Exploiting Extrinsic Information for Serial MAP Detection by Utilizing Estimator in Holographic Data Storage Systems

Abstract

In the big data era, data are created in huge volume. This leads to the development of storage devices. Many technologies are proposed for the next generation of storage fields. However, among them, holographic data storage (HDS) has attracted much attention and has been introduced as the promising candidate to meet the increasing demand for capacity and speed. For signal processing, HDS faces two major challenges: inter-page interference (IPI) and two-dimensional (2D) interference. To access the IPI problem, we can use balanced coding, which converts user data into an intensity level with uniformly distributed values for each page. For 2D interference, we can use the equalizer and detection to mitigate the 2D interference. However, the often-used equalizer and detection are methods in wireless communication and only handle the one-dimensional (1D) signal. Thus, we can combine the equalizer, detection, and estimator to reduce 2D interference into 1D interference. In this paper, we proposed a combined model using serial maximum a posteriori (MAP) detection and estimator to improve the detection of HDS systems. In our proposed model, instead of using an estimator with the Viterbi algorithm to predict the upper–lower interference (UPI) or left–right interference (LRI) and converting the received signal into 1D ISI, we used the estimator to predict the extrinsic information for serial MAP detection. This preserves the 2D information in the received signal in serial MAP detection and improves the detection of serial MAP detection by extrinsic information. The simulation results demonstrate that our proposed model significantly improves the bit-error rate (BER) performance compared to previous studies.

1. Introduction

Nowadays, with the huge amount of data created from many devices in internet networks, conventional storage systems are approaching their limits. In addition, with wide application in many fields such as security [1,2,3], image processing [4], wireless communication [5], etc., artificial intelligence (AI) and machine learning (ML) models require storage devices with large volume capacity. Thus, many new technologies are proposed as the next generation for storage systems. Particularly, with the development of demand for capacity and speed rate, holographic data storage (HDS) systems have attracted much attention from researchers and are becoming a promising technology for the next generation [6,7]. In HDS, the composition material is an electro-optic crystal, Iron-doped Lithium Niobate (LiNbO₃:Fe). HDS stores the data by recording the interference between the wave-fronts of a modulated optical field, containing the data, and a reference optical field, as a refractive index variation inside the storage media [8]. Because of storing information as holographs, when the signal is retrieved from HDS, it is disturbed by inter-page interference (IPI) and two-dimensional (2D) interference [9]. To reduce IPI, we can use a balanced code to ensure an equal number of 0 and 1 bits [10]. In 2D interference, the desired signal is affected by horizontal interference (HI) according to the horizontal direction and vertical interference (VI) according to the vertical direction [11].

To solve the 2D interference in HDS systems, a modulation code with non-isolated patterns of codewords or detection techniques can be used. For modulation code, in [12], Chen and Chiueh introduced 6/8 modulation code in HDS systems. To improve the code rate and reduce the isolated patterns for the codewords, in [13], Tarng, Tseng, and Chen proposed a 2D modulation code for HDS systems. In addition, because the interference in bit-patterned magnetic recording (BPMR) is similar to the interference of HDS, we can apply the modulation code in BPMR systems to HDS systems. In [14], Nguyen and Lee presented 9/12 modulation code to mitigate 2D interference. Warisarn, Arrayangkool, and Kovintavewat proposed 5/6 modulation code to eliminate the patterns that lead to severe ITI [15]. Finally, recently, with development of AI and ML, the combination of modulation code and ML algorithms has also been proposed. In [16], a K-mean algorithm was used to decode 4/6 modulation code for a BPMR channel. Nguyen and Lee proposed non-isolated modulation code with a minimum Hamming distance of 3 for BPMR to reduce interference and improve the error correction of BPMR systems.

