## 3 Materials and Methods  
The bidirectional satellite-ground link can be modeled using the carrier phase as a medium for information transfer. Several factors can affect the transmission and reception of a carrier-phase signal generated by a high-precision time and frequency reference at a ground station. These include the changing relative position between the satellite and the ground, phase variations in the onboard oscillator, ionospheric effects, and relativistic effects. All such factors must be considered. When a ground station transmits a signal, it possesses an initial phase. Subsequently, the signal undergoes phase variations during its propagation through the uplink and downlink non-dispersive media. Such phase changes are proportional to both the carrier frequency and the pseudo range. Because the ionosphere is a dispersive medium, its effect requires separate consideration in the equations. The signal forwarded by the synchronized orbit satellite and subsequently received on the ground can be described by the following relationship [25; 27]:  
\[ \begin{split}&\phi_{AR}(t)=\phi_{AT}(t)-\frac{\omega_{uA}\rho_{as }(t)}{c}-\phi_{5}-\frac{\omega_{dA}\rho_{as}(t)}{c}\\ &+K_{\text{iono}}\left(\frac{T\text{C}_{A}}{\omega_{uA}}+\frac{T \text{C}_{A}}{\omega_{dA}}\right)+(\omega_{uA}-\omega_{dA})\gamma_{A}\Sigma_{A},\end{split}  \tag{1}\]  
Figure 2: Architecture of the frequency synchronization and comparison system. A, B and C represent multiple future ground stations. The circle symbol represents a mixer. where \(\phi_{AT}\) denotes the signal's phase at the earth station's transmitting end, \(\phi_{S}\) denotes the additional phases introduced by the satellite transponder, \(\rho_{as}(t)\) denotes the pseudo range from Station A to the satellite, and \(\omega_{uA}\) and \(\omega_{dA}\) respectively denote the uplink and downlink carrier angular frequencies. The correction coefficient for Station A's proper time is denoted by \(\gamma_{A}\), whereas the link asymmetry in Minkowski spacetime is denoted by \(\Sigma_{A}\). \(TEC_{A}\) denotes the total electron content along Station A's transmission path, and \(K_{\text{iono}}\) denotes a constant in the expression for the phase delay caused by the phase refractive index affecting the propagation of electromagnetic waves in the ionosphere, given by  
\[K_{\text{iono}} =\frac{40.28}{c}\times 4\pi^{2} \tag{2}\]  
The final two terms of Equation (1) describe the nonlinear time delay introduced by the electromagnetic waves traversing the ionosphere from the viewpoint of the carrier phase and the path asymmetry of the two-way satellite-Earth station link, respectively, which can be analyzed on the basis of special relativity. This is discussed in the following section. The phase of the signal received by the satellite can be expressed as follows:  
\[\phi_{SR}(t)=\phi_{AT}(t)-\frac{\omega_{uA}\rho_{as}(t)}{c}+\frac{K_{\text{ iono}}\ TEC_{A}}{\omega_{uA}}+\omega_{uA}\gamma_{A}\Sigma_{A} \tag{3}\]  
where \(\phi_{SR}\) denotes the phase of the signal received by the satellite. The three terms described above must be corrected in real-time at the transmitter to synchronize the signal phase at the ground station with the phase received at the satellite. Consequently, we proposed a frequency-synchronization scheme based on active carrier-phase compensation. The fundamental principles are illustrated in Figure 3.  
The two phase shifters, as shown in the \"Phase-shift Uplink\" and \"Phase-shift Downlink\" blocks in Figure 3, can be added before and after the carrier-phase receiver, ensuring that the phase-shifted values of the two phase shifters are maintained at all times by a specific ratio of the uplink and downlink carrier angular frequencies. This linkage, which  
Figure 3: Frequency recovery schematic based on the GEO communication satellite’s phase-locked transponder.