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Binary SOVA and Nonbinary LDPC Codes for Turbo Equalization in Magnetic Recording Channels Seungjune Jeon and B. V. K. Vijaya Kumar, Fellow, IEEE Data Storage Systems Center (DSSC), Carnegie Mellon University, Pittsburgh, PA 15213 USA Nonbinary low-density parity check (LDPC) codes can provide coding gains over binary LDPC codes at the cost of increased complexity. For magnetic recording channels, optimal detectors for the nonbinary codes are symbol-based, which are significantly more complex than binary detectors. In this paper, we investigate the feasibility of using much less complex bit-based soft output Viterbi algorithm (SOVA) as a channel detector for nonbinary LDPC codes and performing turbo iterations between the SOVA and the nonbinary LDPC decoder. We show that performance gains can still be obtained by the turbo iterations by using the bit-based SOVA, although the performance gains are suboptimal to the symbol-based detectors. This scheme can be useful in low-complexity nonbinary LDPC coding systems. Index Terms—Low complexity, magnetic recording channel, nonbinary low-density parity check (LDPC) code, soft-output Viterbi algorithm (SOVA), turbo equalization.
I. INTRODUCTION ONBINARY low-density parity check (LDPC) codes can provide coding gains over binary LDPC codes by clustering multiple bits into multibit symbols that can mitigate the violation of conditional independence assumption in the message passing decoding algorithms. In Gallager’s 1963 monograph [1, Ch. 5], LDPC codes over nonbinary symbol or arbitrary alphabet size were discussed for the first time including probabilistic decoding algorithms and upper bounds on error probabilities for low column weight codes. In the beginning of the renaissance of the LDPC codes, Davey and MacKay [2], [3] investigated nonbinary LDPC codes over and showed by simulations that the nonbinary LDPC codes can provide performance gains over binary LDPC codes. For magnetic recording channels, Song and Cruz [4] showed that nonbinary LDPC codes can provide coding gains over binary LDPC codes. More recently, Chang and Cruz [5]–[7] investigated performance and complexity of nonbinary LDPC codes. The cost for the improved performance of the nonbinary LDPC code is the decoding complexity that grows exponentially in symbol size. There have been research efforts aimed at reducing the decoding complexity. In [4], reduced-complexity decoding algorithms in Fourier domain was demonstrated. Another reduced-complexity algorithm is an approximation corresponding to the min-sum approximation for binary LDPC code decoding. This approximation for nonbinary case was proposed by Declercq and Fossorier [8] and is called extended min-sum (EMS) algorithm. For magnetic recording channels, Risso [9] demonstrated layered decoding for nonbinary LDPC codes that can provide good performance within limited numbers of decoding iterations. Aside from the complexity of the decoder, optimal symbol-based channel detectors for nonbinary codes in
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Manuscript received October 31, 2009; accepted February 02, 2010. Current version published May 19, 2010. Corresponding author: S. Jeon (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2043068
magnetic recording channels are another source of exponential growth of complexity in symbol size. For magnetic recording channels and nonbinary LDPC decoders, optimal detectors are symbol-based since they can generate exact symbol posterior probabilities by using the joint probability of all bits in each symbol rather than the product of marginal probabilities of the bits in each symbol. However, symbol-based detectors have higher complexity than bit-based detectors. The complexity usually grows exponentially with the number of bits per symbol. This detector complexity makes the already complex nonbinary LDPC decoding even more challenging. For binary decoders, the symbol-based detectors are not required because the binary decoders assume that bits in codewords are correlated only by the parity check equations, not by other relations such as intersymbol interference in the channel. That is, even if a symbol-based detector provides symbol reliabilities to a binary decoder, the actual joint probability of bits in each symbol is ignored and only the marginal probabilities of the bits are used by the binary decoder. However, for nonbinary codes, symbol reliabilities are not discarded and optimal detectors must be symbol-based detectors. The Bahl–Cocke–Jelinek–Raviv (BCJR) detector [10] is an optimal symbol-based detector and minimizes the symbol error probability of the outputs. However, the number of operations in BCJR algorithm grows exponentially with the symbol size. Cheng et al. [11] proposed a reduced-complexity symbolbased BCJR detector for Reed–Solomon (RS) codes in magnetic recording channels based on Hoeher’s subblock-by-subblock detector [12]. Recently, Chang and Cruz [13] proposed a symbol-based detector that can accept symbol-based prior probabilities that are desired for channel iterations or turbo iterations in magnetic recording channels. In magnetic recording channels, channel equalization or turbo equalization considerably improves the performance of LDPC decoding by using the output likelihood information of the decoders as the prior probabilities for the channel output samples. Soft-output Viterbi algorithm (SOVA), proposed by Hagenauer and Hoeher [14], has much lower complexity than the BCJR detectors and can be used as a heuristic substitute for the BCJR detector. In this paper, by using bit-based SOVA as a detector for magnetic recording channel, we try to reduce the complexity
