## 4 Unitary Reduced-Dimension Root MUSIC  
In the standard 2D MUSIC, the array observation of both x-axis and y-axis ULAs are combined in such a way that it is possible to estimate the azimuth and elevation angle of each signal source by performing a 2D search on the spatial spectrum. In contrast, the Unitary Reduced-Dimension (R-D) Root MUSIC computes the two ULAs separately by applying a root-finding method that does not require a 2D search on the spatial spectrum. Consequently, that method is substantially faster than standard 2D MUSIC [17]. Moreover, the Unitary transformation turns centro-hermitian matrices into real-valued ones which further decreases the execution time and reduces memory consumption. For the standard 2D MUSIC, let \(\mathbf{z}(t)\) be the combination of the array observation of the x-axis and y-axis in a timestamp \(t\) as defined in Equation (12),  
\[\mathbf{z}(t)= \begin{bmatrix}\mathbf{x}(t)\\ \mathbf{y}(t)\end{bmatrix} . \tag{12}\]  
After collecting \(N\) array observations, the method computes its sample covariance matrix expressed in Equation (13),  
\[\mathbf{R}_{zz}\approx\mathbf{\hat{R}}_{zz}=(\frac{1}{N})\mathbf{Z}\mathbf{Z }^{H}, \tag{13}\]  
such that \(\mathbf{Z}= \begin{bmatrix}\mathbf{z}(t_{1})&\mathbf{z}(t_{2})&\ldots&\mathbf{ z}(t_{N})\end{bmatrix} \in\mathbb{C}^{2M\times N}\). Under some assumptions [24], the covariance matrix, \(\mathbf{R}_{zz}\), is non-singular and its eigenvectors are divided into two subspaces: a noise subspace (\(\mathbf{U}_{N}\)) and a signal subspace (\(\mathbf{U}_{S}\)). More specifically, the signal subspace (\(\mathbf{U}_{S}\)) is composed of eigenvectors corresponding to the \(d\) largest eigenvalues. While the noise subspace (\(\mathbf{U}_{N}\)) is composed of eigenvectors corresponding to the \(N-d\) smallest eigenvalues. Ideally, the \(N-d\) smallest eigenvalues are zeros; however, due to AGWN, they are nonzeros. Moreover, the noise subspace is orthogonal to the combined steering vector as indicated by Equation (14),  
\[\mathbf{U}_{N}^{H} \begin{bmatrix}\mathbf{a}(\mathbf{a}_{i})\\ \mathbf{a}(\mathbf{\beta}_{i})\end{bmatrix} =\mathbf{0},\;i=1,\ldots,d. \tag{14}\]  
From Equation (14), the spatial spectrum of 2D MUSIC method can be denoted in Function (15),  
\[P_{2D-MUSIC}(\alpha,\beta)=\frac{1}{ \begin{bmatrix}\mathbf{a}(\alpha)\\ \mathbf{a}(\beta)\end{bmatrix} ^{H}\mathbf{U}_{N}\mathbf{U}_{N}^{H} \begin{bmatrix} \mathbf{a}(\alpha)\\ \mathbf{a}(\beta)\end{bmatrix} } \tag{15}\]  
and the angles \((\alpha_{i},\beta_{i})\), \(i=1,\ldots,d\) can be found by performing a 2D search of the \(d\) peaks on Equation (15). As previously mentioned, since the 2D search peak finding is prohibitively time consuming, the Unitary R-D Root MUSIC overcomes this problem by performing a root-finding method instead. Before describing the Unitary R-D Root MUSIC, we have to define three matrices that are used in the unitary transformation to convert centro-hermitian matrices into real-valued ones. Let \(\mathbf{\Pi}_{p}\in\mathbb{C}^{p\times p}\) be any anti-diagonal identity matrix in keeping with Definition (16),  
\[\mathbf{\Pi}_{p}\triangleq \begin{bmatrix}0&0&\ldots&0&1\\ 0&0&\ldots&1&0\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 0&1&\ldots&0&0\\ 1&0&\ldots&0&0\\ \end{bmatrix} , \tag{16}\]and \(\mathbf{Q}_{n}\in\mathbb{C}^{n\times n}\) be an unitary transform matrix expressed in Definitions (17) and (18),  
\[\mathbf{Q}_{2n}\triangleq\frac{1}{\sqrt{2}}= \begin{bmatrix}\mathbf{I}_{n}&j \mathbf{I}_{n}\\ \mathbf{\Pi}_{n}&-j\mathbf{\Pi}_{n}\end{bmatrix} \quad\quad\text{or} \tag{17}\]  
\[\mathbf{Q}_{2n+1}\triangleq\frac{1}{\sqrt{2}}= \begin{bmatrix}\mathbf{I}_{n}&0& j\mathbf{I}_{n}\\ \mathbf{0}_{n}^{T}&\sqrt{2}&\mathbf{0}_{n}^{T}\\ \mathbf{\Pi}_{n}&0&-j\mathbf{\Pi}_{n}\end{bmatrix} , \tag{18}\]  
depending if its size is even or odd. The algorithm is outlined below. 1. Collect \(N\) array observations for timestamp \(t_{1},t_{2},\ldots,t_{N}\).