# Joint Azimuth, Elevation and Delay Estimation for Single Base Station Localization in 3D IIoT +
## 2 System Model  
The JAEDE localization method introduces the system model as the basis of our approach. As depicted in Figure 2, the model consists of a mobile station (MS) located in a 3D Cartesian coordinate system at an unknown position denoted as \(\mathbf{P_{\rm{B}}}=[x_{\rm{m}},y_{\rm{m}},z_{\rm{n}}]\). Communication between the MS and the BS involves a set of UCA positioned at a known location marked as \(\mathbf{P_{\rm{b}}}=[x_{\rm{b}},y_{\rm{b}},z_{\rm{b}}]\). The main objective of the BS is to estimate the AoAz, AoAe, and ToA of the modulated signals transmitted from the MS, thereby enabling estimation of the MS's coordinates. In accordance with our proposed approach, we consider the MS to be positioned at a far-field distance from the BS to maintain the generality of our framework. In the analysis, we account for \(N_{\rm L}\) components contributing to multipath propagation. These components consist of a single line-of-sight (LOS) path with a length of \(l_{0}\), as well as \(N_{\rm L}-1\) NLOS paths, namely \(l_{1},l_{2},\ldots,l_{N_{\rm L}-1}\). The i-th path, where \(\texttt{i}=0,1,\ldots,N_{\rm L}-1\), is characterized by its AoAz, AoAe, and ToA represented by \(\theta_{\texttt{i}}\), \(\phi_{\texttt{i}}\), and \(\tau_{\texttt{i}}\), respectively. Notably, we express \(l_{\texttt{i}}=v_{\texttt{c}}\tau_{\texttt{i}}\), where \(v_{\texttt{c}}=3\times 10^{8}\)m/s signifies the speed of light. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Ref.** & **Scenario** & **Technique** & **Description** \\ \hline
[12] & 2D & ESPRIT-based JADE & Low computational cost, but parameter \\  & & & & paring introduces extra errors \\ \hline
[14] & 2D & MUSIC-based JADE & Low complexity, but sensitive to low SNR and \\  & & & & sampling rate \\ \hline
[15] & 2D & MUSIC-based JADE & Two-step algorithm, but computationally \\  & & & & exhaustive in the fine grid searching \\ \hline
[18] & 2D & MUSIC-based JADE & Our previous work with two-step \\  & & & MUSIC-Taylor algorithm, but utilizing ULA \\  & & & & for 2D scenario \\ \hline
[16; 17] & 3D & MUSIC-based AoA & Utilizing UCA for 3D scenario, but estimates \\  & & & AoA only \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of Related Works. Assuming the UCA consists of \(N_{\text{A}}\) identical sensor elements arranged equidistantly along a circle with a radius of \(r\) meters. The circle is positioned in the \(xy\)-plane of a local coordinate system, where the origin coincides with the center of the circle. In the coordinate system model represented in Figure 2, the azimuth angle, denoted as \(\theta\), is defined as the angle within the \(xy\)-plane, starting from the \(x\)-axis and pointing towards the \(y\)-axis. The elevation angle, denoted as \(\phi\), is defined as the angle starting from the \(xy\)-plane and moving towards the \(z\)-axis. The angular separation between any two adjacent elements within the array is \(\Delta_{\text{A}}=2\pi/N_{\text{A}}\). Consequently, the phase difference between the a-th element and the origin can be expressed as \(\Delta\Phi_{\text{a}}=\frac{2\pi r}{\lambda}\sin\phi_{\text{a}}\cos(\theta_{ \text{a}}-\text{a}\Delta_{\text{A}})\), where \(\text{a}=0,1,\ldots,N_{\text{A}}-1\), and \(\lambda\) represents the wavelength of the signal. Given a transmitted signal \(u(t)\), the received signal on the a-th element of the UCA can be expressed as  
\[r_{\text{a}}(t)=\sum_{\text{i}=0}^{N_{\text{A}}-1}g_{\text{i}}u_{\text{i}}(t- \tau_{\text{i,a}})+w_{\text{a}}(t), \tag{1}\]  
where \(g_{\text{i}}=|g_{\text{i}}|e^{i\Phi_{\text{i}}}\) represents the complex attenuation of the i-th multi-path component, with \(\Phi_{\text{i}}\in[0,2\pi]\) denoting the phase, and \(w_{\text{a}}(t)\) representing the additive white Gaussian noise (AWGN) on the a-th array element. In addition, we perform the discrete Fourier transform (DFT) of the signal and consider \(N_{\text{K}}\) sub-carriers with equal frequency increments.