# Optimization of Intelligent-Reflecting-Surface-Enabled Covert Communication in Multi-Sensor Systems
## 7 Numerical Results  
In this section, we first consider scenarios characterized by noise symmetry along the \(x\) and \(y\) axes. Leveraging theoretical results from the referenced paper [8], we validate the efficacy of our proposed KL divergence and mutual information approximation methods. After establishing the accuracy of our approximation results, we proceed to validate the algorithm in two distinct scenarios utilizing our proposed approximation methods. We assess whether the gradient descent algorithm, as articulated, successfully converges to optimal angles \(\theta_{1}^{*}\) and \(\theta_{2}^{*}\). Finally, through a comprehensive exploration of all angles in the target scenarios, we confirm the efficacy and superiority of our proposed algorithm. Figure 3 illustrates the simulation results for scenario 1, with specific parameters detailed in Table 1. As depicted in Figure 3, under common conditions of noise symmetry along the \(x\) and \(y\) axes, the mutual information remains constant at 40 bit/Hz/s as the phase angles \(\theta_{1}\) and \(\theta_{2}\) vary. This observation aligns with the theorem in [8], indicating that altering the initial phase angles does not affect the transmission rate when the variances of noise along the \(x\)-axes and \(y\)-axes are equal. This result validates the correctness and effectiveness of our approximation for mutual information. Figure 3,c provide the main view and top view of the KL divergence concerning the variation of phase angles \(\theta_{1}\) and \(\theta_{2}\). As illustrated, the KL divergence reaches its maximum when \(\theta_{1}\) equals \(\theta_{2}\) (i.e., \(\theta_{1}=\theta_{2}\)), indicating the poorest covertness performance. Conversely, the KL divergence achieves its minimum when \(\theta_{1}\) and \(\theta_{2}\) have a 90-degree separation (i.e., \(\theta_{1}-\theta_{2}=90^{\circ}\)), representing optimal covertness performance. For example, the simulation reveals that the minimum KL divergence occurs at \(\theta_{1}=97^{\circ}\) and \(\theta_{2}=7^{\circ}\). This finding aligns with the conclusion in [8] that, under equal x-axis and y-axis noise variances, the optimal angle for 2-BPSK is 90 degrees. At \(\theta_{1}=\theta_{2}\), the KL divergence is \(6\times 10^{-4}\), while at \(\theta_{1}-\theta_{2}=90^{\circ}\), the KL divergence is \(3\times 10^{-4}\), showing a halving of the KL divergence, consistent with the results in [8]. Therefore, the simulation results validate the effectiveness and accuracy of our proposed approximation method, serving as a basis for further validation of our optimization strategy based on this approximation method. Figure 4 presents the results of covert communication performance versus amplitude gain \(\beta\) for scenarios 2 and 3, with specific simulation parameters and optimization outcomes detailed in Table 1. For performance comparison, we conducted experiments using phase angles adopted in [8; 10], specifically, \(\theta_{1}=45^{\circ}\), \(\theta_{2}=135^{\circ}\) and \(\theta_{1}=0^{\circ}\), \(\theta_{2}=90^{\circ}\). Figure 4a,b depict the KL divergence versus amplitude gain and the transmission rate versus amplitude gain in scenario 2. As shown, based on our proposed gradient descent algorithm, the optimal covertness performance can be achieved under the same amplitude at \(\theta_{1}^{*}=27^{\circ}\) and \(\theta_{2}^{*}=162^{\circ}\). However, at \(\theta_{1}^{*}=27^{\circ}\) and \(\theta_{2}^{*}=162^{\circ}\), the transmission rate is lower than the two comparative results. Figure 4c,d illustrate the KL divergence versus transmission amplitude and the transmission rate versus transmission amplitude in scenario 3. In scenario 3, the phase angle difference \(\Delta_{b,w}\) is 20 degrees, and based on our proposed gradient descent algorithm, optimal covertness performance can be achieved at \(\theta_{1}^{*}=25.2^{\circ}\) and \(\theta_{2}^{*}=176.4^{\circ}\). However, at \(\theta_{1}^{*}=25.2^{\circ}\), \(\theta_{2}^{*}=176.4^{\circ}\) or \(\theta_{1}^{*}=27.1^{\circ}\), \(\theta_{2}^{*}=0^{\circ}\)