## 2 System Model and Problem Formulation
### Problem Formulation  
In our model, while forwarding the information of satellite \(S\), the selected relay \(R_{m}\) opportunistically transmits its information to the destination covertly. However, since the total communication resources are limited, the behavior of covert communication will inevitably occupy the communication resources of the primary link and be unfavorable for its communication. Therefore, the covert communication behavior at \(R_{m}\) is illegal. Based on this, \(S\) must always detect \(R_{m}\) to judge whether it transmits covert information to the user \(U\) or not, so that \(S\) can respond on time. Against this background, we perform the following analysis and present the research problem of this paper.  
The received signal at \(S\) in the second time period can be obtained as:  
\[y_{S}(k)=\left\{\begin{array}{c}\sqrt{P_{R_{m},0}}h_{R_{m}S}x_{R}(k)+n_{S}(k )H_{0}\\ \sqrt{P_{R_{m},1}}h_{R_{m}S}x_{R}(k)+\sqrt{P_{C}}h_{R_{m}S}x_{C}(k)+n_{S}(k)H_{ 1}\end{array}\right., \tag{14}\]  
where \(n_{S}\) is the complex AWGN received by \(S\), and \(n_{S}(k)\sim\mathcal{CN}\big{(}0,\sigma_{S}{}^{2}\big{)}\). \(H_{1}\) denotes \(R_{m}\) transmits with covert information and \(H_{0}\) is the opposite. Besides, due to the channel reciprocity, \(h_{R_{m}S}=h_{SR_{m}}\). The energy detection method is adopted by \(S\) to probe whether \(R_{m}\) is transmitting covert information, which has been proved as the best method in quasi-static fading channels in [32]. Namely, when the average power of the received signal of \(S\) is larger than the preset energy detection threshold, \(S\) considers \(R_{m}\) are transmitting covert information to \(U\)[33]. In order to further analyze the impact of system parameters on the performance of covert communication and obtain the inspiration for optimization, we need to create a model to quantify the covert performance, namely, the detection capability of \(S\). Generally, we first analyze the error detection probability of \(S\).  
The error detection of \(S\) can be divided into two categories as:  
\[\begin{array}{c}P_{fa}=P(P_{Sr}\geq\lambda|H_{0})\\ P_{md}=P(P_{Sr}<\lambda|H_{1})\end{array}, \tag{15}\]  
where \(P_{Sr}=1/n\sum\limits_{k=1}^{n}\left|y_{S}(k)\right|^{2}\), and \(\lambda\) is the preset energy detection threshold. Moreover, \(P_{fa}\) is the false alarm (FA) probability, and \(P_{md}\) is the missed detection (MD) probability.  
**Lemma 1**.: _When the demands set \(\Gamma\) is met, \(R_{m}\) will transmit information successfully and \(S\) stars to detect. Thus, under the condition \(\Gamma\), for a given \(\lambda\), we can derive:_  
\[P_{fa}=\left\{\begin{array}{cc}1,&\lambda<\sigma_{S}{}^{2}\\ FA(\lambda),&\sigma_{S}{}^{2}\leq\lambda\leq\alpha\\ 0,&\lambda>\alpha\end{array},\right. \tag{16}\]  
\[P_{md}=\left\{\begin{array}{cc}0,&\lambda<\beta\\ MD(\lambda),&\beta\leq\lambda\leq\gamma\\ 1,&\lambda>\gamma\end{array}\right., \tag{17}\]  
_where_  
\[\alpha=P_{M}|h_{R_{m}S}|^{2}+\sigma_{S}{}^{2}, \tag{18}\]  
\[\beta=(\gamma_{th}+1)P_{\mathbb{C}}|h_{R_{m}S}|^{2}+\sigma_{S}{}^{2}, \tag{19}\]  
\[\gamma=[P_{M}+(\gamma_{th}+1)P_{\mathbb{C}}]|h_{R_{m}S}|^{2}+\sigma_{S}{}^{2}, \tag{20}\]_Moreover,_  
\[FA(\lambda)=\frac{\left[1-\exp\left(-\frac{\gamma_{th}\omega_{1} ^{2}|h_{Rn}|^{2}}{\lambda-\sigma_{5}^{2}}\right)\right]^{N}-\left[1-\exp\left(- \frac{\gamma_{th}\omega_{1}^{2}}{P_{M}}\right)\right]^{N}}{1-\left[1-\exp\left(- \frac{\gamma_{th}\omega_{1}^{2}}{P_{M}}\right)\right]^{N}}, \tag{21}\]  
\[MD(\lambda)=\frac{1-\left[1-\exp\left(-\frac{\gamma_{th}\omega_{1} ^{2}|h_{Rn}|^{2}}{\lambda-(\gamma_{th}+1)P_{c}|h_{Rn}|^{2}-\sigma_{5}^{2}} \right)\right]^{N}}{1-\left[1-\exp\left(-\frac{\gamma_{th}\omega_{1}^{2}}{P_{M }}\right)\right]^{N}}. \tag{22}\]  
Proof.: See Appendix A.