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Abstract
Phase modulated holographic storage offers superior storage capacity and a longer life span compared with other storage technologies. However, its application is limited by its high raw bit error rate. We aimed to introduce low-density parity-check (LDPC) codes for data protection in phase modulated holographic storage systems. However, traditional LDPC codes can not fully exploit data error characteristics, causing inaccurate initial log-likelihood ratio (LLR) information, which degrades decoding performance, thus limiting the improvement degree of data reliability in phase modulated holographic storage. Therefore, we propose a reliable bit aware LDPC optimization method (RaLDPC) that analyzes and employs phase demodulation characteristics to obtain reliable bits. More accurate initial LLR weights are assigned to these reliable bits. Hence, the optimized initial LLR can reflect the reliability of the demodulated data more accurately. Experimental results show that RaLDPC can reduce the bit error rate by an average of 38.89% compared with the traditional LDPC code, improving the data reliability of phase modulated holographic storage.
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1. Introduction
In the era of explosive data growth, holographic storage, which is a three-dimensional storage technology that uses optical information to record data, has become one of the storage technologies with the most potential [1–4]. Compared with other storage technologies, such as hard disk drives, Blu-ray disks, and solid-state drives, holographic storage offers a large storage capacity and high data transfer rate [5,6]. There are two types of holographic storage, which depend on the type of light information that is used, such as the amplitude-modulated holographic storage [7–12] and phase-modulated holographic storage [13–17]. Amplitude-modulated holographic storage records data by utilizing information about the amplitude of light waves, whereas phase modulated holographic storage records data using information regarding the phase of the light. Compared with amplitude-modulated holographic storage, phase modulated holographic storage has advantages, such as homogeneous recording [18] and high encoding rate [4], indicating its potential to become the most promising method for implementing holographic storage. Nevertheless, the data recording is affected by the complex noise of the system and the limitation of phase reconstruction methods [4,17,19–22]. Therefore, the received images and the reconstructed phase are prone to errors. The high data error rate has a negative impact on the development of phase modulated holographic storage. Therefore, improving the data reliability of phase modulated holographic storage has attracted significant attention from researchers.
Although various studies have focused on improving the signal-to-noise ratio or reducing the noise of the phase modulated holographic storage [19–21,23–25], employing error correction code technologies is still essential to ensure data reliability. This has led to the extensive application of low-density parity-check (LDPC) codes [26,27] in amplitude-modulated holographic storage [28–32]. However, in their current forms, these schemes are not suitable for phase modulated holographic storage. This motivated us to introduce LDPC codes to phase modulated holographic storage to provide strong protection for user data. LDPC decoding algorithms can be divided into two categories: hard-decision decoding and soft-decision decoding. Compared with hard-decision decoding, soft-decision decoding has a higher error correction capability. However, the initial log-likelihood ratio (LLR) corresponding to each bit used in soft-decision decoding depends on the probability density function of the modulated state. No study has yet provided an accurate channel model and the probability density function of each modulated phase for the phase-modulated holographic storage system due to various complex noises, such as electrical noise, medium noise, optical noise, inter-page crosstalk, inter-pixel crosstalk, image misalignment, and lens aberration [21,23,33,34]. As a result, it is not viable to execute soft-decision decoding using the relevant formula to explicitly compute initial LLR information. If the conventional LDPC is used directly in the phase-modulated holographic storage system, only hard-decision decoding can be performed, and the initial LLR can only be obtained by the demodulated data bit. The initial LLR computed in this manner does not account for the data characteristics during the reading process and is thus insufficiently precise. The inaccurate initial LLR degrades the decoding performance, thus preventing the significant improvement in data reliability.
