# Simplified Two-Dimensional Generalized Partial Response Target of Holographic Data Storage Channel

^{*}

## Abstract

**:**

## 1. Introduction

^{−3}compared to the 1D GPR target in [16].

## 2. 2D GPR Target and Serial Detection

#### 2.1. 2D GPR Target

**G**with a size of 3 × 3, and the equalizer

**F**with a size of 5 × 5; these are presented as in the below matrices.

**G**and

**F**, we can achieve the signals d[j,k] and z[j,k], as follows:

**g**and

**f**in order that z[j,k] is close to d[j,k]. To implement this, we apply the minimum mean square error (MMSE) algorithm. This problem is presented as follows:

**f**=

**g**= 0, we must impose the constraint on

**f**or

**g**. The constraint is represented by the below expression.

**f**,

**g**, and $\lambda $ to zero vectors, we achieve the answers to (15) as follows:

**A**= E{

**aa**

^{T}},

**R**= E{

**yy**

^{T}}, and

**T**= E{

**ya**

^{T}}.

#### 2.2. Proposed Serial Detection

_{−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

**G**as given below.

_{1}and l

_{2}are the general parameters to fit the form of the GPR target

**G**. In other words, the 2D GPR target

**G**with 2D ISI is decomposed into a series of 1D vertical direction interference vector

**v**= [p l

_{1}p]

^{T}and 1D horizontal direction interference vector

**b**= [r l

_{2}r]. Then,

**v**and

**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.

**v**and

**b**, we achieve the following expression:

_{2}p)(rl

_{1}) = (l

_{1}l

_{2})(rp), the parameters of the matrix

**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}≈ g

_{−1,0}g

_{0,1}, which is close to the above condition. Therefore, we can match the parameters between the GPR target

**G**and the matrix

**vb**to determine the elements of the vector

**v**and

**b**. To simplify, we choose the parameter l

_{1}= 1 and determine other parameters with the formulas:

_{1}, and l

_{2}, we can achieve the vector

**v**and

**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

**v**and

**b**to recover the original data.

**v**and

**b**have the asymmetric form as below.

## 3. Proposed Model

**v**and

**b**.

_{x}and m

_{y}are x and y axis misalignments, respectively.

**G**). With the GPR target estimated earlier, the signal z[j,k] is detected by the serial detection and restored into the signal $\widehat{a}$[j,k]. Finally, the signal $\widehat{a}$[j,k] is decoded into the original signal $\widehat{u}$[k].

## 4. Simulation Results

#### 4.1. Results of Proposed Model

**v**and

**b**as in Table 1 and Table 2.

#### 4.2. Complexity of Proposed Model

## 5. Conclusions

^{−3}compared to the 1D GPR target [16] with 1.8 blur and 0% and 10% misalignments. In addition, our proposed method can still achieve the best performance when the blur increases.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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SNR | G | v | b |
---|---|---|---|

10 dB | $\left[\begin{array}{ccc}0.2& 0.4& 0.2\\ 0.4& 0.85& 0.4\\ 0.2& 0.4& 0.2\end{array}\right]$ | $\left[\begin{array}{c}0.4709\\ 1\\ 0.4709\end{array}\right]$ | $\left[\begin{array}{ccc}0.4023& 0.85& 0.4023\end{array}\right]$ |

11 dB | $\left[\begin{array}{ccc}0.199& 0.4& 0.199\\ 0.4& 0.85& 0.4\\ 0.199& 0.4& 0.199\end{array}\right]$ | $\left[\begin{array}{c}0.4743\\ 1\\ 0.4743\end{array}\right]$ | $\left[\begin{array}{ccc}0.4047& 0.85& 0.4047\end{array}\right]$ |

12 dB | $\left[\begin{array}{ccc}0.196& 0.4& 0.196\\ 0.4& 0.85& 0.4\\ 0.196& 0.4& 0.196\end{array}\right]$ | $\left[\begin{array}{c}0.4744\\ 1\\ 0.4744\end{array}\right]$ | $\left[\begin{array}{ccc}0.4053& 0.85& 0.4053\end{array}\right]$ |

SNR | G | v | b |
---|---|---|---|

10 dB | $\left[\begin{array}{ccc}0.21& 0.41& 0.2\\ 0.41& 0.85& 0.4\\ 0.2& 0.4& 0.199\end{array}\right]$ | $\left[\begin{array}{c}0.4877\\ 1\\ 0.4732\end{array}\right]$ | $\left[\begin{array}{ccc}0.41& 0.85& 0.4\end{array}\right]$ |

11 dB | $\left[\begin{array}{ccc}0.214& 0.417& 0.2\\ 0.417& 0.85& 0.4\\ 0.2& 0.4& 0.197\end{array}\right]$ | $\left[\begin{array}{c}0.4905\\ 1\\ 0.4705\end{array}\right]$ | $\left[\begin{array}{ccc}0.417& 0.85& 0.4\end{array}\right]$ |

12 dB | $\left[\begin{array}{ccc}0.212& 0.4189& 0.2038\\ 0.4189& 0.85& 0.4045\\ 0.2038& 0.4045& 0.1925\end{array}\right]$ | $\left[\begin{array}{c}0.4928\\ 1\\ 0.4758\end{array}\right]$ | $\left[\begin{array}{ccc}0.4189& 0.85& 0.4045\end{array}\right]$ |

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Nguyen, T.A.; Lee, J.
Simplified Two-Dimensional Generalized Partial Response Target of Holographic Data Storage Channel. *Appl. Sci.* **2022**, *12*, 4070.
https://doi.org/10.3390/app12084070

**AMA Style**

Nguyen TA, Lee J.
Simplified Two-Dimensional Generalized Partial Response Target of Holographic Data Storage Channel. *Applied Sciences*. 2022; 12(8):4070.
https://doi.org/10.3390/app12084070

**Chicago/Turabian Style**

Nguyen, Thien An, and Jaejin Lee.
2022. "Simplified Two-Dimensional Generalized Partial Response Target of Holographic Data Storage Channel" *Applied Sciences* 12, no. 8: 4070.
https://doi.org/10.3390/app12084070