## 3 Mathematical Model for L-Shaped Uniform Array  
Figure 3 shows the structure of the L-shaped uniform array. It is composed of two orthogonal uniform linear arrays of \(M\) antennas in the X-Y plane in which the distance between two adjacent antennas is \(\Delta\). All antennas are assumed to be identical, isotropic, and omnidirectional. Suppose there are \(d\) (\(d<M\)) independent far-field narrowband stationary signals, \(s_{i}(t)\) such that \(i=1,\ldots,d\), incident on the array plane at 2D angle \((\theta_{1},\phi_{1}),(\theta_{2},\phi_{2}),\ldots,(\theta_{d},\phi_{d})\) in which \(\theta_{i}\) is the azimuth and \(\phi_{i}\) is the elevation angle. Let us also assume the signals propagate in an AWGN channel with linear and isotropic transmission medium. DOA methods compute the broadside angle, which is the signal direction measured relative to the line perpendicular to the array. Since that array is composed of two ULAs, there are two broadside angles, \(\alpha_{i}\), and \(\beta_{i}\), as shown in Figure 3, which correspond to the x-axis and y-axis ULAs, respectively. From geometric properties, the relation between \(\alpha_{i}\) and \(\beta_{i}\) with azimuth and elevation are shown in Equations (1) [23],  
\[\cos\alpha_{i}=\cos\theta_{i}\sin\phi_{i},\] \[\cos\beta_{i}=\sin\theta_{i}\sin\phi_{i}. \tag{1}\]  
Assume that the L-shaped array is not subject to nonidealities such as mutual coupling and cross-polarization effect. Additionally, consider that all antennas are identical and have  
Figure 3: Depiction of L-shaped array with its antennas (black dots), angles, and the signal direction. omnidirectional gain functions, i.e., \(g(\theta,\phi)=1\). The model of the signal samples received by x-axis and y-axis arrays at a timestamp \(t\) can be expressed as Equations (2) and (3) [24],  
\[\mathbf{x}(t)=\mathbf{A}_{x}\mathbf{s}(t)+\mathbf{n}_{x}(t), \tag{2}\]  
\[\mathbf{y}(t)=\mathbf{A}_{y}\mathbf{s}(t)+\mathbf{n}_{y}(t), \tag{3}\]  
where \(\mathbf{x}(t)= \begin{bmatrix}x_{1}(t)&x_{2}(t)&\ldots&x_{M}(t)\end{bmatrix} ^{T}\) is the array observation at timestamp \(t\) of the x-axis ULA which is a vector of the signal samples for each individual antenna in the x-axis ULA, such that \(x_{i}(t)\) corresponds to a single signal sample received from the antenna \(i\) at timestamp \(t\). Likewise, for \(\mathbf{y}(t)= \begin{bmatrix}y_{1}(t)&y_{2}(t)&\ldots&y_{M}(t)\end{bmatrix} ^{T}\) in the case of y-axis ULA. Moreover, \(\mathbf{s}(t)\in\mathbb{C}^{d\times 1}\) is a vector of signals of \(d\) sources, \(\mathbf{n}_{x}(t),\mathbf{n}_{y}(t)\in\mathbb{C}^{M\times 1}\) are the additive white Gaussian noise (AWGN) and \(\mathbf{A}_{x},\mathbf{A}_{y}\in\mathbb{C}^{M\times d}\) are the ideal steering matrices of the x-axis array and y-axis array, respectively, as shown in Equations (4) and (5),  
\[\mathbf{A}_{x}= \begin{bmatrix}\mathbf{a}(\alpha_{1})&\mathbf{a}(\alpha_{2})& \ldots&\mathbf{a}(\alpha_{d})\end{bmatrix}  \tag{4}\]  
\[\mathbf{A}_{y}= \begin{bmatrix}\mathbf{a}(\beta_{1})&\mathbf{a}(\beta_{2})& \ldots&\mathbf{a}(\beta_{d})\end{bmatrix}  \tag{5}\]  
where the ideal array responses are defined in Equations (6) and (7),  
\[\mathbf{a}(\alpha_{i})^{T}= \begin{bmatrix}1&e^{j\mu_{a_{i}}}&e^{j2\mu_{a_{i}} }&\ldots&e^{j(M-1)\mu_{a_{i}}}\end{bmatrix} , \tag{6}\]  
\[\mathbf{a}(\beta_{i})^{T}= \begin{bmatrix}1&e^{j\mu_{\beta_{i}}}&e^{j2\mu_{ \beta_{i}}}&\ldots&e^{j(M-1)\mu_{\beta_{i}}}\end{bmatrix} , \tag{7}\]  
in which \(\mu_{a_{i}}=-\frac{2\pi f_{c}}{c}\Delta\cos\theta_{i}\sin\phi_{i}=-\frac{2\pi f _{c}}{c}\Delta\cos\alpha_{i}\) and \(\mu_{\beta_{i}}=-\frac{2\pi f_{c}}{c}\Delta\sin\theta_{i}\sin\phi_{i}=-\frac{2 \pi f_{c}}{c}\Delta\cos\beta_{i}\), and \(c\) is the speed of light, \(f_{c}\) is the carrier frequency  
Moreover, in our analyses, it would be useful in some cases to consider the model signal samples for the whole L-shaped array instead of two separate ULAs. Let us define \(\mathbf{h}(t)= \begin{bmatrix}h_{1}(t)&h_{2}(t)&\ldots&h_{2M-1}(t)\end{bmatrix}