## 2 Background
### Generalized Sidelobe Canceler (GSC)  
The GSC algorithm was originally proposed by [PERSON] [17]. For adaptive algorithms, the beamformers should follow the principle of suppressing the interference and noise signal while maintaining the desired signal. This principle can be express as a mathematical constraint as follow [23]:  
\[\text{min}\mathbf{\omega}^{H}\mathbf{R}\mathbf{\omega},\text{ subject to }\mathbf{\omega}^{H} \mathbf{a}=1 \tag{4}\]  
where \(\mathbf{R}\) is the covariance matrix of the interference and noise signal, and \(\mathbf{a}\) denotes a steering vector. In the present study, \(\mathbf{a}\) becomes a vector of ones because the echo data has been delayed and focused. Equation (4) can be solved by utilizing the Lagrange method, and the optimal weight vector is given by [24]  
\[\mathbf{\omega}=\frac{\mathbf{R}^{-1}\mathbf{a}}{\mathbf{a}^{H}\mathbf{R}^{-1} \mathbf{a}} \tag{5}\]  
Equation (5) is usually used as the adaptive weight of MV beamformer, but GSC will decompose it further. Figure 1 shows the schematic diagram of the GSC. It can separate the linear constraint with an adaptive filter, thereby converting the constrained optimization problem of Equation (4) into the unconstrained optimization problem. The weight vector \(\mathbf{\omega}\) is decomposed into an adaptive weight vector \(\mathbf{\omega}_{a}\) and a non-adaptive weight vector \(\mathbf{\omega}_{q}\). Moreover, the \(\mathbf{\omega}_{a}\) locates in the constrained subspace while the \(\mathbf{\omega}_{q}\) is orthogonal to the constrained subspace. This structure can overcome the problem of signal cancellation.  
As we can see in Figure 1, the upper part is the non-adaptive branch where all the signals can pass and determine the non-adaptive weight \(\mathbf{\omega}_{q}\). Moreover, we set \(\mathbf{\omega}_{q}\) parallel to the desired signal \(\mathbf{S}(k)\) to ensure that the desired signal can pass through the branch without loss. Meanwhile, the bottom half is the adaptive branch where blocking matrix \(\mathbf{B}\) is provided to block the desired signal [25]. Thus, only interference and noise signals can pass and determine the adaptive weight \(\mathbf{\omega}_{a}\). Finally, by subtracting the adaptive branch output \(\text{Y}_{b}(k)\) from the non-adaptive branch output \(\text{Y}_{a}(k)\), we can remove the interference and noise signals from the non-adaptive branch while retaining the desired signal. The weight vector of GSC can be expressed as:  
\[\mathbf{\omega}=\mathbf{\omega}_{q}-\mathbf{B}\mathbf{\omega}_{a} \tag{6}\]  
Then, we can redefine the constraints of Equation (4) using Equation (6) as  
\[\mathbf{\omega}_{a}=\text{arg}\min_{\mathbf{\omega}_{a}}\Bigl{(}\bigl{(}(\mathbf{\omega} _{q}-\mathbf{B}\mathbf{\omega}_{a})^{H}\mathbf{R}\bigl{(}\mathbf{\omega}_{q}-\mathbf{ B}\mathbf{\omega}_{a}\bigr{)}\Bigr{)} \tag{7}\]  
Figure 1: Structure of generalized sidelobe canceler.  
the expressions of \(\mathbf{\omega}_{\rm q}\) and \(\mathbf{\omega}_{\rm a}\) are:  
\[\mathbf{\omega}_{\rm q}=\left(\mathbf{\rm aa}^{\rm H}\right)^{-1}\mathbf{\rm a} \tag{8}\]  
\[\mathbf{\omega}_{\rm a}=\left(\mathbf{\rm B}^{\rm H}\mathbf{\rm R}\mathbf{\rm B}\right)^{-1}\!\! \mathbf{\rm B}^{\rm H}\mathbf{\rm R}\mathbf{\omega}_{\rm q} \tag{9}\]  
where the blocking matrix \(\mathbf{\rm B}\) is an M*(M\(-1\)) dimensional matrix and is orthogonal to the desired signal [26; 27]. In the present study, \(\mathbf{\rm B}\) can be set as:  
\[\mathbf{\rm B}^{\rm H}\mathbf{\rm a}=0 \tag{10}\]  
\[\mathbf{\rm B}=\left[\begin{array}{cccccc}1&-1&0&0&\cdots&0\\ 0&1&-1&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&\cdots&0&1&-1\end{array}\right]^{\rm T} \tag{11}\]