# Cooperative Jamming with AF Relay in Power Monitoring and Communication Systems for Mining
## 3 Optimization Problem Transformation
\begin{bmatrix}0&\mathbf{\Delta}_{D }^{H}\\ \mathbf{\Delta}_{D}&\mathbf{0}\end{bmatrix}  \tag{42}\]  
To eliminate the \(\mathbf{\Delta}_{D}\), the sign-definiteness lemma is applied. Definition 2: Defined matrix \(\mathbf{U}\) and \(\{\mathbf{P}_{i},\mathbf{Q}_{i}\},\ i\in\{1,2,\ldots,N\}\) with \(\mathbf{U}=\mathbf{U}^{H}\), the semi-infinite Linear Matrix Inequality (LMI) of the form  
\[\mathbf{U}\succ\sum_{i}^{N}\left(\mathbf{P}_{i}^{H}\mathbf{Y}_{i}\mathbf{Q}_ {i}+\mathbf{Q}_{i}^{H}\mathbf{Y}_{i}^{H}\mathbf{P}_{i}\right),\ \|\mathbf{Y}_{i}\|\leq\delta_{i} \tag{43}\]  
Holds if and only if there exist nonnegative real numbers \(\lambda_{1},\lambda_{2},\ldots,\lambda_{N}\) such that  
\[ \begin{bmatrix}\mathbf{U}-\sum_{i=1}^{N}\lambda_{i}\mathbf{Q}_{i}^{H}\mathbf{ Q}_{i}&-\delta_{1}\mathbf{P}_{1}^{H}&\cdots&-\delta_{N}\mathbf{P}_{N}^{H}\\ -\delta_{1}\mathbf{P}_{1}&\delta_{1}\mathbf{E}&\cdots&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots\\ -\delta_{N}\mathbf{P}_{N}&\mathbf{0}&\cdots&\delta_{N}\mathbf{E}\end{bmatrix}  \succ\mathbf{0} \tag{44}\]  
Appropriately choose the parameters below  
\[\mathbf{U}_{D}= \begin{bmatrix}\beta_{D}&\bar{\phi}_{D}^{H}\\ \bar{\phi}_{D}&\mathbf{I}\end{bmatrix}  \tag{45}\]  
\[\mathbf{Q}_{D1}=\mathbf{Q}_{D2}=\mathbf{Q}_{D3}=[-\mathbf{10}] \tag{46}\]  
\[\mathbf{P}_{D1}= \begin{bmatrix}\mathbf{0}&\mathbf{\Omega}_{DSR,1}^{H}\end{bmatrix}  \tag{47}\]  
\[\mathbf{P}_{D2}= \begin{bmatrix}\mathbf{0}&\mathbf{\Omega}_{DSD,2}^{H}\end{bmatrix}  \tag{48}\]  
\[\mathbf{P}_{D3}= \begin{bmatrix}\mathbf{0}&\mathbf{\Omega}_{DRR,1}^{H}\end{bmatrix}  \tag{49}\]Apply Lemma 2 to transform (42) as  
\[\left[ \begin{array}{ccc}\left[\begin{array}{cc}\beta_{D}\!-\!\lambda_{D1}\!-\! \lambda_{D2}\!-\!\lambda_{D3}&\tilde{\phi}_{D}^{H}\\ \tilde{\phi}_{D}&\mathbf{E}\end{array}\right]&\mathbf{\Theta}_{D}^{H}\\ \mathbf{\Theta}_{D}&\text{diag}(\lambda_{D1}\mathbf{E},\lambda_{D2}\mathbf{I}, \lambda_{D3}\mathbf{I})\end{array} \right]\succ 0 \tag{50}\]  
where \(\mathbf{\Theta}_{D}=-[\delta_{DSR,1}\mathbf{P}_{D1}^{T},\delta_{DSD,2}\mathbf{P }_{D2}^{T},\delta_{DRR,1}\mathbf{P}_{D3}^{T}]^{T}\). Similarly, the other constraint \(\text{tr}(\mathbf{K}_{i}\mathbf{N}_{i})\leq\beta_{i}\) is written as follows. \[\left[ \begin{array}{ccc}\left[\begin{array}{cc}\beta_{i}-\sum\limits_{k= i}^{j}\lambda_{k}&\tilde{\phi}_{i}^{H}\\ \tilde{\phi}_{i}&\mathbf{E}\end{array}\right]&\mathbf{\Theta}_{i}^{H}\\ \mathbf{\Theta}_{i}&\mathbf{\Theta}_{i}&\left[\lambda_{1}\mathbf{E},\cdots, \lambda_{j}\mathbf{E}\right]\end{array} \right]\succ 0 \tag{51}\]  
By assembling all the components, the problem can now be written as  
\[\max_{\mathbf{L},\mathbf{J}_{1},\mathbf{J}_{2},\mathbf{G},\mathbf{F}_{i} \succ 0,\mathbf{T}_{i},\lambda_{i}\geq 0,\beta_{i}}h(\mathbf{L},\mathbf{J}_{1}, \mathbf{J}_{2},\mathbf{G},\mathbf{F}_{i},\mathbf{T}_{i},\lambda_{i},\beta_{i}) \tag{52}\]  
\[\text{s.t. \eqref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eqsecrecy rate and the BCD algorithm converges faster with larger uncertainty. Additionally, the proposed scheme achieves a better average secrecy rate with more ports of legitimate users and fewer ports of eavesdroppers, which is especially obvious in small uncertainty scenarios. It can be explained that the number of ports suggests the ability to receive or intercept the information. Figure 3 shows the impact of a different transmit power of the proposed scheme. It can be observed that the average secrecy rate increases with the increase of transmitting power. In addition, it is observed that the security rate does not improve significantly when the transmitted power is more than 10 dB under the condition of more ports of eavesdroppers and greater CSI uncertainty. It can be explained that the increase in transmitting power increases the capacity of not only legitimate users but also eavesdroppers, resulting in a slight change in the security rate.