To ensure the data reliability of phase modulated holographic storage, we propose the reliable bit aware LDPC optimization method (RaLDPC). Specifically, our initial analysis of the phase demodulation process revealed that the read patterns obtained by phase reconstruction can be divided into four categories. The data errors mainly originate from one of these four categories. Then, we analyze all state shifts of this category from the record patterns to the read patterns. Although we cannot accurately determine the phase data of this category, we can correctly determine the logical values of some de-mapped data bits when the read pattern of the specific category is in some states. These bits are referred to as reliable bits. The reliable bits, which come from the analysis of demodulated data of the phase modulated holographic storage, can serve as a prior information to assist LDPC decoding. Based on the characteristics of reliable bits, we design RaLDPC to assign higher initial LLR weights to those reliable bits, which is equivalent to introducing additional soft information for the initial LLR of each bit. Therefore, the initial LLR can reflect the reliability of demodulated bit data more accurately. Finally, we implement and evaluate RaLDPC on a real phase modulated holographic storage system. The experimental results show that RaLDPC can significantly reduce the bit error rate of the phase modulated holographic storage system.
2. Data reliability of phase modulated holographic storage
2.1 Phase modulated holographic storage theory
Figure 1 shows the schematic diagram of a typical phase modulated holographic storage system. In the recording process, user data are converted into signal data by phase modulation. The signal and known reference data are uploaded on the spatial light modulator (SLM) to generate signal and reference beams [20]. Then, the signal beam interferes with the reference beam in the holographic media to form gratings [12] carrying data, completing the data recording process. In the reading process, the above reference beam is used to irradiate the holographic media, and the gratings are diffracted to generate a reconstruction beam. Subsequently, the Fourier intensity information of the reconstruction beam is captured by a complementary metal-oxide-semiconductor transistor (CMOS) image sensor [20]. The obtained information is exploited to extract the user data through phase demodulation.
2.2 Phase modulation and demodulation
Phase modulation is the process of transforming user data into phase data. Holographic data storage systems usually use 4-level phase modulation [21,24] to improve the code rates. The four phase values with $P1=0$, $P2=\frac {{\pi }}{2}$, $P3=\pi$, and $P4=\frac {{3\pi }}{2}$ correspond to four binary symbols $C1=11$, $C2=01$, $C3=00$, and $C4=10$, respectively. Before recording data, the user data are converted into phase data, which are organized into two-dimensional phase data pages. Then, these pages are uploaded on the SLM to record data. On the SLM, a pixel group that consists of multiple pixels is utilized to present one-phase data to meet the sampling requirements of the CMOS image sensor [4,20,25]. To facilitate description, we regarded the phase type of a pixel group before uploading it to the SLM as the record pattern. Figure 2 shows the types of record patterns for a pixel group containing $2\times 2$ pixels.
Phase demodulation is the process of reconstructing phase data from the reconstruction beam and retrieving user data. Phase reconstruction algorithms include interferometry methods [17,24,35,36] and non-interferometric methods [4,20,25,37]. Systems that use non-interferometric methods are more stable than those using interferometry methods. The iterative Fourier transform algorithm (IFA) [4,20,25] is widely used in non-interferometric methods. Nevertheless, the IFA requires extensive iteration to reconstruct the phase data. Lin et al. [4] proposed an optimized IFA (OIFA) based on embedded data. The OIFA greatly reduces the need for iteration, and it is one of the most practical phase reconstruction methods. However, errors inevitably occur in reconstructed pixel groups, owing to noise from the system and the limitation of phase reconstruction methods. The phase types of reconstructed pixel groups are known as read patterns. There are 256 read patterns with 4-level phase modulation (i.e., ${{\textrm {4}}^{\textrm {4}}}$) because errors may occur at each pixel in a pixel group. Figure 3 shows several read patterns. After reconstructing the phase data page, all read patterns on this page are changed into phase data using the phase downsampling method. Phase downsampling methods commonly select the phase values with the largest number of pixels from the pixel group as the reconstructed phase. If the number of phase values with the largest number of pixels is greater than 1, the reconstructed phase data are any of these phase values. Then, the phase data are converted into user data by phase demapping. For example, one of the read patterns on a reconstructed phase data page is the same as the read pattern presented in Fig. 3(e) (i.e., $(\frac {{3\pi }}{2}, \frac {{3\pi }}{2}, 0, \frac {{3\pi }}{2})$), in which case the phase value of this read pattern is $\frac {{3\pi }}{2}$ after downsampling. The user data after phase demapping are the binary symbol “10.”
