# Simplified Two-Dimensional Generalized Partial Response Target of Holographic Data Storage Channel
## 2 2D GPR Target and Serial Detection
### Proposed Serial Detection  
After the coefficients of the GPR target are estimated by the MMSE algorithm, the GPR target has a matrix form like (1). In the HDS channel, the read-back signal is the Gaussian beam. Therefore, the interference coefficients from the adjacent pixels can be given by the values from the Gaussian function. Based on these coefficients, we assume the conditions: \(g_{-1,0}=g_{1,0}=g_{0,1}=g_{0,-1}\) and \(g_{-1,-1}=g_{-1,1}=g_{1,-1}=g_{1,1}\). Then, we can decompose \(\mathbf{G}\) as given below.  
\[\mathbf{G}=\left[\begin{array}{ccc}g_{-1,-1}&g_{-1,0}&g_{-1,1}\\ g_{0,-1}&g_{0,0}&g_{0,1}\\ g_{1,-1}&g_{1,0}&g_{1,1}\end{array}\right]=\left[\begin{array}{ccc}p\\ l_{1}\\ p\end{array}\right]\left[\begin{array}{ccc}r&l_{2}&r\end{array}\right]. \tag{20}\]  
where \(r\) and \(p\) are the horizontal and vertical interferences, respectively; \(l_{1}\) and \(l_{2}\) are the general parameters to fit the form of the GPR target \(\mathbf{G}\). In other words, the 2D GPR target \(\mathbf{G}\) with 2D ISI is decomposed into a series of 1D vertical direction interference vector \(\mathbf{v}=[p~{}l_{1}~{}p]^{T}\) and 1D horizontal direction interference vector \(\mathbf{b}=[r~{}l_{2}~{}r]\). Then, \(\mathbf{v}\) and \(\mathbf{b}\) act like the outer and inner encoders, respectively. Therefore, we detect the horizontal direction first and the vertical direction next. The serial detection of the ideal channel model is illustrated in Figure 2.  
With the proposed form \(\mathbf{v}\) and \(\mathbf{b}\), we achieve the following expression:  
\[\mathbf{G}=\left[\begin{array}{ccc}rp&l_{2}p&rp\\ rl_{1}&l_{1}l_{2}&rl_{1}\\ rp&l_{2}p&rp\end{array}\right], \tag{21}\]  
Since \((l_{2}p)(rl_{1})=(l_{1}l_{2})(rp)\), the parameters of the matrix \(\mathbf{G}\) must satisfy the condition \(g_{0,0}g_{-1,1}=g_{-1,0}g_{0,1}\). When we implement the estimation of the GPR target for the HDS channel, the coefficients of the GPR target shows \(g_{0,0}g_{-1,1}\approx g_{-1,0}g_{0,1}\), which is close to the above condition. Therefore, we can match the parameters between the GPR target \(\mathbf{G}\) and the matrix \(\mathbf{v}\mathbf{b}\) to determine the elements of the vector \(\mathbf{v}\) and \(\mathbf{b}\). To simplify, we choose the parameter \(l_{1}=1\) and determine other parameters with the formulas:  
\[r=g_{0,1}, \tag{22}\]  
\[p=\frac{g_{1,0}}{g_{0,0}}, \tag{23}\]  
\[l_{2}=g_{0,0}. \tag{24}\]  
With the parameters \(r\), \(p\), \(l_{1}\), and \(l_{2}\), we can achieve the vector \(\mathbf{v}\) and \(\mathbf{b}\). Therefore, we can apply two 1D Viterbi detectors for the serial detection [18; 20; 21] with the interference coefficient supplied from the target vector \(\mathbf{v}\) and \(\mathbf{b}\) to recover the original data.  
Figure 2: Ideal channel and detection model.  
In addition, when misalignment occurs in the HDS channel, the vectors \(\mathbf{v}\) and \(\mathbf{b}\) have the asymmetric form as below.  
\[\mathbf{G}=\left[\begin{array}{ccc}\mathcal{S}_{-1,-1}&\mathcal{S}_{-1,0}& \mathcal{S}_{-1,1}\\ \mathcal{S}_{0,-1}&\mathcal{S}_{0,0}&\mathcal{S}_{0,1}\\ \mathcal{S}_{1,-1}&\mathcal{S}_{1,0}&\mathcal{S}_{1,1}\end{array}\right]= \left[\begin{array}{c}p_{1}\\ l_{1}\\ p_{2}\end{array}\right]\left[\begin{array}{ccc}r_{1}&l_{2}&r_{2}\\ r_{1}l_{1}&l_{1}l_{2}&r_{2}\\ r_{1}p_{2}&l_{2}p_{2}&r_{2}p_{2}\end{array}\right]. \tag{25}\]  
Similarly, we can achieve the approximated parameters as below.  
\[l_{1}=1;l_{2}=g_{0,0}, \tag{26}\]  
\[r_{1}=g_{0,-1};r_{2}=g_{0,1}, \tag{27}\]  
\[p_{1}=\frac{\mathcal{S}_{-1,0}}{l_{2}};p_{2}=\frac{\mathcal{S}_{1,0}}{l_{2}}. \tag{28}\]