# Covert Communication for Integrated Satellite-Terrestrial Relay Networks with Cooperative Jamming
## 3 Performance Analysis
### Statistical Properties of Channels  
Considering the actual scenario and referring to the relevant study [5], it is generally believed that terrestrial channels obey the independent identically distributed (i.i.d.) Rayleigh fading. Thus, the probability distribution function (PDF) and cumulative distribution function (CDF) of \(\gamma_{p}\) (\(p\in\{RE,RB\}\)) are obtained, respectively, as  
\[f_{\gamma_{p}}(x)=\frac{1}{\tilde{\gamma}_{p}}e^{-x/\tilde{\gamma}_{p}},x\geq 0, \tag{6}\]  
\[F_{\gamma_{p}}(x)=1-e^{-x/\tilde{\gamma}_{p}},x\geq 0, \tag{7}\]  
where \(\gamma_{p}\) is the received signal-to-noise ratio (SNR), \(\tilde{\gamma}_{p}\) represents the average gain of channel, and \(\gamma_{p}=\tilde{\gamma}_{p}\big{|}h_{p}\big{|}^{2}=P_{R}\big{|}h_{p}\big{|}^{ 2}/\sigma^{2}\). Moreover, it is inevitable to use multi-beam technology to improve the spectral efficiency in SatCom, which should be considered in the establishment of a channel model [33]. However, in the previous system description, we assumed that the relay and the eavesdropper are in close proximity and under the same beam coverage. Therefore, the subsequent formula derivation and analysis are based on the specific single beam. In addition, the channels of satellite-terrestrial links are generally modeled as shadowed Rician (SR) fading channels. Therefore, the coefficient \(h_{q}\) (\(q\in\{SR,SE\}\)) can be shown as  
\[h_{q}=C_{q}g_{q}, \tag{8}\]  
where \(g_{q}\) is the SR coefficient, and \(C_{q}\) represents the loss that depends on the radio frequency, including the free space loss and antenna gain. Therefore, \(C_{q}\) can be written as  
\[C_{q}=\frac{\lambda}{4\pi}\frac{\sqrt{G_{S}G_{R/E}}}{\sqrt{s^{2}+R_{e}}^{2}}, \tag{9}\]  
where \(\lambda\) represents the carrier wavelength of the transmitted signal, \(s\) denotes the distance from the terrestrial station to the center of the satellite beam, and \(R_{e}\approx\) 35,786 km. \(G_{S}\) is the satellite beam gain, and \(G_{R/E}\) is the relevant antenna gain. In addition, for parabolic antenna [34], the \(G_{R/E}\) can be expressed as  
\[G_{R/E}(dB)\simeq\left\{ \begin{array}{c}G_{MR/E},0^{\circ}<\psi<1^{\circ}\\ 32-25\log\psi,1^{\circ}\leq\psi<48^{\circ}\\ -10,48^{\circ}<\psi\leq 180^{\circ}\end{array} \right., \tag{10}\]  
where \(G_{MR/E}\) represents the upper limit of the relevant antenna gain, and \(\psi\) denotes the deflection angle of \(S\). Moreover, \(\vartheta\) is the angle from the terrestrial station to the center of the satellite beam, and \(\vartheta_{3 dB}\) denotes the satellite beam's 3 dB angle. Thus, the beam gain of \(S\) is shown as [35]  
\[G_{S}(dB)\simeq G_{MS}\bigg{(}\frac{I_{1}(\tau)}{2\tau}+36\frac{I_{3}(\tau)}{ \tau^{3}}\bigg{)}^{2}, \tag{11}\]where \(\tau=2.07123\sin\theta/\sin\theta_{3 dB}\), \(G_{MS}\) denotes the maximum of \(G_{S}\), \(I_{1}(x)\), and \(I_{3}(x)\) are the order 1 and 3 first-kind Bessel functions, respectively. Generally, to gain the system's best performance, \(\theta\to 0\)--namely, \(G_{S}=G_{MS}\), and further, \(h_{q}=C_{q}^{Max}g_{q}\). From [36], the PDF of \(\gamma_{q}=\bar{\gamma}_{q}\left|C_{q}^{Max}g_{q}\right|^{2}\) is given by  
\[f_{\gamma_{q}}(x)=\frac{\alpha_{q}}{\bar{\gamma}_{q}}e^{-\frac{ \beta_{q}x}{\bar{\gamma}_{q}}}{}_{1}F_{1}\left(m_{q};1;\frac{\delta_{q}x}{\bar {\gamma}_{q}}\right),x>0, \tag{12}\]  
where \(m_{q}\) denotes the fading severity parameter, \(\Omega_{q}\) represents the line of sight (LOS) link's average power, and \(b_{q}\) is half of the average power of the multi-path links. In addition, \(\alpha_{q}=\frac{1}{2b_{q}}\left(\frac{2b_{q}m_{q}}{\left(\frac{2b_{q}m_{q}}{ \left(2b_{q}m_{q}+\Omega_{q}\right)}\right)}\right)^{m_{q}},\delta_{q}=\frac{ \Omega_{q}}{\left[2b_{q}\left(2b_{q}\left(2b_{q}\left(2b_{q}\right)+\Omega_{q }\right)\right]}\right)\), and \(\beta_{q}=\frac{1}{2b_{q}}\). Generally, \(m_{q}\) is an integer.