## 5 Discussion  
According to the simulation experiment of point targets and cyst target, as well as the real data experiment, we prove that due to the more reliable covariance matrix estimation, GSC-CROSS can not only improve the lateral resolution but also lead to better contrast. In the simulated point targets experiment shown in Figure 3; Figure 4, we can see that GSC-CROSS has the narrowest main lobe width compared with DS, SA, and GSC, indicating that GSC-CROSS sightly improves the lateral resolution of the medical ultrasound system. Meanwhile, GSC-CROSS also has outstanding performance in suppressing sidelobe energy. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Computation Time (s)** \\ \hline GSC & 325 \\ GSC-cross & 8395 \\ GSC-CROSS & 146 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Imaging speed with covariance matrix and cross-covariance matrix. Figure 8: Lateral variation of point at (**a**) z = 75.6 mm; (**b**) z = 88 mm. Figure 7: Images of actual echo data with different algorithms. (**a**) DS; (**b**) SA; (**c**) GSC (L = 16, \(\Delta\) = 20); (**d**) GSC-CROSS (L = 16, \(\Delta\) = 20). The dynamic range of image is 80 dB. It can inhibit the PSL to \(-88.7\) dB, which is much lower than the other three algorithms. Thus, when we set the dynamic range at 80 dB, the artifact caused by the sidelobe can hardly be observed in Figure 3d, while it is obvious in Figure 3a-c. The cyst target simulation experiment shows GSC-CROSS also performs well in improving contrast. Compared with DS, SA, and GSC, the contrast with GSC-CROSS is improved by about 20 dB. Observing the beamforming response processed by GSC and SA, we can see that though the lateral resolution is improved, the contrast is not improved correspondingly. It is because the filtering effect of GSC is associated with the degree of freedom of the system, and it will have good performance with the condition that the number of interference sources is less than L-2. However, the degree of freedom after subaperture averaging is L-1, so the beamformer will be weakened [7]. Therefore, GSC is usually combined with processing methods to improve the performance of contrast [30; 35]. These processing methods will increase the complexity and computational demand for GSC. However, GSC-CROSS uses the echo signals from different subarrays to estimate the cross-covariance matrix, which is more reliable than the traditional covariance matrix. Furthermore, the number of sub-matrices involved in averaging the cross-covariance matrix is much larger than that in the traditional covariance matrix, which makes the interference and noise suppressed better. Thus, compared with GSC, GSC-CROSS dramatically improves the contrast as well as the lateral resolution for the medical ultrasound system. The cross-covariance matrix introduces more matrix multiplications, which will slow down the imaging speed. In a previous study [22], only partial subarrays are extracted by sampling to estimate the cross-covariance matrix, limiting the performance of the beamformer. In our method, by calculating the mean-subarray first, the cross-covariance matrix can be obtained with a onetime matrix multiplication. It allows us to use all the subarrays to estimate the matrix and takes less time. Thus, our method can improve the imaging speed and it is even faster than the traditional GSC beamformer. For adaptive beamforming, some parameters of the processing algorithms have a great influence on the performance of the beamformer. Figure 9 shows the echo images with GSC-CROSS using different parameters, and the contrast of each image is listed in Table 5. Parameter L determines the number of subarrays P. For traditional GSC, the smaller the L is, the more subarrays P there are, the worse beamformer will be. However, because of a more stable covariance matrix estimation, GSC-CROSS shows high robustness to the change of L(P). In Table 5, we can see that when L changes, the contrast hardly changes.