# Optimization of Intelligent-Reflecting-Surface-Enabled Covert Communication in Multi-Sensor Systems
## 6 Problem Reformulation  
In Sections 4 and 5, we derive KL divergence and mutual information in our problem. Combined with (17) and (43), the covertness constraint can be expressed as  
\[\frac{A^{4}\beta^{4}n}{16\sigma_{w,x}^{4}\sigma_{w,y}^{4}}\mathcal{G}(\theta_{ 1},\theta_{2})\leq\epsilon, \tag{59}\]  
where\[\mathcal{G}(\theta_{1},\theta_{2}) =\sigma_{w,x}^{4}\cos(\theta_{1})^{4}-4\sigma_{w,x}^{4}\cos(\theta_{ 2})^{2}-4\sigma_{w,x}^{4}\cos(\theta_{1})^{2}+\sigma_{w,x}^{4}\cos(\theta_{2})^ {4}\] \[\quad+\sigma_{w,y}^{4}\cos(\theta_{1})^{4}+\sigma_{w,y}^{4}\cos( \theta_{2})^{4}+4\sigma_{w_{x}}^{4}+2\sigma_{w_{x}}^{2}\sigma_{w_{y}}^{2}\cos (\theta_{1})^{2}+2\sigma_{w_{x}}^{2}\sigma_{w_{y}}^{2}\cos(\theta_{2})^{2}\] \[\quad-2\sigma_{w_{x}}^{2}\sigma_{w_{y}}^{2}\cos(\theta_{1})^{4}-2 \sigma_{w_{x}}^{2}\sigma_{w_{y}}^{2}\cos(\theta_{2})^{4}+2\sigma_{w_{x}}^{4} \cos(\theta_{1})^{2}\cos(\theta_{2})^{2}\] \[\quad+2\sigma_{w_{y}}^{4}\cos(\theta_{1})^{2}\cos(\theta_{2})^{2} +4\sigma_{w_{x}}^{2}\sigma_{w_{y}}^{2}\cos(\theta_{1})\cos(\theta_{2})\sin( \theta_{1})\sin(\theta_{2}). \tag{60}\]  
Then, we can obtain the maximum amplitude gain as  
\[\beta=\frac{2\sigma_{w,x}\sigma_{w,y}}{A}\left(\frac{e}{n\mathcal{G}(\theta_{ 1},\theta_{2})}\right)^{\frac{1}{4}}. \tag{61}\]  
Combined with (57) and (58), the mutual information (16) can be expressed as  
\[\mathbb{I}(\mathbf{Y}_{b}^{n};\mathbf{c}^{n})=\frac{\sigma_{w,x}^{2}\sigma_{ w,y}^{2}\mathcal{T}(\theta_{1},\theta_{2})}{A^{2}}\left(\frac{n\epsilon}{ \mathcal{G}(\theta_{1},\theta_{2})}\right)^{\frac{1}{2}}, \tag{62}\]  
where  
\[\mathcal{T}(\theta_{1},\theta_{2})=\sum_{j=1}^{2}\bigg{(}\frac{B^{2}\text{cos }(\Delta_{b,w}+\theta_{j})^{2}}{\sigma_{b,x}^{2}}+\frac{B^{2}\text{sin}( \Delta_{b,w}+\theta_{j})^{2}}{\sigma_{b,y}^{2}}\bigg{)}. \tag{63}\]  
Combined with (59)-(63), we can reformulate the problem as  
\[\text{P2}:\quad\max_{\{\theta\}} \frac{\sigma_{w,x}^{2}\sigma_{w,y}^{2}\mathcal{T}(\theta_{1}, \theta_{2})}{A^{2}}\left(\frac{n\epsilon}{\mathcal{G}(\theta_{1},\theta_{2}) }\right)^{\frac{1}{2}}\] (64) s.t. \[0\leq\theta_{1}\leq\pi, \tag{65}\] \[\mathcal{G}(\theta_{1},\theta_{2}) =\sigma_{w,x}^{4}\cos(\theta_{1})^{4}-4\sigma_{w,x}^{4}\cos(\theta_{ 2})^{2}-4\sigma_{w,x}^{4}\cos(\theta_{1})^{2}+\sigma_{w,x}^{4}\cos(\theta_{2})^ {4}\]0  
Then, we define the objective function as \(\mathcal{R}(\theta_{1},\theta_{2})\triangleq\frac{\sigma_{w,x}^{2}\sigma_{w,y }^{2}\mathcal{T}(\theta_{1},\theta_{2})}{A^{2}}\Big{(}\frac{n\epsilon}{ \mathcal{G}(\theta_{1},\theta_{2})}\Big{)}^{\frac{1}{2}}\). To solve the problem (64), we adopted a gradient descent optimization algorithm with random initialization; the details are summarized in Algorithm 1. In Algorithm 1, the final output \(\mathbb{R}^{t+1}\) represents the ultimate transmission rate, while the outputs \(\theta_{1}^{t+1}\) and \(\theta_{2}^{t+1}\) denote the optimal values for the phase angles. Combined with the outputs \(\theta_{1}^{t+1}\) and \(\theta_{2}^{t+1}\) and (61), we can obtain the optimal amplitude gain as \(\beta^{*}=\frac{2\sigma_{w,x}\sigma_{w,y}}{A}\Big{(}\frac{\epsilon}{n\mathcal{ G}(\theta_{1}^{t+1}\theta_{2}^{t+1})}\Big{)}^{\frac{1}{4}}\).