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JEON AND KUMAR: BINARY SOVA AND NONBINARY LDPC CODES FOR TURBO EQUALIZATION IN MAGNETIC RECORDING CHANNELS
of turbo iteration for nonbinary LDPC codes rather than pursuing optimal performances by using symbol-based detectors. Bit-based SOVA has much lower complexity than symbol-based and bit-based BCJR-type detectors, although the performance will be suboptimal. By using SOVA, it will be shown that coding gains by turbo iteration can still be obtained. In Section II, we will discuss methods of conversion between bit reliability of SOVA and symbol reliability of nonbinary LDPC codes. We will also discuss a quantitative measure of bit correlations within a symbol and the complexity of SOVAs and BCJR detectors. In Section III, we will show simulation results of using SOVAs and nonbinary LDPC codes for turbo iterations in perpendicular magnetic channels. In Section IV, we will provide our conclusions. II. SYMBOL RELIABILITY AND BIT RELIABILITY A. Reliability Conversion In this subsection, we will discuss the conversion between a -bit symbol reliability in LDPC decoder and bit reliability in SOVA. Suppose that the log probability ratio for a random for the th bit in a -bit symbol is denoted as variable (1) where a random variable represented as
. Similarly, a log probability ratio for a -bit symbol to have a value
for is
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Fig. 1. Setup for turbo iteration with binary SOVA and nonbinary LDPC codes. Interleavers and deinterleaver may be optional.
As shown in Fig. 1, a bit-level interleaver and deinterleaver can be used between the SOVA and the nonbinary LDPC decoder to distribute adjacent bits to different symbols such that the dependency among the bits within a symbol decreases. For the conversion from nonbinary LDPC decoder output to SOVA input, (5) becomes over-determined and do not have an exact solution in general. Simple ways to handle this situation include choosing only independent equations out of the in (5) and dropping (i.e., ignoring) other equations, or using the pseudoinverse. However, information is lost in these methods making them suboptimal. Note that a set of bit probability ratios, which are independent real numbers, cannot capture -tuple symbol probability ratio the full information in a vector without loss of information. By dropping equations except the equations, we have
(2)
(7)
is one of the nonzero values over and . If the bits in a -bit symbol are independent, the log probability ratios
where is a matrix consisting of linearly independent , and is a -tuple vector of elements from rows taken from of corresponding rows of . For example, if , we may choose
(3)
(8) (9)
where
in a -bit symbol can be converted to the log probability ratio vector for the -bit symbol (4)
Another approach is to use the Moore–Penrose pseudoinverse such that (10)
by (5) where is a binary matrix whose rows are binary representation of 1,2, , and . For example, for , the binary is the 7-by-3 matrix matrix (6) Please note that (5) is over real numbers as the log probability ratios are real. For the conversion from SOVA output to nonbinary LDPC decoder input, a set of bit log probability ratios uniquely deelements of a symbol log probability ratio termine the vector. No information is lost in going from SOVA outputs to LDPC inputs. However, the dependence among the adjacent bits of the SOVA output violates the assumption of (5) and makes the conversion suboptimal.
which has higher complexity than the equation drop method. Equation (7) is a multiplication of a matrix and a -element matrix vector whereas (10) is a multiplication of a and a -element vector. B. Correlation Among Bits Within Each Symbol A quantitative measure of independence of multiple random variables is the Kullback–Leibler (KL) divergence below
(11)
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This KL divergence measures the difference between the joint probability mass function of the bits in each symbol and the product of the marginal probability mass functions of the bits. The KL divergence is zero when all the bits are independent. Otherwise, the KL divergence has a positive value. Higher values indicate higher correlation among the bits. If we measure the KL divergence for each -bit symbol, we can quantify the degree of independence of the bits in each symbol. If a majority of symbols have zero or low KL divergence, using SOVA for turbo iterations may be feasible. However, if only a small portion of symbols have zero or low KL divergence, using SOVA may degrade the performance of turbo iterations. C. Complexity In [5, Table I], the number of floating-point operations per bit per iteration for symbol-based BCJR is provided. In [6, Table I], the number of floating-point operations per bit per iteration for binary SOVA is also provided. The complexity depends on the constraint length of the partial response (PR) target. According to these references, the numbers of additions are versus for symbol-based BCJR and SOVA, respectively. The number of log-sum table look-up while the number of min for symbol-based BCJR is , where is a parameter that operations in SOVA is depends on the length of a buffer, from 1 to 50. The number of multiplications and the number of divisions for the two detectors are the same. Based on this complexity analysis, we can see the complexity reduction of SOVA over symbol-based BCJR is exponential in , the number of bits in each symbol, except the multiplication and division.
