# Optimization of Intelligent-Reflecting-Surface-Enabled Covert Communication in Multi-Sensor Systems
## 5 Performance Analysis of the Covert Transmission  
In (9), we have rewritten the mutual information as  
\[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{1}{2}\mathbb{I}(Y_{b,1,1},\ldots,Y_{b,1,n};c_{1,1},\ldots,c_{1,n})+\frac{1}{2}\mathbb{I}(Y_{b,2,1}, \ldots,Y_{b,2,n};c_{2,1},\ldots,c_{2n}). \tag{44}\]  
Following from the chain rule, the mutual information \(\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})\) can be expressed as  
\[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]  
For \(j=1\) or \(j=2\), we have  
\[\mathrm{I}(Y_{b,j};c_{j})=\mathrm{H}(Y_{b,j})-\mathrm{H}(Y_{b,j} |c_{j}) \tag{46}\] \[=-\iint Q_{b,j}(x,y)\log Q_{b,j}(x,y)\mathrm{d}x\mathrm{d}y+\iint \sum_{t=1}^{2}\frac{1}{2}Q_{b,j,t}(x,y)\log Q_{b,j,t}(x,y)\mathrm{d}x\mathrm{ d}y\] (47) \[=-\iint\sum_{s=1}^{2}\frac{1}{2}Q_{b,j,s}(x,y)\log\Big{[}\sum_{t =1}^{2}\frac{1}{2}Q_{b,j,t}(x,y)\Big{]}\mathrm{d}x\mathrm{d}y\] (48) \[\quad+\iint\sum_{s=1}^{2}\frac{1}{2}Q_{b,j,s}(x,y)\log Q_{b,j,s}( x,y)\mathrm{d}x\mathrm{d}y\] (49) \[=-\iint\sum_{s=1}^{2}\frac{1}{2}Q_{b,j,s}(x,y)\log\Big{[}\frac{ \sum_{t=1}^{2}\frac{1}{2}Q_{b,j,t}(x,y)}{Q_{b,j,s}(x,y)}\Big{]}\mathrm{d}x \mathrm{d}y, \tag{50}\]  
with  
\[Q_{b,j,1}(x,y) =\frac{1}{2\pi\sigma_{b}^{2}}\Bigg{[}\exp\bigg{(}-\frac{(x+B\beta \cos(\Delta_{b,w}+\theta_{j}))^{2}}{2\sigma_{b,x}^{2}}-\frac{(y+B\beta\sin( \Delta_{b,w}+\theta_{j}))^{2}}{2\sigma_{b,y}^{2}}\bigg{)}\Bigg{]}, \tag{51}\] \[Q_{b,j,2}(x,y) =\frac{1}{2\pi\sigma_{b}^{2}}\Bigg{[}\exp\bigg{(}-\frac{(x-B\beta \cos(\Delta_{b,w}+\theta_{j}))^{2}}{2\sigma_{b,x}^{2}}-\frac{(y-B\beta\sin( \Delta_{b,w}+\theta_{j}))^{2}}{2\sigma_{b,y}^{2}}\bigg{)}\Bigg{]}, \tag{52}\]  
where \(\sigma_{b,x}^{2}\) and \(\sigma_{b,y}^{2}\) represent noise variances at Bob on the \(x\)-axis and \(y\)-axis, and \(\Delta_{b,w}\) is the phase difference value between \(h_{RA}h_{AW}S_{R,i}\) and \(h_{RA}h_{AB}S_{R,j}\). Performing Taylor expansion, we have  
\[\log\Big{[}\frac{\sum_{i=1}^{2}\frac{1}{2}Q_{b,j,t}(x,y)}{Q_{b,j,1}(x, y)}\Big{]}=\beta\Big{(}\frac{B\text{cos}(\Delta_{b,w}+\theta_{j})x}{\sigma_{b,x}^{2}}+ \frac{B\text{sin}(\Delta_{b,w}+\theta_{j})y}{\sigma_{b,y}^{2}}\Big{)}\] \[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]0 \[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]1  
and  
\[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]2 \[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]3 \[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]4  
With some calculations, we can obtain  
\[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]5 \[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]6 \[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]7 \[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]8  
Combined with (46)-(56), the term \(\text{I}(Y_{b,j};c_{j})\) can be expressed as  
\[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{\epsilon}^{n})=\frac{n}{2}\mathbb{I}(Y_{b,1 };c_{1})+\frac{n}{2}\mathbb{I}(Y_{b,2};c_{2}). \tag{45}\]9  
Then, we can obtain the mutual information as  
\[\mathrm{I}(Y_{b,j};c_{j})=\mathrm{H}(Y_{b,j})-\mathrm{H}(Y_{b,j} |c_{j}) \tag{46}\]0