For detection, we used the Viterbi [17] and maximum a posteriori (MAP) [18] algorithms to detect the received signal. However, with the Viterbi and MAP algorithms, we only handle one-dimensional (1D) interference because these algorithms are designed for the 1D signal in wireless communication. Thus, to apply the Viterbi and MAP algorithms to HDS, we need to modify these algorithms to handle 2D signals. In addition, when applying the Viterbi and MAP algorithms, it is essential to accurately estimate the coefficients of the HDS channel. Thus, we used the method in [19] to compensate and predict the coefficients of the HDS channel. This method is inspired by the generalized partial response (GPR) target from [20]. In addition, we can use the serial or parallel structures in [21,22] to split the 2D interference into 1D interference, which is suitable for the Viterbi and MAP algorithms. Recently, interference estimators were introduced by Nguyen and Lee in [11] to predict vertical interference (VI) and horizontal interference (HI). Then, we can remove these interferences from the received signal to obtain the signal with 1D interference. Finally, we can apply deep learning (DL) models to the interference estimator to improve the performance of conventional estimator [11], utilizing the Viterbi algorithm to find out the interference. In [23], the authors proposed the interference estimator using the recurrent neural network (RNN) to improve the accuracy of estimation. Beyond the Viterbi algorithm, other detection methods have been derived from the BCJR algorithm. One such method is the iterative row–column soft-decision feedback algorithm (IRCSDFA), which was proposed to mitigate 2D interference but is computationally intensive [24]. To address this issue, Zheng et al. proposed a lower-complexity variant using a Gaussian approximation (IRCSDFA-GA). This version reduces computational complexity but is still more demanding than the Viterbi algorithm (VA) [25].

In this paper, a new scheme for serial MAP detection is introduced. In the proposed model, instead of using a parallel structure to exploit extrinsic information, we utilized the interference estimator in [11] to exploit the extrinsic information for serial MAP detection. Firstly, the received signal is split into two parts. One was fed to the interference estimator to extract the interference symbols and convert into the extrinsic information for serial MAP detection. To predict the probability of the six-level signals and mitigate HI, another received signal and the extrinsic information were processed through the horizontal MAP detection. To convert the interference into extrinsic information, we introduced the parameter $\alpha$ as the ratio of extracting the information into extrinsic information. Then, the probability of the six-level signal was detected by vertical MAP detection to remove VI and estimate the probability of the original signal. These probabilities can be used to create the iterative algorithm between the horizontal and vertical MAP detections. Finally, the probability of the original signal was compared to a certain threshold to obtain the original user data. The results from the simulation clearly indicate that our proposed model significantly improves the bit-error rate (BER) performance of HDS systems compared to previous studies.

Our contributions are as follows:

-

Proposed the method to convert the estimated interference into extrinsic information.

-

Found out the optimized value of $\alpha$ (the ratio of extracting the interference into extrinsic information).

-

Analyzed and explained the combination of the interference estimator and the serial MAP detection.

-

Presented comprehensive simulation results to validate the effectiveness of the proposed model. The results demonstrate significant improvements in BER performance compared to existing approaches, confirming the robustness of our method.

The rest of this paper is structured as follows: Section 2 provides a brief introduction to the background of the GPR target, interference estimator, and serial MAP detection. Section 3 provides a detailed discussion of the channel modeling for HDS systems, along with a comprehensive explanation of the proposed scheme. In Section 4, we present and discuss the simulation results. Lastly, Section 5 concludes the paper.

2. Background

2.1. 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.

In this system, the equalizer F and target G are assigned the following forms. The parameters of G are provided to the detection process to facilitate the removal of interference from z[j,k].

$$G=\left[\begin{array}{ccc}{g}_{-1,-1}& g_{-1,0}& g_{-1,1}\\ g_{0,-1}& g_{0,0}& g_{0,1}\\ g_{1,-1}& g_{1,0}& g_{1,1}\end{array}\right],$$

(1)

$$F=\left[\begin{array}{ccccc}{f}_{-2,2}& f_{-2,-1}& f_{-2,0}& f_{-2,1}& f_{-2,2}\\ f_{-1,-2}& f_{-1,-1}& f_{-1,0}& f_{-1,1}& f_{-1,2}\\ f_{0,-2}& f_{0,-1}& f_{0,0}& f_{0,1}& f_{0,2}\\ f_{1,-2}& f_{1,-1}& f_{1,0}& f_{1,1}& f_{1,2}\\ f_{2,-2}& f_{2,-1}& f_{2,0}& f_{2,1}& f_{2,2}\end{array}\right].$$

(2)

Then, the matrices G and F are converted into vector forms g and f as below. We have rearranged the elements of the matrix, row by row, into a column vector.

$$g={\left[\begin{array}{cccc}{g}_{-1,-1}& g_{-1,0}& \dots & g_{1,1}\end{array}\right]}^T,$$

(3)

$$f={\left[\begin{array}{cccc}{f}_{-2,-2}& f_{-2,-1}& \dots & f_{2,2}\end{array}\right]}^T.$$

