## 2 Background  
### Generalized Partial Response (GPR) Target and Equalizer  
To facilitate the analysis of the proposed model in this paper, we introduce the GPR target and the equalizer. These components are commonly used in channel estimation, particularly in scenarios involving 2D interference. The GPR target defines the coefficients for detection, while the equalizer compensates for channel distortions and converts the received signal into the desired signal. In Figure 1, the positions and roles of the equalizer and target of the HDS channel are presented. With 2D interference, directly converting the received signal back to the original signal is challenging. This difficulty arises because applying the equalizer directly amplifies white noise, turning it into colored noise, which degrades performance. To address this issue, instead of converting the received signal directly into the original signal, we first convert it into a desired signal where the interference coefficients are known. This approach enables effective detection using algorithms such as Viterbi or MAP. Figure 1: Equalizer and target in HDS systems. In this system, the equalizer \(\mathbf{F}\) and target \(\mathbf{G}\) are assigned the following forms. The parameters of \(\mathbf{G}\) are provided to the detection process to facilitate the removal of interference from \(z[j,k]\). \[\mathbf{G}= \begin{bmatrix}g_{-1,-1}&g_{-1,0}&g_{-1,1}\\ g_{0,-1}&g_{0,0}&g_{0,1}\\ g_{1,-1}&g_{1,0}&g_{1,1}\end{bmatrix} , \tag{1}\]  
\[\mathbf{F}= \begin{bmatrix}f_{-2,2}&f_{-2,-1}&f_{-2,0}&f_{-2,1}&f_{-2,2}\\ f_{-1,-2}&f_{-1,-1}&f_{-1,0}&f_{-1,1}&f_{-1,2}\\ f_{0,-2}&f_{0,-1}&f_{0,0}&f_{0,1}&f_{0,2}\\ f_{1,-2}&f_{1,-1}&f_{1,0}&f_{1,1}&f_{1,2}\\ f_{2,-2}&f_{2,-1}&f_{2,0}&f_{2,1}&f_{2,2}\end{bmatrix} . \tag{2}\]  
Then, the matrices \(\mathbf{G}\) and \(\mathbf{F}\) are converted into vector forms \(\mathbf{g}\) and \(\mathbf{f}\) as below. We have rearranged the elements of the matrix, row by row, into a column vector. \[\mathbf{g}= \begin{bmatrix}g_{-1,-1}&g_{-1,0}&\dots&g_{1,1}\end{bmatrix} ^{T}, \tag{3}\]  
\[\mathbf{f}= \begin{bmatrix}f_{-2,-2}&f_{-2,-1}&\dots&f_{2,2}\end{bmatrix} ^{T}. \tag{4}\]  
Here, \({}^{T}\) is the transpose operator. Simultaneously, the 2D signals of the input \(a[j,k]\) and output \(y[j,k]\) are rearranged row by row and transformed into column vectors, as shown below:  
\[ \begin{bmatrix}a[j-1,k-1]&a[j-1,k]&a[j-1,k+1]\\ a[j,k-1]&a[j,k]&a[j,k+1]\\ a[j+1,k-1]&a[j+1,k]&a[j+1,k+1]\end{bmatrix} , \tag{5}\]  
\[\Rightarrow\mathbf{a}= \begin{bmatrix}a[j+1,k+1]&a[j+1,k]&\dots&a[j-1,k-1] \end{bmatrix} ^{T}\]  
\[ \begin{bmatrix}y[j-2,k-2]&y[j-2,k-1]&y[j-2,k]&y[j-2,k+1]&y[j-2,k+2]\\ y[j-1,k-2]&y[j-1,k-1]&y[j-1,k]&y[j-1,k+1]&y[j-1,k+2]\\ y[j,k-2]&y[j,k-1]&y[j,k]&y[j,k+1]&y[j,k+2]\\ y[j+1,k-2]&y[j+1,k-1]&y[j+1,k]&y[j+1,k+1]&y[j+1,k+2]\\ y[j+2,k-2]&y[j+2,k-1]&y[j+2,k]&y[j+2,k+1]&y[j+2,k+2]\end{bmatrix} . \tag{6}\]  
We observe that the vectors \(\mathbf{a}\) and \(\mathbf{y}\) have been rearranged in the reverse order of their original matrices. This rearrangement occurs because the convolution operation in the equalizer \(\mathbf{F}\) inverts the data as the signal passes through it. Consequently, during the estimation process, we must also invert the vectors \(\mathbf{a}\) and \(\mathbf{y}\). The equations below were used to calculate the signals \(d[j,k]\) and \(z[j,k]\). \[d[j,k]=\mathbf{g}^{T}\mathbf{a}, \tag{7}\]  
\[z[j,k]=\mathbf{f}^{T}\mathbf{y}. \tag{8}\]  
To choose the parameters of \(\mathbf{g}\) and \(\mathbf{f}\), we optimized the below expression. \[e[j,k]=E\Big{\{}(z[j,k]-d[j,k])^{2}  
where E{.} denotes the expectation (9), which is expanded as follows:  
\[ \begin{split} e[j,k]&=E\bigg{\{}\Big{(}\mathbf{f}^{T} \mathbf{y}-\mathbf{g}^{T}\mathbf{a}\Big{)}^{2}\bigg{\}}\\ &=\mathbf{f}^{T}E\{\mathbf{y}\mathbf{y}\}\mathbf{f}-2\mathbf{f}^{T }E\{\mathbf{ya}^{T}\}\mathbf{g}+\mathbf{g}^{T}E\{\mathbf{aa}^{T}\}\mathbf{g} \end{split} , \tag{10}\]  
To avoid \(\mathbf{f}=\mathbf{g}=0\), we added constraints as below to (9).