2.3 Data reliability and error correction codes
The data of each phase of a phase data page only has one record pattern, but there are 256 types of read patterns in its corresponding reconstructed phase data. Among these 256 read patterns, 255 read patterns show errors of different levels. User data obtained through these error patterns may also have errors. Although the number of errors can be reduced by increasing the signal-to-noise ratio or degrading the noise [21,24,25], data errors still exist and are non-negligible. Error correction code technologies, such as LDPC codes, can correct errors and ensure data reliability. Previous studies focused on the effective use of LDPC codes in amplitude-modulated holographic storage to improve data reliability. However, these schemes are not suitable for phase modulated holographic storage. We aim to effectively introduce LDPC codes into phase modulated holographic storage to provide strong protection for user data.
Both encoding and decoding of LDPC codes are based on the parity check matrix H. Figure 4 shows a parity check matrix H with four rows and eight columns. This matrix can also be represented by a Tanner graph [27]. ${V_j} (1 \le j \le 8)$ is the variable node, representing the data bits of each LDPC code word. ${C_i} (1 \le i \le 4)$ is the check node and also known as the check equation, denoting the constraint relationship between bits in each LDPC code word. LDPC encoding is accomplished by generating parity data bits for user data bits (i.e., information bits) using H. LDPC decoding is realized by circularly passing LLR information between variable and check nodes [26] of the Tanner graph. Before the beginning of LDPC decoding, it is necessary to assign each variable node an initial LLR. The accuracy of the initial LLR determines the decoding performance of LDPC codes. A more accurate initial LLR can accelerate the convergence of decoding to the correct state. An inaccurate initial LLR significantly limits the error correction capability of LDPC codes and even causes decoding failure. Directly using traditional LDPC codes in phase modulated holographic storage systems enables the initial LLR information to be obtained only by the phase-demodulated bit data. Common practice is to set the initial LLR information to +1 for bit data of 1; otherwise, it is set to -1. Here, “+” and “-” indicate the polarity of the initial LLR, and “1” represents the weight of the initial LLR. The weights of all initial LLR in the traditional LDPC code are the same, which inaccurately reflects the reliability variation of each demodulated bit data. Therefore, the initial LLR is inaccurate, which degrades the error correction performance of LDPC codes and negatively impacts upon the data reliability of the system. The improvement of the data reliability of phase modulated holographic storage systems makes it necessary to propose an optimized LDPC scheme that fully considers phase modulation and phase demodulation characteristics.
3. Phase data characteristics
3.1 Read pattern
We analyzed 256 read patterns and established that they can be divided into four types: case1, case2, case3, and case4, respectively. Case1 only has one phase type ${P_i}$. Case2 contains two phase types ${P_i}$ and ${P_j}$. Case3 comprises three phase types ${P_i}$, ${P_j}$, and ${P_k}$. Case4 includes four phase types ${P_i}$, ${P_j}$, ${P_k}$, and ${P_l}$. The subscripts i, j, k, and l belong to the set $\{1, 2, 3, 4\}$ and are not equal to each other. Figure 5 shows examples of case1, case2, case3 and case4 respectively. We experimentally count the proportion of each type in all phase errors of a page group. A total of 1000 page groups were tested, and each page group consists of 18 phase data pages, i.e., 18000 phase data pages. The experimental setup is described in Section 5. As shown in Fig. 6, case2 has the highest proportion of errors. Therefore, case2 is the main source of errors. Next, we analyzed the data characteristics of case2.
3.2 State shift
We analyzed all state shifts of case2 from the record to the read patterns. As shown in Fig. 7, case2 has four types of state shift, such as SS1, SS2, SS3, and SS4, each of which has a different probability of occurrence. We counted the percentage of these four state shift using 18,000 phase data pages via experiments. The average proportions of SS1, SS2, SS3, and SS4 are 98.06%, 1.89%, 0.025%, and 0.025%, respectively, with the proportions of SS3 and SS4 are extremely small. Therefore, we can roughly determine that the recorded phase is ${P_i}$ or ${P_j}$ when the read pattern is case2.