(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:

$$\begin{array}{l}\left[\begin{array}{ccc}a\left[j-1,k-1\right]& a\left[j-1,k\right]& a\left[j-1,k+1\right]\\ a\left[j,k-1\right]& a\left[j,k\right]& a\left[j,k+1\right]\\ a\left[j+1,k-1\right]& a\left[j+1,k\right]& a\left[j+1,k+1\right]\end{array}\right]\\ \Rightarrow a={\left[\begin{array}{cccc}a\left[j+1,k+1\right]& a\left[j+1,k\right]& \dots & a\left[j-1,k-1\right]\end{array}\right]}^T\end{array},$$

(5)

$$\begin{array}{l}\left[\begin{array}{ccccc}y\left[j-2,k-2\right]& y\left[j-2,k-1\right]& y\left[j-2,k\right]& y\left[j-2,k+1\right]& y\left[j-2,k+2\right]\\ y\left[j-1,k-2\right]& y\left[j-1,k-1\right]& y\left[j-1,k\right]& y\left[j-1,k+1\right]& y\left[j-1,k+2\right]\\ y\left[j,k-2\right]& y\left[j,k-1\right]& y\left[j,k\right]& y\left[j,k+1\right]& y\left[j,k+2\right]\\ y\left[j+1,k-2\right]& y\left[j+1,k-1\right]& y\left[j+1,k\right]& y\left[j+1,k+1\right]& y\left[j+1,k+2\right]\\ y\left[j+2,k-2\right]& y\left[j+2,k-1\right]& y\left[j+2,k\right]& y\left[j+2,k+1\right]& y\left[j+2,k+2\right]\end{array}\right]\\ \Rightarrow y={\left[\begin{array}{cccc}y\left[j+2,k+2\right]& y\left[j+2,k+1\right]& \dots & y\left[j-2,k-2\right]\end{array}\right]}^T\end{array}.$$

(6)

We observe that the vectors a and y have been rearranged in the reverse order of their original matrices. This rearrangement occurs because the convolution operation in the equalizer F inverts the data as the signal passes through it. Consequently, during the estimation process, we must also invert the vectors a and y. The equations below were used to calculate the signals d[j,k] and z[j,k].

$$d\left[j,k\right]=g^{T}a,$$

(7)

$$z\left[j,k\right]=f^{T}y.$$

(8)

To choose the parameters of g and f, we optimized the below expression.

$$e\left[j,k\right]=E\left\{{\left(z\left[j,k\right]-d\left[j,k\right]\right)}^2\right\},$$

(9)

where E{.} denotes the expectation (9), which is expanded as follows:

$$\begin{array}{l}e\left[j,k\right]=E\left\{{\left(f^{T}y-g^{T}a\right)}^2\right\}\\ =f^{T}E\left\{yy\right\}f-2f^{T}E\left\{y{a}^T\right\}g+g^{T}E\left\{{aa}^T\right\}g\end{array},$$

(10)

To avoid f = g = 0, we added constraints as below to (9).

$$C^{T}g=\alpha ,$$

(11)

where $\alpha$ is constant in, and

$$C^T=\left[\begin{array}{ccccccccc}0& 0& 0& 0& 1& 0& 0& 0& 0\end{array}\right],$$

(12)

With (10) and (12), we can present the optimized problem as below.

$$\begin{array}{l}\underset{f,g}{\arg \min }\left(e\left[j,k\right]\right)\\ \mathrm{s}.\mathrm{t}. C^{T}g=\alpha \end{array}$$

(13)

To solve (13), the Lagrange function is used as below.

$$L=e\left[j,k\right]-2\lambda \left(C^{T}g-\alpha \right),$$

(14)

where $\lambda$ is a vector containing the Lagrange multipliers. The answers of (14) are presented as follows:

$$\lambda ={\left(C^T{\left(E\left\{a{a}^T\right\}-E{\left\{ya\right\}}^{T}E{\left\{yy\right\}}^{-1}E\left\{ya\right\}\right)}^{-1}C\right)}^{-1}\alpha ,$$

(15)

$$g={\left(E\left\{a{a}^T\right\}-E{\left\{y{a}^T\right\}}^{T}E\left\{y{y}^T\right\}E\left\{y{a}^T\right\}\right)}^{-1}C\lambda , \mathrm{and}$$

(16)

$$f=E{\left\{y{y}^T\right\}}^{-1}E\left\{ya\right\}g.$$

(17)

After using (15)–(17) to obtain g and f, we can analyze the GPR target as below.

