Two high-density recording methods with run-length limited turbo code for holographic data storage system

To further improve data capacity, the size of the hologram recorded in the medium must be reduced. A schematic diagram of a conventional HDSS is shown in Fig. 3. The size of the hologram depends on the diffraction angle from a pixel on the SLM. The hologram size is given as

$$\begin{equation} (f\cdot\alpha)^{2},\quad\alpha = \frac{\lambda}{\Delta}, \end{equation} \tag{ 1 }$$

where α is the diffraction angle, f is the focal length of the lens, λ is the wavelength of the laser, and Δ is the pixel pitch of the SLM. The equation indicates that the angle spreads inversely with increasing pixel pitch. Accordingly, a larger pixel pitch makes a smaller hologram. However, the data capacity of the SLM can be written as

$$\begin{equation} \pi\left(f\cdot\frac{\mathit{NA}}{\Delta}\right)^{2}{}\cdot R, \end{equation} \tag{ 2 }$$

where NA is the numerical aperture of the lens and R is the code rate. Thus, the data capacity of the SLM decreases with increasing pixel pitch. From Eqs. (1) and (2), the recording capacity of the HDSS is given as

$$\begin{equation} \frac{\text{SLM data capacity}}{\text{Hologram size}} = \pi\left(\frac{\mathit{NA}}{\lambda}\right)^{2}{}\cdot R. \end{equation} \tag{ 3 }$$

Eventually, it is evident that the recording density cannot be increased by only changing the pixel pitch.

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According to Eq. (3), increasing NA can improve the recoding capacity; however, at the same time, the size and cost of the HDSS will be increased. It is thus difficult to attain high-density recording without modifying the optics of the HDSS.

In light of the above-mentioned difficulty, RLL coding, a modulation method that limits lengths of runs of repeated pixels, was focused on.^12,13) RLL modulation is generally described with $\text{RLL}(d,k)$, where d is the minimum run-length of "0"s, and k is the maximum run-length of "0"s. An example of a data pattern modulated with a non-return-to-zero (NRZ) code is given in Fig. 4(a). If the data pattern is modulated with an $\text{RLL}(1,\infty )$ modulation, the data is constrained from the repetition of "1"s, e.g., "101" satisfies the modulation rule; in contrast, "11" is restricted, as illustrated in Fig. 4(b). Moreover, modulation with a non-return-to-zero invert (NRZI) code makes it possible to constrain the run length of modulated data, e.g., "0110" satisfies the rule; in contrast, "010" is restricted, as illustrated in Fig. 4(c).¹⁴⁾ Hence, the minimum run length of $\text{RLL}(1,\infty )$-modulated NRZI data is expanded, which corresponds to the same effect as enlarging the pixel pitch.

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Images on the SLM, Fourier plane, and camera, which is on a Fourier plane equivalent to the position of the spatial filter shown in Fig. 3, are shown in Fig. 5. The SLM image is converted to a Fourier image, which shows the spatial frequency of the data pattern. The red rectangular line of the Fourier image indicates the Nyquist area, and the spatial filter is sized somewhat larger than that area. The filter size thus corresponds to the hologram size. In this HDSS, $\text{RLL}(1,\infty )$ modulates the data in the horizontal direction but not in the vertical direction. This rule doubles the effective horizontal pixel pitch because the minimum run length is 2. The Nyquist size of the data with RLL modulation is half as large as that without modulation. As a result, the filter and hologram sizes can be reduced by half. Consequently, raw recording density can be doubled by halving the filter size.^15,16)

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The above discussion clarified that the RLL high-density recording method can increase the raw recording density by using a filter with half the size. However, at the same time, compared with a conventional filter, a half-sized filter limits the lower spatial frequency of the data pattern. Thus, the IPI of the reproduced image on the camera occurs. The IPI, generated by the pixel response of a reproduced image as shown in Fig. 6, degrades the signal quality of the image.

