# Covert Communication for Integrated Satellite-Terrestrial Relay Networks with Cooperative Jamming
## 3 Performance Analysis
### Probability of Detection Error  
In the first time slot, the likelihood ratio test is utilized by \(E\), which further employs radiometry to perform the binary detection [21]. Moreover, we obtain the signal received at \(E\) as  
\[y_{E}(k)=\left\{\begin{array}{cc}\sqrt{P_{R}}h_{RE}x_{j}(k)+n_{E}(k),&H_{0} \\ \sqrt{P_{S}}h_{SE}x_{S}(k)+\sqrt{P_{R}}h_{RE}x_{j}(k)+n_{E}(k),&H_{1}\end{array}\right. \tag{16}\]  
where \(n_{E}(k)\sim\mathcal{CN}\left(0,\sigma^{2}\right)\). Furthermore, it should be noted that \(E\) is usually unaware of the code book for the signal \(x_{S}\).  
Based on (16), \(T(n)\) denotes the average received power, which is used as the test statistic by \(E\), namely  
\[T(n)\overset{\Delta}{=}\frac{1}{n}\sum_{k=1}^{n}\left|y_{E}(k)\right|^{2} \underset{D_{0}}{\overset{D_{1}}{\overset{D_{1}}{\overset{D_{1}}{\overset{ <}{\sim}}}}}\
u, \tag{17}\]  
where \(\
u\) represents the detection threshold of \(E\). Relying on the strong law of large numbers and (2), the false alarm probabilities of \(E\) are written as  
\[P_{fa}=\Pr\left(P_{R}|h_{RE}|^{2}+\sigma^{2}\geq\
u\right)=\int_{\
u_{0}}^{ \infty}f_{{}_{TEE}}(x)dx, \tag{18}\]  
where \(\
u_{0}=\left(\
u-\sigma^{2}\right)/\sigma^{2}\).  
By means of (6), we can obtain the \(P_{fa}\) as  
\[P_{fa}=\left\{\begin{array}{c}1,\
u<\sigma^{2}\\ \Gamma,\
u\geq\sigma^{2}\end{array}\right., \tag{19}\]  
where \(\Gamma=\exp\left(-\frac{\
u 0}{\tau_{RE}}\right)=\exp\left(-\frac{\
u-\sigma^{2}}{P _{R}}\right)\).  
Moreover, the probability of missed detection at \(E\) is derived as  
\[P_{md}=\Pr\left(P_{R}|h_{RE}|^{2}+P_{S}|h_{SE}|^{2}+\sigma^{2}\leq\
u\right). \tag{20}\]  
With the help of (6) and (14), the \(P_{md}\) can be rewritten as  
\[P_{md}=\int_{0}^{\
u_{0}}\left(\int_{0}^{\
u_{0}-y}f_{\gamma_{SE}}(x)dx\right) f_{\gamma_{RE}}(y)dy. \tag{21}\]  
By utilizing the specific combination function of exponential and rational in [37], namely  
\[P_{md}=\left\{\begin{array}{c}0,\
u<\sigma^{2}\\ 1-\Gamma-\Xi,\
u\geq\sigma^{2}\end{array}\right., \tag{22}\]  
where  
\[\Xi=\phi\frac{\exp\left(-\
u_{0}/\hat{\gamma}_{RE}\right)}{\hat{\gamma}_{RE}} \frac{\Upsilon(t+1,\omega\
u_{0})}{\omega^{t+1}} \tag{23}\]  
and \(\hat{\sigma}=\alpha_{SE}\sum\limits_{i=0}^{m_{SE}-1}\sum\limits_{t=0}^{i} \frac{(1-m_{SE})_{i}(-\delta_{SE})^{i}}{\hat{\gamma}_{RE}\left(\hat{\gamma}_{ SE}\right)^{i+1}t!\Delta_{SE}t^{-t+1}}\), \(\omega=(\Delta_{SE}-1/\hat{\gamma}_{RE})>0\), \(\Upsilon(t+1,\omega\
u_{0})=t!-\exp\left(-\omega\
u_{0}\right)\sum\limits_{j=0} ^{i}\frac{t!}{j!}\frac{\upsilon_{j}^{j}}{\omega^{2-j+1}}\).  
Therefore, the PDE for a preset \(\
u\) can be given by  
\[P_{e}=\left\{\begin{array}{c}1,\
u<\sigma^{2}\\ 1-\Xi,\
u\geq\sigma^{2}\end{array}\right.. \tag{24}\]  
According to \(\omega>0\), a new constraint can be obtained, which can be expressed as  
\[\eta<\frac{\beta_{SE}-\delta_{SE}}{1+\beta_{SE}-\delta_{SE}}=\eta_{\max}, \tag{25}\]  
which implies that there is a new range for the \(\eta\).  
On the other hand, since \(E\) aims to minimize the PDE, it will never select the case of \(\
u<\sigma^{2}\), which represents an absolute incorrect detection. Therefore, the optimal detection threshold for \(E\), defined as \(\
u_{opt}\), occurs only when \(\
u\geq\sigma^{2}\). It can be calculated by \(\frac{\partial P_{e}}{\partial\
u}=0\), which has been solved and shown in numerical simulation. Furthermore, under the prescribed optimal detection threshold \(\
u_{opt}\), the minimum PDE at \(E\) should be expressed as  
\[\left(P_{e}\right)_{\min}=\left.P_{e}\right|_{\
u=\
u_{opt}}. \tag{26}\]