# Joint Azimuth, Elevation and Delay Estimation for Single Base Station Localization in 3D IIoT +
## 3 A Three Step MUSIC-Taylor-Based Localization Approach
### Step 2, Taylor-Series Based Enhancement
\tag{17}\]  
The augmented matrix of (17) is given by  
\[\begin{bmatrix}\mathbf{A}^{\mathrm{H}}\mathbf{V}\mathbf{V}^{\mathrm{H}}\mathbf{A}&\mathbf{T}^{ \mathrm{H}}\mathbf{V}\mathbf{V}^{\mathrm{H}}\mathbf{A}&\mathbf{E}^{\mathrm{H}}\mathbf{V}\mathbf{V}^{ \mathrm{H}}\mathbf{A}&-\mathbf{s}^{\mathrm{H}}\mathbf{V}\mathbf{V}^{\mathrm{H}}\mathbf{A}\\ \mathbf{A}^{\mathrm{H}}\mathbf{V}\mathbf{V}^{\mathrm{H}}\mathbf{E}&\mathbf{E}^{\mathrm{H}}\bm\[\Delta\phi=\mathbb{R}\bigg{\{}\frac{(a_{31}a_{23}-a_{21}a_{33})(a_{21}a_{14}-a_{11}a _{24})-(a_{21}a_{13}-a_{11}a_{23})(a_{31}a_{24}-a_{21}a_{34})}{(a_{31}a_{23}-a_{21 }a_{33})(a_{21}a_{13}-a_{11}a_{23})-(a_{21}a_{13}-a_{11}a_{23})(a_{31}a_{22}-a_{ 21}a_{32})}\bigg{\}}, \tag{21}\]  
\[\Delta\tau=\mathbb{R}\bigg{\{}\frac{(a_{22}a_{31}-a_{32}a_{21})(a_{22}a_{14}-a _{12}a_{24})-(a_{22}a_{11}-a_{12}a_{21})(a_{22}a_{34}-a_{32}a_{24})}{(a_{22}a_{ 31}-a_{32}a_{21})(a_{22}a_{13}-a_{12}a_{23})-(a_{22}a_{11}-a_{12}a_{21})(a_{22} a_{33}-a_{32}a_{23})}\bigg{\}}. \tag{22}\]  
After obtaining the values of \(\Delta\theta\), \(\Delta\phi\), and \(\Delta\tau\), these values can be used to calculate the new pair of estimated points in the next iteration. Let the iteration begin with the point \(\hat{\mathbf{p}}_{0}\). Assuming that \(\hat{\mathbf{p}}_{\text{t}}\) represents the current state, the current steering function can be obtained by replacing \(\hat{\mathbf{p}}_{0}\) with \(\hat{\mathbf{p}}_{\text{t}}\) in Equation (12). Consequently, \(\Delta\theta_{\text{t}}\), \(\Delta\phi_{\text{t}}\), and \(\Delta\tau_{\text{t}}\) can be calculated step-by-step. Therefore, the estimation of the next state is given by:  
\[ \begin{split}\tilde{\mathbf{\theta}}_{\text{t+1}}&= \tilde{\mathbf{\theta}}_{\text{t}}+\Delta\theta_{\text{t}},\\ \tilde{\mathbf{\phi}}_{\text{t+1}}&=\tilde{\mathbf{\theta}}_ {\text{t}}+\Delta\phi_{\text{t}},\\ \tilde{\mathbf{\tau}}_{\text{t+1}}&=\tilde{\mathbf{\tau}}_{ \text{t}}+\Delta\tau_{\text{t}}.\end{split}  \tag{23}\]  
Figure 3 illustrates a simplified demonstration of the enhancement process for grid bias reduction through iterations. The ball represents the ground truth position of the target MS. The initial coarse estimation falls on the grid point at coordinates \([36,44,64]\), where the grid constraints (i.e., the distance between the coarse estimation and the ground truth) are observable. The proposed Taylor-series-based enhancement method is then applied, which effectively eliminates the mis-distance through iterations. Please note that the figure provides a condensed view of the iterative process, and the actual simulation may involve denser iteration results. If the maximum number of iterations reaches \(N_{\text{T}}\) or \(\Delta_{\text{t+1}}\) is less than a predefined threshold \(\Delta_{\text{th}}\), the iteration will come to the end. The summarized implementation of the Taylor-series-based refinement can be found in lines 8 to 20 of Algorithm 1. Figure 3: A demo of the Step 1 coarse estimation, and Step 2 enhancement process.