# A Doppler Frequency-Offset Estimation Method Based on the Beam Pointing of LEO Satellites
## 3 Model Optimization  
The Doppler frequency-offset model based on satellite beam pointing does not account for the effects of satellite orbital inclination, Earth's rotation, and other factors related to the frequency-offset estimation. In this section, the model is optimized and refined accordingly. In an Earth-centered inertial coordinate system (ECI), the angular velocity is constant. However, due to the rotation of the Earth, the angular velocity of the satellite in an Earth-centered fixed coordinate system (ECF) varies with latitude. As shown in Figure 3, \(\beta\) is the geocentric latitude value. \(V_{S}\) represents the tangential velocity of the satellite in the ECI coordinate system, and \(V_{E(\beta)}\) represents the velocity at latitude \(\beta\) due to the rotation of the Earth. \(i\) represents the inclination of the satellite orbit, \(i^{\prime}\) represents the angle between \(V_{S}\) and \(V_{E(\beta)}\), \(V_{S}\) represents the tangential velocity of the satellite in the ECI coordinate system, and \(V_{E(\beta)}\) represents the velocity at latitude \(\beta\) due to the rotation of the Earth. \(r\) is the orbital radius of the satellite. The latitude \(\beta\) plane is parallel to the equatorial plane, and \(r^{\prime}\) is the projection of \(r\) within the latitude \(\beta\) plane. Figure 3: Relationship between Earth’s rotation and the satellite’s speed. When the satellite orbital inclination is less than \(90^{\circ}\), the direction of the Earth's rotation speed \(\mathbf{V}_{E(\beta)}\) is the same as that of the satellite's motion speed \(\mathbf{V}_{S}\). The Earth's rotation speed \(\mathbf{V}_{E(\beta)}\) will reduce the relative velocity between satellite \(S\) and ground terminal \(M\). Therefore, when the satellite is located at latitude \(\beta\), its velocity in the ECF coordinate system is as follows:  
\[\mathbf{V}_{\beta}=\mathbf{V}_{S}-\mathbf{V}_{E(\beta)} \tag{10}\]  
Both sides of Equation (8) are squared, and we obtain the following:  
\[\mathbf{V}_{\beta}^{2}=\left(\mathbf{V}_{S}-\mathbf{V}_{E(\beta)}\right)^{2} \tag{11}\]  
Then, we obtain the following:  
\[\left|\mathbf{V}_{\beta}\right|^{2}=\left|\mathbf{V}_{S}\right|^{2}+\left|\mathbf{V}_{E( \beta)}\right|^{2}-2\mathbf{V}_{S}\cdot\mathbf{V}_{E(\beta)} \tag{12}\]  
Since the angle between vector \(\mathbf{V}_{S}\) and vector \(\mathbf{V}_{E(\beta)}\) is \(i^{\prime}\), it follows that  
\[\mathbf{V}_{S}\cdot\mathbf{V}_{E(\beta)}=\left|\mathbf{V}_{S}\right|\left|\mathbf{V}_{S} \right|\cos i^{\prime} \tag{13}\]  
Then, Equation (12) yields the following:  
\[\left|\mathbf{V}_{\beta}\right|^{2}=\left|\mathbf{V}_{S}\right|^{2}+\left|\mathbf{V}_{E( \beta)}\right|^{2}-\left|\mathbf{V}_{S}\right|\left|\mathbf{V}_{E(\beta)}\right|\cos i ^{\prime} \tag{14}\]  
Since \(\left|\mathbf{V}_{\beta}\right|\)=\(\left|\mathbf{V}_{\beta}\right|\), \(\left|\mathbf{V}_{E(\beta)}\right|\)=\(\left|\mathbf{V}_{E(\beta)}\right|\), Equation (14) yields  
\[\left|\mathbf{V}_{\beta}\right|^{2}=\left|\mathbf{V}_{S}\right|^{2}+\left|\mathbf{V}_{E( \beta)}\right|^{2}-2\left|\mathbf{V}_{S}\right|\left|\mathbf{V}_{E(\beta)}\right|\cos i ^{\prime} \tag{15}\]  
In the ECI coordinate system, \(\omega_{S}\) is the angular velocity of the satellite; then, \(\left|\mathbf{V}_{S}\right|\) is  
\[\left|\mathbf{V}_{S}\right|=r\omega_{S} \tag{16}\]  
Let \(r^{\prime}\) be the radius of the latitude \(\beta\) plane and \(\omega_{E}\) be the angular velocity of Earth's rotation.