# AOA-Based Three-Dimensional Multi-Target Localization in Industrial WSNs for LOS Conditions
## 5 Simulation Results
### Multi-Target Localization  
In this subsection, we randomly deploy 10 MSNs and estimate their positions using ATML. The estimation errors in the \(x\), \(y\), and \(z\) coordinates are plotted in Figure 7.  
Figure 5: The network coordinate system and topology when there are \(K=10\) MSNs. Figure 6: AOA estimation results for various number of MSNs, SNR, and antenna array dimension. Figure 7a shows the coordinate estimation errors of the 10 MSNs when the antenna array dimension is \(4\times 4\) and the received SNR is 0 dB. The \(x\)-label is the index of the MSNs, from the 1 st to the 10 th MSN. It can be seen that the errors are mostly about 0.5 m and the maximum value is 1.5 m.  
In Figure 7b, the received SNR is set as 0 dB and the number of MSNs is \(K=40\). The antenna array dimension is set from \(2\times 2\) to \(8\times 8\). The plotted results are the average of the position estimation errors of all MSNs. To obtain the average result, we sum up the \(x\)-coordinate errors of all the \(K\) MSNs together and then divide the summation by \(K\). The average results of the \(y\) and \(z\)-coordinates are obtained in the same way. When the array dimension is \(2\times 2\), the large position estimation errors are caused by the large AOA estimation errors due to insufficient antenna numbers, as shown in Figure 6c. When the array dimension is greater than \(3\times 3\), we can obtain accurate estimation results. In Figure 7c, the number of MSNs varies from \(K=10\) to \(70\) with a step size of 10. The received SNR is set as 0 dB and the array dimension is \(4\times 4\). The results are the average of the position estimation errors of the 40 MSNs. The estimation errors increase with the number of MSNs when \(K\) is larger than 40 due to severe inter-MSN interference. For a large number of MSNs to locate, we can use larger antenna arrays on ANs (to increase \(M\)) or longer PN-sequences with more chips (to increase \(N\)). When the SINR is increased, more MSNs can be located simultaneously and accurately. Furthermore, the square root errors (RMSs) of the MSN positions are plotted in Figure 8.  
We performed simulations to compare the proposed ATML scheme with other algorithms in the literature. In [28], the localization scheme, AOA Localization with RSSI Differences of Directional Antennas (ALRD), was presented. ALRD fits RSSI values into a parabola function of AOA between \(0^{\circ}\) and \(90^{\circ}\) by quadratic regression analysis. Furthermore, the paper proposes two methods, maximum-point-minimum-diameter (MPMD) and maximum-point-minimum-rectangle (MPMR), to reduce ALRD localization errors by gathering more beacon signals. Figure 8: Multi-target position estimation RMSs for various SNR and antenna array dimension. Figure 7: Multi-target position estimation for various numbers of MSNs, SNR, and antenna array dimension. According to the scenario described in [28], a total of 81 MSNs and two ANs are distributed in a plane of \(100\times 100\) m\({}^{2}\). The MSNs are organized in a \(9\times 9\) matrix with the intervals of 10 m. The position coordinates of the two ANs are \((0,0)\) and \((0,10)\) m. The deployment of the network is shown in Figure 9a. The SNR is set as 0 dB and the antenna array dimension of the ANs is \(4\times 4\). Following the same approach as in [28], we also take the average of the localization results of 10 simulation trails. The cumulative distribution functions (CDFs) of the MSN position errors by the four schemes are plotted in Figure 9b. We can see in Figure 9b that, for very small localization error (such as smaller than 0.7 m), the ALRD-MPMD and ALRD-MPMR have higher probability (better performance) than ATML. However, for the localization error between 0.7 and 1.25 m, the probability of ATML is much larger than the other methods. The localization errors of almost all nodes are smaller than 1.25 m by ATML, but the errors of about 25% nodes are larger than 1.25 m by the other methods.