$$G=\left[\begin{array}{ccc}{g}_{-1,-1}& g_{-1,0}& g_{-1,1}\\ g_{0,-1}& g_{0,0}& g_{0,1}\\ g_{1,-1}& g_{1,0}& g_{1,1}\end{array}\right]=\left[\begin{array}{ccc}pr& p{l}_2& pr\\ l_{1}r& l_1{l}_2& l_{1}r\\ pr& p{l}_2& pr\end{array}\right]=\left[\begin{array}{c}p\\ l_1\\ p\end{array}\right]\left[\begin{array}{ccc}r& l_2& r\end{array}\right],$$

(18)

where r and p are the horizontal and vertical interferences, respectively, and l₁ and l₂ are the general parameters to fit the form of the GPR target G. We set l₁ = 1 and obtain the parameters as below.

$$r=g_{0,1},$$

(19)

$$p={\displaystyle \frac{g_{1,0}}{g_{0,0}}},$$

(20)

$$l_2=g_{0,0}.$$

(21)

2.2. Interference Estimator

The received signal y[j,k] was supplied in two parts. The first is fed to the VI/HI estimator to predict VI and HI. The second is the equalizer part, which converts y[j,k] into z[j,k]. Signal z[j,k] is approximate to the desired signal d[j,k], which is interfered with by the 2D target G. After equalizing the received signal y[j,k] into z[j,k], we can present the signal z[j,k] as follows:

$$\begin{array}{ll}z[j,k]& \approx d\left[j,k\right]\\ & \approx a\left[j+1,k+1\right]rp+a\left[j+1,k\right]p{l}_2+a\left[j+1,k-1\right]rp\\ & +a\left[j,k+1\right]r{l}_1+a\left[j,k\right]l_1{l}_2+a\left[j,k-1\right]r{l}_1\\ & +a\left[j-1,k+1\right]rp+a\left[j-1,k\right]p{l}_2+a\left[j-1,k-1\right]rp\end{array}.$$

(22)

The interferences [rl₁ l₁l₂ rl₁] and [pl₂ l₁l₂ pl₂] are from the horizontal and vertical main tracks, respectively. The interference [rp pl₂ rp] is from the upper and lower tracks. In this study, we define [rp pl₂ rp] as VI. The interference [rp rl₁ rp]^T is from the left and right tracks and assigned as HI. Then, we define a vector v as follows:

$$v=\left[\begin{array}{ccc}v\left[j,k+1\right]& v\left[j,k\right]& v\left[j,k-1\right]\end{array}\right],$$

(23)

where

$$v\left[j,k+1\right]=a\left[j+1,k+1\right]rp+a\left[j,k+1\right]r{l}_1+a\left[j-1,k+1\right]rp,$$

(24)

$$v\left[j,k\right]=a\left[j+1,k\right]p{l}_2+a\left[j,k\right]l_1{l}_2+a\left[j-1,k\right]p{l}_2,$$

(25)

$$v\left[j,k-1\right]=a\left[j+1,k-1\right]rp+a\left[j,k-1\right]r{l}_1+a\left[j-1,k-1\right]rp,$$

(26)

Using the values in

Table 1. Values of v.

v[j,k − 1]	v[j,k]	v[j,k + 1]
−2rp − pl2	−2rl1 − l1l2	−2rp − pl2
−pl2	−l1l2	−pl2
2rp − pl2	−2rl1 + l1l2	2rp − pl2
−2rp + pl2	2rl1 − l1l2	−2rp + pl2
pl2	l1l2	pl2
2rp + pl2	2rl1 + l1l2	2rp + pl2

, we can detect v to obtain HI by employing a trellis with 36 states for [v[j,k + 1], v[j,k]], where each state has six output branches for v[j,k − 1]. The Viterbi algorithm is then applied to this trellis to detect the values of v[j,k]. This process is referred to as hard detection or hard output. We further modified these detected values into the probabilities of v[j,k], which are analogous to the extrinsic information in the MAP algorithm. The modified method is detailed in Section 3.3. Similarly, the vector v can be grounded as shown below to detect VI.

$$v=\left[\begin{array}{ccc}v\left[j+1,k\right]& v\left[j,k\right]& v\left[j-1,k\right]\end{array}\right],$$

(27)

where

$$v\left[j+1,k\right]=a\left[j+1,k+1\right]rp+a\left[j+1,k\right]p{l}_2+a\left[j+1,k-1\right]rp,$$