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To confirm this signal-quality degradation, the performance of the error correction was evaluated. The results of a simulation on error-correction performance are shown in Fig. 7. The horizontal axis represents E_b/N₀, namely, SNR per bit, whereas the vertical axis represents bit error rate (bER). The two lines indicate the conventional LDPC code and an RLL modulation + LDPC code, respectively. When RLL modulation is used, a half aperture size is applied in the simulation. Moreover, E_b/N₀ is especially useful when comparing the bER performance characteristics of different digital-modulation schemes without taking bandwidth into account. bER is the ratio of errors to the reproduced user data, and it must be less than 10⁻⁶, which means no errors in the simulation results. This result indicates that the RLL + LDPC code degrades the performance by −2.9 dB as compared with the conventional LDPC code because of the IPI. It is therefore necessary to devise a new error-correction method to improve the performance more than the conventional ECC, even though the IPI is caused by the half-sized filter.

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Mutual information is extensively used to analyze the capability of a decoder of the ECC.¹⁷⁾ The decoding capability can be evaluated by comparing the input and output mutual information of the decoder. Moreover, an EXIT chart is useful for illustrating the characteristics of the mutual information.^18–20)

An EXIT chart for the LDPC decoder is shown in Fig. 8. This figure indicates the relationship between input mutual information (I_a: a priori information) and output mutual information (I_e: extrinsic information) of the decoder. The higher mutual information means the higher signal quality. When the output mutual information reaches point $(1,1)$, the user data can be decoded completely. In accordance with this figure, a signal with a higher I_a of 0.64 can be perfectly decoded. However, if RLL modulation is used, the IPI degrades the input mutual information to I_a of 0.53. As a result, the output mutual information cannot reach point $(1,1)$, meaning that the user data cannot be decoded.

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In the field of information theory, turbo codes, well-known as high-performance error-correction codes developed in the 1990s, were the first practical codes to closely approach the theoretical maximum code rate.²¹⁾ Flow diagrams of signal processing with a conventional turbo code are shown in Fig. 9. The crucial point concerning a turbo code is twofold: the two decoders are soft-in/soft-out decoders that accept and deliver probabilities or soft values, and the extrinsic part of the soft-output of one decoder is passed on to the other decoder to be used as a priori input.

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Examples of EXIT charts for decoders 1 and 2 are shown in Fig. 10. For the parallel decoder, the axes are swapped and the iteration alternates between the curves of decoders. The difference meaning the difference to the diagonal is the average information gain measured in bits per half iteration. The iteration of the mutual information between decoders is described as trajectories in the figures. In the case of a satisfactory code, as shown in Fig. 10(a), the iterated information will reach point $(1,1)$ and decode virtually error-free. On the other hand, in the case of an unsatisfactory code or a low-SNR channel, as shown in Fig. 10(b), the iteration stops before the end point if no information gain is achievable.²²⁾

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For this reason, to improve the capability of a turbo code, an encoder and a decoder must be designed so their EXIT curves do not cross each other.

3.3.1. Architecture.

Flow diagrams of signal processing using the developed RLL turbo code are shown in Fig. 11. This RLL turbo code consists of an RLL modulator and a convolutional encoder, and the iteration alternates information between the demodulator and the decoder. The remarkable idea concerning this code is using the RLL modulator and demodulator as parts of error correction. RLL modulation is widely used in conventional optical discs, e.g., CDs, DVDs, and Blu-ray Discs™.²³⁾ However, it does not function as an error correction methods. In contrast, this idea enables the effective use of RLL modulation.

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In the case of general turbo-signal processing, a Bahl-Cocke-Jelinek-Raviv (BCJR) decoder is widely used as a maximum a posteriori decoder for turbo decoding.²⁴⁾ A BCJR decoder needs a priori input and a posteriori output; however, a decoder as a part of a turbo decoder requires a priori input and extrinsic output. Thus, the composition shown in Fig. 12 was used in the present study. The extrinsic information $(L_{\text{ie}},L_{\text{ce}})$ is calculated by subtracting a posteriori output $(L_{\text{ip}},L_{\text{cp}})$ from a priori input $(L_{\text{ia}},L_{\text{ca}})$ of the BCJR decoder. A priori information of the RLL BCJR decoder is obtained from the channel LLR (L_ch). Moreover, the BCJR decoding algorithm for calculating a posteriori information is based on a Markov property.²⁵⁾ The developed RLL modulator must therefore have a Markov property.

