## 4 NR Navigation System
### EKF for LEO Navigation  
The extended Kalman filter (EKF) is a nonlinear estimation technique and demonstrates particular efficacy in handling the real-time positioning of GNSS navigation receivers [2]. To ensure compatibility with subsequent research, the receiver's state vector incorporates position, velocity, and clock error parameters:  
\[\mathbf{x}(k)=\left[\mathbf{r}(k),\dot{\mathbf{r}}(k),\delta t(k),\dot{\delta t }(k)\right]^{\mathsf{T}} \tag{24}\]  
where \(\mathbf{r}(k)=[x(k),y(k),z(k)]\) denotes the receiver position at epoch \(k\) in Earth-centered, Earth-fixed (ECEF) coordinates, and \(\dot{\mathbf{r}}(k)\) represents receiver's velocity. \(\delta t\) and \(\dot{\delta t}\) are the receiver's clock bias and drift rate, respectively. The system dynamic model is constructed based on receiver motion characteristics:  
\[\mathbf{x}(k+1)=\mathbf{F}\mathbf{x}(k)+\mathbf{w}(k) \tag{25}\]  
where \(\mathbf{F}\) is the state transition matrix and its definition is as follows:  
\[\mathbf{F}=\left( \begin{array}{cccc}\mathbf{I}_{3}&\mathbf{I}_{3}\tau_{s}& \mathbf{0}_{3,1}&\mathbf{0}_{3,1}\\ \mathbf{0}_{3}&\mathbf{I}_{3}&\mathbf{0}_{3,1}&\mathbf{0}_{3,1}\\ \mathbf{0}_{1,3}&\mathbf{0}_{1,3}&1&0\\ \mathbf{0}_{1,3}&\mathbf{0}_{1,3}&0&1\end{array} \right) \tag{26}\]  
Figure 4: Different correlation method: (**a**) conventional CP-based method with integration interval \(T_{\text{sym}}\) (replica signal includes CP); (**b**) effective integration interval of proposed ZPC method is \(T_{\text{u}}\) (replica signal utilizes zero-padding). where \(\tau_{s}\) denotes the EKF state propagation interval, equal to the coherent integration duration \(T_{\rm coh}\) in this study. \(\mathbf{I}_{n}\) represents an \(n\times n\) identity matrix, while \(\mathbf{0}_{m\times n}\) indicates an \(m\times n\) zero matrix. \(\mathbf{w}(k)\) is the process noise vector and its covariance matrix \(\mathbf{Q}\) models system uncertainties:  
\[\mathbf{Q}=\left( \begin{array}{cccc}\frac{1}{3}\mathbf{S}_{a}\tau_{3}^{3}& \frac{1}{2}\mathbf{S}_{a}\tau_{s}^{2}&\mathbf{0}_{3,1}&\mathbf{0}_{3,1}\\ \frac{1}{2}\mathbf{S}_{a}\tau_{s}^{2}&\mathbf{S}_{a}\tau_{s}&\mathbf{0}_{3,1}& \mathbf{0}_{3,1}\\ \mathbf{0}_{1,3}&\mathbf{0}_{1,3}&S_{c\phi}\tau_{s}+\frac{1}{3}S_{cf}\tau_{s} ^{3}&\frac{1}{2}S_{cf}\tau_{s}^{2}\\ \mathbf{0}_{1,3}&\mathbf{0}_{1,3}&\frac{1}{2}S_{cf}\tau_{s}^{2}&S_{cf}\tau_{s} \end{array} \right) \tag{27}\]  
where \(\mathbf{S}_{a}\) denotes the PSD vector of receiver acceleration in ECEF coordinates, \(S_{cf}\) represents clock frequency drift PSD, and \(S_{c\phi}\) corresponds to clock phase noise PSD. In this paper, \(\mathbf{S}_{a}\), \(S_{cf}\), and \(S_{c\phi}\) are, respectively, set to diag(10, 10, 10), 1, and 1. Hence, \(\mathbf{Q}\) remains constant for specified motion states. The pseudorange and pseudorange rate are used as observations in this paper, and these observations are measured by the DLL and FLL, respectively. In addition to real geometric distance and velocity, these measurements also contain various error components. For \(i\)-th LEO satellite, its pseudorange measurement \(\rho_{i}(k)\) can be expressed as  
\[\rho_{i}(k)=d_{i}(k)+c[\delta t(k)+\delta t_{\text{sat},i}(k)+I_{i}(k)+T_{i}(k )]+\varepsilon_{i}(k) \tag{28}\]  
where \(d_{i}(k)=\|\mathbf{r}_{s,i}(k)-\mathbf{r}(k)\|\) denotes the true geometric distance between receiver and satellite, with \(\mathbf{r}_{s,i}(k)\) representing the satellite's ECEF position coordinates. \(c\) denotes the speed of light while \(\delta t(k)\) and \(\delta t_{\text{sat},i}(k)\) correspond to receiver and satellite clock offsets, respectively. The ionospheric and tropospheric delays are modeled as \(I_{i}(k)\) and \(T_{i}(k)\), while \(\varepsilon_{i}(k)\) is the ranging measurement error caused by receiver noise.