## 3 Error Introduced by LEO Satellite Assistance
### Clock Stability of Sattelite and Receiver  
In addition to the clock accuracy, the clock stability of the satellite and receiver can also cause the frequency bias, resulting in signal attenuation when a long coherent integration is adopted. The frequency stability of the clock refers to the relative change in frequency caused by the oscillator noise. This noise is usually expressed in the form of power-law noise, and its power spectral density can be expressed as:  
\[S_{y}(f)=h_{\alpha}f^{\alpha} \tag{7}\]  
where \(f\) represents frequency, \(h_{\alpha}\) indicates the intensity factor, and \(\alpha\) is the index of power-law noise. When \(\alpha=0\), \(\alpha=-1\), and \(\alpha=-2\), it stands for white noise, flicker noise, and random walk noise, respectively. In applications, the Allan variance is often used to evaluate the time-domain frequency stability of the frequency. It can be determined by [25]:  
\[\delta_{y}^{2}(\tau)=\frac{S_{f}}{\tau}+\frac{S_{g}\tau}{3}. \tag{8}\]  
\(S_{f}\) and \(S_{g}\) indicate the phase and frequency drift strength of the clock. \(S_{f}\sim h_{0}/2\), \(S_{g}\sim 2\pi^{2}h_{-2}\). The frequency stability of the clock is related to the selected oscillator type. In general, LEO satellites often choose the Oven Controlled Crystal Oscillator (OCXO) with better stability [23], while GNSS receivers often use the Temperature Compensated Crystal Oscillator (TCXO) with general performance. In the LEO-assisted GNSS system, when the SNR is improved through a long coherent integration time, the frequency deviation caused by the clock instability can result in the attenuation of the integrated signal. To analyze the influence of clock noise, this paper uses OCXO and TCXO typical parameters to generate clock noises for simulation of LEO satellites and receivers, respectively. In the simulation, the parameters of the TCXO are set to be \(h_{0}=2\times 10^{-19}\), \(h_{-2}=2\times 10^{-20}\), and the parameters for the OCXO are chosen as \(h_{0}=2.51\times 10^{-26}\), \(h_{-2}=2.51\times 10^{-22}\). The clock model used in the simulation is as follows:  
\[\left[ \begin{array}{c}\delta t(k+1)\\ \delta f(k+1)\end{array} \right]=\left[ \begin{array}{cc}1&\Delta t_{k}\\ 0&1\end{array} \right]\left[ \begin{array}{c}\delta t(k)\\ \delta f(k)\end{array} \right]+\left[ \begin{array}{cc}\sqrt{S_{f}\Delta t_{k} +S_{g}\Delta t_{k}{}^{3}/3}&\sqrt{S_{g}\Delta t_{k}{}^{2}/2}\\ \sqrt{S_{g}\Delta t_{k}{}^{2}/2}&\sqrt{S_{g}\Delta t_{k}}\end{array} \right]w_{k} \tag{9}\]  
where \(\Delta t_{k}\) is the time interval, and \(w_{k}\) is an identity matrix. Figure 6 is the Allan deviation of the simulated clock noise sequences of OCXO and TCXO. It can be seen from the figure that the performance of the clock generated by this clock model is consistent with the theoretical analysis. The Allan deviation of OCXO caused by the oscillator noise is not higher than \(10^{-10}\) within 10 s. The stability of TCXO reaches its lowest value around 500 ms. As the time interval is further lengthened, the frequency deviation caused by noise begins to increase gradually. When Doppler positioning is performed on a GNSS receiver through an LEO satellite, the GNSS receiver will absorb the LEO satellite's clock error. Therefore, the impact of the LEO satellite's clock and GNSS receiver's clock on signal processing should be considered at the same time. For the GNSS receiver, the clock frequency error caused by instability will cause the attenuation of the coherent gain. The power attenuation caused by coherent integration can be given by:  
\[G_{d}=10 lg\tilde{sinc}^{2}(f_{e}T_{coh}) \tag{10}\]  
where \(f_{e}\) is the frequency deviation, and \(T_{coh}\) represents the coherent integration time.