## 3 Results and Discussion
### Experimental Results
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Proposed Method with** & **Proposed Method with** & **ESPRIT with Fixed** & **ESPRIT with Fixed** & **Proposed Method with** & **Proposed Method with** \\
**Copper Plate** & **Bone-Mimicking Plate** & **Threshold (–40 dB)** & **Threshold (–30 dB)** & \(\omega_{m}=0.1\) MHz & \(\omega_{m}=0.75\) MHz \\ \hline \(108\) & \(121\) & \(184\) & \(143\) & \(144\) & \(118\) \\ \hline \end{tabular}
\end{table} 
Table 1: Root mean square errors (RMSEs) of the experimental results with different settings. The units are rad/m. Figure 6: Wavenumbers of the 4-mm-thick bone-mimicking plate estimated using the conventional SVD method. The color intensity map shows the results obtained using the conventional SVD method. Solid white lines show the theoretical curves. Figure 7: Wavenumbers of the 4-mm-thick bone-mimicking plate estimated using the proposed method and the conventional SVD method. Red dots show the estimates obtained using the proposed method. The color intensity map shows the results obtained using the conventional SVD method. Solid white lines show the theoretical curves. The wavenumber or phase velocity estimation steps that have high computational costs in the proposed method are eigenvalue decomposition of the covariance matrix with a size of \(N_{\text{sub}}\times N_{\text{sub}}\), \(\mathcal{O}(N_{\text{sub}}^{3})\) and the SVD of the matrix of size \(N_{\text{sub}}-1\times 2M\), \(\mathcal{O}(\text{min}\left\{(N_{\text{sub}}-1)\times 4M^{2},(N_{\text{sub}}-1)^{2} \times 2M\right\})\). The calculation steps are shown in detail in [30]. The corresponding steps in the conventional SVD method, the Radon transform method, and the S-SVD method are the SVD of the matrix of size \(N\times N_{\text{sub}}\), \(\mathcal{O}(N^{2}\times N_{\text{sub}})\)[30], the inversion of the matrix of size \(N_{\text{c}}\times N_{\text{c}}\), \(\mathcal{O}(N_{\text{c}}^{3})\), and the inversion of the matrix of size \(N_{\text{k}}\times N_{\text{k}}\), \(\mathcal{O}(N_{\text{k}}^{3})\), respectively, where \(N_{\text{c}}\) and \(N_{\text{k}}\) are the numbers of sampling points in the phase velocity and wavenumber axes, which is normally larger than 64. Therefore, the SVD method showed the smallest computational cost with the lowest resolution. The computational cost of the proposed method was significantly lower than the corresponding costs of the Radon and S-SVD methods. The calculation time (not including loading time) for the experimental data was less than 0.5 s when using a commercial processor (Core(TM) i7-7500U; Intel, Santa Clara, CA, USA) on a laptop computer with MATLAB (R2017a, Mathworks, Natick, MA, USA) platform. The calculation time taken for the conventional Radon transform was less than 60 s, according to [12]. While noting here that we did not use the same processor or computer, and in fact used a faster processor, the reduction in the calculation time of 99% is interesting. In addition, the computational cost of the S-SVD method is greater than that of the Radon method. The calculation time of the conventional SVD method is around 0.1 s. Note that, as mentioned above, the conventional SVD method has considerably lower computational complexity than the proposed method. Because we ran the SVD method and the proposed method on MATLAB platform, the calculation time does not directly reflect the theoretical complexity. To examine the effectiveness of the proposed method in determination of \(M\), we compared the performance of the proposed method with that of the ESPRIT method with a fixed threshold. Figures 9 and 10 show the results obtained for a bone plate when we used eigenvalues that exceed \(-40\) and \(-30\) dB of the maximum eigenvalue, respectively. The red circles represent the results obtained using the proposed method. The blue cross marks show the estimates obtained using the ESPRIT algorithm with a fixed threshold. The solid gray lines show the theoretical curves.