor, and \(\epsilon_{k}\) is the innovation vector. The calculation of the fading weighting factor is a recursive process. The update method is expressed as follows [31]:  
\[\beta_{k}=\frac{\beta_{k-1}}{\beta_{k-1}+b} \tag{27}\]  
where \(\beta_{0}=1\) is the initial value, and \(b\) is a preset constant with a value range of 0.9-0.999. \(\hat{V}_{k}\) is approximately equal to the theoretical innovation variance matrix, given by  
\[\hat{V}_{k}\approx H_{k}\Big{(}\Phi_{k/k-1}P_{k-1}\Phi_{k/k-1}^{T}+Q_{k-1} \Big{)}H_{k}^{T}+R_{k} \tag{28}\]  
where \(H_{k}\) is the measurement matrix, \(\Phi_{k/k-1}\) is the state transition matrix, \(P_{k-1}\) is the state covariance matrix, \(Q_{k-1}\) is the system noise matrix, and \(R_{k}\) is the measurement noise matrix. The channel prefilter contains three state variables. For filters with multiple state variables, if the uncertainty of the model has dissimilar effects on different states, multiple fading factors should be used to adaptively adjust the different state components. The multiple fading factor matrix \(\lambda_{k}\) is defined as  
\[ \begin{array}{c}A_{k}=\sqrt{c_{k}}\mathbf{\alpha}\\ \mathbf{\alpha}=\text{diag}\Big{(}\sqrt{\alpha_{1}},\sqrt{\alpha_{2}},\cdots, \sqrt{\alpha_{n}}\Big{)}\end{array}  \tag{29}\]  
where \(c_{k}\) is the coefficient to be calculated, \(\text{diag}(\cdot)\) denotes a diagonal matrix composed of elements in the parentheses, and \(\alpha_{i}\geq 1\) is the fading proportional coefficient corresponding to the \(i\)-th component. The coefficient \(c_{k}\) is a scalar parameter, expressed as follows:  
\[c_{k}=\frac{\text{tr}(N_{k})}{\text{tr}(M_{k})} \tag{30}\]where \(\text{tr}(\cdot)\) represents the matrix trace operation. Detailed expressions of \(\mathbf{N}_{k}\) and \(\mathbf{M}_{k}\) are provided as follows:  
\[ \begin{array}{c}\mathbf{N}_{k}=\hat{\mathbf{V}}_{k}-\mathbf{H}_{k}\mathbf{Q}_{k-1}\mathbf{H}_{k}^{ T}-\mathbf{R}_{k}\\ \mathbf{M}_{k}=\mathbf{H}_{k}\mathbf{\Phi}_{k/k-1}(\mathbf{\alpha}\mathbf{P}_{k-1}\mathbf{\alpha})\mathbf{ \Phi}_{k/k-1}^{T}\mathbf{H}_{k}^{T}\end{array} . \tag{31}\]  
Note that the lower limit of the fading factor is 1. \(\lambda_{k}\) is a diagonal matrix. The diagonal element \(\lambda_{k}(i,i)\) is the fading factor corresponding to the \(i\)-th state component, defined as follows:  
\[\lambda_{k}(i,i)=\max(1,\sqrt{c_{k}}\mathbf{\alpha}). \tag{32}\]  
The difference between the adaptive Kalman filter with multiple fading factors and the classic Kalman filter is the time update of the state covariance matrix.