(28)

$$v\left[j,k\right]=a\left[j,k+1\right]r{l}_1+a\left[j,k\right]l_1{l}_2+a\left[j,k-1\right]r{l}_1,$$

(29)

$$v\left[j-1,k\right]=a\left[j-1,k+1\right]rp+a\left[j-1,k\right]p{l}_2+a\left[j-1,k-1\right]rp,$$

(30)

3. Proposed Model

3.1. HDS Channel

In this channel environment, the blurring effect and misalignment are represented by a point spread function (PSF), while the noise is characterized as additive white Gaussian noise (AWGN). The digital user signal u[k] is modulated into 2D data a[j,k] to ensure compatibility with the medium used in HDS systems. The output of the HDS channel is captured by a camera sensor and converted into a 2D array, forming a digital image of the data. The received signal y[j,k] is distorted in the following manner:

$$y\left[j,k\right]=a\left[j,k\right]\otimes h\left[j,k\right]+w\left[j,k\right],$$

(31)

where $⨂$ is the 2D convolution operator. h[j,k] is the discrete signal as a point spread function (PSF) of 2D interference in the HDS channel and can be calculated as below. Equation (32) represents the energy of h(x,y) in the bit region, which is detected by the camera sensor.

$$h\left[j,k\right]={\displaystyle \underset{j-\frac{1}{2}}{\overset{j+\frac{1}{2}}{\int }}{\displaystyle \underset{k-\frac{1}{2}}{\overset{k+\frac{1}{2}}{\int }}h\left(x,y\right)dxdy}},$$

(32)

where h(x,y) is a continuous PSF as the below definition.

$$h\left(x,y\right)={\displaystyle \frac{1}{{\sigma }_b^2}}\sin c^2\left({\displaystyle \frac{x-m_x}{{\sigma }_b}},{\displaystyle \frac{y-m_y}{{\sigma }_b}}\right),$$

(33)

where sin c(x,y) = (sin($\pi$x)/$\pi$x)(sin($\pi$y)/$\pi$y); ${\sigma }_b$ is the blur grade; and m_x and m_y are the misalignments in the horizontal and vertical directions, respectively. In this paper, the PSF h[j,k] has a size of 3 $\times$ 3 array pixels. Finally, w[j,k] is the additive white Gaussian noise (AWGN) with zero mean and variance ${\sigma }_w^2$. In addition, SNR is defined as follows.

$$\mathrm{SNR}=10{\log }_{10}\left({\displaystyle \frac{1}{{\sigma }_w^2}}\right),$$

(34)

3.2. Serial MAP Detection

With the parameters r, p, l₁, and l₂, we can use the serial detection in Figure 2 for the received signal. The MAP algorithm is applied to horizontal detection and vertical detection as follows.

The signal z[j,k] is similar to the signal influenced by the vertical interference [p l₂ p]^T and the horizontal interference [r l₁ r]. Thus, serial detection consists of horizontal detection followed by vertical detection. After applying horizontal detection to z[j,k], we obtain the signal b[j,k], which is similar to a[j,k] $⨂$ [p l₂ p]^T. Since b[j,k] is the output of horizontal detection, it no longer contains the horizontal interference factor r but retains the vertical interference factor p. Given a[j,k] $\in$ {$-$1;1}, the possible values of a[j,k] $⨂$ [p l₂ p]^T are {$-$l₂ $-$ 2p; $-$l₂; $-$l₂ $+$ 2p; l₂ $-$ 2p; l₂; l₂ + 2p}. After horizontal detection, only vertical interference remains in the received signal. Similarly, vertical detection can then be applied to b[j,k] to recover $\widehat{a}$[j,k], which closely approximates the original signal a[j,k].

• Horizontal detection

The signal b[j,k] can be estimated as below without noise in channel.

$$b\left[j,k\right]=a\left[j,k\right]\otimes {\left[\begin{array}{ccc}p{l}_2& l_1{l}_2& p{l}_2\end{array}\right]}^T,$$

(35)

where $⨂$ is the convolution operator. Due to a[j,k] being modulated and represented as {−1/1}, the signal b[j,k] $\in$ {−2p − l₁; −l₁; 2p − l₁; −2p + l₁; l₁; 2p + l₁}. We assigned the values of b[j,k] as the symbols {R₁ = −2p − l₁; R₂ = −l₁; R₃ = 2p − l₁; R₄ = −2p + l₁; R₅ = l₁; R₆ = 2p + l₁}. With six symbols, horizontal detection was performed using a trellis with thirty-six states, each having six branches. The conditional probability P(R_i|z) for the six symbols was calculated using the equation below.