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3.3.2. Formula of RLL trellis modulation.

As mentioned, RLL modulation is widely used for conventional optical discs, and the conventional modulation uses translation tables to convert input data into modulated data. Nevertheless, the modulation does not necessarily have the Markov property. Accordingly, in the following, the modulation is explained as a trellis modulation because a code of a trellis obviously has the Markov property.

The theoretical limitation of the code rate of $\text{RLL}(1,\infty )$ is 0.69.²⁶⁾ When the actual code rate is increased to the theoretical limitation, the code with a 2-bit input and a 3-bit output because of its rate thereby becomes 2/3 = 0.66, which is close to the limitation, and its complexity will be relatively low.

An $\text{RLL}(1,\infty )$ code with a rate of 2/3 under the condition of NRZ data is constructed as follows. Under this condition, successive "1"s (e.g., "11") are restricted, so the state transition diagram of the $\text{RLL}(1,\infty )$ sequence can be described as shown in Fig. 13(a). Moreover, this diagram should be expanded to the third extension of the sequence shown in Fig. 13(c), via the second extension shown in Fig. 13(b), because the third extension can be described as the transition of the 3-bit output of the RLL encoder. Accordingly, the connection matrix is written as

$$\begin{equation} D^{3} = \begin{bmatrix} 1 & 2\\ 2 & 3 \end{bmatrix} . \end{equation} \tag{ 4 }$$

To form the 2-bit input of the encoder, four (i.e., 2²) branches from each state $(\text{S0},\text{S1})$ are required; however, there are only three branches from the states, as shown in Fig. 13(c). Accordingly, the branches and states were rearranged. An eigenvector inequality is given by

$$\begin{align} &D^{m}\boldsymbol{{v}}\geq 2^{n}\boldsymbol{{v}}, \end{align} \tag{ 5 }$$

$$\begin{align} &D^{3}\boldsymbol{{v}} = \begin{bmatrix} 1 & 2\\ 2 & 3 \end{bmatrix} \begin{bmatrix} 1\\ 2 \end{bmatrix} \geq 2^{2}\boldsymbol{{v}}, \end{align} \tag{ 6 }$$

where v is an approximate eigenvector, m is the number of output bits (three in this code), and n is the number of output bits (two in this code).²⁷⁾ The approximate eigenvector $\boldsymbol{{v}} = [1,2]^{\text{T}}$ indicates that state S1 will split into two states, while state S0 will not split. Thus, S1 splits into states S1¹ and S1², as shown in Fig. 14(a), and states S1² and S0 merge as shown in Fig. 14(b). As a result, a state-transition diagram can be constructed to satisfy the four branches from each state.

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A state-transition table for NRZ data is given as Table I. The state transition of output bits is shown in Fig. 14(b), and the table describes the correspondence between input and output bits. In the table, the left side of "/" indicates an output, and the right side indicates a transition destination state.

Table I. State-transition table for NRZ data.

State	Input
	00	01	10	11
S0	000/S0	000/S1	010/S1	010/S0
S1	101/S0	100/S1	100/S1	001/S0

^*Output/Next state.

In the discussions so far, the code we have mentioned is modulated with the NRZ code. However, bits must be modulated with the NRZI code. When NRZI modulation is taken into consideration, two cases of the states depending on just the previous bit are applicable. A state-transition table expanded from S0 to S3 to deal with these cases is given in Table II. Moreover, a table in which the output bits are modulated with NRZI modulation is given in Table III. This table for NRZI data can be rewritten as a trellis chart, as shown in Fig. 15. According to the results shown in Fig. 15, the RLL trellis modulation of $\text{RLL}(1,\infty )$ was formed.

Table II. State-transition table for NRZ data, which includes pre-bit.

State	Pre-bit	Input
		00	01	10	11
S0	0	000/S0	000/S1	010/S2	010/S3
S1		101/S0	100/S2	100/S3	001/S3
S2	1	101/S3	100/S1	100/S0	001/S0
S3		000/S3	000/S2	010/S1	010/S0

^*Output/Next state.

