# Optimization of Intelligent-Reflecting-Surface-Enabled Covert Communication in Multi-Sensor Systems
## 7 Numerical Results
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Scenario** & \(A\) & \(\sigma_{w,x}\) & \(\sigma_{w,y}\) & \(B\) & \(\sigma_{b,x}\) & \(\sigma_{b,y}\) & \(\Delta_{b,w}\) & \(n\) & \(\theta_{1}^{*}\) & \(\theta_{2}^{*}\) \\ \hline
1 & 0.7 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0.5 & 0\({}^{\circ}\) & 100,000 & null & null \\
2 & 0.5 & 0.7 & 0.3 & 0.6 & 0.7 & 0.3 & 0\({}^{\circ}\) & 100,000 & 27\({}^{\circ}\) & 162\({}^{\circ}\) \\
3 & 0.5 & 0.7 & 0.3 & 0.6 & 0.7 & 0.3 & 20\({}^{\circ}\) & 100,000 & 25.2\({}^{\circ}\) & 176.4\({}^{\circ}\) \\  & & & & & & & & & & 27.1\({}^{\circ}\) & 0\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for comparative experiments. Figure 3: Covert communication performance in scenario 1. **(a)** Mutual information versus phase angles. **(b)** Main view of KL divergence versus phase angles. **(c)** Top view of KL divergence versus phase angles. the transmission rate is lower than the two comparative results. To further compare the effectiveness of the optimization results, a re-evaluation is needed under the same covert communication constraint \(\epsilon\). Figure 5 illustrates the covert transmission rate versus the covert constraint \(\epsilon\) in scenarios 2 and 3. As depicted in Figure 5a, in scenario 1, leveraging our proposed gradient descent algorithm, the covert transmission rate is maximized at \(\theta_{1}^{*}=27^{\circ}\) and \(\theta_{2}^{*}=162^{\circ}\) under the same covert constraint \(\epsilon\). In scenario 2, although the transmission rate is not maximized at \(\theta_{1}^{*}=27^{\circ}\) and \(\theta_{2}^{*}=162^{\circ}\) for the same transmission amplitude, considering the covert constraint, this phase angle pair is optimal. For Scenario 3, our algorithm yields two optimal phase configurations: \(\theta_{1}^{*}=25.2^{\circ}\), \(\theta_{2}^{*}=176.4^{\circ}\) and \(\theta_{1}^{*}=27.1^{\circ}\), \(\theta_{2}^{*}=0^{\circ}\). As shown in Figure 5b, both optimal configurations yield the same covert transmission rate under the same covert constraint, confirming their optimality in scenario 3. To further validate the effectiveness of the algorithm, we exhaustively traverse all phase angle options to verify if the algorithm obtains all optimal results. Figure 4: Covert communication performance in different scenarios. (**a**) KL divergence versus amplitude gain in scenario 2. (**b**) Transmission rate versus amplitude gain in scenario 2. (**c**) KL divergence versus amplitude gain in scenario 3. (**d**) Transmission rate versus amplitude gain in scenario 3. Figure 6a,b present the main and top views of the maximum covert transmission rate versus the phase angles \(\theta_{1}\) and \(\theta_{2}\) variations in scenario 2. In Figure 6, it is evident that there are two options for achieving the maximum covert transmission rate. Considering the symmetry of phase angles, our proposed algorithm has successfully identified all optimal phase angle options, validating the convergence and effectiveness of the algorithm. When employing conventional phase angles for covert communication, namely, at the moments when \(\theta_{1}=45^{\circ}\) and \(\theta_{2}=135^{\circ}\), the covert transmission rate is 35 bits/Hz/s. However, after optimization, the maximum covert rate reaches 45 bits/Hz/s, representing a nearly 30% increase in the transmission rate, which is a substantial improvement. Figure 7a,b depict the main and top views of the maximum covert transmission rate versus the phase angles \(\theta_{1}\) and \(\theta_{2}\) variations in scenario 3. It can be observed that there are four options for achieving the maximum covert transmission rate, reduced to two when considering the symmetry of phase angles. Through multiple random initializations, the proposed algorithm effectively converges and identifies all optimal phase angle pairs. The validity and convergence of the proposed algorithm have been thoroughly validated through the numerical simulations mentioned above.