## 3 Capacity Analysis of the IWMSNs
### The Linear MMSE Capacity
 \tag{36}\]  
After substituting Equations (32)-(36) into Equation (31), we derive the linear MMSE capacity of the return link for the IWMSNNs as:  
\[\begin{split}&\lim_{N\to\infty}\mathbf{C}_{\text{MMSE}}((\mathbf{T }\otimes\mathbf{1}_{K})\mathbf{R};\alpha,\beta,\gamma)\\ &\approx-K\log\Big{(}1-\tfrac{\gamma}{N^{2}K}\tfrac{(-\gamma)^{- \frac{1}{2}N(N-1)}\prod_{n=1}^{N}(k-1)! }{\gamma(\mathbf{Z})\mathcal{V}( \mathbf{\Theta})\mathbf{G}(u)}\\ &\quad\cdot\left(-\tfrac{1}{2\gamma}N(N-1)\cdot\det\mathbf{Y}(u )\cdot\text{tr}\left(\mathbf{Z}(u)\cdot\left(\log(1+\gamma\theta_{i}\zeta_{j} )\cdot(1+\gamma\theta_{i}\zeta_{j})^{u+N-1}\right)_{i,j}\right)\right.\\ &\quad+\det\mathbf{Y}(u)\cdot\text{tr}\left(\mathbf{Z}(u)\cdot \left(\tfrac{(u+N-1)\theta_{i}\zeta_{j}}{1+\gamma\theta_{i}\zeta_{j}}\cdot(1+ \gamma\theta_{i}\zeta_{j})^{u+N-1}\right)_{i,j}\right)\\ &\quad\cdot\text{tr}\left(\mathbf{Z}(u)\cdot\left(\log(1+\gamma \theta_{i}\zeta_{j})\cdot(1+\gamma\theta_{i}\zeta_{j})^{u+N-1}\right)_{i,j} \right)\\ &\quad+\det\mathbf{Y}(u)\cdot\text{tr}\left(\mathbf{Z}(u)\cdot \left(\log(1+\gamma\theta_{i}\zeta_{j})\cdot\frac{(u+N-1)\theta_{i}\zeta_{the 1D beam model and \(N=L\times M\) beams for the 2D beam model. There are \(K\) sinks with a single antenna as the sensor nodes or connecting the sensor fields, which can sense information in each beam, and \(J\) terrestrial interferences in each beam. \(K\) sinks in each beam transmit signals to the satellite, and \(N\) antennas or feeds and a super-receiver on board the satellite are assumed to receive all of the signals, which can be considered as SIMO networks. The L/S band is assumed for the return link, and the signals experience Rician fading. Either the satellite decodes the received signals or the gateway station decodes the signals, and then, the gateway station sends sensing information to the remote monitoring hosts. This paper mainly investigates the optimal capacity and linear MMSE capacity for the IWMSNNs under Rician fading. \(\alpha\) is defined as the equivalent inter-beam interference from inter-beam and \(0\leq\alpha\leq 1\), while \(\beta\) is defined as the equivalent intra-beam interference from TCNs and \(0\leq\beta\leq 1\). We mainly investigate the effects of \(\alpha\) and \(\beta\) on the capacity. The numerical simulations are plotted by using MATLAB. In the simulations, the number of hybrid 1D beams is \(N=300\); the number of hybrid 2D beams is \(L=50\) and \(M=50\); and the Rician fading factor is chosen as \(\kappa=0\), \(\kappa=10\) and \(\kappa=50\). Figures 4-6 respectively compare the optimal capacity with MMSE capacity for the hybrid 1D beams, the hybrid 2D beams and modified hybrid 1D beams; in the higher SNR \(\gamma=10\), the number of sinks is \(K=1\), versus interference factors \(\alpha\) and \(\beta\). Obviously, the linear MMSE capacities are lower than the optimal capacity with the same Rician fading factor. The capacities of three models are approximately 2.38 nats/beam when \(\alpha=0\), \(\beta=0\), \(K=1\) and \(\kappa=50\) in the higher SNR. The performance of this paper is fit to [33] when \(\beta=0\). Figures 4 and 5 show that the interference factor \(\beta\) from the TCNs decreases the capacity of \(\bar{\mathbf{T}}_{\rm 1D}\) and \(\bar{\mathbf{T}}_{\rm 2D}\) and increases the capacity as \(\alpha\) increases to one. That is to say, the capacity decreases first and then increases with \(\alpha\), while the coordinate values of inflection points \((\alpha,\beta)\) decrease as \(\alpha\) and \(\beta\) increase. The optimal capacity and linear MMSE capacity respectively increase as the Rician fading factor increases. Figure 4: The optimal capacity and MMSE capacity of the hybrid 1D beams with \(\gamma=10\) and \(N=100\) (\(\kappa=0\), \(\kappa=10\) and \(\kappa=50\)) versus \(\alpha\) and \(\beta\). the optimal capacity and MMSE capacity respectively differ more in the lower region of \(\alpha\) and \(\beta\) and differ less in the higher region from \(\kappa=0\) to \(\kappa=10\) and \(\kappa=50\).