## 2 System Model and Problem Formulation
### Covert Communication Model  
In the secure satellite communication scenario, an eavesdropping GEO satellite aims to detect whether there is covert data from the received signal, while CUEs try to transmit more covert data without being eavesdropped. The eavesdropping GEO satellite aims to detect the transmission activities from the following binary hypothesis test:  
\[\left\{\begin{array}{l l}\mathcal{H}_{0}:&\left[y_{eJ}(t)\right]_{1\leq l \leq L}=\left[x_{mu\varepsilon}(t)+n_{eJ}\right]_{1\leq l\leq L}\\ \mathcal{H}_{1}:&\left[y_{eJ}(t)\right]_{1\leq l\leq L}=\left[x_{mu\varepsilon }(t)+x_{cueJ}(t)+n_{eJ}\right]_{1\leq l\leq L}\end{array}\right. \tag{6}\]  
where \(\left[y_{eJ}(t)\right]_{1\leq l\leq L}\) denotes the received signal samples and \(L=\delta B\) is the number of samples collected by the eavesdropping GEO satellite. \(\mathcal{H}_{0}\) represents the case that UEs only send normal data, and \(\mathcal{H}_{1}\) represents the case that UEs send both normal data and covert data. \(n_{eJ}\) is the noise at the eavesdropping GEO satellite and it follows the complex circularly symmetric Gaussian distribution with mean 0 and variance \(\sigma^{2}\). \(x_{mu\varepsilon}(t)\) is the received signal from NUEs at time slot \(t\), and \(x_{cue,l}(t)\) is the received signal from CUEs at time slot \(t\), which can be expressed as  
\[x_{nue,l}(t)=\sum_{i\in\mathcal{I},\mathcal{J}}\sum_{m\in\mathcal{M}}\sqrt{h_{e,j }P_{i,t}^{m}}x_{i,m,l} \tag{7}\]  
\[x_{cue,l}(t)=\sum_{j\in\mathcal{J}}\sum_{m\in\mathcal{M}}\sqrt{h_{e,j}P_{j,t}^ {m}}x_{j,m,l} \tag{8}\]  
where \(P_{i,t}^{m}\) and \(P_{j,t}^{m}\) denote the transmit power of NUE \(i\) and CUE \(j\) on the band \(m\) at time slot \(t\). \(x_{i,m,l}\) denotes the \(l\)-th symbol transmitted by NUE \(i\) on the band \(m\) and it follows \(x_{i,m,l}\sim\mathcal{CN}(0,1)\), while \(x_{j,m,l}\) denotes the \(l\)-th symbol transmitted by CUE \(j\) on the band \(m\) and it also follows \(x_{j,m,l}\sim\mathcal{CN}(0,1)\). The eavesdropper detection can produce two error decisions: false alarm probability \(\alpha\) and missed detection probability \(\beta\). The covert performance of the system is usually measured by the total detection error probability: \(p_{e}=\alpha+\beta\). Similar to the existing studies [22], the covert transmission requirements of CUEs can be guaranteed if \(p_{e}\geq 1-\varepsilon\), where \(\varepsilon\) is a small positive constant.  
According to [18], when the eavesdropper adopts the optimal decision threshold, the lower bound of the total detection error probability is given as \(1-\sqrt{\mathcal{D}(\mathbb{P}_{1,m}||\mathbb{P}_{0,m})}/2\), where \(\mathbb{P}_{1,m}\) is the probability distribution of \(\left[y_{e,l}(t)\right]_{1\leq l\leq L}\) when \(\mathcal{H}_{1}\) is true, and \(\mathbb{P}_{0,m}\) is the probability distribution of \(\left[y_{e,l}(t)\right]_{1\leq l\leq L}\) when \(\mathcal{H}_{0}\) is true. \(\mathcal{D}(\mathbb{P}_{1,m}||\mathbb{P}_{0,m})\) is the Kullback-Leibler (KL) divergence of these two distributions. Given the parameter \(\varepsilon\), we can meet the covert transmission requirements of CUEs as long as  
\[\sum_{m\in\mathcal{M}}\mathcal{D}(\mathbb{P}_{1,m}||\mathbb{P}_{0,m})\leq \frac{2\varepsilon^{2}}{\delta B} \tag{9}\]  
Based on the above discussion, we propose a covertness constraint and a optimization problem of the multi-dimensional resource scheduling strategy in the following subsection.