## 3 Capacity Analysis of the IWMSNs
### The Optimal Joint Decoding Capacity  
\(r_{n,k}\) is denoted as the Rician fading coefficient for the \(k\)-th sink in the \(n\)-th beam, and \(\mathbf{R}=\text{diag}(r_{1,1},...,r_{1,K},...,r_{N,1},...,r_{N,K})\). The Rician fading coefficient \(r_{n,k}\) is assumed to be complex, independent, strictly stationary and ergodic and normalized as \(\mathbb{E}[r_{n,k}r_{n,k}^{*}]=1\). Because the distance between the sink and receiving antennas (feeds) on board the satellite is far longer than the distance between the receiving antennas (feeds) on board the satellite, thus the signal of the sink to the satellite would experience the same fading at every receiving antenna on board the satellite. It is quite clear that the same fading to the receiving antenna is different from the TCNs where each sink experiences independent fading at every receiving antenna of the base stations. Sinks from all beams can be optimally jointly decoded by a super-receiver on-board the gateway station or on-board the satellite to be as the optimal joint decoding. The optimal capacity per beam can be expressed as the log-determinant formula [49]:  
\[ \begin{array}{l}\lim\limits_{N\rightarrow\infty}\mathbf{C}_{\text{opt}}( \mathbf{HR};\alpha,\beta,\gamma)\\ =\lim\limits_{N\rightarrow\infty}\mathbf{C}_{\text{opt}}((\mathbf{T}\otimes \mathbf{1}_{K})\mathbf{R};\alpha,\beta,\gamma)\\ =\frac{1}{N}\mathbb{E}\left[\text{logdet}(\mathbf{I}_{N}+\gamma(\mathbf{T} \otimes\mathbf{1}_{K})\mathbf{RR}^{\dagger}(\mathbf{T}\otimes\mathbf{1}_{K})^ {\dagger}\right]\\ =\frac{1}{N}\mathbb{E}\left[\text{logdet}(\mathbf{I}_{N}+\gamma\mathbf{T} \mathbf{\Theta}\mathbf{T}^{\dagger}\right]\\ =\frac{1}{N}\mathbb{E}\left[\text{logdet}(\mathbf{I}_{N}+\gamma\mathbf{T}^{ \dagger}\mathbf{\Theta}\right]\end{array}  \tag{15}\]  
where \(\mathbf{\Theta}=\text{diag}\{\theta_{1},...,\theta_{N}\}\in\mathbb{C}^{N \times N}\), and \(\theta_{n}=\sum_{k=1}^{K}\left(r_{n,k}r_{n,k}^{*}\right)\). \(\gamma=P/\sigma^{2}\) is the signal-to-noise ratio (SNR) for the receiver, while the interference and Rician fading will be analyzed in the matrices \(\mathbf{H}\) and \(\mathbf{R}\). From Equations (6)-(8), the interference matrix \(\mathbf{T}\) can be considered as a fixed matrix for \(N\); \(\alpha\) and \(\beta\) are constant as fixed. The fading matrix \(\mathbf{R}\) is distributed as i.i.d. c.c.s. Gaussian with independent real and imaginary parts. \(\mathbf{T}\) and \(\mathbf{R}\) generally are not unitary matrices. The work in [48] has shown that the Haar distributed approximations achieved the asymptotic performance with the theoretical performance. Thus, we adopt the Haar distributed approximations to solve the problem, where the Haar matrix is employed to replace the fixed matrix \(\mathbf{T}\). The capacity approximations will converge as \(N\) increases. \(\mathbf{T}^{\dagger}\mathbf{T}\) can be decomposed as \(\mathbf{T}^{\dagger}\mathbf{T}=\mathbf{Q}\mathbf{\Sigma}\mathbf{Q}^{\dagger}\), where \(\mathbf{Q}\) is a unitary matrix and \(\mathbf{\Sigma}\) is a diagonal matrix. A Haar matrix \(\mathbf{W}\) is used to replace the fixed matrix and \(\mathbf{T}^{\dagger}\mathbf{T}{\approx}\mathbf{W}\mathbf{\Sigma}\mathbf{W}^{\dagger}\)[50] and \(\mathbf{T}^{\dagger}\mathbf{T}\mathbf{\Theta}{\approx}\mathbf{W}\mathbf{ \Sigma}\mathbf{W}^{\dagger}\mathbf{\Theta}=\mathbf{\Sigma}\mathbf{W}\mathbf{ \Theta}\mathbf{W}^{\dagger}\). The approximation will be more precisely asymptotic as the beam number \(N\) increases. Thus, the Haar approximation of capacity is given by:  
\[ \begin{array}{l}\lim\limits_{N\rightarrow\infty}\mathbf{C}_{\text{OPT}} \left((\mathbf{T}\otimes\mathbf{1}_{K})\mathbf{R};\alpha,\beta,\gamma\right)\\ \approx\frac{1}{N}\mathbb{E}\left[\text{logdet}(\mathbf{I}_{N}+\gamma\mathbf{ \Sigma}\mathbf{W}\mathbf{\Theta}\mathbf{W}^{\dagger})\right]\end{array}  \tag{16}\]  
From [51], the moment-generating function (MGF) \(\mathbf{F}_{MGF}(u)\) of \(f(X)\) is given by \(\mathbb{E}[\exp(u\cdot f(X))]\).