## 3 System Model and Problem Formulation  
### System Model  
As shown in Figure 1, we focus on the downlink of the massive MIMO LEO satellite multigroup multicast communication system, and the LEO satellite uses a UPA, which is composed of \(N=N_{x}\times N_{y}\) antennas and \(L(L\leq N)\) RF links and covers \(L\) multicast groups and \(K\) active users, where \(K\geq L,K\geq N\). Let the multicast group covered by the \(l\)th beam be \(U_{l}\) and the number of users in this multicast group be \(|U_{l}|\), assuming that the number of users that can be accommodated in each DVB-S2X frame is \(U_{s}\). It should be noted that each user terminal belongs to only one multicast group, i.e., \(U_{i}\cap U_{j}=\emptyset,\forall i,j\in\{1,\ldots,L\},i\
eq j\). Meanwhile, we assume that the user terminal is equipped with a single antenna capable of data stream demodulation. According to the DVB-S2X standard, in a transmission slot, multiple users' data in a multicast group are multiplexed into a specific forward error correction (FEC) codeword to provide services for more users. The service process is shown in Figure 2.  
Figure 1: Transmission model of the massive MIMO LEO satellite multigroup multicast communication system. The received signal \(y_{k,l}\) of the \(k\)th user in the \(l\)th multicast group can be expressed as:  
\[y_{k,l}=h_{k,l}^{H}\mathbf{F}_{RF}\mathbf{F}_{BB}[:,l]s_{l}+\sum_{j=1,j\
eq l }^{L}h_{k,l}^{H}\mathbf{F}_{RF}\mathbf{F}_{BB}[:,j]s_{l}+n_{k,l},k\in\{1,\ldots,|U_{l}| \},l\in\{1,\ldots,L\}, \tag{1}\]  
where the first term in (1) represents the expected received signal of the \(k\)th user in the \(l\)th multicast group, the second term represents the interference of other multicast groups and the third term represents the additive Gaussian white noise; \(\mathbf{h}_{k,l}\in\mathbb{C}^{N\times 1}\) represents the channel vector of the \(k\)th user in the \(l\)th multicast group, \(\mathbf{F}_{BB}\in\mathbb{C}^{L\times L}\) represents the digital beamforming matrix, \(\mathbf{F}_{BB}[:,l]\in\mathbb{C}^{L\times 1}\) represents the digital beamforming vector of the \(l\)th multicast group, and \(\mathbf{F}_{RF}\in\mathbb{C}^{N\times L}\) represents the analog beamforming matrix, where each element of \(\mathbf{F}_{RF}\) should meet the unit modulus element [19], i.e., \(\left|\left(\mathbf{F}_{RF}\right)_{ij}\right|=1\). In addition, \(s_{l}\) represents the signal of the multicast group \(U_{l}\), which meets the unit power constraint, i.e., \(E\left\{\left|s_{l}\right|^{2}\right\}=1\), \(n_{l}\sim CN(0,\sigma^{2})\) represents the additive Gaussian white noise, which is related to the Boltzmann constant \(\kappa\), system bandwidth \(B\) and the noise temperature \(T\). For the convenience of analysis, we set \(\mathbf{F}\in\mathbb{C}^{N\times L}=\mathbf{F}_{RF}\mathbf{F}_{BB}=\left[\mathbf{f}_{1},\mathbf{f} _{2},\ldots,\mathbf{f}_{L}\right]\) as the hybrid beamforming matrix, and \(\mathbf{f}_{l}\in\mathbb{C}^{N\times 1}=\mathbf{F}_{RF}\mathbf{F}_{BB}[:,l]\) is the hybrid beamforming vector of the \(l\)th multicast group. Therefore, (1) can be rewritten as:  
\[y_{k,l}=\mathbf{h}_{k,l}^{H}\mathbf{f}_{l}s_{l}+\sum_{j=1,j\
eq l}^{L}\mathbf{h}_{k,j}^{H} \mathbf{f}_{j}s_{l}+n_{k,l},k\in\{1,\ldots,|U_{l}|\},l\in\{1,\ldots,L\} \tag{2}\]  
Due to the high orbital speed of LEO satellites and the long transmission delay, it is difficult to obtain the precise instantaneous CSI.