# The Optimization of Covert Communication in Asymmetric Jammer - Assist Systems
## 5 Numerical Results  
In this section, we initiate the validation of our proposed KL divergence and mutual information approximation methods, operating within two distinct scenarios that encompass diverse \(h_{w}\) and \(h_{b}\) channel coefficients, along with varying interferer total power levels. Following the confirmation of the precision of these approximation methods, we conduct a thorough performance analysis, assessing both covertness and transmission rate using our proposed methodology. Subsequently, we ascertain the optimal noise allocation ratio for interferers. Finally, through a comparative performance evaluation between the optimal interference noise allocation ratio and alternative ratios, we establish the efficacy and superiority of the optimization method we have introduced. In Figure 1(a), the graph illustrates the KL divergence versus amplitude gain (\(\beta\)). We compare two scenarios with distinct parameters: \(h_{w,x,0}=h_{w,y,0}=0.5\), \(P_{I}=3\) and \(h_{w,x,0}=h_{w,y,0}=0.3\), \(P_{I}=5\). As depicted in Figure 1(a), with the increase in amplitude gain (\(\beta\)), KL divergence gradually increases, indicating a deterioration in covertness. The Taylor expansion results obtained for KL divergence exhibit high fidelity with the precise values as \(\beta\) increases, affirming the precision and effectiveness of our proposed method in characterizing KL divergence in covert communication. Figure 1(b) presents the KL divergence versus amplitude gain (\(\beta\)) graph. We compare two scenarios with parameters \(h_{b,x,0}=h_{b,y,0}=0.5\), \(P_{I}=3\), \(\eta=0.3\) and \(h_{b,x,0}=h_{b,y,0}=0.2\), \(P_{I}=5\), \(\eta=0.3\). As illustrated in Figure 1(b), with the increase in amplitude gain (\(\beta\)), mutual information and transmission rate gradually increase. The Taylor expansion results for transmission rate exhibit high fidelity with the precise values as \(\beta\) increases, demonstrating the precision and effectiveness of our proposed method in characterizing transmission rate in covert communication. In Figures 3 and 4, we design three distinct scenarios for comparative experiments. The parameters for the three scenarios are provided in Table 1.  
In Figure 3a, Scenarios 1, 2, and 3 achieve optimal covertness (\(\alpha=0.48\), \(\alpha=0.58\), \(\alpha=0.45\), respectively) with minimal KL divergence values. In Figure 3b, Scenarios 1, 2, and 3 achieve maximum transmission rates (\(\alpha=1\), \(\alpha=0\), \(\alpha=0\), respectively). To obtain the globally optimal value for \(\alpha\) that maximizes transmission rate under the same amplitude gain and covertness constraints, we utilize our proposed method and obtain the results of transmission rate versus noise allocation ratio \(\alpha\) for the three scenarios. Figure 3: Performance analysis of covert communication in jammer\(-\)assist system. (**a**) KL divergence versus jammer noise ratio. (**b**) Transmission rate versus jammer noise ratio. Figure 2: Performance analysis of covert communication in jammer\(-\)assist system. (**a**) KL divergence versus amplitude gain. (**b**) Transmission rate versus amplitude gain. As depicted in Figure 4, the results show the transmission rate versus noise allocation ratio \(\alpha\) for \(\beta=0.1\). In Scenario 1, the maximum covert transmission rate is achieved at \(\alpha=0.78\); in Scenario 2, it is at \(\alpha=0.7\), and in Scenario 3, it is at \(\alpha=0.3\). Subsequently, based on the parameters of Scenario 3 and the obtained optimal noise allocation ratio \(\alpha^{*}=0.3\), we conduct a theoretical validation. Figure 4(a) presents the KL divergence versus amplitude gain (\(\beta\)) results. We compare the results with the globally optimal allocation ratio \(\alpha^{*}=0.3\) against samples with \(\alpha=0.5\) and \(\alpha=0.8\). As \(\beta\) increases, \(\alpha^{*}=0.3\) consistently maintains optimal covertness (lower KL divergence values). Figure 4(b) displays the mutual information versus amplitude gain (\(\beta\)) results.