3.3 Reliable bit
Table 1 lists our analysis of all values of ${P_i}$ and ${P_j}$ in case2. ${P_i}$ and ${P_j}$ form the phase pair $({P_i},{P_j})$. ${C_i}$ and ${C_j}$ represent the binary symbols demodulated by ${P_i}$ and ${P_j}$, respectively. ${C_i}$ and ${C_j}$ constitute the binary symbol pair $({C_i},{C_j})$. In the table, MSB and LSB represent the most significant and least significant bits of a binary symbol, respectively. The “x” indicates that the correct value of the current bit is uncertain and may be 0 or 1. Although ${P_i}$ and ${P_j}$ are different, their ${C_i}$ and ${C_j}$ have the same MSB or LSB in most situations. For example, both LSBs of their binary symbols ${C_i}$ and ${C_j}$ are 1 when $({P_i},{P_j})$ belongs to Situation1 in Table 1. We refer to the bits that take the same value for both binary symbols as reliable bits. That is, regardless of whether the recorded phase is $P i$ or $P j$, the reliable bit occurs in most situations when the read pattern is case2. Reliable bits are more responsible than other bits because of their deterministic values. Similarly, reliable bits also exist in larger pixel groups. We exploited the characteristics of reliable bits to optimize the LDPC code for enhancing the error correction capability of LDPC codes, thus improving the reliability of phase modulated holographic storage.
Table 1. All values of phase pair $({P_i},{P_j})$ in case2
4. Proposed RaLDPC
4.1 Overview of RaLDPC
Figure 8 presents a data flow diagram of phase modulated holographic storage systems that use LDPC codes as error correction codes. The introduction of LDPC codes into phase modulated holographic storage requires the data to be processed by LDPC encoding and decoding modules in the record and read process, respectively. The green dashed box indicates the submodules of traditional LDPC decoding, in which the initial LLR only depends on the logical value of the demodulated bit data that produces an inaccurate initial LLR. The blue dashed box encloses the phase demodulation submodules. The analysis described in Section 3 ensures that the demodulated bit data have the reliable bit characteristic. Based on this characteristic, we proposed RaLDPC, in which the submodules are enclosed within the red dashed box in Fig. 8. Compared to traditional LDPC codes, RaLDPC optimizes the initial LLR information. Specifically, the LLR polarity is determined by the logical values of the demodulated bit data, but the LLR weight depends on the reliable bit characteristic. To identify the reliable bits in the demodulated bit data, RaLDPC adds two submodules, pattern identification, and reliable bit recognition. The initial LLR of those reliable bits has greater weights than that of other bits because of the greater reliability of the reliable bits. Then, the accurate initial LLR information is utilized to optimize LDPC decoding.
4.2 Pattern identification
As mentioned in Section 3.1, read patterns can be divided into four categories, each with different phase types. In particular, the read pattern of case2 contains two types of phases, in which case, a reliable bit exists in the phase-demodulated binary symbol. Therefore, to recognize pixel groups whose read patterns are case2, we need to identify the read patterns of these pixel groups. For each pixel group obtained by phase reconstruction, we recorded the number of phase types. When the number of phase types is 1, 2, 3, and 4, their corresponding read patterns are case1, case2, case3, and case4, respectively. After completing the pattern identification, reliable bits can be confirmed by those two-phase types of case2, as shown in Table 1.
4.3 Reliable bit recognition
We recognized the reliable bits by first obtaining $({P_i},{P_j})$ and $({C_i},{C_j})$ of case2. If the MSBs or LSBs of ${C_i}$ and ${C_j}$ are the same, the corresponding bit is a reliable bit. Figure 9 shows an example of the process of recognizing a reliable bit. After reconstructing a phase data page by the OIFA, the read patterns that belong to case2 are first selected through pattern identification. In Fig. 9, the read patterns of ②, ⑤, ⑥, and ⑦ are case2. Then, the phase pairs of ②, ⑤, ⑥, and ⑦ can be obtained as $({P_2},{P_3})$, $({P_1},{P_2})$, $({P_1},{P_3})$, and $({P_3},{P_4})$, respectively. Their corresponding binary symbol pairs (01,00), (11,01), (11,00), and (00,10) can be obtained by phase demapping. Finally, the MSBs and LSBs of each binary symbol pair are compared. The MSB of ②, the LSB of ⑤, and the LSB of ⑦ are the reliable bits, and the other bits are non-reliable.
