# Joint Azimuth, Elevation and Delay Estimation for Single Base Station Localization in 3D IIoT +
## 3 A Three Step MUSIC-Taylor-Based Localization Approach
### Step 1, MUSIC-Based Coarse JAEDE  
The covariance matrix, \(\mathbf{C}(\theta,\phi,\tau)\), of the received signal, \(\mathbf{R}(\theta,\phi,\tau)\), in (4) is given by  
\[\mathbf{C}(\theta,\phi,\tau)=\mathbb{E}\{\mathbf{R}(\theta,\phi,\tau)\mathbf{R}(\theta, \phi,\tau)^{\rm H}\}=\mathbf{H}\mathbb{E}\{\mathbf{S}(\theta,\phi,\tau)S(\theta,\phi, \tau)^{\rm H}\}\mathbf{H}^{\rm H}+\mathbb{E}\{\mathbf{W}\mathbf{W}^{\rm H}\} \tag{7}\]  
Given that the noise is assumed to be a white Gaussian process with zero mean and variance \(\sigma^{2}\), i.e., \(\mathbb{E}\{\mathbf{W}\mathbf{W}^{\rm H}\}=\sigma^{2}\mathbf{I}\), and that it is independent among both the array elements and the signal, we can simplify the covariance matrix as follows:  
\[\mathbf{C}(\theta,\phi,\tau)=\mathbf{H}\mathbf{C}_{S}(\theta,\phi,\tau)\mathbf{H}^{\rm H}+\mathbf{ C}_{W} \tag{8}\]  
where \(\mathbf{C}_{S}(\theta,\phi,\tau)\) and \(\mathbf{C}_{W}\) are the covariance matrix of the signal and the noise, respectively. Additionally, let \(\mathbf{v}_{S}\) and \(\mathbf{v}_{W}\) represent diagonal matrices consisting of the eigenvalues of \(\mathbf{C}_{S}(\tau,\theta)\) and \(\mathbf{C}_{W}\), respectively. Given this, the covariance matrix can be decomposed as follows:  
\[\mathbf{C}(\theta,\phi,\tau)=\mathbf{V}_{S}\mathbf{v}_{S}\mathbf{V}_{S}^{\rm H}+\mathbf{V}\mathbf{v}_{ W}\mathbf{V}_{W}^{\rm H} \tag{9}\]  
where, \(\mathbf{V}_{S}\) and \(\mathbf{V}_{W}\) denote the signal and noise subspaces, respectively. It is important to note that \(\mathbf{v}_{S}\) contains the top \(N_{\mathbb{L}}\) eigenvalues, while the remaining eigenvalues are included in \(\mathbf{v}_{W}\). Since MUSIC-based approaches concentrate solely on the noise subspace,in the subsequent discussion, \(\mathbf{V}\) refers to \(\mathbf{V}_{W}\) unless specified otherwise. Consequently, the spectral function can be formulated as follows:  
\[\mathbf{\hat{f}}(\theta,\phi,\tau)=\frac{1}{S(\theta,\phi,\tau)^{\mathrm{H}}\mathbf{V} \mathbf{V}^{\mathrm{H}}S(\theta,\phi,\tau)} \tag{10}\]  
Since these two subspaces are orthogonal, the JAEDE results can be determined by the following object function,  
\[[\mathbf{\tilde{\theta}},\mathbf{\tilde{\phi}},\mathbf{\tilde{\tau}}]=\underset{\theta, \phi,\tau}{\arg\max}\mathbf{\hat{f}}(\theta,\phi,\tau), \tag{11}\]  
where the vector of the estimated AoAz, AoAe and ToA correspond to the \(N_{\mathrm{L}}-1\) multipath components, and formed by \(\mathbf{\tilde{\theta}}=[\tilde{\theta}_{0},\tilde{\theta}_{1},\ldots,\tilde{ \theta}_{N_{\mathrm{L}}-1}]\), \(\mathbf{\tilde{\phi}}=[\tilde{\phi}_{0},\tilde{\phi}_{1},\ldots,\tilde{\phi}_{N_ {\mathrm{L}}-1}]\), and \(\mathbf{\tilde{\tau}}=[\tilde{\tau}_{0},\tilde{\tau}_{1},\ldots,\tilde{\tau}_{N_ {\mathrm{L}}-1}]\). One approach commonly used for obtaining the pseudo-spectrum \(\mathfrak{f}(\theta,\phi,\tau)\) is grid searching. This involves constructing a three-dimensional mesh grid, \([\mathbf{\hat{\theta}},\mathbf{\hat{\phi}},\mathbf{\tilde{\tau}}]\), which divides the predefined range of AoAz (\([\mathbf{\hat{\theta}}_{\texttt{min}},\mathbf{\hat{\theta}}_{\texttt{max}}]\)), AoAe (\([\mathbf{\hat{\tau}}_{\texttt{min}},\mathbf{\hat{\tau}}_{\texttt{max}}]\)), and ToA (\([\mathbf{\hat{\theta}}_{\texttt{min}},\mathbf{\hat{\theta}}_{\texttt{max}}]\)) with equal spacing. The accuracy and complexity of the estimation depend on the number of grid points and the spacing between them. A smaller number of grids with larger spacing results in lower complexity but may lead to poorer accuracy. Therefore, in the initial stage, a coarse estimation is performed for computational efficiency. Later, an enhancement approach based on the Taylor-series is introduced to improve the estimation accuracy while maintaining low complexity. A summarized step-by-step implementation of Step 1, MUSIC-based coarse JAEDE is shown in Line 1 to 7, Algorithm 1. ```
1:Initialize \(N_{\mathrm{A}},\Delta_{\mathrm{A}},N_{\mathrm{K}},N_{\mathrm{T}},\mathbf{P}_{ \mathrm{b}}\).