## 2 Network Architecture and Signal Model
### Signal Model  
Figure 2 shows the 2D beam arrangement [33]. We define it as a hybrid 2D beam arrangement, which can be regarded as the arrangement of the multiple beams of a satellite in Figure 1. \(L\) and \(M\) denote the rows and columns of the 2D beam, respectively. The IWMSNN mainly considers a multibeam satellite equipped with \(N\) antennas (feeds) covering \(N\) beams, \(K\) sinks (\(K\) sensor fields) per beam and \(J\) TCNs contributing to the terrestrial interference. \(K\) sinks per beam and the total number of sinks result in \(NK\) transmitting signals to the satellite, while \(N\) antennas or feeds and a super-receiver on board the satellite are assumed to receive all of the signals that can be considered as single input multiple output (SIMO) networks. The RMHs receive the sensing data via the multibeam satellite. Due to the characteristics of the satellite channel, the satellite sinks are equipped with a single antenna. The sinks transmit signals (sensing data) to the satellite, while signals transmitted by TCNs are considered as the terrestrial interference to the satellite. Figure 1: Integrated wireless sensor and multibeam satellite networks’ architecture. The return link of the satellite is regarded as cognitive and cooperative, which is due to the limits of satellite payloads. This paper assumes the perfect symmetry for the IWMSMSN; meanwhile, the IWMSN is also considered under the Rician fading and channel interference. \(K\) sinks per beam transmit signals to the satellite, and \(J\) terrestrial interferences also exist per beam. Thus, the antennas (feeds) on board the satellite receiving signals from sinks can be modeled as:  
\[\mathbf{y}=\mathbf{HRs}+\mathbf{n} \tag{1}\]  
where \(\mathbf{y}\) is the received signal vector on board the satellite and \(\mathbf{y}\in\mathbb{C}^{N\times 1}\). \(\mathbf{H}\) is the channel matrix and \(\mathbf{H}\in\mathbb{C}^{N\times NK}\), which includes the hybrid interference matrix \(\mathbf{T}\) and will be discussed later. \(\mathbf{R}\) is the fading matrix and \(\mathbf{R}\in\mathbb{C}^{NK\times NK}\), \(\mathbf{s}\) is the transmitted signal vector from \(K\) sinks in each beam, \(\mathbf{s}\in\mathbb{C}^{NK\times 1}\), and \(\mathbb{E}[\mathbf{ss^{*}}]=P\mathbf{I}_{NK}\). \(\mathbf{n}\) is the vector of complex circularly symmetric (c.c.s.) Gaussian noise with zero mean, covariance \(\mathbb{E}\left[\mathbf{nn^{\dagger}}\right]=\sigma^{2}\mathbf{I}_{N}\) and \(\mathbf{n}\in\mathbb{C}^{N\times 1}\). The Rician model is usually adopted for the satellite communication networks as they are mainly under the direct LOS propagation channel. The Rician fading coefficient \(\kappa\) is independent and identically distributed (i.i.d.) c.c.s. Gaussian with independent real and imaginary parts, which is distributed as \(\mathcal{N}(\mu_{R}/\sqrt{2},\delta_{R}^{2})\) and can be defined as [44]:  
\[\kappa=\frac{\mu_{R}^{2}}{2\delta_{R}^{2}} \tag{2}\]  
Moreover, the Rician fading can be converted to the nonfading channel as \(\kappa\rightarrow\infty\), while the Rician fading can also be converted to the Rayleigh fading as \(\kappa=0\). The signals transmitted from sinks are assumed to be subjected to the Rician fading, and the receiver on board the satellite or on board a gateway station knows the fading coefficient of each sink from the return link of the IWMSNs. The gain coefficient from \(K\) sinks in each beam to the satellite are assumed approximately equal, because the distances between sinks are about 1% compared with the distance from the sinks to the satellite, as the beam radius may range from 100-300 km, except in the extreme case of sinks distributed on the beam edge. This assumption could be seen in [33]. Then, the channel gain matrix can be given by:  
\[\mathbf{H}=\mathbf{T}\otimes\mathbf{1}_{K} \tag{3}\]  
where \(\mathbf{T}\) is defined as the interference matrix for the hybrid 1D and 2D beam models. Thus, the receiving signals under the fading channel can be rewritten as:  
Figure 2: The hybrid 2D beam arrangement.