## 3 Results and Discussion  
Figure 4 presents analytic simulations of two-way PSFs using unfocused FT-FR and FT-XR in the azimuthal (\(x\)-\(z\) plane in Figure 1c) and the diagonal (\(xy\)-\(z\) plane in Figure 1c) planes. Almost identical responses were found in the azimuthal plane for both the FT-FR and the FT-XR (Figure 4a) with comparable magnitudes of side-lobes (i.e., \(-\)24.72 dB for FR vs. \(-\)23.32 dB for XR). However, the PSF of the FT-XR in the diagonal plane (Figure 4b) significantly increased the side-lobe level by up to 32 dB compared with the FT-FR (i.e., \(-\)38.32 dB for FR vs. \(-\)6.12 dB for XR). The main lobe (i.e., largest field strength) was hardly recognizable in the two-way PSF of the unfocused FT-XR (Figure 4b) as the difference between the main-lobe and the side-lobe levels was less than 6 dB. Thus, the X-shaped array was found to be ineffective for unfocused ultrasound imaging, particularly when the imaging plane was rotated at a certain angle (i.e., \(\alpha\
eq 0\) and \(\beta\
eq 0\) in Equations (1) and (2)). Figure 5 presents the simulated images using FT-FR (Figure 5a,b) and FT-XR (Figure 5c,d) for diagonal planes. Due to the transmission of a plane wave using the whole aperture, the beam width of the wave was almost equal to the size of the aperture of the array. Thus, the point targets outside of the beam width (i.e., 16 mm in the azimuthal plane and 22 mm in the diagonal plane) were hardly imaged as demonstrated in Figure 5. While the unfocused FT-FR in the diagonal plane (Figure 5c) displayed the point targets, the image in the diagonal plane for the FT-XR (Figure 5d) was significantly degraded, making the point targets almost unrecognizable. This result clearly demonstrates that the unfocused FT-XR was considerably degraded in the diagonal direction, which was also observed in the simulation of the PSFs of the unfocused FT-XR (Figure 4b). Therefore, it is conceivable that the unfocused FT-XR may hardly be used for volumetric imaging in the diagonal plane. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **FT-FR** & **FT-XR** & **FT-XR w/DTxF** \\ \hline Transmit elements (\(N_{t}\)) & \(N\times N\) & \(N\times N\) & \(N\times N\) \\ Receive elements (\(N_{r}\)) & \(N\times N\) & \(2N\) & \(2N\) \\ Beams per frame (\(N_{beam}\)) & \(1\) & \(1\) & \(\kappa\) \(\sim\)\(\kappa^{2}\) \\ Frame rate & \(1\) & \(1\) & \(1/\kappa^{2}\) \(\sim\)\(1/\kappa\) \\ SNR gain (\(\surd N_{t}\surd N_{r}\surd N_{beam}\)) & \(N^{2}\) & \(\surd 2N^{1.5}\) & \(\surd 2N^{1.5}\kappa^{0.5}\) to \(\surd 2N^{1.5}\kappa\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Quantitative summary of FT-FR and FT-XR for \(N\times N\) element array. Figure 4: Two-way point spread functions (PSFs) of unfocused FT-FR and FT-XR in (**a**) azimuthal and (**b**) diagonal planes. Figure 6 shows azimuthal and diagonal images reconstructed by DTxF for unfocused FT-XR. Figure 6a,b resulted from the optimum angular increment (31 \(\times\) 31 plane waves) and Figure 6c,d resulted from the reduced number of plane waves (11 in the diagonal direction). Overall, the point targets in both the azimuthal (Figure 6a,c) and the diagonal planes (Figure 6b,d) were clearly visible in comparison with those of a single plane wave transmit (Figure 5). Although the point targets reconstructed from 11 plane waves (Figure 6c,d) were hardly clear compared with those from the optimum parameters (Figure 6a,b), it was still conceived that the reduced beams in the degraded plane were feasible to improve the quality of the volumetric images. Since smaller areas of the beams were overlapped as the point targets were located deeper and farther from the center, it was observed that the PSFs improved at the closer and shallower depths. In fact, the observation was more noticeable in the images using the reduced beams (Figure 6c,d). In particular, the point targets in the diagonal plane (Figure 6b,d) were dramatically sharpened (i.e., less blurred) compared with Figure 5d.