# A Doppler Frequency-Offset Estimation Method Based on the Beam Pointing of LEO Satellites
## 2 Model Establishment
### Establishment of the Frequency-Offset Model  
Equation (3) indicates that the Doppler frequency offset \(F_{d}\) is dependent on the angle \(\alpha\). Within the satellite body's coordinate system, \(\alpha\) can be expressed in terms of the satellite's azimuth angle \(\theta\), which can be expressed in terms of the satellite's elevation angle \(\varphi\). Figure 1: Simple satellite–ground Doppler frequency-offset model. The satellite body's coordinate system is defined as follows:  
* Origin (S): Located at the satellite's center of mass. * Z-axis: Directed from the satellite's center of mass toward the center of the Earth. * X-axis: Lies in the orbital plane, perpendicular to the Z-axis, and points in the direction of the satellite's velocity. * Y-axis: Determined according to the right-hand rule with respect to the X-axis and Z-axis. The satellite body's coordinate system and the Doppler frequency-offset model based on beam pointing (DFS-BP) are illustrated in Figure 2.  
In this paper, the satellite antenna's coordinate system is defined consistently with the satellite body's coordinate system; therefore, no coordinate transformation between the two systems is required. According to the characteristics of the satellite body's coordinate system, we divide the satellite into three two-dimensional planes: plane \(XSY\), plane \(XSZ\), and plane \(YSZ\). The satellite \(S\), ground terminal \(M\), and geocentric o form a plane \(SOM\), and the satellite's \(Z\)-axis is in the plane \(SOM\). Therefore, if solving the radial velocity between satellite \(S\) and ground terminal \(M\) is required, this can be divided into two steps:  
* First, the satellite velocity \(V_{S}\) is projected onto the plane \(SOM\). According to the included angle \(\theta\) between \(V_{S}\) and the plane \(SOM\), \(\theta\) is the azimuth in the satellite antenna's coordinate system. The following can be obtained: \[V_{\theta}=V_{S}\cos\theta\] (4) where \(V_{\theta}\) is on the plane \(SOM\). * Then, the velocity \(V_{\theta}\) is projected onto the straight line \(SM\). According to the included angle \(\varphi\) between \(V_{\theta}\) and the straight line \(SM\), \(\varphi\) is the elevation angle in the satellite antenna's coordinate system. The following can be obtained: \[V_{\varphi}=V_{\theta}\cos\varphi\] (5) where \(V_{\varphi}\) is the radial velocity of the satellite's velocity \(V_{S}\) along the straight line \(SM\), and the following is obtained: \[V_{\varphi}=V_{S}\cos\theta\cos\varphi\] (6) Since both \(V_{\varphi}\) and \(|V_{SM}|\) are the relative velocities between satellite \(S\) and ground terminal \(M\), we obtain \(|V_{SM}|=V_{\theta}\). Figure 2: The satellite body coordinate system and the Doppler frequency-offset model based on satellite beam pointing. Therefore, we substitute Equation (6) into Equation (1), and the Doppler frequency offset based on satellite beam pointing is as follows:  
\[F_{d}=\frac{F_{c}}{c}V_{S}\cos\theta\cos\varphi \tag{7}\]  
where \(F_{c}\) is the working frequency of the signal, \(c\) is the signal's propagation speed, \(V_{S}\) is the scalar value of \(V_{S}\), \(\theta\) is the azimuth of the satellite antenna, and \(\varphi\) is the elevation angle of the satellite antenna. In Equation (7), the motion speed \(V_{S}\) of the satellite is as follows:  
\[V_{S}=\sqrt{\frac{GM}{r}},r=R+h \tag{8}\]  
where \(G\) is the universal gravitation constant, \(M\) is the mass of the Earth, h is the orbit altitude of the satellite, \(r\) is the orbit radius of the satellite, and \(R\) is the radius of the Earth. The Doppler frequency offset \(F_{d}\) of Equation (8) is obtained as follows:  
\[F_{d}=\frac{F_{c}}{c}\sqrt{\frac{GM}{r}}\cos\theta\cos\varphi \tag{9}\]  
In this model, \(F_{c}\), \(c\), \(G\), \(M\), and \(r\) are constants, and \(\theta\) and \(\varphi\) are the pointing angle parameters of the satellite antenna, which can be calculated through the pointing of the satellite antenna to the ground terminal.