## 3 Results  
### Numerical Results: Point Scatterer  
#### 3.1.1 Receive Aperture  
Figure 4a,b shows the PSF for varying receive aperture (Nr) between 1 and 20 elements and shifting steps (Ns) of 3 and 4. A full-aperture transmission (transmission with all the elements) is  
Figure 3: **(a)** The daughterboard with the transducer array before applying the ground foil. **(b)** The finalized \(4\times 2\) matrix array. **(c)** Schematic of the measurement setup. **(d)** The imaging phantom: a foam pad with two needles. The distance between the foam and the transducer surface is about 26 mm. **(e)** The imaging phantom: a foam pad with five needles, positioned in different depths (from 10 to 20 mm). The distance between the foam and the transducer surface is about 28 mm. The numbers next to the needles are used for target localization in Section 3.  
used (Nt = 80) and 20 shots are given. For a receive aperture equal to one element (Nr = 1) and step size of three elements (Ns = 3), the total covered aperture is 8.7 mm resulting in a main beam width of about 2 mm at 10 mm distance. The grating lobe level is \(-\)20 dB as expected from the 1.5 cycle excitation and 20 shots and positioned at a lateral distance of \(-\)5 mm caused by the large step (Ns = 3, effective pitch = 450 \(\mu\)m, \(\lambda\) = 200 \(\mu\)m). The mean beam width remains about the same for larger Nr up to 5 (see Figure 4e) since two opposite effects play a role. The first one is the slowly increasing covered aperture (e.g., for Nr = 5, the covered aperture is 9.3 mm). The opposite one is the increasing directivity of the receiving apertures (for Nr = 5 the directivity is about 10\({}^{\circ}\)), which makes the contribution of the outer elements less. For larger receive apertures, Nr = 10 the main beam width is increased to about 3 mm and it further increases to 4.2 mm for Nr = 20. For Nr = 20, the covered aperture is admittedly the largest, but very directive (2\({}^{\circ}\)), and therefore only the middle part of the covered aperture is contributing in image reconstruction. For Ns = 4 (see Figure 4b), the grating becomes stronger, making the grating lobes appear closer to the mainlobe, at \(-\)4 mm lateral distance (see the rectangles in Figure 4a,b). Figure 4: (**a-d**) The point spread functions (PSFs) obtained for different Nr and Ns when a full-aperture transmission is used. A point scatterer was positioned at the depth of 10 mm and the lateral distance of 0 mm. The angular weighting (AW) technique was used in the bottom row. The black and red rectangular regions were used to calculate the clutter level (Figure 5a) and effects of grating lobes (Figure 5b). The red circles show the effects of grating lobes. (**e**) The lateral variation for Ns = 3 when the AW is on. Since the effective aperture can be very large and directive, angular weighting, as described in [40], can be applied in reception. This results in reduced grating lobes (compare the first and second rows in Figure 4) and about 16 and 13 dB lower clutter level for Ns of 3 and 4, respectively (see Figure 5a where the pixel intensities within the red rectangular regions shown in Figure 4a were used for statistical analysis). The level of grating lobes (for different Nr) obtained with the AW technique is presented in Figure 5b. The AW technique hardly works for Nr of 1 up to 5 due to the large angular view of the receive aperture. Figure 6 shows the FWHM as function of aperture settings. It indicates that the smaller the Nr, the lower the FWHM (i.e., better lateral resolution). This is mainly because we use the dynamic focusing technique in reception, and increment in Nr results in a narrower directivity in each receive aperture, which reduces the effective number of elements (i.e., aperture size) contributing in the reconstruction of a given pixel (voxel in 3D). An analysis of Figures 4 and 6 shows that the AW technique helps with suppression of the grating lobes at the expense of resolution.