## 3 Proposed Strongest AoA Components Estimation Algorithm
### The Design of Quantization Tiresholds  
In this subsection, we detail the algorithm proposed for the strongest AoAs estimation with 1-bit A/Ds equipped at both transceivers. We also use the Monte Carlo simulations to statistically find appropriate quantization thresholds for different SNRs to reduce the estimation overhead. Then, the channel can be estimated with a relatively smaller number of beam searching repetitions for the case of limited hardware in the hybrid architecture with 1-bit A/Ds. For our proposed AoAs estimation algorithm, we firstly obtain the ordered detection probability output vector through periodic beam scanning at the BS after sorting in descending order as \(\left[\widehat{p}_{(1),n},\cdots,\widehat{p}_{(M+1),n}\right]\), where \(\widehat{p}_{(i),n}\geqslant\widehat{p}_{(i+1),n}\), \(\forall i\). Then we calculate the Wald-type confidence interval which will be used to determine when the beam scanning stops. The confidence interval can be represented as  
\[\left\{ \begin{aligned} &\widehat{p}_{(i),n}-\zeta_{(i),n}\leqslant \widehat{p}_{(i),n}\leqslant\widehat{p}_{(i),n}+\zeta_{(i),n},\,\forall i,\\ &\zeta_{(i),n}=\alpha\sqrt{\frac{\widehat{p}_{(i),n}\left(1- \widehat{p}_{(i),n}\right)}{L}},\end{aligned} \right. \tag{9}\]  
where \(\alpha>0\) is from the standard normal distribution table. In the iterative search process, a reward function \(\lambda\) is introduced in low SNRs to decide when to stop searching, which can be expressed as  
\[\left\{ \begin{aligned} &\widehat{p}_{(i),n}>\widehat{p}_{(i+1),n},& \lambda=\lambda+1,\\ &\widehat{p}_{(i),n}-\zeta_{(i),n}\geqslant\widehat{p}_{(i+1),n}+ \zeta_{(i+1),n},&\lambda=\lambda+1.\end{aligned} \right. \tag{10}\]  
After the two aforementioned conditional judgments, the number of detection repetitions increases, \(L=L+1\). Then, the proposed algorithm recalculate the corresponding \(\zeta_{(i),n}\) and update the parameter \(\lambda\). The beam search can be stopped when \(\lambda\) reaches to a pre-determined reward function threshold \(T\) (\(T>0\)). The beam with the largest \(\widehat{p}_{(i),n}\) upon termination will be selected as the best beam. This proposed algorithm is summarized in Algorithm 1. From Algorithm 1, we calculate the number of detection repetitions required for the estimation of strongest AoA components with a given quantization threshold \(\tau\) and SNR. In general, it is possible to obtain SNR via different methods [26]. From Equation (6), the quantization threshold, \(\tau\), is the only parameter during the 1-bit quantization. Though a higher quantization threshold may reduce the impact of environmental noise in low SNR regimes, it may lead to loss of useful-information since larger fractions of the received signals are quantized to 0. On the contrary, a lower quantization threshold may lead to an increase of false alarm probability (FAP) and a decrease in estimation accuracy. In order to obtain appropriate quantization thresholds to minimize the number of detection repetitions, \(L_{i}\), for different SNRs, we further introduce the SNR as a parameter in simulation to characterize the connection between \(\tau\) and \(L_{i}\). Our main idea is to perform statistical experiments through Monte Carlo simulation. The process for selecting specific quantization thresholds based on Monte Carlo simulation is demonstrated in Algorithm 2. With Algorithms 1 and 2, we can exhaustively search for an appropriate quantization threshold, \(\tau\), to minimize the required detection repetitions, \(L_{\text{min}}\), for a certain SNR via a large mount of digital simulations, which satisfies  
\[L_{\text{min}}\leqslant[L_{1},\cdots,L_{i},\cdots,L_{N}], \tag{11}\]  
where \(L_{i}\) is the number of detection repetitions when the quantization threshold is set as \(\tau_{i}\), \(i\in\{1,\ldots,N\}\). To the end, with aforementioned parameters and the minimum number of detection repetitions, we can obtain the strongest AoA estimation results, \(\hat{\theta}_{k,l}\).