4.4 Initial LLR configuration
Configuration of the initial LLR includes setting the polarity and weight of the initial LLR. The initial LLR polarity depends on phase-demodulated bit data, and the initial LLR weight relies on the reliable bit characteristic. Figure 10 shows the process in which the initial LLR is configured. The branches on the left and right are used to set the LLR polarity and to determine the weight of the LLR, respectively. In setting the initial LLR polarity, if the logical value of a bit is 1, its LLR polarity is positive and indicated by “+,” as shown in Fig. 10. Otherwise, its initial LLR polarity is negative and indicated by the “-” sign. When setting the initial LLR weight, if a bit is a reliable bit, its initial LLR weight is $k$; otherwise, it is 1. Noticeably, $k>1$ is required because a reliable bit has much higher reliability than a non-reliable bit. When $k>1$ is satisfied, the difference in reliability between the reliable and non-reliable bits can be reflected. Therefore, the value of k has no strict restriction, except for $k>1$. In this study, $k$ is set to 2. Compared with traditional LDPC codes, the initial LLR information obtained by RaLDPC can reflect the reliability of each demodulated bit more accurately.
4.5 Decoding process
LDPC decoding is an iterative process in which the LLR information between variable nodes and check nodes is cyclically updated. The initial LLR is used to activate the decoding process. RaLDPC can obtain more accurate initial LLR information using the reliable bit characteristic. The accurate initial LLR can improve the decoding accuracy. The RaLDPC decoding algorithm is as follows:
- Step1: Obtain the initial LLR polarity $\vec {LP}$ of $\vec {D}$
$\vec {D}=(D_1,D_2,\ldots,D_{n-1},D_n)$ is a demodulated bit sequence and $n$ is the length of a LDPC code word. Using phase demodulation, the logical value of each bit in the LDPC code word can be obtained. If the logical value ${D_j}$ of the $jth$ bit is 1, then its initial LLR polarity ${LP_j}$ is "+"; otherwise, its ${LP_j}$ is “-."
- Step2: Obtain the initial LLR weights $\vec {LW}$ of $\vec {D}$
The reliable-bit flag sequence $\vec {S}=(S_1,S_2,\ldots,S_{n-1},S_n)$ can be obtained through pattern identification and reliable bit recognition. If the reliable-bit flag ${S_j}$ of the $jth$ bit is 1, then the initial LLR weight ${LW_j}$ of the $jth$ bit is set at $k$; otherwise, its ${LW_j}$ is set at 1.
- Step3: Obtain the optimal initial LLR $\vec {L}$ of $\vec {D}$
Through ${LP_j}$ and ${LW_j}$ of the $jth$ bit, the optimal initial LLR ${L_j}$ of the $jth$ bit can be obtained using the formula below.