$$\begin{array}{l}P\left(R_i|z\right)={\displaystyle \sum _{R_i}P\left(s^{\prime },s|z\right)}={\displaystyle \sum _{S_i}\frac{P\left(s\prime ,s|z\right)}{P\left(z\right)}}\\ ={\displaystyle \sum _{S_i}\frac{P\left(s\prime ,z_{<k}\right)P\left(z_k,s|s\prime \right)P\left(z_{>k}|s\right)}{P\left(z\right)}}\\ ={\displaystyle \sum _{S_i}\frac{{\alpha }_{k-1}\left(s\prime \right){\gamma }_k\left(s\prime ,s\right){\beta }_k\left(s\right)}{P\left(z\right)}}\\ \mathrm{with} i=1, 2, \dots , 6\end{array},$$

(36)

where s′, s, and z are the previous state, current state, and received symbol of the MAP algorithm, respectively. ${\alpha }_{k-1}\left(s\prime \right)=$ P($s\prime$, z_<k) is the joint probability of the previous state s′ and sequence received in the past z_<k. ${\gamma }_k\left(s^{\prime },s\right)=$ P(z_k, s|$s\prime$) represents the conditional probability of the current state s and the current received symbol z_k given the previous state $s\prime$, and ${\beta }_k\left(s\right)=$ P(z_>k|s) is the conditional probability of the future received sequence z_>k under the current state s. Next, we calculate the branch transition probability as below.

$${\gamma }_k\left(s\prime ,s\right)=P\left(z|Z_i\right)P\left(R_i\right)=\exp \left(-{\displaystyle \frac{{\left(z-Z_i\right)}^2}{2{\sigma }^2}}\right)P_e\left(R_i\right),$$

(37)

where $\mathrm{z}$ represents the value of z[j,k] and ${\mathrm{Z}}_{\mathrm{i}}$ represents the discrete value of $\mathrm{z}$ in the noiseless scenario (216 symbols on the trellis). The probability distribution P(z|Z_i) is modeled using a Gaussian function. Additionally, P_e(R_i) is the extrinsic information obtained from vertical detection and used to create the iterative algorithm in serial MAP detection.

• Vertical detection

In vertical detection, the probability of values in signal a[j,k] is estimated. As a result, vertical detection employed the trellis with 4 states, where each state has 2 branches. The branch transition probability ${\gamma }_k\left(s^{\prime },s\right)$ is estimated as follows.

$${\gamma }_k\left(s\prime ,s\right)={\displaystyle \frac{P\left(R_i|z\right)}{{\displaystyle \sum _{i}P\left(R_i|z\right)}}},$$

(38)

The probabilities of values {−1, 1} in signal a[j,k] are calculated by using R_i as follows:

$$P\left(a=\pm 1|R\right)={\displaystyle \sum _{\begin{array}{l}\left(s\prime ,s\right)\\ a=\pm 1\end{array}}P\left(s\prime ,s,R\right)},$$

(39)

where a is the value of the signal a[j,k].

3.3. Extracting the Extrinsic Information for Serial MAP Detection

According to Section 2.2, the output of the VI/HI estimators is the vector with the below forms.

HI estimator:

$$v=\left[\begin{array}{ccc}v\left[j,k+1\right]& v\left[j,k\right]& v\left[j,k-1\right]\end{array}\right],$$

(40)

VI estimator:

$$v=\left[\begin{array}{ccc}v\left[j+1,k\right]& v\left[j,k\right]& v\left[j-1,k\right]\end{array}\right],$$

(41)

In this study, because we exploit the extrinsic information for horizontal MAP detection in serial MAP detection, we only use the results of the HI estimator in (38). With (38), we can collect all the values of v[j,k] $\in$ {R₁ = −2p − l₁; R₂ = −l₁; R₃ = 2p − l₁; R₄ = −2p + l₁; R₅ = l₁; R₆ = 2p + l₁}. Then, to calculate the extrinsic information when the value of v[j,k] = R_i, we used the below equation to convert the values of v[j,k] into probabilities of R_i and other symbols.

$$\left\{\begin{array}{c}{P}_0\left(R_i\right)=\alpha 0\le \alpha \le 1\\ P_0\left(R_q\right)={\displaystyle \frac{1-\alpha }{5}} \mathrm{with} q\ne i\end{array}\right.$$

(42)

From (35), we give the value of P₀(R_i) to P_e(R_i) (P₀(R_i) = P_e(R_i)). Regularly, P_e(R_i), in conventional methods without feedback from vertical detection, is equal to 1 or the same probabilities for all symbols. This leads to the low performance of horizontal detection in an initialized loop when there is no feedback from vertical detection. However, with our proposed model, we can supply the extrinsic information by using P₀(R_i) for the horizontal detection in the initialized loop and improve the performance of system. In addition, $\alpha$ is the ratio to extract the extrinsic information. Simultaneously, to find out the value of $\alpha$, we implemented the survey as described in Section 4.

