## 4 Unitary Reduced-Dimension Root MUSIC
Afterward, we can estimate the covariance matrix of the x-axis and y-axis separately in line with Equations (19) and (20), \[\mathbf{R}_{xx}\approx\mathbf{\hat{R}}_{xx}=(\frac{1}{N})\mathbf{X}\mathbf{X}^ {H},\] (19) \[\mathbf{R}_{yy}\approx\mathbf{\hat{R}}_{yy}=(\frac{1}{N})\mathbf{Y}\mathbf{Y}^ {H},\] (20) where \(\mathbf{X}= \begin{bmatrix}\mathbf{x}(t_{1})&\ldots&\mathbf{x}(t_{N})\end{bmatrix}  \in\mathbb{C}^{M\times N}\) and \(\mathbf{Y}= \begin{bmatrix}\mathbf{y}(t_{1})&\ldots&\mathbf{y}(t_{N})\end{bmatrix}  \in\mathbb{C}^{M\times N}\) are the \(N\) array observations of x-axis and y-axis respectfully. 2. Apply forward-backward averaging on the covariance matrix, that is, \(\mathbf{\hat{R}}_{xx}^{fb}=\mathbf{\hat{R}}_{xx}+\mathbf{\Pi}_{M}\mathbf{ \overline{R}}_{xx}\mathbf{\Pi}_{M}\) to deal with coherent signals due to the multipath reflections [25]. Since \(\mathbf{\hat{R}}_{xx}^{fb}\) is a centro-hermitian matrix, we apply the unitary transformation to convert that complex-valued matrix into a real-valued one, that is, \(\mathbf{\hat{C}}_{x}=\{\mathbf{Q}_{M}^{H}\;\mathbf{\hat{R}}_{xx}^{fb}\; \mathbf{Q}_{M}\}=\text{Re}\{\mathbf{Q}_{M}^{H}\;\mathbf{\hat{R}}_{xx}\; \mathbf{Q}_{M}\}\in\mathbb{R}^{M\times M}\). The same goes for the covariance matrix of the y-axis, \(\mathbf{\hat{C}}_{y}=\mathbf{Re}\{\mathbf{Q}_{M}^{H}\;\mathbf{\hat{R}}_{yy}\; \mathbf{Q}_{M}\}\in\mathbb{R}^{M\times M}\). 3. Apply the real-valued EVD in \(\mathbf{C}_{x}\) and \(\mathbf{C}_{y}\) to construct the noise subspace \(\mathbf{U}_{N,x}\) and \(\mathbf{U}_{N,y}\) of the x-axis and y-axis, respectively. It is a time-consuming operation that is optimized in this paper. 4. As previously mentioned, the Unitary R-D Root MUSIC estimates the DOAs based on the roots of two polynomials, so it avoids the exhaustive search of standard MUSIC. Defining \(z=e^{-j\frac{2\pi}{\lambda}\Delta\cos\alpha}\) for the x-axis, the array response in Equation (6) can be redefined as a function of \(z\), as shown in Equation (21), \[\mathbf{a}(\alpha)^{T}=\mathbf{a}(z)^{T}= \begin{bmatrix}1&z&z^{2}&\ldots&z^{M- 1}\end{bmatrix} ,\] (21) likewise for the y-axis if we define \(z=e^{-j\frac{2\pi}{\lambda}\Delta\cos\beta}\). In addition, since \(a(z)^{H}=a^{T}(\frac{1}{2})\), the MUSIC spectrum can be viewed as a polynomial function whose DOA information is contained in some of its roots. The polynomial of the Unitary R-D Root MUSIC for the x-axis and y-axis are defined in Equations (22) and (23) [11], \[p_{x}(z) =z^{M-1}\mathbf{a}^{T}(\frac{1}{z})\mathbf{Q}_{M}\mathbf{U}_{N,x}( \mathbf{U}_{N,x})^{H}\mathbf{Q}_{M}^{H}\mathbf{a}(z),\] (22) \[p_{y}(z) =z^{M-1}\mathbf{a}^{T}(\frac{1}{z})\mathbf{Q}_{M}\mathbf{U}_{N,y}^ {y}(\mathbf{U}_{N,y})^{H}\mathbf{Q}_{M}^{H}\mathbf{a}(z),\] (23)whose degree is \(2M-2\). By defining \(\mathbf{G}=\mathbf{Q}_{M}\mathbf{U}_{N,x}(\mathbf{U}_{N,x})^{H}\mathbf{Q}_{M}^{H}\), we can derive Equations (24),  
\[ \begin{split} p_{x}(z)&=z^{M-1}\Big{[}1\quad z^{-1} \quad z^{-2}\quad\ldots\quad z^{-(M-1)}\Big{]}\mathbf{G}\big{[}1\quad z\quad z ^{2}\quad\ldots\quad z^{M-1}\big{]}^{T}\\ &=\big{[}z^{M-1}\quad z^{M-2}\quad z^{M-3}\quad\ldots\quad 1 \big{]}\begin{bmatrix}\begin{matrix}\delta_{1,1}&\cdots&\delta_{1,M}\\ \vdots&\ldots&\vdots\\ \delta_{M,1}&\cdots&\delta_{M,M}\end{matrix}\end{bmatrix}\begin{bmatrix}1\\ z\\ z\\ \vdots\\ z^{M-1}\end{bmatrix}\\ &=a_{0}+a_{1}z+\cdots+a_{2M-2}z^{2M-2},\end{split}  \tag{24}\]  
whose coefficients are defined in the relation (25),  
\[a_{k-1}= \begin{cases}\sum\limits_{i=1}^{k}s_{i,M-k+i}&\text{if}&k=1,2,\ldots,M \\ \sum\limits_{i=1}^{(2M-1)-k+1}&\text{if}&k=M+1,M+2,\ldots,2M-1. \end{cases}  \tag{25}\]  
The same procedure goes to \(p_{y}(z)\). 5. The \(d\) complex roots of \(p_{x}(z)\) and \(p_{y}(z)\) that are inside of a unit circle and closest to it, namely, \(\hat{z}_{x,1},\hat{z}_{x,2},\ldots,\hat{z}_{x,d}\) and \(\hat{z}_{y,1},\hat{z}_{y,2},\ldots,\hat{z}_{y,d}\), respectively, contain information about the \(d\) DOAs. The azimuth and elevation angles can be found as indicated by the set of equation in (26), \[