## 3 Proposed Method  
With the help of subaperture averaging, GSC can be applied to process medical ultrasound data. However, it will cause the degradation of the sample covariance matrix \(\overset{\mathbf{\rm A}}{\mathbf{\rm R}}\). By traditional subaperture averaging, the main diagonal components of the sub-covariance matrix \(\mathbf{\rm G}_{\rm p}(\rm k)\mathbf{\rm G}_{\rm p}(\rm k)^{\rm H}\) correspond to the square of the echo signal \(\left[\mathbf{\rm x}_{\rm i}(\rm k)\right]^{2}\) received by the  
Figure 2: Schematic diagram of subaperture averaging.  
same element, which is always positive, while the rest of the sub-covariance matrix consists of \(\mathbf{x}_{\mathrm{i}}(\mathrm{k})\mathbf{x}_{\mathrm{j}}(\mathrm{k})\), which are positive or negative randomly. Therefore, after averaging according to Equation (13), the elements on the diagonal of the covariance matrix are much larger than those in other positions, which makes the sample covariance matrix \(\mathbf{R}\) approach the identity matrix \(\mathbf{I}\). Observing Equations (3) and (5), GSC gradually degenerates into SA, as the number of subarrays increases.  
To improve the accuracy and stability of the estimation of the covariance matrix, we put forward a new cross sub-superprime averaging generalized sidelobe canceler approach (GSC-CROSS) for medical ultrasound systems. This method uses the echo data from different subarrays to estimate the covariance matrix, and the new matrix is called the cross-covariance matrix. With the more stable and accurate cross-covariance matrix, rather than the traditional covariance matrix obtained from sub-superture averaging, the beamformer response improves significantly.  
The cross sub-superture averaging considers not only the covariance matrix of the same sub-array but also the covariance matrix between different subarrays. The expression of the cross-covariance matrix is:  
\[\mathbf{\overset{\wedge}{R}}=\frac{1}{\mathrm{P}^{2}}\sum_{\mathrm{i=1}}^{ \mathrm{P}}\sum_{\mathrm{j=1}}^{\mathrm{P}}\mathbf{G}_{\mathrm{i}}(\mathrm{k })\mathbf{G}_{\mathrm{j}}^{\mathrm{H}}(\mathrm{k}) \tag{14}\]  
where \(\mathrm{i}\), \(\mathrm{j}\) denote subarray numbers. By introducing the cross-covariance matrix, the main diagonal components of the sub-covariance matrix may not always be positive, so the increase of diagonal components in \(\mathbf{\overset{\wedge}{R}}\) will be avoided, which makes the estimation of the covariance matrix more stable and reliable.  
Observing Equation (14), the cross-covariance matrix requires \(\mathrm{P}^{2}\) matrix multiplications while the conventional covariance matrix requires \(\mathrm{P}\) times. This will lead to a huge increase in the computational cost to obtain images. Here, we propose to calculate the mean-subarray \(\bar{\mathbf{G}}\) first, then the cross-covariance matrix can be express as:  
\[\bar{\mathbf{G}}=\frac{1}{\mathrm{P}}\sum_{\mathrm{i=1}}^{\mathrm{P}}\mathbf{ G}_{\mathrm{i}}(\mathrm{k}) \tag{15}\]  
\[\mathbf{\overset{\wedge}{R}}=\bar{\mathbf{G}}\bar{\mathbf{G}}^{\mathrm{H}} \tag{16}\]  
According to Equations (15) and (16), we can obtain the cross-covariance matrix with a one-time matrix multiplication. Therefore, the computational cost is reduced significantly, which always means the faster imaging speed.  
In the present study, diagonal loading is applied to ensure that the matrix can be inverted correctly. It will add a tiny noise signal \(\varepsilon\) into the sample covariance matrix \(\mathbf{\overset{\wedge}{R}}\)[31]. The \(\varepsilon\) can be defined as:  
\[\varepsilon=\frac{1}{\Delta*\mathrm{I}}\;\mathrm{tr}\!\left(\mathbf{\overset{ \wedge}{R}}\right) \tag{17}\]  
\[\mathbf{\overset{\wedge}{R}}=\mathbf{\overset{\wedge}{R}}+\varepsilon\mathbf{I} \tag{18}\]  
where \(\mathrm{tr}(\cdot)\) is the trace of the matrix and \(\Delta\) is a constant whose value ranges from 10 to 100. In general, smaller \(\varepsilon\) provide the image better lateral resolution, but the computation may fail due to numerical instability [8; 9].