## 4 Simulation Results and Discussions  
In this section, we verify the effectiveness of our proposed algorithms via simulation results. First, we demonstrate the relationship between the quantization threshold, \(\tau\), and the minimum required number of detection repetitions, \(L_{\text{min}}\), for different SNRs. In addition, we also illustrate the RMSE performance versus SNR with different detection repetitions for different scenarios. Without further clarification, the detailed parameters in simulations are set as follows: \(\text{K-factor}=10\), the number of RF-chains equipped at the BS is \(N_{\text{RF}}=N\), the number of users, \(N=5\), the number of ULA antennas, \(M=64\), the reward function threshold, \(T=3\), and the receive SNR before beamforming ranges from \(-25\) dB to \(-5\) dB. Figure 2 illustrates the number of detection repetitions \(L_{\text{min}}\) required for our proposed beamforming direction search algorithm versus different quantization thresholds \(\tau\) for \(M=64\) in low SNR regimes. We also set \(\alpha=1.96\) in Equation (9) to calculate the 95% confidence intervals as a trade-off between estimation accuracy and overhead. According to characteristics of mmWave communication systems, we choose \(\text{SNR}=-15\) dB as the conversion point, i.e., \(\text{SNR}\leqslant-15\) dB will be considered as low SNR regimes. In Figure 2, it can be seen that the the required \(L_{i}\) changes with different \(\tau_{i}\) for the strongest AoA components estimation. Thus, it is interesting to choose a proper \(\tau\) to minimize the number of required detection repetitions \(L_{\text{min}}\) via comprehensive Monte Carlo simulations, which are illustrated in Table 1 for different SNRs with \(M=64\) antennas. Meanwhile, according to the results presented in Table 1, we can note that both \(\tau\) and \(L_{\text{min}}\) decrease with an increasing SNR. Figure 3 illustrates the variation of \(\tau\) versus SNR for different numbers of antennas, e.g., \(M\in\{16,32,48,64\}\). We can observe that, \(\tau\) decreases from 0.98 to 0.50 in low SNR regimes. However, it stabilizes around 0.50 for medium-to-high SNR regimes, e.g., \(\tau\approx 0.50\). Furthermore, for a given receive SNR, the quantization threshold, \(\tau\), decreases with an increasing number of antennas M as extra beamforming gain can be provided. Figure 3: Quantization threshold, \(\tau\), versus SNR [dB] for different number of antennas. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline \multicolumn{12}{l}{**Case 1: Low SNR Regimes**} \\ \hline SNR [dB] & \(-25\) & \(-24\) & \(-23\) & \(-22\) & \(-21\) & \(-20\) & \(-19\) & \(-18\) & \(-17\) & \(-16\) & \(-15\) \\ \(\tau\) & 0.98 & 0.89 & 0.80 & 0.89 & 0.78 & 0.72 & 0.64 & 0.61 & 0.57 & 0.55 & 0.54 \\ \(L_{\min}\) & 10.40 & 10.36 & 10.34 & 9.97 & 9.78 & 9.49 & 9.05 & 8.52 & 7.89 & 7.21 & 6.51 \\ \hline \multicolumn{12}{l}{**Case 2: Medium-to-High SNR Regimes**} \\ \hline SNR [dB] & \(-14\) & \(-13\) & \(-12\) & \(-11\) & \(-10\) & \(-9\) & \(-8\) & \(-7\) & \(-6\) & \(-5\) \\ \(\tau\) & 0.50 & 0.50 & 0.50 & 0.50 & 0.50 & 0.50 & 0.50 & 0.50 & 0.50 & 0.50 \\ \(L_{\min}\) & 4.55 & 3.82 & 3.20 & 2.73 & 2.41 & 2.13 & 1.83 & 1.63 & 1.55 & 1.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The results of the Monte Carlo simulation for different SNRs. Figure 2: A demonstration for quantization threshold, \(\tau\), and detection repetitions, \(L\), versus SNR [dB] with \(M=64\). Based on simulation results given in Figure 3, we can further approximate the relationship between \(\tau\), SNR, and \(M\) as follows:  
\[\tau\approx\left\{ \begin{array}{ll}0.98,&\text{SNR}<-19-\frac{M}{8},\\ (-0.033-\frac{24}{125M})\cdot\text{SNR}+\frac{48}{25M},&-19-\frac{M}{8}\leqslant \text{SNR}\leqslant-7-\frac{M}{8},\\ 0.50,&\text{SNR}>-7-\frac{M}{8}.\end{array} \right. \tag{18}\]  
We note here, results given in Equation (18) will be exploited in the following simulations.