## 3 Numerical Experiment and Discussion
### Simulation Results  
For easy reading and comprehension, we considered two simulation cases. The first scenario is termed _CASE ONE_, where a single source is considered and positioned 1.5 m from the antenna array in one of the 24 different directions provided by possible combinations of \(\in\left\{70^{0},90^{0}\right\}\), \(\phi\in\left\{0^{0},30^{0},\ldots,330^{0}\right\}\). For all test conditions, four signals were employed, using 96 test samples for each condition. The second scenario is named _CASE TWO_, where 4 different sources were positioned 1.5 m away from the antenna array at a 50\({}^{0}\) interval in the azimuth direction and switching between 70\({}^{0}\) and 90\({}^{0}\) elevation. Ensuring that the sources' positions were not affecting the outputs, twelve possible orientations of sources were examined using the azimuths rotation of the four sources. For all orientations of the four sources, eight signal combinations were collected, i.e., 96 test samples in total for a condition.  
We considered two uncorrelated EM signals impinging SAA from (40, 32) and (32, 64) degree sources, and at 10 dB SNR. The root mean square errors (RMSEs) decayed as snapshots increased, as depicted in Figure 2 using Cramer-Rao bound (CRB). This shows the accuracy level of the proposed PV and SPV methods as compared to the PWD-SRP and PWD-SRP baseline methods.  
Using _CASE ONE_, the plots of angular RMSE in DoA estimation for all combinations of SNRs and algorithms are presented in Figure 3. Generally, the RMSE rises as the SNR increases, and the PWD-SRP exhibits a better performance than the DPD-MUSIC. The SPV performs better than the PV by about 0.3\({}^{0}\), and this is more obvious in worse scenarios, where the SNR is 10 dB. This implies that, for DoA estimation with a single source, the PV approach is more efficient.  
Figure 4 shows the number of detected sources versus SNR under _CASE TWO_. Both the PV and PWD-SRP methods are more affected by noise and interference, which impact the capacity of sources' localization. The SPV method achieved a better performance than the PV, DPD-MUSIC and PWD-SRP methods, but the DPD-MUSIC is clearly not impacted by noise and other interferences.  
Figure 2: RMSEs estimated against SNR using two sources (**a**) elevation and (**b**) azimuth.  
Figure 5 shows the performance of each method using an RMSE plot versus SNR for trials where all four sources were found. The PV achieved a better performance than the SPV and PWD-SRP. Generally, the error in CASE TWO is significantly greater than that in CASE ONE (except for the scenario where the PWD-SRP is not perfect). The SPV method showed a reduced performance comapred to the PWD-SRP (heuristically viewed to be half of the error) and a more significant performance than PV. In the worse scenario, i.e., when the SNR was 10 dB, the RMSE was about 3.7\({}^{0}\). When compared with the DPD-MUSIC, the rise in SPV error is between 0.9\({}^{0}\) (at 25 dB SNR) and 2\({}^{0}\) (40 dB).  
Figure 4: Number of sources detected against SNR, with four sources.  
Figure 3: RMSEs estimate versus SNR for different methods using one source.  
In general, the results for CASE ONE and the computational benefits show, that for a single-source DoA estimation scenario, the PV method shows more robustness than the SPV, PWD-SRP and the DPD-MUSIC methods. Conversely, under CASE TWO, subspace method (i.e., the SPV) provides an improved performance when multiple sources are concurrently active. Considering TF-regions and noise subspace with one dominating source, the DPD-MUSIC exhibits a better degree of correctness than the SPV. At dense resolutions, since the SPV has a lower computational cost than the DPD-MUSIC, it is more appropriate for the DoA estimation under a scenario that involves multiple sources.