For extrinsic information from the vertical detection, we consider

Table 2. Estimating of P_e(R_i).

Ri	Pe(Ri)
−l1 − 2p	Pa(−1)Pa(−1)Pa(−1)
−l1	Pa(1)Pa(−1)Pa(−1) + Pa(−1)Pa(−1)Pa(1)
−l1 + 2p	Pa(1)Pa(−1)Pa(1)
l1 − 2p	Pa(−1)Pa(1)Pa(−1)
l1	Pa(1)Pa(1)Pa(−1) + Pa(−1)Pa(1)Pa(1)
l1 + 2p	Pa(1)Pa(1)Pa(1)

to convert the output of the vertical detection into the probabilities of symbols R_i.

After detecting the initialized loop, we can obtain extrinsic information from the vertical detection. Thus, at the first loop, we supplied the results of P_e(R_i) from Table 2 to (35) to improve the performance of horizontal MAP detection.

In Figure 3, the signal z[j,k], contains both the original signal and 2D interference. The algorithm in Section 2.2 helps isolate the signal affected only by 1D interference, which closely resembles b[j,k]. Next, we modified the output of the “VI/HI Estimator” to obtain probability values, which were then supplied as extrinsic information for the horizontal MAP detection.

4. Results and Discussions

In this study, we simulated the proposed model in Figure 3. A sequence of bits representing user data, u[k], of size 1,440,000, is generated randomly. In this sequence, the probabilities of 0 and 1 bits are the same. To obtain a page of data, u[k] is transformed into a[j,k], which has a size of 1200 $\times$ 1200. This transformation resembles the data format used in HDS system media. A data page is then fed into the channel to generate y[j,k]. To estimate the coefficients of F and G, a[j,k] and y[j,k] are supplied to Equations (15)–(17). To evaluate the BER performance of the proposed model, 10 data pages were used to go through the system. The output signal from HDS channel y[j,k] was fed to equalizer F to obtain z[j,k]. With the previous estimated coefficients, the forms of z[j,k] and d[j,k] were among the same. Thus, z[j,k] could be handled by detection with parameters from target G. In detection, we used horizontal and vertical MAP. This is serial MAP detection with extrinsic information to create the iterative algorithm. The output of horizontal MAP is b[j,k]. This signal included the probabilities of the symbols when the original is only affected by VI. Simultaneously, to improve the performance for horizontal MAP detection, we also used z[j,k] as the input to the VI/HI estimator to extract extrinsic information. The output of detection is compared with 0.5 as a threshold to detect bits 0 and 1 and obtained $\widehat{a}$[j,k]. Finally, we used demodulation to restore the original signal $\widehat{u}$[k]. $\widehat{u}$[k] is compared with u[k] to calculate BER.

In the first experiment, we varied the values of $\alpha$ from 0 to 1 with the step of 0.1 to choose the optimized value of $\alpha$. The results were shown in Figure 4. We can see that when $\alpha$ = 0.3, the system achieves the best performance. Thus, we choose $\alpha$ = 0.3 for our proposed model.

In the second experiment, we focused on evaluating the two-dimensional (2D) interference reduction capability of the proposed model. To achieve this, we configured the system with 0% misalignment and a blur factor of 1.85. In HDS systems, as the blur level increased, a greater portion of the light energy was scattered in various directions, reducing the concentration of energy at the intended pixels. The blur level in the system was a key variable associated with the HDS channel, directly affecting the distribution of light energy on the projection screen. Additionally, the simultaneous scattering of light caused 2D ISI, where the signal from one symbol interfered with adjacent symbols, further complicating the decoding process and reducing signal clarity.

