## 3 Numerical Experiment and Discussion  
In this section, the proposed DoA methods are analyzed and compared with the PWD-SRP [8; 41] and DPD-MUSIC [15; 42] baseline approaches from the literature in a simulated environment. The methods were also compared using measured data from the experiment, which is a generally acceptable grounds to evaluate any method.  
At first, an SAA of radius, \(r=1.8\) cm, and 32 elements working at 8.4 GHz (as shown in Figure 1) that was distributed uniformly on a rigid sphere was used for the simulations, as in [3; 4; 43; 44]. The distance between two repeated samples in the uniform sampling scheme remained constant. The uniform sampling scheme led to a reduction in platonic solids. The phenomenon appeared for a particular number of sensors [44]. Four (4) was the highest order of the SAA. About 600 independent Monte Carlo simulations were performed using a 2019b version of matlab software. We employed a search step of 0.2\({}^{0}\) for DPD-MUSIC, PWD-SRP, and the developed PV, and SPV methods. Two iterations were utilized during the simulations. To avoid aliasing problems, we ensured that \(kr\) was less than the order N. Furthermore, narrowband amplitude modulation (AM) signals in the far field were utilized in all the simulation scenarios.  
The discrete time signals were transformed into the short-time Fourier transform (STFT) domain via a 77% overlapping Hamming window. A stacked vector of signals can be formulated as \(\mathrm{x}(\varpi,\mathcal{I})\), where \(\mathcal{I}\) and \(\varpi\), respectively, denote the STFT frame index and frequency index. After a sample examination, we found that the frames from 6 to 10 ms provided the largest PVs concentration in the neighborhood of the actual DoA estimates compared to frames \(\geq 18\) ms. This small frame provides a probability enhancement of the window disjoint orthogonality that is assumed [39] to be real. In addition, using additional TF-bins causes the directional distribution of the vectors to be highly regular.  
The \(\mathrm{PVs},\tilde{\mathbf{I}}(\varpi,\mathcal{I})\) were computed using Equation (8), where \(k\) is substituted with the time-frequency index. The \(\mathrm{SVs},\tilde{\mathbf{I}}_{\mathbf{ss}}(\varpi,\mathcal{I})\) were computed by employing \(\tilde{\mathbf{R}}_{\tilde{\kappa}_{lm}}(\varpi,\mathcal{I})\) as a substitute for \(\mathbf{R}_{\tilde{\kappa}_{lm}}\), which approximates in the surrounding frame index \(\mathcal{I}\) and frequency index \(\varpi\) by  
\[\tilde{\mathbf{R}}_{\tilde{\kappa}_{lm}}(\varpi,\mathcal{I})=\frac{1}{J_{\varpi} \mathcal{I}_{\mathcal{I}}}\sum_{j_{\varpi}=0}^{I_{\varpi}-1}\sum_{j_{\mathcal{ I}}=0}^{I_{\varpi}-1}\tilde{\mathbf{\kappa}}_{lm}(\varpi+j_{\mathcal{I}},\ \mathcal{I}+j_{\mathcal{I}})\times\mathbf{\kappa}_{lm}^{H}(\varpi+j_{\mathcal{I}},\ \mathcal{I}+j_{\mathcal{I}}) \tag{14}\]  
where \(J_{\mathcal{I}}\) and \(J_{\varpi}\) represent timeframes and frequency bins, respectively, added in average form. Averaging over the frequency can be achieved, since the steering vector in the SH domain does not depend on frequency [10]. After a comprehensive study, calculating Equation (14) with a time of 35 ms and a 10 ms STFT frame length provided acceptable results, even though a frame length of 5-20 ms gave identical results. In each simulation, a source was positioned 1.5 m from SAA in one of the twenty-four directions provided by each combination of \(\in\left\{70^{0},90^{0}\right\}\), \(\phi\in\left\{0^{0},30^{0},\ldots,330^{0}\right\}\). For each test conditions, four EM signals were employed, providing 96 test samples per condition.