# Angle-of-Arrival Estimation Using Difference Beams in Localized Hybrid Arrays
## 4 Simulation Results  
In this section, we present the simulation results to evaluate the proposed approach, compared with the state of the art [5]. Denote the average received SNR per antenna as \(\gamma_{a}\). The training symbols, \(\tilde{s}(t)\), are generated following complex Gaussian distributions with zero mean. Assuming uniformly distributed AoA within \([-\pi,\pi]\), simulation results are obtained by averaging over 50,000 trials. Here, we define \(P_{d}\) as the probability of correctly finding the index \(m^{\prime}\) at Step 6 of Algorithm 1.  
Figure 3 compares the MSEs of \(e^{i\widetilde{Nu}}\) versus \(\gamma_{a}\) with \(Q=1\) and 2. As shown in the figure, the proposed phase shift design outperforms that of [5] in terms of MSE of \(e^{i\widetilde{Nu}}\), since a higher SNR for \(\widetilde{Nu}\) can be achieved at Step 9 of Algorithm 1. The MSE curve of \(e^{i\widetilde{Nu}}\) becomes increasingly tight to its asymptotic lower bound with the increase of \(\gamma_{a}\), where the asymptotic lower bound is the lower bound of proposed approach produced under the assumption of \(P_{d}=1\). There is more gain with \(Q=2\) than \(Q=1\) in comparison with [5], which indicates that the proposed scheme is more applicable to narrow beams, i.e., large \(N\). This is because multiple narrower single beams cannot provide desirable AoA coverage, resulting in estimation performance loss, which, however, can be compensated by our phase shift design.  
Figure 4 shows \(P_{d}\) versus \(\gamma_{a}\) with different values of \(Q\). We can see that the proposed scheme generally has better performance. At high SNRs, it achieves higher \(P_{d}\) with \(Q=1\), while it is inferior to that of [5] when SNR is low. Note that, compared with that in [5], our proposed phase shift design improves the capability of identifying the correct \(m^{\prime}\), thus effectively suppressing the noise and indirectly improving the SNR of estimation. When  
Figure 3: MSE of \(e^{i\widetilde{Nu}}\) versus \(\gamma_{a}\) (\(M=K=8\) and \(T=16\)).  
\(\gamma_{a}\) is greater than 5 dB, the proposed one leads to a higher \(P_{d}\) with a smaller \(Q\). This is because, when the number of beams is fixed within one symbol period, a smaller \(N\), and hence a wider beam, leads to a better coverage of the directions of interest. Therefore, it is easier to find the correct \(m^{\prime}\).  
The MSEs of \(\hat{u}\) are shown as a function of \(Q\) in Figure 5. It can be seen that its MSEs generally increase with \(Q\) attributed to the decreasing number of subarrays. The proposed approach generally achieves better performance than [5] when \(Q\) is an even number. When \(Q\) is odd, the proposed method results in larger estimation errors since the signals are only averaged over \(K-2\) product terms (see Step 22 of Algorithm 2), less than \(K-1\) in [5]. The corresponding asymptotic lower bound of MSEs of \(\hat{u}\) are displayed for comparison. To evaluate the credibility of estimation errors, we calculate 95% confidence intervals (CIs) for \(\hat{u}\). When \(\gamma_{a}=5\) dB and \(Q=2\), 4 and 6, the CIs are given by [-0.0207, 0.0111], [-0.0204, 0.0114] and [-0.0244, 0.0075] for the proposed one, [-0.0219, 0.0099], [-0.0258, 0.0060] and [-0.0224, 0.0096] for [5], [-0.0205, 0.0113], [-0.0232, 0.0086] and [-0.0213, 0.0104] for the asymptotic lower bound, respectively. The MSEs of \(\hat{u}\) in Rician fading channels [12] are also provided to show the impact of multipath channels on the proposed approach, where the Rician factor is assumed to be 10 dB. Figure 6 presents the MSE of \(\hat{u}\) versus \(\gamma_{a}\) with even values of \(Q\). From the figure, we can see that the proposed approach outperforms [5] by 1.4 dB at the MSE of 0.1, 0.4 dB at the MSE of 0.01 and 0.7 dB at the MSE of 0.001, respectively, when \(Q=6\), 4 and 2.  
Figure 4: \(P_{d}\) versus \(\gamma_{a}\) (\(M=K=8\) and \(T=16\)).