## 3 Results and Discussion  
In the proposed method, the total number of operations is not affected by the number of transmit plane-waves but is related to the maximum value of \(k\) because additional delays to be shifted must also be calculated. Therefore, the total number is (\(N_{de}+2\cdot k_{m}\)) in the proposed method while the total number is (\(N_{ele}\times N_{PW}\)) in the conventional method, where \(N_{PW}\) is the number of transmit plane-waves with different steering angles, and \(k_{m}\) is related to \(\theta_{PW}\) and \(d\theta\) as follows:  
\[ \begin{array}{l}\theta_{PW}=k\,d\theta\\ \therefore\,k_{m}=\theta_{PW_{max}}\end{array}  \tag{3}\]  
where \(\theta_{PW_{max}}\) is the maximum steering angle. Since \(k_{m}\) is not related to \(N_{ele}\) and \(N_{PW}\), the computational complexity of the proposed method can be determined by the variable \(k_{m}\) irrespective of \(N_{PW}\). To assess the efficiency of the proposed method, the total number of computations for the conventional method and the proposed method were calculated for various conditions and summarized in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Variables** & \multicolumn{2}{c}{**Conventional**} & \multicolumn{2}{c}{**Proposed**} \\ \hline \(N_{de}\) & \(N_{PW}\) & \(k_{m}\) & Number of computations & Number of computations \\ \hline \multirow{4}{*}{64} & 9 & 32 & 576 & 128 \\  & 64 & 576 & 192 \\  & 65 & 32 & 4160 & 128 \\  & 64 & 4160 & 192 \\ \hline \multirow{4}{*}{128} & 9 & 64 & 1152 & 256 \\  & 9 & 128 & 1152 & 384 \\  & 65 & 64 & 8320 & 256 \\  & 128 & 8320 & 384 \\ \hline \multirow{4}{*}{192} & 9 & 96 & 1728 & 384 \\  & 192 & 1728 & 576 \\ \cline{1-1}  & 96 & 12,480 & 384 \\ \cline{1-1}  & 192 & 12,480 & 576 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Total number of computations of Equation (1) for the conventional method and proposed method. Figure 6: Delay values for various steering angles (i.e., \(k_{min}\cdot d\theta\) to \(k_{max}\cdot d\theta\)) acquired from delay values for a steering angle of 0 degree. As listed in Table 1, the proposed method has a substantially lower total number of computations than the conventional method (e.g., 256 vs. 8320, respectively, with \(k_{m}=64\) when the number of elements and transmit plane-waves are 128 and 65, respectively; these are the same conditions shown in Figure 5b). Although the optimal value of \(k_{m}\) is 64 in the condition, the value can be set higher to facilitate a larger range of steering angles, increasing the lateral resolution at the outer part of the image. Moreover, in the case of \(k_{m}>N_{ele}\) to increase steering angles, \(\theta_{PW_{max}}\) can be exceeded for the maximum scan angle of the ROI; it does not allow for improved image quality due to the defined scanning area and the inherent element directivity function [20]. Therefore, the proposed method will be at least \(N_{pw}/3\) times more efficient than the conventional method when \(k_{m}\) is lower than \(N_{ele}\) (i.e., the total number of computation is \(N_{ele}+2\cdot k_{m}\leq 3\times N_{ele}\)). To evaluate the real-time performance of the proposed method, the transmit delay calculation was implemented on a NVIDIA Computer Unified Device Architecture (CUDA) platform (GTX 660 Ti, NVIDIA, Santa Clara, CA, USA) for every case listed in Table 1 and the computational time was measured as summarized in Table 2.  
As listed in Table 2, for \(N_{ele}\) of 128, \(N_{PW}\) of 65, and \(k_{m}\) of 64, the proposed method yielded \(8.4\pm 0.7\)\(\upmu\)s compared to the conventional method, \(78.6\pm 0.5\)\(\upmu\)s. The computational efficiency of the proposed method was significantly improved in a larger number of transmit beams. Therefore, the efficient transmit delay calculation method in curved arrays will improve the performance of real-time implementation for ultrafast curved array applications.