- Step4: Initialize variable nodes
The variable nodes can be initialized by
where ${v_j}$ is the $jth$ variable node and ${c_i}$ is the $ith$ check node. $BL_{{v_j} \to {c_i}}^{0}$ represents the LLR information passed by the variable node to the check node before iterative decoding, and 0 indicates that the iterative decoding has not started. - Step5: Update LLR information of check nodes
Set the maximum number of decoding iterations as $Kmax$ and start the decoding process. The check node information $CL_{{c_i} \to {v_j}}^{t + 1}$ can be updated by
$$CL_{{c_i} \to {v_j}}^{t + 1} = \alpha \prod_{{v_x} \in E({c_i})\backslash {v_j}} {{\mathop{\textrm sgn}} (BL_{{v_x} \to {c_i}}^{t})} \mathop {\min }_{{v_x} \in E({c_i})\backslash {v_j}} (BL_{{v_x} \to {c_i}}^{t}),$$where $CL_{{c_i} \to {v_j}}^{t + 1}$ is the LLR information passed from the $i$th check node to the $j$th variable node in the $(t+1)$th iteration, and $0 \le t \le Kmax$. $BL_{{v_x} \to {c_i}}^{t}$ represents the LLR information passed from the $x$th variable node to the $i$th check node in the $t$th iteration. $E(i)$ is the set of all variable nodes that have the connection relationship with the $ith$ check node. The connection relationship is determined by the parity check matrix $H$ [26] of LDPC codes. $E(i)\backslash j$ denotes the removal of the $jth$ variable node from $E(i)$. - Step6: Update the LLR information of variable nodes
After updating the LLR information of all check nodes through step5, the variable node can be updated using
$$BL_{{v_j} \to {c_i}}^{t + 1} = {L_j} + \sum_{{c_y} \in M({v_j})\backslash {c_i}} {(CL_{{c_y} \to {v_j}}^{t+1})},$$where $M({v_j})$ is the set of all check nodes that have a connection relationship with the $jth$ variable node. $M({v_j})\backslash i$ denotes the removal of the $ith$ check node from $M({v_j})$. - Step7: Obtain the posterior LLR information
After calculating the LLR information of the check and variable nodes in the $(t+1)$th iteration, the posterior LLR $P_{{D_j}}$ is used to check whether all bit data in the LDPC code word are corrected successfully. In the $(t+1)$th iteration, $P_{{D_j}}$ can be calculated by
where $P_{{D_j}}^{t + 1}$ represents the maximum posterior LLR information of the $j$th modulated bit ${D_j}$ in the LDPC code word at the $(t+1)$th iteration. - Step8: Decoding decision
If $P_{{D_j}}^{t+1} > 0$ is true, the logical value $DV_j^{t+1}$ of the $jth$ bit data after LDPC decoding is 1; otherwise, $DV_j^{t+1}$ is 0. If $H \times \overrightarrow {D{V^{t+1}}} = 0$ is true, then all bit data are corrected successfully, and $\overrightarrow {D{V^{t+1}}}$ is the correct user data. Otherwise, the iterative decoding continues by repeatedly executing step5 until all bit data are successfully corrected, or the number of iterative cycles reaches $Kmax$ and the decoding failure occurs.
5. Experiment
5.1 Experimental setup
Figure 11 illustrates experimental platform of the phase modulated holographic storage system that was designed for this study. The laser generates light with a wavelength of 532 nm. The attenuator adjusts the beam power. The diameter of the beam can be expanded using the beam expander (BE). The polarizer is used to attain linearly polarized light. The aperture can limit the diameter and shape of the beam uploaded onto the SLM. The aperture, Lens1, Lens2 and SLM form an optical 4f system. The half-wave plate (HWP) can change the polarization state of the beam. The beam splitter (BS) can perform beam-splitting operations. The SLM is used to generate signal and reference beams. The media is used to record gratings that carry user data information. Lens3 and Lens4 form another 4f system. The mirror can be used to adjust the propagation direction of the beam. Lens5 is used to perform an optical Fourier transform on the reconstructed beam. The CMOS image sensor captures the Fourier intensity spectrum information of the reconstructed beam, and the Fourier intensity information can be used to reconstruct the phase data after noise reduction. In our experiment, the pure-phase SLM (X10468-04, HAMAMATSU) has a resolution of 792$\times$600 and a pixel pitch of 20 ${\mathrm \mu}$m. The holographic media is a PQ/PMMA photopolymer [38] with a thickness of 1.5 mm. The CMOS image sensor (DCC3260M, Thorlabs) has a resolution of 1936$\times$1216 and a pixel pitch of 5.86 ${\mathrm \mu}$m. The focal lengths of Lens1–Lens4 and Lens5 were 150mm and 300 mm, respectively.
