## 2 System Model and Problem Formulation
### Transmission Model  
In our system, a partial relay selection scheme is considered by us, which is expressed as that the relay with the highest instantaneous average signal-to-noise ratio (SNR) at \(U\), denoted by \(R_{m}\) (\(m=1,\ldots,N\)), is selected as the special relay to transmit the source information from \(S\)[31]. Thus, for the DF relaying mentioned above, we have:  
\[\left|h_{R_{m}U}\right|^{2}=\max_{i\in\{1,\ldots,N\}}\left|h_{R_{i}U}\right|^{ 2}. \tag{1}\]  
Therefore, the received SNR at \(U\) can be given by:\[\gamma_{U}=\frac{P_{R}|h_{R_{m}U}|^{2}}{\sigma_{U}u^{2}}. \tag{2}\]  
where \(P_{R}\) is the transmit power of the selected relay, \(n_{U}\) denotes the noise at \(U\) with \(\mathcal{CN}(0,\sigma_{U}{}^{2})\). Moreover, since the system realizes communication from \(S\) to \(U\) by relays, the information transfer process includes two time periods. In the first time period, \(S\) transmits the source information to \(U\) with a fixed rate, denoted by \(r_{SU}\)[29]. The received signal at \(R_{m}\) is given by:  
\[y_{R_{m}}(k)=\sqrt{P_{ST}}h_{SR_{m}}x_{S}(k)+n_{R_{m}}(k), \tag{3}\]  
where \(P_{ST}\) is the transmit power of \(S\), \(k=1,2,\ldots,n\) is the symbol subscript, \(x_{S}\) denotes the source transmission signal of \(S\), which satisfies \(E\Big{\{}|x_{S}(k)|^{2}\Big{\}}=1\), and \(n_{R_{m}}\) is the complex AWGN received by \(R_{m}\) with \(n_{R_{m}}(k)\sim\mathcal{CN}\big{(}0,\sigma_{R_{m}}{}^{2}\big{)}\). In the second time period, whether \(R_{m}\) is able to transmit information successfully is decided by two necessary conditions. One of them is that the \(R_{m}\) has enough ability to decode \(x_{S}\) correctly, which is expressed by:  
\[\gamma_{R_{m}}=\frac{P_{ST}|h_{SR_{m}}|^{2}}{\sigma_{R_{m}}{}^{2}}\geq\gamma_ {th}, \tag{4}\]  
where \(\gamma_{R_{m}}\) represents the received SNR at \(R_{m}\), and \(\gamma_{th}=2^{2r_{SU}}-1\) is the threshold [21]. Moreover, another one is that \(U\) can decode the signal from \(R_{m}\) successfully:  
\[\gamma_{U,M}=\frac{P_{M}|h_{R_{m}U}|^{2}}{\sigma_{U}{}^{2}}\geq\gamma_{th}, \tag{5}\]  
Figure 1: Covert communication system based ISMRN with partial relay selection. where \(\gamma_{U,M}\) is the maximum received SNR at \(U\), \(P_{M}\) is the maximum transmit power of \(R_{m}\), and \(\sigma_{U}{}^{2}\) represents the average noise power at \(U\). Meanwhile, the satisfaction of the above conditions is monitored by \(R_{m}\) at any time. According to Equations (4) and (5), the condition for keeping the communication link uninterrupted is obtained as:  
\[\Gamma=\{\min(\gamma_{R_{m}},\gamma_{U,M})\geq\gamma_{th}\}, \tag{6}\]  
when the demands set \(\Gamma\) is met, \(R_{m}\) can transmit information successfully. Furthermore, the two cases need to be considered, including transmitting without or with covert information. If \(R_{m}\) only transmits source information to \(U\) without covert information, the received signal at \(U\) is given by:  
\[y_{U_{0}}(k)=\sqrt{P_{R_{m},0}}h_{R=U}x_{R}(k)+n_{U}(k), \tag{7}\]  
where \(P_{R_{m},0}\) is the transmit power of \(R_{m}\) without covert information, \(x_{R}\) denotes transmission signal of source information of \(R_{m}\), which satisfies \(E\Big{\{}\left|x_{R}(k)\right|^{2}\Big{\}}=1\), and \(n_{U}\) is the complex AWGN received by \(U\) with \(n_{U}(k)\sim\mathcal{CN}(0,\sigma_{U}{}^{2})\). Since the transmission rate of \(S\) to \(U\) is fixed, \(R_{m}\) must ensure \(\gamma_{R_{m}}=\gamma_{th}\) at least.