## 2 Network Architecture and Signal Model
### Hybrid Interference Matrix
\right] \tag{7}\]  
Thus, the received signals intra-beam can be equivalent to \((1+\beta)\), and the normalized desired signal can be equivalent to \(1/(1+\beta)\). The interference from adjacent beams is equivalent to \((\alpha+\alpha\beta)\). The hybrid 1D interference matrix \(\overline{\mathbf{T}}_{\mathrm{1D}}\) is an \(N\times N\) Toeplitz block Toeplitz (TBT) matrix as:  
\[\overline{\mathbf{T}}_{\mathrm{1D}}\!=\!\frac{1}{1\!+\!\beta}\!\left[ \begin{array} []{cccc}1&\alpha(1\!+\!\beta)&&&\\ \alpha(1\!+\!\beta)&1&\ddots&\\ &\ddots&\ddots&\alpha(1\!+\!\beta)\\ &&&\alpha(1\!+\!\beta)&1\end{array} \right] \tag{8}\]  
Figure 3: The relationships between the substitution and the practical parameter. (**a**) \(\alpha\) versus beam isolation parameter \(\rho\); (**b**) Normalized radiation pattern of \(\beta\) versus elevation angle \(\varphi\). If the terrestrial interference and beam interference from all beams are considered, then the modified hybrid 1D interference matrix \(\mathbf{\hat{T}}_{\mathrm{1D}}\) is also constructed as an \(N\times N\) TBT matrix:  
\[\mathbf{\hat{T}}_{\mathrm{1D}}=\left[ \begin{array}{cccc}\frac{1}{(1+\beta)}& \alpha&\cdots&\alpha^{N-1}\\ \alpha&\frac{1}{(1+\beta)}&\cdots&\alpha^{N-2}\\ \vdots&\vdots&\ddots&\vdots\\ \alpha^{N-1}&\alpha^{N-2}&\cdots&\frac{1}{(1+\beta)}\end{array} \right] \tag{9}\]  
The arrangement of a hybrid 2D beam is \(L\times M\), and the total beam number is \(N=L\times M\) as shown in Figure 2, where \(L\) and \(M\) are respectively the rows and columns of the hybrid 2D beam. While the hybrid 2D interference matrix just considers the TCNs interference and beam interference from the adjacent beams, thus the hybrid 2D interference matrix \(\mathbf{\overline{T}}_{\mathrm{2D}}\) can be constructed as a \((L\times M)\times(L\times M)\) TBT matrix, constituted by \(L\times L\) submatrices, and the submatrix is \(M\times M\), given by:  
\[\mathbf{\overline{T}}_{\mathrm{2D}}=\left[ \begin{array}{cccc}\mathbf{ \overline{T}}_{\mathrm{1D}}&\mathbf{\overline{S}}&&&\\ \mathbf{\overline{S}}^{H}&\mathbf{\overline{T}}_{\mathrm{1D}}&\ddots&\\ &\ddots&\ddots&\mathbf{\overline{S}}\\ &&&\mathbf{\overline{S}}^{H}&\mathbf{\overline{T}}_{\mathrm{1D}}\end{array} \right] \tag{10}\]  
where \(\mathbf{\overline{T}}_{\mathrm{1D}}\) is defined in Equation (8) and considers the interference from the adjacent beam of the same row in the 2D beam model. \(\mathbf{\hat{S}}\) considers the interference from the adjacent beam of the same column in the 2D beam model and can be constructed as:  
\[\mathbf{\hat{S}}{=}\frac{1}{(1+\beta)}\left[ \begin{array}{cccc}\alpha+\alpha \beta&&&\\ \alpha+\alpha\beta&\alpha+\alpha\beta&&\\ &\ddots&\ddots&\\ &&&\alpha+\alpha\beta&\alpha+\alpha\beta\end{array} \right] \tag{11}\]  
Then, we obtain the eigenvalues of the hybrid matrix, which will be used in the capacity approximation. The eigenvalues of \(\mathbf{\overline{T}}_{\mathrm{1D}}\) for the hybrid 1D model are derived as [46]:  
\[\lambda_{i}(\mathbf{\overline{T}}_{\mathrm{1D}})=\frac{1}{1+\beta}(1{+}2\alpha (1+\beta)\cos(\frac{i\pi}{N+1})).\,i{=}\,1,...,\,N. \tag{12}\]  
The eigenvalues of \(\mathbf{\hat{T}}_{\mathrm{1D}}\) for the modified hybrid 1D model are derived as [46]:  
\[\lambda_{i}(\mathbf{\hat{T}}_{\mathrm{1D}})=(\frac{1}{1+\beta}-1)+\frac{1- \alpha^{2}}{1+2\alpha\cos(\frac{i\pi+\epsilon}{N+1})+\alpha^{2}} \tag{13}\]  
The eigenvalues of \(\mathbf{\overline{T}}_{\mathrm{2D}}\) for the hybrid 2D model are derived as [47]:  
\[\lambda_{(l-1)M+m}(\mathbf{\overline{T}}_{\mathrm{2D}})=\frac{1}{1+\beta}(1{+ }2\alpha(1+\beta)(\cos(2\pi l/L)\] \[\quad+\cos(2\pi m/M)+\cos(2\pi l/L+2\pi m/M))). \tag{14}\]  
\(l=1,...,\,L,\ m{=}\,1,...,\,M\).