## 3 Proposed System Model
### Three-Way GPR Target and Three-Way MVA with Feedback  
In this section, we consider a channel represented by the following matrix form \(\mathbf{H}\):  
\[\mathbf{H}=\left[ \begin{array}{ccc}h_{-1,-1}&h_{-1,0}&h_{-1,1}\\ h_{0,-1}&h_{0,0}&h_{0,1}\\ h_{1,-1}&h_{1,0}&h_{1,1}\end{array} \right] \tag{43}\]  
The channel output (in the case of zero TMR) can be written as listed below, where \({}^{*}\) denotes convolution, the \(\mathbf{h}_{0,k}\) coefficients are \([h_{0,-1},h_{0,0},h_{0,1}]\), the \(\mathbf{h}_{-1,k}\) coefficients are \([h_{-1,-1},h_{-1,0},h_{-1,1}]\), and the \(\mathbf{h}_{1,k}\) coefficients are \([h_{1,-1},h_{1,0}\,h_{1,1}]\):  
\[ \begin{array}{rl}y[j,k]=&a[j,k]*\mathbf{h}_{0,k}\\ &+a[j-1,k]*\mathbf{h}_{-1,k}+a[j+1,k]*\mathbf{h}_{1,k}+n[j,k]\end{array}  \tag{44}\]  
The channel coefficients at the edges of \(\mathbf{H}\) (i.e., \(h_{-1,-1}\), \(h_{-1,1}\), \(h_{1,-1}\), and \(h_{1,1}\)) often have very small values. Similarly, for the equalizer, the interference coefficients at the edge can be zero (\(h_{-1,-1},h_{-1,1},h_{1,-1},h_{1,1}\approx 0\)). Therefore, we can rewrite the output of the channel as follows:  
\[y[j,k]\approx a[j,k]*\mathbf{h}_{0,k} \tag{45}\] \[+a[j-1,k]h_{-1,0}+a[j+1,k]h_{1,0}+n[j,k]\]  
In our proposed system, we use a three-way GPR target with the matrix form in (46) or (47). These targets are achieved when applying (40)-(42) with the appropriate constraints. The reason we did not choose a two-way GPR target is that it ignores the ITI information. Meanwhile, the four-way GPR target would create a large alias, which degrades the performance. We will explain this based on simulations in Section 4. If the TMR effect is \(+\Delta_{off}\), we use the form in (46). If the TMR effect is \(-\Delta_{off}\), we use the form in (47). In other words, based on \(g_{-1,0}\) and \(g_{1,0}\), we choose the form in (46) if \(g_{-1,0}\geq g_{1,0}\) (\(\alpha_{1}\geq\alpha_{2}\)) and we choose the form in (47) if \(g_{-1,0}<g_{1,0}\) (\(\alpha_{1}<\alpha_{2}\)). In this analysis, we assume the TMR effect is \(+\Delta_{off}\). The analysis is similar when the TMR effect is \(-\Delta_{off}\). \[\mathbf{G}=\left[ \begin{array}{ccc}0&\alpha_{1}&0\\ r&1&r\\ 0&0&0\end{array} \right] \tag{46}\]  
\[\mathbf{G}=\left[ \begin{array}{ccc}0&0&0\\ r&1&r\\ 0&\alpha_{2}&0\end{array} \right] \tag{47}\]  
The output signal \(d_{u}[j,k]\) of the GPR target is expressed as follows:  
\[d_{u}[j,k]=a[j,k]*\mathbf{g}_{0,k}+a[j-1,k]\alpha_{1} \tag{48}\]  
where \(\mathbf{g}_{0,k}\) coefficients are \([r\;1\;r]\). After performing the minimum mean square error (MMSE) algorithm, the output signal of equalizer \(z[j,k]\) will be close to the output signal of the GPR target. \[z[j,k]\approx d_{u}[j,k] \tag{49}\]  
We can rewrite (49) as follows:  
\[y[j,k]*\mathbf{F}=a[j,k]*\mathbf{g}_{0,k}+a[j-1,k]\alpha_{1}+v[j,k] \tag{50}\]  
where \(v[j,k]\) is the difference between \(z[j,k]\) and \(d_{u}[j,k]\). To increase the accuracy of the expression in (50), we must accurately estimate component \(v[j,k]\). To estimate \(v[j,k]\), we will utilize Equations (45) and (48). The ISI component \(a[j,k]^{*}\mathbf{h}_{0,k}\) in (45) is represented as \(a[j,k]^{*}\mathbf{g}_{0,k}\) in (48), and the ITI component \(a[j-1,k]h_{-1,0}\) in (45) is converted to \(a[j-1,k]\alpha_{1}\) in (48). Ideally, the remaining components, \(a[j+1,k]h_{1,0}\) and \(n[j,k]\) in (45), can be suppressed to zero. However, in practice, these components are not zero. In other words, \(v[j,k]\) includes the ITI component \(a[j+1,k]\) and the noise component \(n[j,k]\). Thus,  
\[v[j,k]\approx\epsilon a[j+1,k]+\beta n[j,k] \tag{51}\]  
Alternatively, we can disregard \(\beta n[j,k]\), which is relatively small (\(\beta\approx 0\)). Using (51), we replace \(v[j,k]\) in (50). \[y[j,k]*\mathbf{F}\approx a[j,k]*\mathbf{g}_{0,k}+a[j-1,k]\alpha_{1}+a[j+1,k]\varepsilon \tag{52}\]  
For the \(a[j-1,k]\alpha_{1}\) term, we design a feedback path with a factor of \(\alpha_{1}\) to eliminate this component from detection.