5.2 Experimental method
Experiments were performed by randomly generating the user data. Before conducting actual experiments, we simulated the bit error rates of RaLDPC and conventional LDPC codes at three different coding rates with changing signal-to-noise ratios in order to observe the performance enhancement of RaLDPC. In the simulation, Gaussian noise with varying signal-to-noise ratios is added. As shown in Fig. 12, RaLDPC outperforms the conventional LDPC code by 0.2 dB at all three code rates. It has a lower bit error rate than traditional LDPC codes at the same raw bit error rate, regardless of the code rate used. Based on both storage capacity and data reliability considerations, we chose a code with a 0.89 coding rate for experiments on the actual platform. The experimental process is divided into four stages: data encoding, data recording, data reading, and data decoding. Data encoding and decoding were conducted using a computer, whereas data recording and reading were carried out on the experimental platform shown in Fig. 11.
Fig. 12. The bit error rates of RaLDPC and conventional LDPC codes at three different coding rates with changing signal-to-noise ratios.
Data encoding: LDPC encoding is first implemented in each 2-kilobyte (KB, 2KB = $2 \times 1024 \times 8 =16384$ bits) user data block to produce data frames with a length of 2.25 KB. The code rate of LDPC is 0.89. In other words, the length of an LDPC code word is 18,432 bits (i.e., $2.25 \times 1024 \times 8$), where the lengths of information bits and parity bits are 16,384 and 2048, respectively. Then, each data frame is evenly divided into 18 data segments with a length of 1024 bits, and each segment is subjected to 4-level phase modulation to gain two-dimensional signal data arranged by $32 \times 16$ phase data (i.e., each phase data can represent two bits: $1024/2 = 512$ and $512 = 32 \times 16$).
Data recording: each $32\times 16$ signal data and the $32\times 16$ reference data form a $32\times 32$ phase data page, as shown in Fig. 13(a). The reference data are used to generate the reference beam, and its phase values are known. Phase data pages that belonged to the same data frame constitute a page group. Then, each phase data page is uploaded onto the SLM shown in Fig. 11 to generate the signal beam and reference beam, and each phase data is represented by $4\times 4$ pixels in the SLM. The signal beam interferes with the reference beam to record data in the holographic media.
Data reading: the reference data is uploaded onto the SLM to generate the reference beam, which is irradiated onto the holographic media to produce a reconstruction beam. The CMOS image sensor obtains the Fourier intensity information of the reconstruction beam, as shown in Fig. 13(b). To meet the sampling requirement, only two Nyquist sizes of the captured Fourier intensity information are retained.
Data decoding: the OIFA is exploited to reconstruct the phase data. The maximum number of iterative cycles of OIFA is set to 20. Then, the reconstructed phase data are demodulated into bit data, and reliable bits are identified. Finally, all bit data in the same data frame are combined into an LDPC code word (i.e., a data frame). The traditional LDPC and proposed RaLDPC are used to perform decoding operations on the obtained data frames. The maximum number of iterative cycles of LDPC decoding is set to 30.
5.3 Experimental results and analysis
We recorded 1000 data frames, which consisted of 18,000 phase data pages. For each data frame, we counted the bit error rate after using conventional LDPC decoding and RaLDPC via the experimental method described in Section 5.2. Figure 14 shows the attained results as follows: 668 and 508 data frames were completely corrected by RaLDPC and traditional LDPC decoding, respectively. The number of data frames fully corrected by RaLPDC exceeds those corrected by the conventional method by 1.31 times. A fully corrected data frame means that its bit error rate is 0. For ease of presentation of the results, we used ${10^{ - 5}}$ to present 0. For those data frames that are not fully corrected by both methods, their bit error rate after RaLDPC is mostly lower than that of the conventional LDPC code. In addition, we calculated the average bit error rates of 1000 data frames after employing the RaLDPC and the conventional LDPC code. The average bit error rates of conventional LDPC and RaLDPC are 0.0054 and 0.0033, respectively. In other words, RaLDPC can reduce the bit error rate by an average of 38.89% compared to the conventional LDPC code.
