## 3 Time-Frequency Compensation and Measurement Model  
### Clock Model  
Under the constraints of weight and power consumption, the extender cannot be equipped with an atomic clock with high time-frequency accuracy and only a high-stability crystal oscillator can be used. The original high-stability crystal oscillator output reference frequency is 10 MHz. The frequency accuracy of each measurement is calculated using the following formula:  
\[\Delta f_{r}=\frac{f-f_{0}}{f_{0}}. \tag{1}\]  
Here, \(\Delta f_{r0}\) is the frequency accuracy, \(f\) is the frequency-measurement value, and \(f_{0}\) is the reference frequency.  
After continuous measurement for two hours using a high-temperature crystal oscillator, the clock was in a stable state with a linear trend in frequency accuracy, and there was also a jitter change. By using linear fitting to analyze frequency accuracy data, frequency drift values can be obtained and fitting residuals can be given. Based on the above analysis, the clock frequency variation can be described using the following equation.  
\[\Delta f_{r}=\Delta f_{r0}+Kt+\delta f_{r}. \tag{2}\]  
Here, \(\Delta f_{r0}\) is the initial value of the frequency accuracy, \(K\) is the frequency drift rate and \(\delta f_{r}\) is a random jitter that conforms to a normal distribution.  
The relationship between clock uncertainty and clock frequency accuracy can be represented by the following equation [18; 19; 20]:  
\[\begin{cases}\delta t=m_{0}+m_{1}t+m_{2}t^{2}+\xi t,\\ m_{1}=\Delta f_{r0},\\ m_{2}=\frac{1}{2}K,\\ \xi t=\int_{t_{0}}^{t}\delta f_{r}dt.\end{cases} \tag{3}\]  
Here, \(\delta t\) is the clock bias at time \(t\), \(m_{0}\) is the clock bias at time \(t_{0}\), \(m_{1}\) is the clock speed term, \(m_{2}\) is the clock drift term and \(\xi t\) is the clock bias jitter at time \(t\).  
The observation model for the pseudorange observation value of navigation satellite \(s\) is as follows [21; 22; 23]:  
\[p_{f}^{s}=r^{s}+\delta t-\delta t^{s}-\Delta_{via}^{s}+\sigma^{u}+\sigma^{s}+ \varepsilon_{p_{f}}. \tag{4}\]  
In Equation (4), \(p_{f}^{s}\) represents the pseudorange observation value at frequency \(f\); \(r^{s}\) represents the geometric distance from the Ka antenna phase center of the Beidou satellite to the extended user's Ka antenna phase center; \(\delta t\) and \(\delta t^{s}\), respectively, represent the true clock bias of the extended user and the navigation satellite; \(\Delta_{via}^{s}\) represents the relativistic effect correction error caused by the relativistic delay, which can be ignored after correction; \(\sigma^{u}\) and \(\sigma^{s}\), respectively, represent the channel delay for extended users and navigation satellites; \(\varepsilon_{p_{f}}\) represents the pseudorange observation noise at frequency \(f\), whose initial values are pre-set.  
The clock bias can be expressed as follows:  
\[\delta t=p_{f}^{s}-(r^{s}-\delta t^{s}-\Delta_{via}^{s}+\sigma^{u}+\sigma^{s} +\varepsilon_{p_{f}}). \tag{5}\]  
Neglecting the random jitter \(\xi t\) caused by clock error, we construct the system of equations:\[\begin{cases}\delta t_{1}=m_{0}+m_{1}\Delta t_{1}+m_{2}\Delta t_{1}^{2},\\ \ldots\\ \delta t_{i}=m_{0}+m_{1}\Delta t_{i}+m_{2}\Delta t_{i}^{2},\ \ (i=2,\ldots,n-1)\\ \ldots\\ \delta t_{n}=m_{0}+m_{1}\Delta t_{n}+m_{2}\Delta t_{n}^{2}.\end{cases} \tag{6}\]  
According to the least squares algorithm, the clock difference parameters \(m_{0}\), \(m_{1}\) and \(m_{2}\) can be fitted by the linear model. The time capture capability of the extended user equipment is at the millisecond (ms) scale, and the frequency capture capability is at the kilohertz (kHz) scale. After capturing the unidirectional signal, the clock bias \(m_{0}\), clock velocity term \(m_{1}\) and clock drift term \(m_{2}\) calculated from the group of unidirectional measurements are used to adjust the time-frequency in orbit, which can achieve microsecond- and hundred-hertz-level time-frequency synchronization requirements and meet the conditions for bidirectional link building with the navigation constellation.