## 3 Numerical Experiment and Discussion
### Estimating DoA from PVs and SPVs  
Different DoA estimation methods have been developed in the literature that consider both single- [11; 40] and multiple-source [34; 36] scenarios. The direction of the computed PVs and SPVs concentrates around the DoAs of the dominating source. The methods estimate the maximums of the cost function that deals with the approximation of the vectors' probability distribution over uniform directions of the 2D grid.  
In practical applications, we initially calculated the dictionary as having a Gaussian kernel in the middle of each direction, spreading on a uniformly spaced \(N_{\mathcal{R}_{\phi}}\times N_{\mathcal{R}_{\phi}}\) grid. The element of both \(j_{\phi}\), \(j_{\phi}\) can be expressed as  
\[K_{j_{\phi},\ j_{\phi}}(\varphi)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{ \angle\left(\varphi,\Psi_{j_{\phi},\ j_{\phi}}\right)^{2}}{2\sigma^{2}}\right) \tag{15}\]where \(\varphi\) represents the look direction, \(\sigma\) denotes Gaussian kernel's standard deviation, \(j_{\theta}\in\left\{0\ldots N_{K_{\varphi}}-1\right\}\) and \(j_{\phi}\in\left\{0\ldots N_{K_{\varphi}}-1\right\}\) represents indices of elevation and azimuth of DoA \(\Psi_{j_{\theta},\ j_{\phi}}=(j_{\theta}\pi/N_{K_{\theta}},j_{\phi}2\pi/N_{K_{ \varphi}}\), respectively. The dictionary (sparse) is computed using lesser entries than \(\lambda\), and reaching down to zero, as  
\[\hat{\mathbf{K}}_{j_{\theta},\ j_{\phi}}(\varphi)=\left\{\begin{matrix}0&\quad K_{ j_{\theta},\ j_{\phi}}(\varphi)<\lambda\\ K_{j_{\theta},\ j_{\theta}}(\varphi)&\quad otherwise\end{matrix}\right. \tag{16}\]  
Describing \(\chi(\varpi,\mathcal{I})\triangleq(\theta_{\chi},\phi_{\chi})\) as the direction associated with each region computed via PV or SPV, the dictionary indices are computed based on \(J_{\theta}(\chi(\varpi,\mathcal{I}))=\left\lfloor\theta_{\chi}N_{K_{\varphi}} /\pi+0.5\right\rfloor\) and \(J_{\phi}(\chi(\varpi,\mathcal{I}))=\left\lfloor\phi_{\chi}N_{K_{\varphi}}/0.5 +2\pi\right\rfloor\), with \(\lfloor\cdot\rfloor\), denoting the floor operator. Hence, the resulting function can be expressed as  
\[\mathbf{H}(\varphi)=\sum_{(\varpi,\mathcal{I})\in\mathcal{T}}\hat{\mathbf{K}}_{j_{ \theta}(\chi(\varpi,\mathcal{I}),J_{\theta}(\chi(\varpi,\mathcal{I}))}(\varphi) \tag{17}\]  
where \(\mathcal{T}\) represents the TF-bins within the interval of observation. The directions-of-arrival were then computed from the corresponding directions towards the entire peaks in the \(\mathbf{H}\). The associated parameters of the developed techniques are \(N_{K_{\varphi}}=175\) and \(N_{K_{\varphi}}=92\), \(\sigma=5^{0}\), resolution is \(2^{0}\) in both elevation and azimuth, and \(\lambda=0.002/(\sigma\sqrt{2\pi})\), which eliminates entries >15\({}^{0}\) from the look direction.  
For each approach under consideration (i.e., the PWD-SRP, SPV, DPD-MUSIC, and PV), the spatial form of cost function that corresponds to all the approaches was calculated over a 2D grid in elevation and azimuth. The number of sources was assumed to be known a priori. A single DoAs was obtained for each simulation, placing the observation interval at full length of the signal.