Finally, we used a more visual manner to demonstrate the effectiveness of the traditional LDPC and RaLDPC to correct data errors. Figure 15 shows the phase error distribution of three typical phase data pages under three conditions: after phase reconstruction, after traditional LDPC decoding, and after RaLDPC decoding. Figures 15(a1), 15(b1), and 15(c1) present the three error distributions of phase data page 1. Both the conventional LDPC code and RaLDPC could fully correct the phase errors of phase data page 1, indicating that it is useful to introduce LDPC codes in phase modulated holographic storage for data protection. Figures 15(a2), 15(b2), and 15(c2) show the three error distributions of phase data page 2. RaLDPC can completely correct the phase errors of phase data page 2, whereas the conventional LDPC code can only reduce the number of phase errors. This is because the number of phase errors in the data frame corresponding to phase data page 2 exceeds the error correction capability of conventional LDPC codes while remaining within the error correction capability of RaLDPC. Figures 15(a3), 15(b3), and 15(c3) show the three error distributions of phase data page 3. Both the conventional LDPC code and RaLDPC can reduce the phase errors of phase data page 3, and RaLDPC reduces more data errors than traditional LDPC codes. Note that there are also phase data pages where the bit error rate of RaLDPC is larger than that of the conventional LDPC code. However, these kinds of phase data pages are extremely rare, accounting for only 2.7% of all phase data pages. This can be attributed to the enhanced error correction capacity of RaLDPC by acquiring a more accurate initial LLR using the reliable bit characteristic. The above results confirm that RaLDPC can effectively improve the data reliability of phase-modulated holographic data storage and promote the practical process of holographic storage.
Fig. 15. Phase error distributions of three typical phase data pages. (a) Phase error distribution after phase reconstruction; (b) phase error distribution after traditional LDPC decoding; (c) phase error distribution after RaLDPC decoding.
5.4 Discussion
Even though RaLDPC codes can fully correct more data frames and reduce the average bit error rate compared to traditional LDPC codes in the above experiments, there are still certain data frames with a higher bit error rate. This is due to the fact that the raw bit error rate magnitude of the data frames retrieved from the actual system varies. At a code rate of 0.89, the raw bit error rates of some data frames surpass the error correction capability of RaLDPC and substantially surpass the error correction capability of traditional LDPC codes. There are two approaches to resolving this issue. The first approach is to select a lower code rate based on the maximum raw bit error rate in the system with the criterion that RaLDPC can fully correct data errors. Since traditional LDPC codes are weaker than RaLDPC in accommodating raw bit error rate at the same code rate, RaLDPC will always outperform conventional LDPC codes regardless of the code rate used. This is because, at the used code rate, the bit error rate of RaLDPC is 0, while the bit error rate of conventional LDPC codes is greater than 0. The second approach is to combine other techniques such as signal noise reduction to improve the signal-to-noise ratio of the read data and reduce the raw bit error rate. For example, if the raw bit error rate is reduced to near the order of magnitude of the error-free output achieved by RaLDPC at a code rate of 0.89, the bit error rate of RaLDPC will be 0, while traditional LDPC will still have errors. Compared to conventional LDPC codes, RaLDPC reduces the system requirements for other techniques such as noise reduction.
6. Conclusion
In this study, we thoroughly analyzed the reconstructed phase data and the phase demodulation process of phase modulated holographic storage, resulting in an interesting discovery. Although the reconstructed phase data could not be used to accurately determine the recorded phase data, we identified reliable bits (i.e., bits with a reliable value) in several read patterns. Based on this characteristic, this study proposed RaLDPC for the actual phase modulated holographic storage. RaLDPC uses the determination of reliable bits to optimize the initial LLR information for LDPC decoding, making the initial LLR more accurate and enhancing the decoding performance. The experimental results showed that RaLDPC can effectively reduce the bit error rate, improving the data reliability of the phase-modulated holographic storage system. Simultaneously, RaLDPC provides a good example for optimization studies of LDPC codes. In other fields such as communication and traditional storage, decoding and demodulation are usually designed and optimized independently. In contrast, the decoding of RaLDPC fully incorporates the information provided by demodulation. For further study, we will continue studying the co-optimization of demodulation and decoding to further improve the data reliability of phase modulated holographic storage.
Funding
National Key Research and Development Program of China (2018YFA0701800); Project of Fujian Province Major Science and Technology (2020HZ01012).
Acknowledgments
This work is supported by Key Laboratory of Information Storage System, Ministry of Education of China, and the Engineering Research Center of Data Storage Systems and Technology.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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