# Disguised Full-Duplex Covert Communications
## System Model
### Covert Message Detection  
At the same time, the warden node receives  
\[y_{W}=h_{SW}\sqrt{P_{S}}x_{p}+h_{DW}\sqrt{P_{D}}x_{C}+z_{W}. \tag{6}\]  
It first excludes public messages from \(y_{W}\) to obtain the effective residual signal \(\tilde{z}_{W}\triangleq y_{W}-h_{SW}\sqrt{P_{S}}x_{p}\), assuming it perfectly knows \(h_{SW}\) and \(P_{S}\)[38]. We then have the following two hypotheses:  
\[ \begin{array}{ll}H_{0}:&\tilde{z}_{W}=z_{W},\\ H_{1}:&\tilde{z}_{W}=h_{DW}\sqrt{P_{D}}x_{C}+z_{W},\end{array}  \tag{7}\]  
where the null hypothesis \(H_{0}\) assumes that there does not exist a covert message, and the alternative hypothesis \(H_{1}\) presumes that the source node did not send a covert message. This work considers a radiometer [39] as a detection means at the warden, and the sufficient test statistic \(T\) for (7) after observing \(N\rightarrow\infty\) number of signals reduces to the average residual power \(\mathbb{E}[|\tilde{z}_{\text{w}}|^{2}]\) as [40]  
\[H_{0}: T=\sigma_{W}^{2}, \tag{8}\] \[H_{1}: T=|h_{D\text{W}}|^{2}P_{D}+\sigma_{W}^{2}.\]  
The warden node makes a decision that a covert transmission exists when \(T\geq\tau\) and otherwise when \(T<\tau\) with some threshold \(\tau\). We consider the uncertainty in the noise variance \(\sigma_{W}^{2}\) at the warden node as in [39; 41]. Concretely, \(\sigma_{W,\text{dB}}^{2}\sim U(\sigma_{W,\text{dB}}^{2}-\zeta_{\text{dB}}, \sigma_{W,\text{dB}}^{2}+\zeta_{\text{dB}})\) in decibel scale with \(\sigma_{W,\text{dB}}^{2}\) and \(\zeta_{\text{dB}}\geq 0\) set to the mean and bounded range, respectively. We then derive the DEP \(\Pr(\text{e})\) that consists of the false alarm and miss probabilities as  
\[\Pr(\text{e}) =\underbrace{\Pr(T\geq\tau|H_{0})}_{\text{False alarm}}\Pr(H_{0})+ \underbrace{\Pr(T<\tau|H_{1})}_{\text{Miss}}\Pr(H_{1}), \tag{9}\]  
in which the warden node conjectures that the covert transmission randomly takes place with \(\Pr(H_{0})=\Pr(H_{1})=0.5\)[42]. Making use of the cumulative distribution function (CDF) of \(\sigma_{W}^{2}\) (Appendix A),  
\[F_{\sigma_{W}^{2}}(\
u)=\frac{1}{2\ln\zeta}\left(\ln\
u-\ln\frac{\bar{\sigma _{W}^{2}}}{\zeta}\right),\
u\in\left[\frac{\bar{\sigma}_{W}^{2}}{\zeta},\zeta \bar{\sigma}_{W}^{2}\right], \tag{10}\]  
the false alarm and miss probability are specifically written by  
\[\Pr(T\geq\tau|H_{0}) =1-F_{\sigma_{W}^{2}}(\tau),\tau\in\mathcal{T}_{1}, \tag{11}\] \[\Pr(T<\tau|H_{1}) =F_{\sigma_{W}^{2}}\left(\tau-|h_{D\text{W}}|^{2}P_{D}\right), \tau\in\mathcal{T}_{2}, \tag{12}\]  
respectively. Here, \(\mathcal{T}_{1}\triangleq[\bar{\sigma}_{W}^{2}/\zeta,\zeta\bar{\sigma}_{W}^{2}]\) and \(\mathcal{T}_{2}\triangleq[|h_{D\text{W}}|^{2}P_{D}+\bar{\sigma}_{W}^{2}/\zeta,|h_{D\text{W}}|^{2}P_{D}+\zeta\bar{\sigma}_{W}^{2}]\). We have two different cases depending on the value of \(|h_{D\text{W}}|^{2}\) and \(P_{D}\). If \(\zeta\bar{\sigma}_{W}^{2}<|h_{D\text{W}}|^{2}P_{D}+\bar{\sigma}_{W}^{2}/\zeta\),  
\[\Pr(\text{e})=\left\{ \begin{array}{ll}\frac{1}{2}\Pr(T\geq\tau|H_{0})&\tau \in\mathcal{T}_{1},\\ 0&\tau\in\mathcal{T}_{3},\\ \frac{1}{2}\Pr(T<\tau|H_{1})&\tau\in\mathcal{T}_{2},\end{array} \right. \tag{13}\]  
where \(\mathcal{T}_{3}\triangleq[\zeta\bar{\sigma}_{W}^{2},|h_{D\text{W}}|^{2}P_{D}+ \bar{\sigma}_{W}^{2}/\zeta]\). In contrast, if \(\zeta\bar{\sigma}_{W}^{2}\geq|h_{D\text{W}}|^{2}P_{D}+\bar{\sigma}_{W}^{2}/\zeta\),  
\[\Pr(\text{e}) =\!\left\{ \begin{array}{ll}\frac{1}{2}\Pr(T\geq\tau|H_{0}),&\tau\in \mathcal{T}_{4},\\ \frac{1}{2}(\Pr(T\geq\tau|H_{0})+\Pr(T<\tau|H_{1})),&\tau\in\mathcal{T}_{5}, \\ \frac{1}{2}\Pr(T<\tau|H_{1}),&\tau\in\mathcal{T}_{6},\end{array} \right. \tag{14}\]  
with \(\mathcal{T}_{4}\triangleq[\bar{\sigma}_{W}^{2}/\zeta,|h_{D\text{W}}|^{2}P_{D} +\bar{\sigma}_{W}^{2}/\zeta]\), \(\mathcal{T}_{5}\triangleq[|h_{D\text{W}}|^{2}P_{D}+\bar{\sigma}_{W}^{2}/\zeta,\zeta\bar{\sigma}_{W}^{2}]\) and \(\mathcal{T}_{6}\triangleq[\zeta\bar{\sigma}_{W}^{2},|h_{D\text{W}}|^{2}P_{D} +\zeta\bar{\sigma}_{W}^{2}]\). The warden node may desire a particular \(\tau\) that can minimize the DEP.