## 3 System Model and Problem Formulation
### System Model
To cope with this problem, we adopt the statistical CSI, and the channel vector between the LEO satellite and the \(k\)th user in the \(l\)th multicast group at instant \(t\) and frequency \(f\) can be modeled as follows [20]:  
\[\mathbf{h}_{k,l}(t,f)=\sum_{p=1}^{P_{k,l}}a_{k,l,p}e^{j2\pi(f_{d(k,l,p)}t-f_{\bar{ \
u}_{k,l,p}})}\times\mathbf{V}_{k,l,p}, \tag{3}\]  
where \(f\) denotes the carrier frequency, \(a_{k,l,p}\),\(f_{d(k,l,p)}\),\(\tau_{k,l,p}\) are the complex channel gain, Doppler shift and propagation delay, respectively, \(P_{k,l}\) denotes the number of propagation paths and \(\mathbf{V}_{k,l,p}\in\mathbb{C}^{N\times 1}\) is the UPA array response vector, which can be given by  
\[\mathbf{V}_{k,l,p}=\mathbf{V}(\sigma_{k,l,p}^{x},\sigma_{k,l,p}^{y})=\mathbf{v}_{N_{x}} \Big{(}\sin\sigma_{k,l,p}^{y}\cos\sigma_{k,l,p}^{x}\Big{)}\otimes\mathbf{v}_{N_{y }}\Big{(}\cos\sigma_{k,l,p}^{y}\Big{)}, \tag{4}\]  
Figure 2: Service process of the LEO communication satellite multigroup multicast transmission based on the DVB-S2X. \[\mathbf{v}_{N_{x}}(\sin\varphi^{y}_{k,l,p}\cos\varphi^{x}_{k,l,p}) =\frac{1}{\sqrt{N_{x}}}\bigg{(}1,e^{-j\frac{2 nd}{\lambda}\sin\varphi ^{y}_{k,l,p}\cos\varphi^{x}_{k,l,p}},\ldots,e^{-j\frac{2 nd}{\lambda}(N_{x}-1) \sin\varphi^{y}_{k,l,p}}\cos\varphi^{x}_{k,l,p}\bigg{)}, \tag{5}\] \[\mathbf{v}_{N_{y}}(\cos\varphi^{y}_{k,l,p}) =\frac{1}{\sqrt{N_{y}}}\bigg{(}1,e^{-j\frac{2 nd}{\lambda}\cos \varphi^{y}_{k,l,p}},\ldots,e^{-j\frac{2 nd}{\lambda}(N_{y}-1)\cos\varphi^{y}_ {k,l,p}}\bigg{)}, \tag{6}\]  
where \(\varphi^{x}_{k,l,p^{\prime}}\), \(\varphi^{y}_{k,l,p}\) represent azimuth angle and pitch angle associated with the propagation path \(p\) of the \(k\)th user in the \(l\)th multicast group, respectively, \(\lambda\) denotes the wavelength and \(d\) represents the spacing of antenna elements, the value of which is usually \(\lambda/2\)[5]. Note that the LEO satellite communication system is usually operated under the line of sight (LOS) transmission, and the channel vector can be modeled using the widely accepted Rician distribution model as follows:  
\[\mathbf{h}_{k,l}=\overline{\mathbf{h}}_{k,l}+\widetilde{\mathbf{h}}_{k,l}, \tag{7}\]  
where \(\overline{\mathbf{h}}_{k,l}=\sqrt{\frac{\kappa_{k,l}T_{k,l}}{\kappa_{k,l}+1}}\times V _{k,l}\) represents the LoS component, \(\widetilde{\mathbf{h}}_{k,l}=\sqrt{\frac{\tau_{k,l}}{\kappa_{k,l}+1}}\times\mathbf{V }_{k,l,c}\times\mathbf{V}_{k,l}^{H}\) represents the multipath component, \(\kappa_{k,l}\) denotes the Rician factor, \(\mathbf{V}_{k,l,c}\in\mathbb{C}^{N_{at}\times 1}\sim\mathbb{C}\mathbb{N}(0,\sum)\) represents Rician component, \(Tr(\sum)=1\) and \(\gamma_{k,l}\) represents the average channel power, which mainly includes the transmit antenna gain \(G_{leo}\), receiver antenna gain \(G_{ut}\) and link power loss. The link power loss is mainly caused by the free space path loss \(LP_{fs}\) and the atmospheric absorption loss \(LP_{at}\). Therefore, the average channel power \(\gamma_{k,l}\) can be written as  
Figure 1:0  
where \(LP_{fs}\) can be given by \(LP_{fs}=20\big{(}\log_{10}(D_{k,l})\big{)}+\log_{10}(f)+\log_{10}(4\pi/c)\), \(c\) is the speed of light, \(D_{k,l}\) represents the transmission distance, \(LP_{at}\) is related to the carrier frequency, temperature \(T(h)\), pressure \(P(h)\) and humidity \(\rho(h)\), which can be given by \(LP_{at}=\int_{h_{at}}^{h_{at}}LP_{at}(f,T(h),P(h),\rho(h))dh\), \(LP_{at}(f,T(h),P(h),\rho(h))\) is the loss per meter, \(h_{ut}\) is the user's height and \(h_{at}\) is the atmosphere thickness. The specific calculation method of \(LP_{at}\) can be found in the literature [21]. For the convenience of analysis, it is assumed that the parameters in the channel vector \(\mathbf{h}_{k,l}\) are constant within coherence time and change over time in a certain ergodic process. In (3), the Doppler shift \(f_{d(k,l,p)}\) and the propagation delay \(\tau_{k,l,p}\) usually cause CSI errors. Next, we focus on analyzing the influence of propagation delay and Doppler shift on CSI errors.