Table III. State-transition table for NRZI data.

State	Input
	00	01	10	11
S0	000/S0	000/S1	011/S2	011/S3
S1	110/S0	111/S2	111/S3	001/S3
S2	001/S3	000/S1	000/S0	110/S0
S3	111/S3	111/S2	100/S1	100/S0

^*Output/Next state.

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3.3.3. Trellis optimization for RLL turbo code.

For evaluating the developed RLL trellis modulation, the EXIT chart of a stand-alone demodulation was measured.²⁰⁾ The setup for measuring mutual information is shown in Fig. 16. By using this setup, the EXIT chart shown in Fig. 17 was obtained. It is clear from the figure that the output mutual information does not reach point $(1,1)$, even though the input information approaches 1. It is reasoned that the trellis demodulator causes errors in the process of demodulation because it has the same bits at different transitions and the same paths connected to the same state in order to obey the RLL constraint, as shown in Table III. Therefore, the RLL trellis was optimized by increasing the number of states from four to six and separating the duplicated transitions. The optimized state-transition table is given as Table IV. Moreover, as for the special feature of this table, the event probability of output bits is balanced as much as possible to avoid decoding error. The table can also be rewritten as the trellis chart shown in Fig. 18.

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Table IV. Optimized state-transition table for NRZI data.

State	Input
	00	01	10	11
S0	001/S4	000/S1	011/S5	000/S2
S1	000/S0	000/S2	011/S3	011/S4
S2	111/S3	110/S0	001/S4	111/S5
S3	000/S2	001/S5	110/S1	000/S0
S4	111/S5	111/S3	100/S2	100/S1
S5	110/S1	111/S4	100/S0	111/S3

^*Output/Next state.

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The measurement results of an EXIT chart for the RLL demodulator are shown in Fig. 19. In accordance with this figure, the output mutual information can reach point $(1,1)$. This result indicates that the proposed $\text{RLL}(1,\infty )$ trellis modulation method can use the BCJR decoder.

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For constructing the turbo code that consists of two encoders, the other encoder whose EXIT curve fits the curve of the RLL demodulator is required to construct the RLL turbo code. To avoid degrading the code rate of ECC compared with the LDPC used, a convolutional encoder with a code rate of 0.66 was developed. For that purpose, the encoder must have a 2-bit input and a 3-bit output. An example of the architecture of the convolutional encoder is shown in Fig. 20.²⁸⁾ The encoder consists of shift registers and exclusive ORs. The number of shift registers is called the "memory number" ν. The further the memory number is increased, the higher the capability of the code is improved.

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For evaluating the convolutional decoder, the EXIT chart of the stand-alone decoder was measured. The evaluation setup for measuring mutual information is shown in Fig. 21.²⁰⁾ This setup made it possible to obtain the EXIT chart shown in Fig. 22. The best-fitting EXIT curve to that of the developed RLL demodulator was searched for, while the memory number of the convolutional decoder was changed. Finally, in the case of ν = 10, the two curves are crossed, as shown in Fig. 22(b). On the contrary, in the case of ν = 1, the curve fits that of the RLL demodulator, as shown in Fig. 22(a).

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The state-transition table is shown as Table V, which can also be rewritten as a trellis chart, as shown in Fig. 23. As shown in this figure, this code can use the BCJR decoder because it has a Markov property.

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A two-dimensional page format was designed for the implementation of the RLL turbo code. The specification of the designed page format is given in Table VI. This page format consists of more than 2.7 M pixels per page, and the code rate is 0.43 (i.e., 0.66 × 0.66). An example of the designed page format is shown in Fig. 24. Known markers arranged in the page data are useful for compensating for the distortions of the reproduced image. Then, the capability of error correction was evaluated in two steps under these conditions.