In Figure 5, we can observe that the user data went through the systems according to the decision in the starting paragraph of Section 4. With “MAP and estimator without iteration”, we removed the feedback path from vertical MAP to horizontal MAP. With “MAP and estimator with iteration”, we simulated the model in Figure 3. We can see that the proposed model without iteration can achieve gains of ~0.6 dB and ~0.5 dB at a BER of 10⁻⁵ compared to a parallel ITI/ISI estimator [11] and serial MAP detection [22], respectively. If we added the iteration to the proposed model, the system could obtain gains of ~1.5 dB and ~5.5 dB at a BER of 10⁻⁵ compared to a parallel ITI/ISI estimator [11] and serial MAP detection [22], respectively.

In the following simulation, we introduce a 10% misalignment into the HDS channel while maintaining a blur value of 1.85. Misalignment in holographic data storage (HDS) refers to any deviation in the precise alignment of the key components involved in reading and writing data. Since HDS relies on the interference patterns of laser beams to store information within a 3D medium, even minor misalignment can degrade data integrity, reduce storage capacity, or lead to read/write errors. The behavior of these systems was simulated in a manner similar to the models shown in Figure 5. The results, presented in Figure 6, demonstrate that our proposed model continues to deliver the best performance under the 10% misalignment condition of the HDS channel. In addition, comparing with 0% misalignment, the BER performance of our proposed model is changed less than that of serial MAP detection [22] and a parallel ITI/ISI estimator [11]. This shows that our proposed model can resist the misalignment effect in HDS systems.

Next, we varied the blur from 1.8 to 3 while setting the SNR to 15 dB to evaluate the BER performance of the HDS channel under media noise conditions. The results, shown in Figure 7 for 0% misalignment and Figure 8 for 10% misalignment, indicate that our proposed model achieves optimal performance when the blur is less than 3. Consequently, our model can effectively resist blur levels up to 3.

To estimate the complexity of the proposed model, we counted the number of operations per detected bit, from the equalizer to the detection stage, for each method. For the equalizer with a size of 5 $\times$ 5, obtaining one element in the signal z[j,k] requires 25 multiplications and 24 additions. In our proposed model, the signal z[j,k] is supplied to the “HI/VI estimator” and “Horizontal MAP”.

For the “HI/VI estimator”, the number of input data points corresponds to the number of state transitions in the trellis diagram. To determine the number of operations per bit, we estimate the operations required for a single state transition. The diagram consists of 36 states, each with 6 input/output branches, resulting in a total of 216 branches. Therefore, calculating the branch metrics involves 432 additions and 216 multiplications.

For the “Horizontal MAP” algorithm, which is based on the trellis diagram, we calculate the operations for a single state transition, similar to the HI/VI estimator. Unlike the Viterbi algorithm, the MAP detection uses Equation (37) to compute γ for the branch metrics, which involves addition, multiplication, and exponential operations. Additionally, Equation (36) is used to determine the probabilities for each symbol on each branch. Consequently, the total number of operations for the “Horizontal MAP” algorithm is 657 additions, 1107 multiplications, and 312 exponential operations.

The “Vertical MAP” algorithm is similar to the “Horizontal MAP” algorithm; however, the number of states and branches is smaller. It consists of four states, with two input/output branches in each state, resulting in a total of eight branches. Consequently, the “Vertical MAP” algorithm requires 48 additions, 51 multiplications, and 21 exponential operations.

Finally, to achieve feedback from the “Vertical MAP” to the “Horizontal MAP”, the method in [22] was used. This process involves 4 additions, 18 multiplications, and 3 exponential operations.

Table 3. Complexity of the proposed model.

Methods	Mul/Div	Add/Sub	Log/Exp
Proposed model	1417	1165	336
Parallel ITI/ISI estimator in [11]	508	1007	0
Serial MAP detection in [22]	1201	733	336

presents the results for the number of operations in each method.

5. Conclusions

In this paper, we proposed a model that exploited extrinsic information using the VI/HI estimator. Firstly, we supplied the received signal to the VI/HI estimator to predict the HI. Then, the HI is converted to the probabilities of the symbols of horizontal MAP detection with the extraction ratio α. Finally, the output of vertical detection is used to predict the extrinsic information for horizontal detection and create the iterative algorithm for serial MAP detection. With the extrinsic information for the initialized loop, our proposed model can improve the performance of HDS systems compared to previous studies. The result simulation showed that our proposed model can obtain gains of ~1.5 and ~2.5 dB at BER of 10⁻⁵ compared with the models in [11,22]. In future research, we will exploit extrinsic information for vertical detection to improve serial MAP detection.