## 3 Capacity Analysis of the IWMSNs
### The Linear MMSE Capacity
We can see that the optimal capacity and MMSE capacity are converging as \(\kappa\) increases from 0-50, due to the fact that the Rician fading is converted to the additive Gaussian noise channel as \(\kappa\rightarrow\infty\) to get the nonfading channel. Figure 6 shows that the capacity of the modified model \(\hat{\mathbf{T}}_{\mathrm{1D}}\) is lower than \(\bar{\mathbf{T}}_{\mathrm{1D}}\) and \(\bar{\mathbf{T}}_{\mathrm{2D}}\) by considering all of the inter-beam interference in the same cases. Thus, the capacity of \(\hat{\mathbf{T}}_{\mathrm{1D}}\) decreases obviously with \(\alpha\) and \(\beta\). \(\beta\) decreases the capacity of \(\hat{\mathbf{T}}_{\mathrm{1D}}\) as \(\alpha\) increases from 0-1, which indicates that the IWSMSNs are interference limited. The capacity decreases to zero when \(\beta=0\) and almost decreases to zero in the lower region of \(\beta\) as \(\alpha=1\). We can see that the linear MMSE capacity differs by about 0.65 nats/beam from \(\kappa=0\) to \(\kappa=10\) in the lower region of the \(\alpha\) and \(\beta\) and no more than 0.2 nats/beam from \(\kappa=10\) to \(\kappa=50\) in the lower region of the \(\alpha\) and \(\beta\). The MMSE capacity gap of \(\kappa\) decreases from the lower region of \(\alpha\) and \(\beta\) to the higher region, and \(\beta\) deceases the capacity as the same \(\alpha\). Comparing the optimal capacity with the MMSE capacity as \(\kappa=50\), we find that the MMSE capacity is very close to the optimal capacity in the lower region of \(\alpha\) and \(\beta\). The optimal and MMSE capacity gap increases first, and it is especially clear from \(\alpha=0.3\) to \(\alpha=0.8\) and decreases to zero from \(\alpha=0.8\) to \(\alpha=1\). Figures 7-9 respectively compare the optimal capacity with MMSE capacity for the hybrid 1D beams, the hybrid 2D beams and modified hybrid 1D beams; in the lower SNR \(\gamma=1\), the number of sink is \(K=1\), versus interference factors \(\alpha\) and \(\beta\). Figure 7: The optimal capacity and MMSE capacity of hybrid 1D beams with \(\gamma=1\) and \(N=100\) (\(\kappa=0\), \(\kappa=10\) and \(\kappa=50\)) versus \(\alpha\) and \(\beta\). The capacity of three models are approximately 0.69 nats/beam when \(\alpha=0\), \(\beta=0\), \(K=1\) and \(\kappa=50\) in the lower SNR. We compare the capacity of the hybrid 1D beams and hybrid 2D beams respectively in the cases of higher SNR and lower SNR between Figures 4 and 7, and Figures 5 and 8. In Figures 7 and 8, \(\alpha\) progressively increases the optimal capacity and MMSE capacity, while \(\beta=0\)  
Figure 8: The optimal capacity and MMSE capacity of the modified hybrid 1D beams with \(\gamma=1\) and \(N=100\) (\(\kappa=0\), \(\kappa=10\) and \(\kappa=50\)) versus \(\alpha\) and \(\beta\). Figure 9: The optimal capacity and MMSE capacity of the hybrid 2D beams with \(\gamma=1\), \(L=50\) and \(M=50\) (\(\kappa=0\), \(\kappa=10\) and \(\kappa=50\)) versus \(\alpha\) and \(\beta\). slightly decreases the capacity with the same \(\alpha\), which is different from the curve trends shown in Figures 4 and 5. There, we see that the capacity gap is approximately 0.12 nats/beam from \(\kappa=0\) to \(\kappa=50\) when \(\alpha=1\), while the capacity gap is approximately 0.32 nats/beam from \(\kappa=0\) to \(\kappa=50\) when \(\alpha=1\). Meanwhile, \(\beta\) affects the capacity clearly in the lower region of \(\alpha\) and in an unclear manner in the higher region of \(\alpha\). Figure 9 shows the performance of the modified hybrid 1D beam with \(\gamma=1\), where the curves approach zero as \(\alpha\) increases to one. In the lower region of \(\alpha\), it is obvious that \(\beta\) decreases the capacity; while in the higher region of \(\alpha\), \(\beta\) affects the capacity in an unclear manner and does not approach zero as \(\beta\) increases to one when \(\alpha=1\). In general, the curves of the modified hybrid 1D beam approximately approach zero as \(\alpha\) and \(\beta\) increase.