# AOA-Based Three-Dimensional Multi-Target Localization in Industrial WSNs for LOS Conditions
## 5 Simulation Results
### Multi-Target AOA Estimation  
The AOA estimation for multiple targets is simulated for relative position determination. We consider an SIWSN comprised of two pre-positioned ANs and a varying number of MSNs. The MT-SIMO signals as described above are used. The positions of MSNs are randomly distributed in a cube of \(100\times 100\times 100\) m\({}^{3}\). For example, when the number of MSNs is \(K=10\), the network coordinate system and the topology are shown in Figure 5. The ANs are placed at the coordinates of \((0,0,0)\) and \((100,0,0)\) m. The two ANs should be placed by the two sides of the deployment area of MSNs to obtain the best performance. The simulation results for this typical scenario are presented in this subsection.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**MSN** & \(\mathbf{\tau_{l}}\) **(Chips)** & \(|\mathbf{\alpha_{l}}|^{2}\) & \(\mathbf{\phi_{l}}\) (\({}^{\circ}\)) & \(\mathbf{\theta_{l}}\) (\({}^{\circ}\)) & \(\mathbf{v_{l}}\) **(KHz)** \\ \hline \(\mathcal{M}_{1}\) & 1 & 20 & 30 & 40 & 1 \\ \(\mathcal{M}_{2}\) & 2 & 18 & 45 & 60 & 5 \\ \(\mathcal{M}_{3}\) & 5 & 14 & 85 & 75 & 2 \\ \(\mathcal{M}_{4}\) & 8 & 12 & 95 & 100 & 3 \\ \(\mathcal{M}_{5}\) & 12 & 8 & 120 & 130 & 8 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Propagation path parameters.  
Figure 4: Iterative ML estimation of azimuth and elevation AOAs of multiple propagation paths.  
Figure 3: Power delay profile and angular power spectra of the five MPCs between the MSN and AN.  
The ML estimation errors of the azimuth and elevation angles are plotted in Figure 6. First, Figure 6a shows the results when the number of MSNs, \(K\), increases from 10 to 70. The received SNR is set as 0 dB and \(4\times 4\) antenna arrays are employed on the ANs. The AOA estimation errors increase slightly because the mutual interference among the MSNs is more severe. However, we can see that the maximum error is about \(2^{\circ}\) even when there are \(K=70\) MSNs. This indicates that the ATML scheme is effective in positioning a large number of MSNs simultaneously in WSNs.  
Second, the estimation errors with varying SNR are plotted in Figure 6b, where the received SNR increases from \(-30\) to \(30\) dB with steps of 10 dB each. With each SNR, \(K=40\) MSNs are placed randomly and the antenna array dimensions are still \(4\times 4\). It is shown that the AOA estimation errors are mostly around \(0.5^{\circ}\). This indicates that the ATML scheme has good robustness and high accuracy for varying SNR. This is mainly due to, as mentioned earlier, exploiting the spread spectrum gain and antenna array gain.  
Third, the estimation errors using different sizes of antenna arrays on ANs are plotted in Figure 6c. The array dimension is changed from \(2\times 2\) (4 antenna elements) to \(8\times 8\) (64 antenna elements). The received SNR is set as 0 dB and number of MSNs is \(K=40\). We can see that the estimation errors are about \(8^{\circ}\) when the array dimension is \(2\times 2\). This is because the aperture antenna arrays is too small and there are not sufficient antenna elements in the azimuth and elevation dimensions. The ML method cannot provide accurate AOA estimation with insufficient information. However, the errors reduce quickly to only \(0.5^{\circ}\) when the array dimension is increased to \(3\times 3\). Therefore, ATML can work even using small antenna arrays on ANs, requiring low hardware complexity and cost.