# Two-Dimensional Interference Estimator with Parallel Structure for Holographic Data Storage Channel
## 2 2D Equalizer and GPR Target  
During the training process, we applied the model shown in Figure 1 to estimate the coefficients of the equalizer and the GPR target. First, the original data \(u[k]\) are modulated into the 2D signal \(a[j,k]\), which is stored in a HDS channel. The received signal \(y[j,k]\) from the output of the HDS channel and the 2D signal \(a[j,k]\) are collected in the buffers. Thereafter, these data are supplied to the minimum mean square error (MMSE) algorithm. As shown in Figure 1, we use equalizer \(\mathbf{F}\) and GPR target \(\mathbf{G}\) with a size of \(5\times 5\) and \(3\times 3\), respectively. \[\mathbf{F}=\left[ \begin{array}{cccc}f_{-2,-2}&f_{-2,-1}&f_{-2,0}&f_{-2,1}&f_{ -2,2}\\ f_{-1,-2}&f_{-1,-1}&f_{-1,0}&f_{-1,1}&f_{-1,2}\\ f_{0,-2}&f_{0,-1}&f_{0,0}&f_{0,1}&f_{0,2}\\ f_{1,-2}&f_{1,-1}&f_{1,0}&f_{1,1}&f_{1,2}\\ f_{2,-2}&f_{2,-1}&f_{2,0}&f_{2,1}&f_{2,2}\end{array} \right], \tag{1}\]  
\[\mathbf{G}=\left[ \begin{array}{cccc}g_{-1,-1}&g_{-1,0}&g_{-1,1}\\ g_{0,-1}&g_{0,0}&g_{0,1}\\ g_{1,-1}&g_{1,0}&g_{1,1}\end{array} \right]. \tag{2}\]  
Signals \(z[j,k]\) and \(d[j,k]\) are presented according to the following expressions:  
\[z[j,k]=\sum_{m=-2}^{2}\sum_{n=-2}^{2}y[j-m,k-n]f_{m,n}, \tag{3}\]  
\[d[j,k]=\sum_{m=-1}^{1}\sum_{n=-1}^{1}a[j-m,k-n]g_{m,n}. \tag{4}\]  
We can expand Equations (3) and (4) as follows:  
\[ \begin{array}{c}z[j,k]=y[j+2,k+2]f_{-2,-2}+y[j+2,k+1]f_{-2,-1}\\ \ldots+y[j-2,k-1]f_{2,1}+y[j-2,k-2]f_{2,2}\end{array} , \tag{5}\]  
\[ \begin{array}{c}d[j,k]=a[j+1,k+1]g_{-1,-1}+a[j+1,k]g_{-1,0}\\ \ldots+a[j-1,k]g_{1,0}+a[j-1,k-1]g_{1,1}\end{array} . \tag{6}\]  
To make Equations (5) and (6) shorter, we assign the below vectors:  
\[\mathbf{y}=\big{[} \begin{array}{cccc}y[j+2,k+2]&\ldots&y[j-2,k-2]\end{array}  \big{]}^{T}, \tag{7}\]  
\[\mathbf{a}=\big{[} \begin{array}{cccc}a[j+1,k+1]&\ldots&a[j-1,k-1]\end{array}  \big{]}^{T}, \tag{8}\]  
\[\mathbf{f}=\big{[} \begin{array}{cccc}f_{-2,-2}&f_{-2,-1}&\ldots&f_{2,2} \end{array} \big{]}^{T}, \tag{9}\]  
\[\mathbf{g}=\big{[} \begin{array}{cccc}g_{-1,-1}&g_{-1,-0}&\ldots&g_{1,1}\end{array}  \big{]}^{T}. \tag{10}\]  
Based on the definitions from Equations (7)-(10), we can rewrite Equations (5) and (6) as follows:  
\[z[j,k]=\mathbf{f}^{T}\mathbf{y}, \tag{11}\]  
\[d[j,k]=\mathbf{g}^{T}\mathbf{a}. \tag{12}\]  
We calculate the mean square error (MSE) according to following expression:  
\[MSE=E\bigg{\{}\Big{(}\mathbf{f}^{T}\mathbf{y}-\mathbf{g}^{T}\mathbf{a}\Big{)} ^{2}\bigg{\}}. \tag{13}\]  
For the HDS channel, regularly, the estimated form of GPR target \(\mathbf{G}\) is symmetric. Thus, we can assign matrix \(\mathbf{G}\) as follows:  
\[\mathbf{G}=\left[ \begin{array}{ccc}t&p&t\\ r&1&r\\ t&p&t\end{array} \right], \tag{25}\]  
where \(r=g_{0,-1}=g_{0,1}\) is the coefficient of interference from the horizontal direction; \(p=g_{-1,0}=g_{1,0}\) is the coefficient of interference from the vertical direction; and \(t=g_{-1,-1}=g_{-1,1}=g_{1,-1}=g_{1,1}\) is the coefficient of interference from the diagonal direction. We can rewrite Equation (24) as follows:  
\[ \begin{array}{c}d[j,k]=a[j+1,k+1]t+a[j+1,k]p+a[j+1,k-1]t\\ +a[j,k+1]r+a[j,k]+a[j,k-1]r\\ +a[j-1,k+1]t+a[j-1,k]p+a[j-1,k-1]t\end{array} . \tag{26}\]  
Based on Equation (26), we define left-right interference (LRI) as the interference from parameter \(\left[ \begin{array}{ccc}t&r&t\\ \end{array} \right]^{T}\); upper-lower interference (ULI) as the interference from parameter \(\left[ \begin{array}{ccc}t&p&t\\ \end{array} \right]\); HI as the interference from parameter \(\left[ \begin{array}{ccc}p&1&p\\ \end{array} \right]^{T}\). In addition, we make a group vector as follows. \[\left[ \begin{array}{ccc}v[j,k+1]&v[j,k]&v[j,k-1]\\ \end{array}