## 3 System Model and Problem Formulation
### System Model
_Doppler shift_: In the LEO satellite communication systems, the Doppler shift is usually large, which is mainly composed of the Doppler shift \(f_{d(k,l,p)}^{leo}\) generated by the LEO satellite motion and the Doppler shift \(f_{d(k,l,p)}^{ut}\) generated by the users' motion [22]. Since the transmission between the LEO satellite and user terminals is mainly under LOS, \(f_{d(k,l,p)}^{leo}\) of different transmission paths can be considered to be the same, and we omit the path index of \(f_{d(k,l,p)}^{leo}\), i.e., \(\left\{f_{d(k,l,p)}^{leo}\right\}_{1}=f_{d(k,l)}^{loo};f_{d(k,l)}^{leo}\) can be calculated using the LEO satellite ephemeris information and the location information of user terminals, as shown in Figure 1:1  
Figure 1:2  
where \(\mu(\theta_{\max})=\cos[\cos^{-1}\left(\frac{\tau_{e}}{\tau}\cos\theta_{\max} \right)-\theta_{\max}]\), \(r_{e}\) denotes the earth radius, \(r\) represents the distance between the LEO satellite track point and the earth center, \((\phi_{t}-\phi_{t_{0}})\) represents the angular distance of the earth's surface along the LEO satellite trajectory from instant \(t\) to instant \(t_{0}\), \(\omega\) represents the angular velocity of the LEO satellite and \(c\) represents the speed of light. The value of \(f_{d(k,l,p)}^{ut}\) in different transmission paths is different, which is mainly caused by the movement of user terminals and surrounding scatterers. The power spectrum of \(f_{d(k,l,p)}^{ut}\) follows the _Jakes_ power spectrum model, and the normalized power spectrum can be expressed as:  
Figure 1:3  
The large Doppler shift in LEO satellite communication systems can make it difficult to receive correctly and result in the degradation of communication performance. In application, to mitigate the impact of Doppler shift, the solution of estimation and compensation is usually adopted [23]. The Doppler shift estimation mainly includes two steps: coarse estimation and fine estimation, which can refer to the literature [24]. When we obtain the estimated value of Doppler shift, \(f_{d}^{e}\), the Doppler shift compensation of size \(f_{d}^{eps}=f_{d}^{e}\) can be implemented at the receivers. It should be noted that due to that the Doppler shift changes rapidly, in addition to the low SINR at the receivers and the limited pilot length in the DVB-S2X frame, the Doppler shift estimation is usually inaccurate, which can result in the incomplete compensation. Then, there would be the residual Doppler shift \(f_{d}^{rad}\), which can cause the sliding of channel phase. According to the Doppler shift estimation theory based the Cramer-Rao bound, the variance in the Doppler shift estimation can be expressed as the Cramer-Rao lower bound (CRLB) [25], i.e.,  
Figure 1:4  
where \(N\) is the pilot length, \(T\) is the sampling time and SNR is the signal-to-noise ratio. According to the properties of variance, after Doppler shift compensation of size \(f_{d}^{cps}\), the variance of residual Doppler shift \(f_{d}^{rad}\) is equal to the variance of \(f_{d}^{e}\), i.e.,  
Figure 1:5  
The influence of residual Doppler shift on channel phase errors can be expressed as \(\phi_{f_{d}^{rad}}=2\pi f_{d}^{rad}\Delta T\), where \(\Delta T\) represents the sum of the downlink propagation delay and the duration of the DVB-S2X frame. Therefore, the variance of \(\phi_{f_{d}^{rad}}\) can be expressed as \(\sigma_{\phi_{f_{d}^{rad}}}^{2}=4\pi^{2}\sigma_{f_{d}^{rad}}^{2}\Delta T^{2}= \frac{1}{SNR}\frac{6\Delta T^{2}}{T^{2}N(N^{2}-1)}\), and \(\phi_{f_{d}^{rad}}\) follows a real-valued Gaussian distribu  
Figure 1:6 Geometric diagram of LEO satellite’s orbital motion relative to the user terminal. tion with mean zero and variance \(\sigma^{2}_{\phi_{f_{d}^{rad}}}\), i.e., \(\phi_{f_{d}^{rad}}\sim N\Big{(}0,\sigma^{2}_{\phi_{f_{d}^{rad}}}\Big{)}\). The channel error vector caused by \(\phi_{f_{d}^{rad}}\) can be expressed as \(\mathbf{v}_{\phi_{f_{d}^{rad}}}=\left[e^{j\phi_{f_{d}^{rad}}},e^{j\phi_{f_{d}^{rad}} },\dots,e^{j\phi_{f_{d}^{rad}}},_{N}\right]^{T}\).