# A Doppler Frequency-Offset Estimation Method Based on the Beam Pointing of LEO Satellites
## 3 Model Optimization
Then, Earth's rotational velocity at latitude \(\beta\) is  
\[\left|\mathbf{V}_{E(\beta)}\right|=r^{\prime}\omega_{E} \tag{17}\]  
Since \(r^{\prime}\) is the projection of \(r\) within the latitude \(\beta\) plane, then \(r^{\prime}\) is  
\[r^{\prime}=r\cos\beta \tag{18}\]  
Thus, \(\left|\mathbf{V}_{E(\beta)}\right|\) is  
\[\mathbf{V}_{\beta}=\mathbf{V}_{S}-\mathbf{V}_{E(\beta)} \tag{10}\]0  
Then, Equation (15) yields  
\[\mathbf{V}_{\beta}=\mathbf{V}_{S}-\mathbf{V}_{E(\beta)} \tag{10}\]1  
According to the three-dimensional cosine theorem in space, we obtain  
\[\mathbf{V}_{\beta}=\mathbf{V}_{S}-\mathbf{V}_{E(\beta)} \tag{10}\]2Thus, \(\left|V_{\beta}\right|^{2}\) is  
\[\mathbf{V}_{\beta}=\mathbf{V}_{S}-\mathbf{V}_{E(\beta)} \tag{10}\]3  
As shown in Figure 3, **N** is the ascending node of satellite \(S\). In the \(SON\) plane, according to the three-dimensional sine theorem in space, we obtain  
\[\mathbf{V}_{\beta}=\mathbf{V}_{S}-\mathbf{V}_{E(\beta)} \tag{10}\]4  
Since \(0^{\circ}\leq\angle NOS<360^{\circ}\), then \(-1\leq\sin\angle NOS\leq 1\). When the satellite orbit is a prograde orbit, \(\sin\beta\) is  
\[\mathbf{V}_{\beta}=\mathbf{V}_{S}-\mathbf{V}_{E(\beta)} \tag{10}\]5  
The value range of latitude \(\beta\) is \(-i\leq\beta\leq i\); then, using Equation (22), we obtain  
\[\left|V_{\beta}\right|^{2}= \begin{cases}r^{2}\omega_{S}^{2}+r^{2}\omega_{E}^{ 2}-2r^{2}\omega_{S}\omega_{E}\cos i&\beta=0^{\circ}\\ r^{2}\omega_{S}^{2}+r^{2}\omega_{E}^{2}\cos i^{2}-2r^{2}\omega_{S}\omega_{E} \cos i&\beta=\pm i\\ r^{2}\omega_{S}^{2}+r^{2}\omega_{E}^{2}\cos\beta^{2}-2r^{2}\omega_{S}\omega_{E }\cos i&\beta\text{ is others}\end{cases}  \tag{25}\]  
Simplifying Equation (24) obtains  
\[\left|V_{\beta}\right|= \begin{cases}r\sqrt{\omega_{S}^{2}+\omega_{E}^{2}-2 \omega_{S}\omega_{E}\cos i}&\beta=0^{\circ}\\ r\omega_{S}-r\omega_{E}\cos i&\beta=\pm i\\ r\sqrt{\omega_{S}^{2}+\omega_{E}^{2}\cos\beta^{2}-2\omega_{S}\omega_{E}\cos i }&\beta\text{ is others}\end{cases}  \tag{26}\]  
Substituting Equation (26) into Equation (7) obtains  
\[F_{d}= \begin{cases}\frac{F_{c}}{c}r\sqrt{\omega_{S}^{2}+\omega_{E}^{2}-2 \omega_{S}\omega_{E}\cos i\cos\theta\cos\varphi}&,\beta=0^{\circ}\\ \frac{F_{c}}{c}r(\omega_{S}-\omega_{E}\cos i)\cos\theta\cos\varphi&,\beta=\pm i \\ \frac{F_{c}}{c}r\sqrt{\omega_{S}^{2}+\omega_{E}^{2}\cos\beta^{2}-2\omega_{S} \omega_{E}\cos i}&\cos\theta\cos\varphi\end{cases}  \tag{27}\]  
When the eccentricity of the satellite orbit is approximately zero, \(\omega_{S}\) is  
\[\omega_{S}=\sqrt{\frac{GM}{r^{3}}} \tag{28}\]  
where \(\omega_{E}\) is  
\[\omega_{E}=\frac{2\pi}{T} \tag{29}\]  
\(T\) is the rotation period of the Earth, which is equal to the time of a stellar day. Then, Equation (27) is  
\[F_{d}= \begin{cases}\frac{F_{c}}{c}r\sqrt{\frac{GM}{r^{3}}+\left(\frac{2\pi}{T} \right)^{2}-\frac{4\pi}{T}\sqrt{\frac{GM}{r^{3}}}\cos i}\ \cos\theta\cos\varphi&,\beta=0^{\circ}\\ \frac{F_{c}}{c}r\left(\sqrt{\frac{GM}{r^{3}}-\frac{2\pi}{T}\cos i}\right)\cos \theta\cos\varphi&,\beta=\pm i\\ \frac{F_{c}}{c}r\sqrt{\frac{GM}{r^{3}}+\left(\frac{2\pi}{T}\right)^{2}\cos \beta^{2}-\frac{4\pi}{T}\sqrt{\frac{GM}{r^{3}}}\cos i}\ \cos\theta\cos\varphi\,\beta\text{ is others}\end{cases}  \tag{30}\]  
\(F_{c},c,G,M,r,i,\) and \(T\) are constants or approximate constants. When \(\beta=0^{\circ}\) or \(\beta=\pm i\), the Doppler frequency shift \(F_{d}\) is only related to the azimuth angle \(\theta\) and elevation angle \(\varphi\) of the satellite antenna's beam; then, \(F_{d,\beta=0^{\circ}}>F_{d,\beta=\pm i}\). When \(0^{\circ}<\beta<i\), the Doppler frequency shift \(F_{d}\) is related not only to the azimuth angle \(\theta\) and elevation angle \(\varphi\) of the satellite antenna beam but also to \(\beta\).