## 5 Simulation Results  
In this section, we first verify the analytical expressions derived above through simulations. Then, the performance of the proposed beamforming scheme Algorithm 1 is simulated for different \(\theta_{e}\) and \(M\) values. The channel parameters are set as \(K_{b}=10\) dB and \(K_{e}=7\) dB and the SNR of the system is \(\text{SNR}=30\) dB for both [PERSON] and Eve. The SINR expectation expression in (29) is tested at first assuming that \(\theta_{b}=45^{\circ}\) and \(\theta_{e}=46^{\circ}\). Without beamforming optimization, we align the signal beam towards [PERSON] and the AN beam towards Eve, i.e., \(\mathbf{w}_{b}=\mathbf{w}(\theta_{b})\) and \(\mathbf{w}_{\text{AN}}=\mathbf{w}(\theta_{e})\). The expectation of \(\text{SINR}_{b}\) and \(\text{SINR}_{e}\) is simulated for different \(M\) and \(g_{\text{AN}}\) values and the results are presented in Figure 1. The theoretical values of the expectation of SINR are also displayed in lines. It can be seen that the theoretical curves perfectly match the simulation results. Additionally, due to the close proximity of [PERSON] and [PERSON], it can be observed that when the number of antennas is small, i.e., \(M\leq 16\), the performance of [PERSON] and [PERSON] is almost the same, regardless of whether AN is used. This is because the beam is wide for these cases. As the number of antennas increases, the beam becomes more focused, and even without using AN, the performance of [PERSON] and [PERSON] becomes distinguishable. When AN is added, the performance of both [PERSON] and [PERSON] decreases, but the performance of [PERSON] decreases more significantly. The simulation results show that, without beam optimization, a large number of antennas is required to differentiate the performance of [PERSON] and [PERSON]. Moreover, based on the above assumptions, the BER performance of [PERSON] is simulated for \(M=64\) and the results are presented in Figure 2 for different \(g_{\text{AN}}\) values. The approximate BERs using GL quadrature sum with \(N_{GL}=300\) are presented as continuous lines, which exhibit good accuracy compared to the simulation results. Moreover, it can be seen that as \(g_{\text{AN}}\) increases, the BER of [PERSON] increases rapidly. Then, we test the performance of the proposed Algorithm 1 in preventing eavesdropping with a small scale of antennas. The locations of [PERSON] and [PERSON] remain unchanged. We set \(\phi_{\phi}=1^{\circ}\), \(\delta_{g}=0.01\) and \(\hat{\gamma}_{b}=10\) dB. We test the performance of [PERSON] at different locations, such that \(\theta_{e}\in[46^{\circ},\cdots,50^{\circ}]\). Assuming \(M=8\) and \(N\in\{1,2,4,8\}\), the expectations of \(\text{SINR}_{b}\) and \(\text{SINR}_{e}\) are simulated and the results are presented in Figure 3. We can see that the performance of [PERSON] remains above the threshold \(\hat{\gamma}_{b}=10\) dB, while the performance of [PERSON] decreases when increasing the distance between [PERSON] and [PERSON]. Moreover, we note that when the number of antennas of [PERSON] increases, the SINR of [PERSON] actually decreases. This is because as the number of antennas \(N\) increases, the power of the desired signal and AN received by [PERSON] both increase proportionally, resulting in a decrease in overall SINR. To clarify this point, the expectation of the power of the desired signal and AN received byEve are displayed in Figure 4. It can be observed that as the number of receiving antennas doubles, the received power is also doubled. However, since the power of the desired signal and AN follow different distributions, the expected SINR changes with the number of receiving antennas. Figure 2: The simulated and approximate BERs of Eve for \(M=64\). The corresponding BERs are also simulated and the results are presented in Figure 5. It is shown that the BER of Bob fluctuates around \(10^{-2}\) and the BER of Eve increases when increasing the distance between Eve and Bob. When there is \(5^{\circ}\) difference between Eve and Bob, the BER of Eve reaches 0.37.