# Cooperative Jamming with AF Relay in Power Monitoring and Communication Systems for Mining
## 2 System Model  
Consider a MIMO wireless system, as shown in Figure 1, where a source, a relay, a user, and two eavesdroppers have \(N_{S}\), \(N_{R}\), \(N_{D}\), and \(N_{E}\) channels, respectively. We assume that there is no direct link between the source and the user for the long-distance path loss. For simplicity, the eavesdroppers represent all the eavesdroppers eavesdropping the same legitimate in the same time phase. More specifically, in the first time phase, eavesdroppers eavesdrop \(E_{1}\) message from the source, and in the second time phase, eavesdropper eavesdrop \(E_{2}\) message from the relay. In wireless communications system, messages are transmitted through MIMO wireless communications channels. We describe each path between two nodes by CSI \(\mathbf{H}_{ij,k}\) as the matrix of channel coefficients, where \(i\in\{S,R\}\), \(j\in\{R,D,E_{1},E_{2}\}\), and \(k=1,2\) denote the transmitter, receiver, and transmission time phases, respectively. It is worth noting that \(\mathbf{H}_{RR,1}\) refers to the self-interference matrix because of self-interference and in the process of transmission \(\mathbf{H}_{ij,k}\) stays constant because of the short transmission time. In this paper, the uncertainty of CSI is taken into consideration, i.e., the CSI of the wireless communications system cannot be perfectly known at the source or the relay due to factors such as the limited capacity of the feedback channel. As a result, the deterministic uncertainty model [18] is introduced to characterize the imperfect CSI, as follows:\[\mathbf{H}_{ijk}\!\in\!\mathcal{H}_{ijk}\!\!=\Big{\{}\mathbf{H}_{ijk}|\mathbf{H}_{ ijk}\!\!=\overline{\mathbf{H}}_{ijk}\!\!+\Delta_{ijk}\!\!,\Big{\|}\Delta_{ijk}\! \!\Big{\|}\leq\delta_{ijk}\Big{\}}, \tag{1}\]  
where \(\Delta_{ijk,k}\) denotes the channel uncertainty as the degree of deviation from the mean CSI \(\overline{\mathbf{H}}_{ijk}\). In Figure 1, during the first time phase, the source sends confidential signals to the relay while \(E_{1}\) eavesdrops on the signals from the source. To interrupt \(E_{1}\), the relay emits jamming signals to \(E_{1}\). More specifically, the message transmitted by the source is secret data symbol \(\mathbf{S}\in\mathcal{CN}(\mathbf{0},1)\) precoded by the precoding vector \(\mathbf{L}\in\mathbb{C}^{N_{S}\times 1}\). Then, we can formulate the progress at the source as follows:  
\[\mathbf{X}_{{}_{S}}=\mathbf{LS}, \tag{2}\]  
Next, we formulate the messages emitted by the relay. Note that the relay in this time phase only emits jamming to disrupt \(E_{1}\) so the messages can be formulated as follows :  
\[\mathbf{X}_{R}=\mathbf{J_{1}}\mathbf{Z}_{1}, \tag{3}\]  
where we utilize the jamming precoding vector \(\mathbf{J_{1}}\in\mathbb{C}^{N_{R}\times 1}\) and jamming symbol \(\mathbf{Z}_{1}\in\mathcal{CN}(\mathbf{0},1)\). Considering the self-interference of the relay, we can formulate the messages received by the relay:  
\[\mathbf{Y}_{R1}=\mathbf{H}_{SR,1}\mathbf{LS}+\mathbf{H}_{I}\mathbf{J_{1}} \mathbf{Z}_{1}+\mathbf{n}_{R1}, \tag{4}\]  
where \(\mathbf{n}_{R1}\) is Additive White Gaussian Noise (AWGN) at the relay and \(\mathbf{H}_{I}\) is the self-interference matrix. Meanwhile, \(E_{1}\) eavesdrops on both of the messages from the source and the relay, so the messages eavesdropped by \(E_{1}\) can be expressed as  
\[\mathbf{Y}_{E1}=\mathbf{H}_{SE,1}\mathbf{LS}+\mathbf{H}_{RE,1}\mathbf{J_{1}} \mathbf{Z}_{1}+\mathbf{n}_{E1}, \tag{5}\]  
where \(\mathbf{n}_{E1}\) is AWGN at \(E_{1}\). Figure 1: Wireless communication system model with a single relay. In the second time phase, the source emits the jamming signals \(\mathbf{X}_{S2}\) to \(E_{2}\) where \(\mathbf{J}_{2},\mathbf{Z}_{2}\in\mathcal{CN}(\mathbf{0},1)\) represent the jamming precoding vector and jamming symbol, respectively.