## 3 Results and Discussion
### Analysis on the Principle of p-DAS  
First of all, the principle of the \(p\)-DAS beamformer is compared to the DAS beamformer, for the case of single plane wave imaging. Namely, their respective PSFs (shown in Figure 3a,b) are investigated in simulation on the resolution phantom described by [15]. Here, \(p\)-DAS is considered as a DAS beamformer with adaptive weighting, considering Equations (11) and (12).  
For the point scatterer placed at (\(x_{0}=0\) mm, \(z_{0}=35\) mm), the PSF obtained with 2-DAS is shown in Figure 3b, and this shows better side-lobe rejection compared with the PSF obtained with DAS, as shown in Figure 3a. To explain this observation, the two beamformers are compared through consideration of two specific pixels: the maximum of the main lobe at (\(x_{0}=0\) mm, \(z_{0}=35\) mm), which corresponds to the point scatterer location, and the peak side-lobe at (\(x_{1}=0.9\) mm, \(z_{0}=35\) mm). First, the pixel reconstruction at the main lobe is investigated. In Figure 3c, the oscillations are successfully rephased for the pixel at (\(x_{0},z_{0}\)) because a wave front actually comes from this location. As a result, a uniform level of amplitude is observed for \(s_{n}(x_{0},z_{0})\) in Figure 3e (blue). For DAS, these delayed samples are simply summed. For 2-DAS, an adaptive weighting window is applied (Figure 3e, red). However, the \(N\) weighting values \(w_{n}(x_{0},z_{0})\) are quasi-identical, as they depend on  
Figure 3: Illustration of the enhanced side-lobe rejection for the point spread function (PSF) with the proposed method with \(p=2\) (2-DAS), compared with conventional DAS. The scatterer placed at (\(x_{0}=0\) mm, \(z_{0}=35\) mm) on the numerical phantom is considered. The B-mode log-compressed images with a 40-dB dynamic range are shown for DAS (**a**) and 2-DAS (**b**). The delayed signals \(s_{n}(x,z)\) are shown for \(x_{0}=0\) mm (**c**) and \(x_{1}=0.9\) mm (**d**); (**e**) the delayed samples \(s_{n}(x_{0},z_{0})\) for the pixel on the main lobe are shown in blue, with their corresponding adaptive weights \(w_{n}(x_{0},z_{0})\) in red; and (**f**) the delayed samples \(s_{n}(x_{1},z_{0})\) for the pixel on the side lobe are shown in blue, with their corresponding adaptive weights \(w_{n}(x_{1},z_{0})\) in red.  
the \(s_{n}(x_{0},z_{0})\) amplitudes, which are similar along the elements. Finally, this pixel reconstructed with 2-DAS is equivalent to that reconstructed with DAS.  
Then, DAS and 2-DAS are compared for the pixel at (\(x_{1}=0.9\) mm, \(z_{0}=35\) mm), which corresponds to the peak side-lobe position for the reconstruction with DAS. The value of the pixel should tend to 0, as no scatterer is present. However, it is corrupted with the energy of the ill-rephased wave front in Figure 3d. As a result, the corresponding \(s_{n}(x_{1},z_{0})\) amplitudes are no longer uniform (Figure 3f, blue). To lower the value of the pixel reached with conventional DAS, the adaptive weights \(w_{n}(x_{1},z_{0})\) of 2-DAS tend to be the strongest for the lowest amplitudes, as demonstrated in Equation (12). This is why side-lobes vanish for 2-DAS (Figure 3b) compared to DAS (Figure 3a).  
Note that, considering Equation (12), the side-lobe rejection increases with the \(p\)-value, as stronger weights are applied. The following subsection presents the impact on the B-mode images that arises from this adaptive approach.