## 2 System Model and Problem Formulation
### Problem Formulation  
In the anti-jamming system, the communication process is divided into two phases. In phase 1, the user transmits messages and selects one relay to help forward messages to BS. When the jammer senses communication signals between the user and relay, it releases jamming signals immediately to destroy normal communication. Let \(P\) and \(J\) denote the user's and jammer's power, where \(P\in\mathcal{P}\) and \(J\in\mathcal{J}\). Inspired by [43], we define \(\alpha_{u,r_{n}}=d_{u,r_{n}}^{-\delta}\) and \(\beta_{j,r_{n}}=d_{j,r_{n}}^{-\delta}\) as the channel gain between the \begin{table}
\begin{tabular}{c c} \hline \hline \(\mathcal{P},\mathcal{J}\) & Power set of the user and jammer \\ \hline \(\mathcal{R},\mathcal{Q}\) & Relay set and all relays’ power set \\ \hline \(P,J\) & Transmitting power of the user and jammer, i.e., \(P\in\mathcal{P},J\in\mathcal{J}\) \\ \hline \(\alpha_{u,r_{n}},\beta_{j,r_{n}}\) & The channel gain between the user, jammer and the \(n\)th relay \\ \hline \(\alpha_{B,r_{n}},\beta_{Bj}\) & The channel gain between the \(n\)th relay, jammer and BS \\ \hline \(N_{r_{n}},N_{B}\) & The back noise power at the \(n\)th relay and BS \\ \hline \(G_{n}\) & The \(n\)th relay’s amplification factor \\ \hline \(X_{u},X_{j},X_{r_{n}}\) & The user’s, jammer’s and \(n\)th relay’s transmitted signal \\ \hline \(Y_{r_{n}},Y_{B}\) & The \(n\)th relay’s and BS’s received signal \\ \hline \(C_{u},C_{j}\) & The transmitting cost of the user and jammer \\ \hline \(\varepsilon_{1},\varepsilon_{2}\) & Feedback error and eavesdropping error \\ \hline \(\gamma\) & The SINR received at BS \\ \hline \(\gamma^{\prime},\gamma^{\prime\prime}\) & The feedback SINR received by the user and eavesdropped by jammer \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summation of used notation. Figure 2: Anti-jamming communication time slot division. user, jammer and the \(n\)th relay, respectively, among which \(\delta\) is the path-loss factor. The backward noise signal at the \(n\)th relay is \(m_{t_{n}}\). \(X_{u}\) and \(X_{j}\) denote the user's and jammer's transmitted signal, respectively. \(Y_{r_{n}}\) denotes the \(n\)th relay's received signal from the user and jammer, which is defined as follows:  
\[Y_{r_{n}}=\sqrt{P}\alpha_{u,r_{n}}X_{u}+\sqrt{\beta}_{j,r_{n}}X_{j}+m_{t_{n}}. \tag{1}\]  
In phase 2, the relay adopts amplifies and forwards (AF) mode. Considering the \(n\)th relay is selected to help the user forward messages to BS, its amplification factor is expressed as follows:  
\[G_{n}=\sqrt{\frac{Q_{n}}{P\alpha_{u,r_{n}}^{2}+\int\beta_{j,r_{n}}^{2}+N_{r_{n} }}}. \tag{2}\]  
Let \(X_{r_{n}}\) denote the \(n\)th relay's retransmitted signal. We define the received signal at BS as \(Y_{B}\), and it is expressed as:  
\[ \begin{array}{l}Y_{B}=\sqrt{Q}_{n}\alpha_{B,r_{n}}X_{r_{n}}+\sqrt{\beta}_{B,j}X_{j}+m_{B}\\ =G_{n}\sqrt{P}\alpha_{B,r_{n}}\alpha_{u,r_{n}}X_{u}+G_{n}\sqrt{\alpha}_{B,r_{n }}\beta_{j,r_{n}}X_{j}+G_{n}\alpha_{B,r_{n}}m_{t_{n}}+\sqrt{\beta}_{B,j}X_{j}+ m_{B},\end{array}  \tag{3}\]  
where \(\alpha_{B,r_{n}}=d_{B,r_{n}}^{-\delta}\) and \(\beta_{B,j}=d_{B,j}^{-\delta}\) denote the channel gain between the \(n\)th relay, jammer and BS, respectively. \(m_{B}\) is the backward noise signal at BS. Thus the SINR received at BS can be defined as follows:  
\[\gamma=\frac{G_{n}^{2}P\alpha_{B,r_{n}}^{2}\alpha_{u,r_{n}}^{2}}{G_{n}^{2} \alpha_{B,r_{n}}^{2}\beta_{j,r_{n}}^{2}+G_{n}^{2}\alpha_{B,r_{n}}^{2}N_{r_{n} }+\int\beta_{B,j}^{2}+N_{B}}. \tag{4}\]  
For the user, it obtains feedback SINR \(\gamma^{\prime}\) at the end of each time slot. Considering there may exists feedback information error, we introduce feedback error \(\varepsilon_{1}=|\gamma-\gamma^{\prime}|/\gamma\), which means the deviation degree of feedback SINR \(\gamma^{\prime}\) with the actual SINR \(\gamma\).