## 3 System Model and Problem Formulation
### Transmission Model of CB  
We consider a CPSS with a social or physical sensing WSN with \(M\) nodes, in which there are \(O\) social sensor nodes and \(F\) physical sensor nodes randomly deployed on a plane, as shown in Figure 2. We assume \(D\) base stations (BSs) are located in the surrounding area of the WSN. We use \(R=\{R_{0},R_{1},\cdots,R_{D}\}\) and \(Z=\{z_{1},z_{2},\cdots,z_{M}\}\) to denote the BSs and the sensor nodes, respectively. The nodes \(\{z_{1},z_{2},\cdots,z_{O}\}\) are referred to as the social sensor nodes. The other nodes denote the physical sensor nodes. The transmission range of each sensor node cannot fully cover the BS due to the limited transmission power. As a result, the sensor nodes need to use CB technique to transmit the monitoring information to the intended BS. Moreover, the polar coordinates \((  
where \(O\) denotes the number of social collaborative nodes, and \(F\) is the number of physical sensor nodes. In addition, \(P_{k}\) refers to the transmitting power of the \(k\)th node \(z_{k}\), while \(\lambda\) is the wavelength of signal carrier. Let the initial phase of the \(k\)th node be:  
\[\Psi_{k}=-\frac{2\pi}{\lambda}d_{k}(\varphi,\theta) \tag{9}\]  
Then, the corresponding array factor \(AF\) at the \(l\)th BS can be rewritten as [11]:  
\[AF(\varphi_{l},\theta,\mathbf{cP})=\sum_{k=1}^{O}c_{k}\sqrt{P_{k}}e^{i\Psi_{k} }e^{j\frac{2\pi}{\lambda}d_{k}(\varphi_{l},\theta)}+\sum_{i=1}^{F}c_{k}\sqrt{P _{i}}e^{i\Psi_{i}}e^{j\frac{2\pi}{\lambda}d_{i}(\varphi_{l},\theta)} \tag{10}\]  
Similar to Mudumbai [11], we focus on the radiation pattern in the far-field region. Thus, we also assume that the far-field condition holds, i.e., the distance between the origin of coordinates and the \(l\)th base station is far greater than the \(r_{k}\), \(A_{l}\gg r_{k}\). Then, \(d_{k}(\varphi_{l},\theta)\) can be approximated as  
\[d_{k}(\varphi_{l},\theta)=\sqrt{A^{2}+r_{k}^{2}-2r_{k}A\sin\theta\cos\left( \varphi-\psi_{k}\right)}\approx A_{l}-r_{k}\cos\left(\varphi_{l}-\psi_{k}\right) \tag{11}\]  
Without loss of generality, \(\theta\) is set to \(\pi/2\), which means the BS is in the same plane with the sensor nodes. When the value of the initial phase in Equation (9) is met by the phase synchronization technique, the amplitude of \(AF\) at the direction of the intended \(BS_{0}\) can be rewritten as  
\[AF(\varphi_{0},\mathbf{cP})=\sum_{k=1}^{O}c_{k}\sqrt{P_{k}}+\sum_{i=1}^{F}c_{ i}\sqrt{P_{i}} \tag{12}\]  
Next, we present the description of the received signal at intended BS and unintended BS. We assume that all collaborative nodes receive the sending data symbol \(m_{i}\) from source node, where \(m_{i}\in\mathbb{C}\) and satisfies \(\mathbb{E}\{m_{i}\}=0\), \(|m_{i}|^{2}=1\), and \(\mathbb{E}\{m_{i}m_{i^{\prime}}\}=0\) for \(i\
eq i^{\prime}\), where \(\mathbb{E}\{\cdot\}\) denotes the expectation operation. Furthermore, the channel coefficient \(h_{kl}\) is assumed to follow a lognormal distribution [23], i.e., \(h_{kl}\sim exp\{M(0,\sigma^{2})\}\), where \(\sigma^{2}\) is the variance of the corresponding distribution. Then, the formula of the received signal at BS can be obtained as  
\[x(\varphi_{l},\mathbf{cP})=\sum_{k=1}^{O}m_{k}c_{k}\sqrt{P_{k}}h_{kl}e^{i\Psi _{k}}e^{j\frac{2\pi}{\lambda}\left(A_{l}-r_{k}\cos\left(\varphi_{l}-\psi_{k} \right)\right)}+\sum_{i=1}^{F}m_{i}c_{i}\sqrt{P_{i}}h_{il}e^{i\Psi_{i}}e^{j \frac{2\pi}{\lambda}\left(A_{l}-r_{i}\cos\left(\varphi_{l}-\psi_{i}\right) \right)}+\omega \tag{13}\]  
Figure 2: Social and physical sensing architecture based on CB for CPSS. where \(\omega\sim CN(0,\sigma_{\omega}^{2})\) refers to the additive white Gaussian noise.