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First, the trajectory of the RLL turbo decoder was confirmed. The results of an HDSS simulator, as shown in Fig. 25, indicate that the actual trajectory mostly matches the stand-alone EXIT curves of the decoders. However, the trajectory and curves differ slightly. The reasons for this difference are twofold. The first reason is the shifting of the histogram shape away from a normal distribution. When the LLR was calculated, the following equation was used under the assumption that the histogram shows a normal distribution.²⁹⁾

$$\begin{equation} \text{LLR}(y) = \ln\left(\frac{\sigma_{0}}{\sigma_{1}}\right) - \frac{1}{2}\left(\frac{y - \mu_{0}}{\sigma_{0}}\right)^{2} {}+ \frac{1}{2}\left(\frac{y - \mu_{1}}{\sigma_{1}}\right)^{2}. \end{equation} \tag{ 7 }$$

Here, σ₀ and σ₁ are standard deviations of "0" and "1", and μ₀ and μ₁ are means of the distributions of "0"s and "1"s, respectively. In fact, the distribution actually has a near-Nakagami-Rice distribution.³⁰⁾ The second reason also concerns the LLR calculation. To calculate the LLR from Eq. (7), reference binary data is required to discriminate between "0"s and "1"s; however, it cannot be known in advance. Therefore, the ideal reference and actual binary data obtained by the binarization of a reproduced signal differ slightly.

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Second, the performance of the RLL turbo decoder was evaluated. The results of the HDSS simulator are shown in Fig. 26. These results indicate that the RLL + LDPC code degrades by −2.9 dB compared with the conventional LDPC code, as mentioned previously. Moreover, the developed RLL turbo code is improved by about +1 dB compared with the conventional LDPC code; nevertheless, the IPI is caused by using the half-size filter in the case of the RLL turbo code.

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The experimental setup for verifying the capability of error correction is shown in Fig. 27. A laser beam from an external-cavity laser diode (ECLD) is divided into signal and reference beams by a polarized beam splitter. The amplitude of the signal beam is spatially modulated using an encoded page data on a 2.1k × 2.1k-pixel SLM with a pixel pitch of 7.8 µm. The incident angle of the reference beam with respect to a holographic medium is controlled by a galvano mirror. The focal point of the reference beam is at the back focal plane of the objective lens, resulting in a collimated reference beam at the holographic medium. The medium is a custom transparent disk with a 1.5-mm-thick recording layer (photopolymer) sandwiched between two 0.7-mm-thick substrate glasses with antireflection coating. A second galvano mirror is placed at the back of the medium to retroreflect the reference beam for data recovery in a quasi-phase conjugate readout geometry. Recovered holograms are imaged using a 2.9k × 2.9k-pixel camera with a pixel pitch of 5.8 µm, and the hologram image is reproduced by oversampling detection. Moreover, encoding and decoding signal processing circuits are embedded in the setup. In addition, the recorded holograms are stacked at the same position on the disc. The other experimental conditions are listed in Table VII. Consequently, the recording density reaches 2.4 Tbit/in.², and the recording speed reaches 4.9 Gbps under the condition of one isolated stack.

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The results of the experimental measurements and HDSS simulator are shown in Fig. 28. The horizontal axis represents SNR, which is given by the following formula, and the vertical axis represents the bER.

$$\begin{equation} \text{SNR} = 20\log_{10}\frac{\mu_{1} - \mu_{0}}{\sigma_{1} + \sigma_{0}}. \end{equation} \tag{ 8 }$$

Here, σ₀ and σ₁ are standard deviations of "0"s and "1"s, and μ₀ and μ₁ are means of the distributions of "0"s and "1"s, respectively. As shown in the figure, the simulation results agree well with the experimental results. Thus, it was experimentally confirmed that the RLL turbo encoder/decoder works as expected in an actual system. In addition, we find that the signal with more than 2.4 dB is completely correctable.

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To confirm the 2.4 Tbit/in.² recording density experimentally, 440 pages were recorded at the same location and reproduced. The power of diffraction from each reproduced page is shown in Fig. 29. This result indicates that 440 peaks are clearly observed. Besides, as the criterion for decoding with no errors, the SNR of the reproduced page data is more than 2.4 dB. An example of the reproduced page data is shown in Fig. 30, and the SNR is 3.3 dB. Consequently, the recording density of 2.4 Tbit/in.² was confirmed experimentally.

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