## 3 Proposed System Model
### Equalizer and GPR Target  
In Figure 7, the equalizer is combined with the GPR target [22]. Here, the equalizer and GPR target are represented by matrix \(\mathbf{F}\) and matrix \(\mathbf{G}\), 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  
The output of equalizer \(z[j,k]\) is the 2D convolution of equalizer \(\mathbf{F}\) and channel output \(y[j,k]\). The output of target \(d[j,k]\) is also the 2D convolution of target \(\mathbf{G}\) and input data \(a[j,k]\). These outputs can be expressed as follows:  
\[z[j,k]=\mathbf{f}^{T}\mathbf{y} \tag{10}\]  
\[d[j,k]=\mathbf{g}^{T}\mathbf{a} \tag{11}\]  
where  
\[\mathbf{f}=\left[ \begin{array}{cccccccc}f_{-2,-2}&f_{-2,-1}&\ldots&0&\ldots &f_{2,1}&f_{2,2}\end{array} \right]^{T} \tag{12}\]  
\[\mathbf{g}=\left[ \begin{array}{cccccccc}g_{-1,-1}&g_{-1,0}&\ldots&0&\ldots &g_{1,0}&g_{1,1}\end{array} \right]^{T} \tag{13}\]  
\[\mathbf{y}=\left[ \begin{array}{cccccccc}y_{j+2,k+2}&y_{j+2,k+1}&\ldots&0& \ldots&y_{j-2,k-1}&y_{j-2,k-2}\end{array} \right]^{T} \tag{14}\]  
\[\mathbf{a}=\left[ \begin{array}{cccccccc}a_{j+1,k+1}&a_{j+1,k}&\ldots&0& \ldots&a_{j-1,k}&a_{j-1,k-1}\end{array} \right]^{T} \tag{15}\]  
We define the error signal \(e[j,k]\) between the equalizer and GPR target as follows:  
\[e[j,k]=z[j,k]-d[j,k] \tag{16}\]  
Using expressions (10), (11), and (16), we can calculate the error as follows:  
\[e[j,k]=\mathbf{f}^{T}\mathbf{y}-\mathbf{g}^{T}\mathbf{a} \tag{17}\]  
Then, the mean square error (MSE) can be expressed as follows:  
\[E\Big{\{}\left(e[j,k]\right)^{2}\Big{\}} =E\Bigg{\{}\left(\mathbf{f}^{T}\mathbf{y}-\mathbf{g}^{T}\mathbf{ a}\right)\left(\mathbf{f}^{T}\mathbf{y}-\mathbf{g}^{T}\mathbf{a}\right)^{T} \Bigg{\}} \tag{18}\] \[=\mathbf{f}^{T}\mathbf{R}\mathbf{f}-2\mathbf{f}^{T}\mathbf{Tg}+ \mathbf{g}^{T}\mathbf{Ag}\]  
where \(\mathbf{R}\) is the auto-correlation matrix of the channel output data, \(\mathbf{T}\) is the cross-correlation between the input data and the channel output data, and \(\mathbf{A}\) is the auto-correlation of the input data. Specifically, \(\mathbf{R}=E\{y\mathbf{y}^{T}\}\), \(\mathbf{T}=E\{ya^{T}\}\), and \(\mathbf{A}=E\{aa^{T}\}\), where \(E\) denotes the expectation, and \(T\) is the transpose operator. To invalidate the trivial condition of \(\mathbf{f}=\mathbf{g}=0\) to minimize the MSE in (18), a different constraint should be imposed on \(\mathbf{g}\) depending on the N-way GPR targets. First, we will set the constraint for the two-way GPR target. The matrix form \(\mathbf{G}\) for the two-way GPR target is given below. \[\mathbf{G}=\left[ \begin{array}{cccccccc}0&0&0\\ g_{0,-1}&1&g_{0,1}\\ 0&0&0\end{array} \right] \tag{19}\]  
\[\mathbf{g}=\left[ \begin{array}{cccccccc}0&0&0&g_{0,-1}&1&g_{0,1}&0&0&0\end{array}  \right]^{T} \tag{20}\]  
The constraint can be expressed as follows:  
\[\mathbf{E}^{T}\mathbf{g}=\mathbf{c} \tag{21}\]  
where  
\[\mathbf{c}=\left[ \begin{array}{cccccccc}0&0&0&1&0&0&0\end{array} \right]^{T} \tag{22}\]and  
\[\mathbf{E}^{T}=\left[ \begin{array}{cccccccccc}1&0&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\\ \end{array} \right] \tag{23}\]  
For the three-way GPR target on the upper \(\Pi\Pi\), the matrix form \(\mathbf{G}_{u}\) is as follows:  
\[\mathbf{G}_{u}=\left[ \begin{array}{cccc}0&g_{-1,0}&0\\ g_{0,-1}&1&g_{0,1}\\ 0&0&0\\ \end{array} \right] \tag{24}\]  
\[\mathbf{g}=\left[ \begin{array}{cccccc}0&g_{-1,0}&0&g_{0,-1}&1&g_{0,1}&0&0&0 \\ \end{array} \right]^{T} \tag{25}\]  
The constraint can be expressed as follows:  
\[\mathbf{E}^{T}\mathbf{g}=\mathbf{c} \tag{26}\]  
where  
\[\mathbf{c}=\left[ \begin{array}{cccccc}0&0&1&0&0&0\\ \end{array} \right]^{T} \tag{27}\]  
and  
\[\mathbf{E}^{T}=\left[ \begin{array}{cccccc}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&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&0&1\\ \end{array}