## 3 Proposed Model
### Equalizer  
To implement a 2D GPR equalizer, we need to use a buffer for processing each data page. In the training process, the parameters of the equalizer and GPR target are estimated by solving Equation (1). We define the equalizer and GPR target as matrices \(\mathbf{F}\) and \(\mathbf{G}\), respectively.  
\[\mathbf{F}=\left[\begin{array}{cccc}f_{-2,-2}&f_{-2,-1}&f_{-2,0}&f_{-2,1}&f_ {-2,2}\\ f_{-1,-2}&f_{-1,-1}&f_{-1,0}&f_{-1,1}&f_{-1,2}\\ f_{0,-2}&f_{0,-1}&f_{0,0}&f_{0,1}&f_{0,2}\\ f_{1,-2}&f_{1,-1}&f_{1,0}&f_{1,1}&f_{1,2}\\ f_{2,-2}&f_{2,-1}&f_{2,0}&f_{2,1}&f_{2,2}\end{array}\right], \tag{12}\]  
\[\mathbf{G}=\left[\begin{array}{cccc}g_{-1,-1}&g_{-1,0}&g_{-1,1}\\ g_{0,-1}&g_{0,0}&g_{0,1}\\ g_{1,-1}&g_{1,0}&g_{1,1}\end{array}\right], \tag{13}\]We convert the matrix \(\mathbf{F}\) and matrix \(\mathbf{G}\) into the vector \(\mathbf{f}\) and vector \(\mathbf{g}\) as below:  
\[\mathbf{f}=\left[\begin{array}{cccccccc}f_{-2,-2}&f_{-2,-1}&\dots&0&\dots&f_{ 2,1}&f_{2,2}\end{array}\right]^{T}, \tag{14}\]  
\[\mathbf{g}=\left[\begin{array}{cccccccc}g_{-1,-1}&g_{-1,0}&\dots&0&\dots&g_{ 1,0}&g_{1,1}\end{array}\right]^{T}, \tag{15}\]  
From (14) and (15), the desired signal \(d[j,k]\) and the equalized signal \(z[j,k]\) can be expressed as follows:  
\[d[j,k]=\mathbf{g}^{T}\mathbf{a} \tag{16}\]  
\[z[j,k]=\mathbf{f}^{T}\mathbf{y}\mathbf{a}=1 \tag{17}\]  
where  
\[\mathbf{a}=\left[\begin{array}{cccccccc}a[j+1,k+1]&a[j+1,k]&\dots&0&\dots&a[j -1,k-1]\end{array}\right]^{T} \tag{18}\]  
\[\mathbf{y}=\left[\begin{array}{cccccccc}y[j+2,k+2]&y[j+2,k+1]&\dots&0&\dots &y[j-2,k-2]\end{array}\right]^{T} \tag{19}\]  
The error signal \(e[j,k]\) is presented as follows:  
\[e[j,k]=z[j,k]-d[j,k]=\mathbf{f}^{T}\mathbf{y}-\mathbf{g}^{T}\mathbf{a}. \tag{20}\]  
Then, the MSE can be expressed as follows:  
\[E\Big{\{}(e[j,k])^{2}\Big{\}} =E\bigg{\{}\Big{(}\mathbf{f}^{T}\mathbf{y}-\mathbf{g}^{T}\mathbf{ a}\Big{)}\left(\mathbf{f}^{T}\mathbf{y}-\mathbf{g}^{T}\mathbf{a}\right)^{T} \bigg{\}}\] \[=\mathbf{f}^{T}\mathbf{R}\mathbf{f}-2\mathbf{f}^{T}\mathbf{T} \mathbf{g}+\mathbf{g}^{T}\mathbf{A}\mathbf{g}.\]  
where \(\mathbf{R}\) is the autocorrelation matrix of the channel output data, \(\mathbf{R}=E[y\mathbf{f}^{T}]\); \(\mathbf{T}\) is the cross-correlation between the input data and the channel output data, \(\mathbf{T}=E[\mathbf{\psi}\mathbf{f}^{T}]\); and \(\mathbf{A}\) is the autocorrelation of the input data, \(\mathbf{A}=E[a\mathbf{f}^{T}]\), where \(E\) denotes the expectation, and \(T\) is the transpose operator.  
In (1), the constraint can be expressed as follows:  
\[\mathbf{E}^{T}\mathbf{g}=\mathbf{b}, \tag{22}\]  
where  
\[\mathbf{b}=\left[\begin{array}{cccccccc}0&0&1&0&0\end{array}\right]^{T}, \tag{23}\]  
and  
\[\mathbf{E}^{T}=\left[\begin{array}{cccccccc}1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&0&1\end{array}\right]. \tag{24}\]  
Based on these constraints, we derive the Lagrange multiplier, optimal target, and equalizer coefficient vectors as follows [18]:  
\[\lambda=\bigg{(}\mathbf{E}^{T}\Big{(}\mathbf{A}-\mathbf{T}^{T}\mathbf{R}^{-1 }\mathbf{T}\Big{)}^{-1}\mathbf{E}\bigg{)}^{-1}\mathbf{b}, \tag{25}\]  
\[\mathbf{g}=\Big{(}\mathbf{A}-\mathbf{T}^{T}\mathbf{R}^{-1}\mathbf{T}\Big{)}^{ -1}\mathbf{E}\lambda, \tag{26}\]  
\[g_{-1,1}=g_{1,-1}=g_{1,1}=g_{-1,-1}=g_{-1,0}.g_{1,-1}, \tag{27}\]  
\[\mathbf{f}=\mathbf{R}^{-1}\mathbf{T}\mathbf{g}, \tag{28}\]  
Equation (27) is a constraint that ensures the form of matrix \(\mathbf{G}\) in [18]. These parameters are used in the testing process.