# Joint Azimuth, Elevation and Delay Estimation for Single Base Station Localization in 3D IIoT +
## 4 Performance Evaluation
The slight upward tails of the J-2M method represent the performance limitation imposed by the grid constraints, which is effectively eliminated by the proposed J-M method. Figure 6 depicts the RMSE for the four candidate algorithms, with respect to AoAz, AoAe, ToA, and the overall location estimation under varying SNR conditions. To simulate the ground truth values of AoAz, AoAe, and ToA, we generate random variables distributed uniformly within a reasonable range, specifically, \(\theta\in[0,180]^{\circ}\), \(\phi\in[0,90]^{\circ}\), and \(\tau\in[3,300]\,\mathrm{ns}\). These ranges ensure that the MS is situated approximately 1 to 100 meters away from the BS. Additionally, the minimum gap between any two components is set as 15 ns and 15\({}^{\circ}\) (equal to three grid spacings), and any two paths closer than the minimum gap could be regarded as one cluster. Analyzing the sub-figures within Figure 6, we observe several common trends. As the SNR increases, all four algorithms consistently exhibit a decrease in RMSE, with the proposed J-MT method consistently outperforming the others. This highlights the superior performance of our proposed joint estimation approach. Moreover, the joint estimation methods (marked with 'J') consistently outperform the separate estimation methods (marked with 'S') in both angle and delay estimation. since the steering matrix adopted by the joint estimation takes advantage of both the antenna array and the sub-carrier frequency array, which is equivalent to an improvement in the antenna capability. Consequently, the joint estimation provided by the algorithms outperforms the separate estimation. However, it is noteworthy that the S-MT method does not exhibit significant superiority over the S-2M method. This limitation primarily arises from the initial MUSIC estimation. While the second-step Taylor enhancement can reduce estimation errors, its effectiveness is limited by the accuracy of the first step's results. If the initial estimations deviate significantly from the ground truth, the second step may not fully compensate for those discrepancies. Additionally, the estimation of AoAz consistently outperforms  
Figure 5: The comparison on the grid bias elimination between J-MT and J-2M [15] for (**a**) AoAz, (**b**) AoAe, and (**c**) ToA estimation. that of AoAe. This can be attributed to two reasons: Firstly, AoAz contributes more to the phase difference at different array elements of the UCA compared to AoAe; Secondly, the elevation angle spread only spans from 0 to 90 degrees, which is half the azimuth angle range, posing additional challenges to the algorithm resolution in our specific setup. Using the estimation results of Aoaz, AoAe, and ToA, along with the LOS judgment principle mentioned earlier, the location of the MS can be calculated. Consequently, the RMSE of the localization is depicted in Figure 6d for the four different algorithms, considering various SNR conditions. As anticipated, the accuracy of localization follows a similar trend to the accuracy of AoAz, AoAe, and ToA estimation. Our proposed algorithm consistently outperforms the other methods across different SNR conditions. The last part of the performance evaluation is the computational complexity analysis. We mainly focus on the two MUSIC-based JAEDE algorithms, J-MT and J-2M [15]. Let \(G\) denote the number of grid points. Specifically, \(G_{\theta}\), \(G_{\phi}\), \(G_{\tau}\) for the grid number of the coarse  
Figure 6: Comparison of the (**a**) AoAz, (**b**) AoAe, (**c**) ToA, and (**d**) the overall location estimation performance under various SNR for J-MT, J-2M [15], S-MT, and S-2M. estimation in the J-MT method; and \(G_{\theta 1},G_{\phi 1},G_{\tau 1},G_{\theta 2},G_{\phi 2}\) and \(G_{\tau 2}\) for the grid number of the two steps in the J-MM method.