# Joint Azimuth, Elevation and Delay Estimation for Single Base Station Localization in 3D IIoT +
## 4 Performance Evaluation  
In this section, the performance of the proposed localization algorithm is evaluated through Monte Carlo simulations, which take into account different SNR conditions and multipath activity patterns. The simulation parameters are detailed in Table 2, following a typical orthogonal frequency division multiplexing (OFDM) system setup. It is important to note that the coordinates of the MS should be calculated using the AoAZ, AoAe, and ToA of the LOS path per Monte Carlo realization. The performance is evaluated by the root mean square error (RMSE) metric averaged over multiple Monte Carlo realizations under different scenarios in terms of two major aspects, the accuracy of JAEDE estimation, and the mis-distance of the localization. For a given estimated result \(\tilde{\gamma}_{\mathbf{n}}\) under one realization and the corresponding ground truth \(\gamma\), with \(N_{\mathbf{M}}\) runs of Monte Carlo realizations, the RMSE is given by  
\[\mathfrak{E}_{\gamma}=\sqrt{\frac{\sum_{\mathbf{n}=1}^{N_{\mathbf{M}}}\mathbb{ D}\{\tilde{\gamma}_{\mathbf{n}},\gamma\}^{2}}{N_{\mathbf{M}}}} \tag{25}\]  
where the operator \(\mathbb{D}\{\tilde{\gamma}_{\mathbf{n}},\gamma\}=||\tilde{\gamma}_{\mathbf{n }}-\gamma||_{2}\) (the \(L-2\) norm) when the case is with regard to the location evaluation, and \(\mathbb{D}\{\tilde{\gamma}_{\mathbf{n}},\gamma\}=\tilde{\gamma}_{\mathbf{n}}-\gamma\) with the angle and delay estimation. The performance of the proposed JAEDE-based localization technique (referred to as _J-MT_) is further compared with three other techniques:  
1. _J-2M_[15]: A two-step MUSIC-based algorithm, with coarse-fine grid spacing. In the following simulations, we extend the original 2D JADE method to 3D JAEDE, and choose coarse, find grid spacing of \([\Delta\theta,\Delta\phi,\Delta\tau]\) as \([1^{\circ},1^{\circ},1\,\mathrm{ns}]\) and \([0.1^{\circ},0.1^{\circ},0.1\,\mathrm{ns}]\), respectively. 2. _S-MT_: Using a MUSIC-Taylor algorithm similar to J-MT, but the angles and delay are estimated separately. The grid spacing for the 1 st step MUSIC is the same as J-MT, \([5^{\circ},5^{\circ},5\,\mathrm{ns}]\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline The BS location & \(P_{\mathrm{B}}=[0,0,0]\,\mathrm{m}\) \\ The carrier frequency & \(f_{c}=2.4\,\mathrm{GHz}\) \\ The bandwidth & \(BW=20\,\mathrm{MHz}\) \\ The number of sub-carriers & \(N_{\mathrm{K}}=64\) \\ The sampling number & \(N_{\mathrm{SP}}=200\) \\ The number of UCA elements & \(N_{\mathbf{A}}=12\) \\ The UCA radius & \(r=\lambda\) \\ The number of multi-path components & \(N_{\mathrm{L}}=3\) \\ The number of Monte Carlo realizations & \(N_{M}=1000\) \\ The number of maximum iterations & \(N_{\mathrm{T}}=500\) \\ The breakout threshold & \(\phi_{\mathrm{th}}=\tau_{\mathrm{th}}=\theta_{\mathrm{th}}=10^{-5}\) \\ The 1 st step grid spacing & \(\Delta_{\theta}=\Delta\phi=5^{\circ}\) \\ \(\Delta_{\mathrm{r}}=5\mathrm{ns}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation Parameters. 3. _S-2M_[15]: A two-step MUSIC-based algorithm, similar to J-2M, but the angles and delay are estimated separately, following the same grid spacing as J-2M. In the first comparison, we evaluate the performance of the proposed J-MT algorithm and compare it with a two-step classic MUSIC-based JAEDE algorithm called J-2M. The objective is to demonstrate the improvement achieved by reducing the grid constraints in the J-MT algorithm. The ground truth values for the three paths are deliberately set off-grid. Specifically, the AoAz, AoAe and ToA are \(l_{0}=[45.411^{\circ},30.402^{\circ},10.637\,\mathrm{ns}]\), \(l_{1}=[-34.222^{\circ},12.511^{\circ},118.46\,\mathrm{ns}]\), and \(l_{2}=[78.538^{\circ},60.816^{\circ},195.86\,\mathrm{ns}]\). As shown in Figure 5, the proposed J-MT method exhibits a stable downward trend. In terms of AoAz and ToA, the J-MT method consistently performs better than the J-2M method. In the case of AoAe, the J-MT method gradually outperforms the J-2M method as the SNR exceeds 15 dB.