## 2 System Model  
### Channel Model  
As depicted in Figure 1, we consider a downlink communication scenario where multiple LEO satellites cooperatively provide services for GUs with a GEO satellite working as a central processing unit (CPU). Inter-satellite links (ISLs) are employed to exchange data among satellites. In the proposed CF-mMIMO-assisted LEO SATCOM system, \(L\) LEO satellites cooperatively serve \(K\) single-antenna GUs, and each GU is served by multiple satellites simultaneously. It is assumed that each satellite contains a uniform planar array (UPA) of \(N\) antenna units, with \(N_{x}\) and \(N_{y}\) antenna units on the x-axis and y-axis, respectively. Antenna spacings are half of the wavelength, and each antenna is equipped with a PA. The UPA adopts a hybrid analog/digital precoding architecture, where the number of RF chains is \(N_{RF}\), and the number of array elements contained in each RF chain is \(N_{g}=N/N_{RF}\). The diagram of the hybrid precoding architecture is shown in Figure 2.  
The channel between the \(k\)th GU and the \(l\)th satellite is denoted by \(\mathbf{h}_{k,l}\in\mathbb{C}^{N\times 1}\), which is modeled as  
\[\mathbf{h}_{k,l}=\alpha_{k,l}\mathbf{v}_{k,l}, \tag{1}\]  
where \(\alpha_{k,l}\) and \(\mathbf{v}_{k,l}\) denote the complex channel gain and the UPA response vector of the path between the \(k\)th GU and the \(l\)th satellite, respectively. Figure 1: System architecture of the multi-satellite cooperative communication network in a CF-mMIMO-assisted LEO SATCOM system with a set of LEO satellites working as APs that jointly serve GUs. The SC for GU \(k\) is further obtained from candidate SC \(C_{k}\) by Algorithm 1. Figure 2: The partially connected hybrid precoding architecture with NPAs at the transmitter. ```
0: minimum elevation angle \(\theta_{\min}\), elevation angle \(\theta_{k,l}\), large-scale fading \(\alpha_{k,l}\) between the \(k\)th GU and the \(l\)th satellite, \(\forall k,l\). 0: Each GU's SC \(S_{k}\) and MSAP\({}_{k}\), \(\forall k\). 1:for\(k=1\) to \(K\)do
2:if\(\theta_{k,l}>\theta_{\min}\)then
3: set \(C_{k}\leftarrow\{\text{satellite }l\}\)
4:endif
5:endfor
6:for\(k=1\) to \(K\)do
7: A MSAP is selected for the \(k\)th GU from \(C_{k}\) such that MSAP\({}_{k}=\{\text{satellite }l:l=\arg\max_{l\in C_{k}}\alpha_{k,l}\}\)
8: the \(k\)th GU sends a serving request to the MSAP\({}_{k}\)
9:if MSAP\({}_{k}\) deny the GU's serving request then
10: set \(C_{k}\gets C_{k}\backslash\{\text{MSAP}_{k}\}\)
11: go to step 7
12:else
13:\(S_{k}\leftarrow\{\text{MSAP}_{k}\}\)
14:endif\(\text{MSAP}_{k}\)
15:endfor
16:for\(k=1\) to \(K\)do
17:if\(C_{k}\
eq\emptyset\)then
18: the \(k\)th GU sends a serving request to all satellites in \(C_{k}\). 19: A SAP is selected for the \(k\)th GU from \(C_{k}\) such that \(SAP_{k}=\{\text{satellite }l:l=\arg\max_{l\in C_{k}}\alpha_{k,l}\}\)
20:if satellite \(l\in C_{k}\) is available then
21: satellite \(l\) will join the GU's SC \(S_{k}\), i.e.,
22:\(S_{k}\gets S_{k}\cup\{\text{satellite }l\}\)
23:\(C_{k}\gets C_{k}\backslash\{\text{satellite }l\}\)
24:else
25:\(C_{k}\gets C_{k}\backslash\{\text{satellite }l\}\)
26: go to step 19
27:endif
28:endif
29:endfor
30: go to step 16 until \(C_{k}=\emptyset\) for all \(k\in\{1,2,\cdots,K\}\). ```  
**Algorithm 1**Initial GU access and SC selection  
The variable \(\alpha_{k,l}\) is affected by large-scale fading and antenna gain, which is expressed as  
\[\alpha_{k,l}=10^{\frac{1}{b}\left(G_{S}+G_{GU}-PL_{k,l}\right)}, \tag{2}\]  
where \(G_{S}\) is the satellite-side antenna gain, \(G_{GU}\) is the GU-side antenna gain and \(PL_{k,l}\) is the large-scale fading between the \(k\)th GU and the \(l\)th satellite.