# The Optimization of Covert Communication in Asymmetric Jammer - Assist Systems
## 4 Analysis of the Transmission Rate
### Problem Reformulation for Two Cases  
To measure the influence of interference, we define  
\[\sigma_{b,x}^{2} \triangleq\sigma_{b,x,0}^{2}+\alpha\Delta P_{J}, \tag{49}\] \[\sigma_{b,y}^{2} \triangleq\sigma_{b,y,0}^{2}+(1-\alpha)\Delta P_{J}. \tag{50}\]  
where \(\Delta P_{J}\) denotes the jammer interference power after interference elimination.  
Case 1: if [PERSON] can perfectly eliminate the influence of interference, \(\Delta P_{J}=0\). Taking the derivative of the function \(F(\mu)\), the minimal value \(F^{*}(\mu)\) can be obtained when  
\[\mu^{*}=\frac{h_{w,x}}{h_{w,y}+h_{w,x}}. \tag{51}\]  
Considering the range of \(\mu\) given in (24), the minimization of the KL divergence can be achieved when  
\[\mu^{*}=\begin{cases}\max\{\frac{\sigma_{w,x,0}^{2}+P_{J}}{\sigma_{w}^{2}}, \frac{\sigma_{w,y0}^{2}+P_{J}}{\sigma_{w}^{2}}\}&\max\{\frac{\sigma_{w,x,0}^{2 }+P_{J}}{\sigma_{w}^{2}},\frac{\sigma_{w,y0}^{2}+P_{J}}{\sigma_{w}^{2}}\}\leq \frac{h_{w,x}}{h_{w,y}+h_{w,x}}\\ \frac{h_{w,x}}{h_{w,y}+h_{w,x}}&\min\{\frac{\sigma_{w,x0}^{2}}{\sigma_{w}^{2}},\frac{\sigma_{w,y0}^{2}}{\sigma_{w}^{2}}\}<\frac{h_{w,x}}{h_{w,y}+h_{w,x}}< \max\{\frac{\sigma_{w,x0}^{2}+P_{J}}{\sigma_{w}^{2}},\frac{\sigma_{w,y0}^{2}+P _{J}}{\sigma_{w}^{2}}\}\\ \min\{\frac{\sigma_{w,x0}^{2}}{\sigma_{w}^{2}},\frac{\sigma_{w,y0}^{2}}{\sigma _{w}^{2}}\}&\frac{h_{w,x}}{h_{w,y}+h_{w,x}}\leq\min\{\frac{\sigma_{w,x0}^{2}} {\sigma_{w}^{2}},\frac{\sigma_{w,y0}^{2}}{\sigma_{w}^{2}}\}\end{cases} \tag{52}\]  
Combining with (35), (36) and the optimal \(\mu^{*}\), the constraint (15) can be rewritten as  
\[\frac{n\beta^{4}}{4}F^{2}(\mu^{*})\leq\epsilon, \tag{53}\]  
and the optimal amplitude gain \(\beta^{*}\) is given by  
\[\beta^{*}=\bigg{(}\frac{4\epsilon}{nF^{2}(\mu^{*})}\bigg{)}^{\frac{1}{4}}. \tag{54}\]Then, we can obtain the maximum mutual information as  
\[\mathbb{I}^{*}(Zb^{n};D^{n})=\sqrt{\frac{4n\epsilon}{F^{2}(\mu^{*})}} \Big{(}\frac{h_{b,x}^{2}}{2\sigma_{b,x,0}^{2}}+\frac{h_{b,y}^{2}}{2\sigma_{b,y,0} ^{2}}\Big{)}. \tag{55}\]  
Case 2: [PERSON] can partially eliminate the influence of interference, \(\Delta P_{I}=\eta P_{I}\) where \(\eta\in(0,1)\) denotes the coefficient of [PERSON]'s interference cancellation capability. Combining with (19) and (20), the ratio \(\alpha\) is given by  
\[\alpha=\frac{\mu\sigma_{a}^{2}-\sigma_{w,x,0}^{2}}{P_{I}}. \tag{56}\]  
Then, the mutual information can be expressed as  
\[\mathbb{I}(Zb^{n};D^{n})=\sqrt{\frac{n\epsilon}{F^{2}(\mu)}} \Bigg{(}\frac{h_{b,x}^{2}}{(\sigma_{b,x,0}^{2}+\alpha\eta P_{I})}+\frac{h_{b,y }^{2}}{(\sigma_{b,y,0}^{2}+(1-\alpha)\eta P_{I})}\Bigg{)}. \tag{57}\]  
Combining with (23), (16), (17) and (24), the problem P2 can be equivalently transformed to  
\[\text{P3}:\underset{\mu}{\text{max}} \mathbb{I}(Zb^{n};D^{n})\] (58) s.t. (24). (59)  
Clearly, the optimal value of \(\mu^{*}\) and the maximization of mutual information can be effortlessly obtained through an iterative exploration algorithm for \(\alpha^{*}\). Then, we can obtain the optimal amplitude gain \(\beta^{*}\), and the original problem P1 has been fully addressed.