GF(2 )
Fig. 2. Block error rates of a (1152, 1024) nonbinary LDPC code over , used in this work, in AWGN channels. Block error rates of a (4608, 4096) biare also plotted for nary LDPC code and a (462,410) RS code over reference. The results for the RS code were calculated analytically.
GF(2 )
III. SIMULATION RESULTS Fig. 2 shows performance of a rate-8/9 (1152, 1024) nonbi, used in this work, in additive nary LDPC code over white Gaussian noise (AWGN) channels. Block error rates (BLERs) of a (4608, 4096) binary LDPC code and a (462,410) are also plotted for reference. The RS code over codes were generated by the progressive-edge-growth (PEG) algorithm [15], [16]. The girth of the bipartite graph was 6 and the column weight was 5. The nonzero elements in the parity check matrix were randomly chosen among nonzero elements . The number of maximum decoding iteration over was 20. The log domain algorithm in [17] was used for the nonbinary decoding. Fig. 3 shows the BLERs of the nonbinary LDPC code over 4-bit symbols in perpendicular magnetic recording channels. Turbo iterations were performed using binary SOVA and nonbinary LDPC decoder. No turbo iteration indicates iteration 1. The equation drop method without interleaving has the same performance for iteration 2, 3, , 6. For the pseudoinverse method with or without interleaving, iteration 2 and iteration 6 were plotted. We can see that the equation drop method does not improve after the second iteration whereas the pseudoinverse method keeps improving up to the sixth iteration. The pseudoinverse method and the equation drop method without interleavers show similar BLERs although the performance of pseudoinverse method was slightly better. The performance gains by turbo equalization gradually decreased in high SNR range.
Fig. 3. Block error rates for turbo iterations between binary SOVA and nonbinary LDPC decoder. No turbo iteration indicates iteration 1. The equation drop , method without interleaving has the same performances for iteration 2, 3, 6. For the pseudoinverse method with or without interleaving, iteration 2 and iteration 6 were plotted.
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With interleaving, additional performance gains were obtained by turbo iterations. The performance gains also decrease in high SNR range. (i.e., the Channel bit density was 2.5 in terms of width of the half maximum of the derivative of the transition (i.e., the width between the half response), or 1.4 in terms of minimum and the half maximum of the transition response). was used as The Gaussian error function the transition response. Transition jitter noise had 80% of the total noise power. For each SNR point, a generalized partial response [18] target of length 4 and an equalizer of length 11 readback samples by simulations. were determined using The maximum number of turbo iterations was 6.
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IV. CONCLUSION It was shown that the turbo equalization with bit-based SOVA and nonbinary LDPC codes provides coding gains with low complexity. The high histogram concentration of the symbols having loosely correlated bits enables the application of bitbased SOVA for the turbo iteration with nonbinary codes. However, the performance of this scheme may be limited by the distortion in the conversion from nonbinary LDPC decoder output to SOVA input. This turbo equalization scheme can be useful in low-complexity nonbinary LDPC coded systems. REFERENCES
Fig. 4. Histogram of KL divergence between the product of marginal probabilities of bits in each 4-bit symbol and the joint probabilities of the four bits. The number of 4-bit symbol samples for each SNR value was 7 10 .
2
The SNR is defined as , where is the energy of the samples of the channel pulse response and is the power of transition jitter noise, corresponding to the first order of the Taylor series approximation of the jitter noise, and is the code rate. In the power spectral density, is is the energy of the the variance of transition jitter noise and samples of derivative of the transition response of the magnetic recording channel. The percentage of transition jitter noise is . The number of block errors collected for each simulation was at least 10 at high SNRs and more than hundred at low SNRs. One exception is the 9.6 dB point for the pseudoinverse case without interleaving, where only one block error was observed. Fig. 4 shows a histogram of KL divergence between the joint probability mass function of the four bits in each symbol and the product of marginal probability functions of the four bit. The number of 4-bit symbol samples for each SNR value was . We can observe that there are high concentration peaks around zero KL divergence. These symbols with zero or low KL divergences have independent or nearly independent bits and make the turbo iteration using SOVA feasible. However, there are also a certain level of population in the side and tail and the population does not decrease much for even high KL divergence values. These symbols with high KL divergence limit the performance gains of the turbo iterations using SOVAs. The behavior for different SNR values was similar and no significant difference was observed among the three SNR values: 7.6, 8.3, and 10.6 dB although the symbols at 10.6 dB are slightly more correlated than the symbols at 7.6 or 8.3 dB. The simulation was performed by C++ implementations on Intel Xeon E5440 2.53-GHz machines. It took about 440 h to obtain one block error for the sixth turbo iteration at the highest SNR 9.6 dB.
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