Reduced-Complexity Decoding of Q-ary LDPC Codes for Magnetic Recording
Hongxin Song, Member, IEEE, and J. R. Cruz, Fellow, IEEE
Abstract—Binary low-density parity-check (LDPC) codes perform very well on magnetic recording channels (MRCs) with additive white Gaussian noise (AWGN). However, an MRC is subject to other impairments, such as media defects and thermal asperities. Binary LDPC codes may not be able to cope with these impairments without the help of a Reed–Solomon code. A better form of coding may be -ary LDPC codes, which have been shown to outperform binary LDPC codes and Reed–Solomon codes on the AWGN channel. In this paper, we report on our investigation of -ary LDPC coded MRCs, both with AWGN and with burst impairments, and we present a new reduced-complexity decoding algorithm for -ary LDPC codes. We show that -ary LDPC codes outperform binary LDPC codes in the presence of burst impairments.
Index Terms—Belief propagation, iterative decoding, low-den- sity parity-check (LDPC) codes, magnetic recording.
I. INTRODUCTION
B
INARY low-density parity-check (B-LDPC) codes have been shown to perform very well on additive white Gaussian noise (AWGN) channels [1], [2], and have been recommended for use on magnetic recording channels (MRCs) [3]–[5]. However, magnetic recording channels have burst impairments due to disk defects and thermal asperities (TAs), which can severely degrade the performance of B-LDPC codes.A disk defect can be modeled as the fading of the readback signal, which in some cases can be completely erased and last for hundreds of bits. A full erasure corresponds to the total loss of the readback signal, while a half erasure corresponds to the readback signal being reduced by a factor of two. When a thermal asperity occurs, the readback signal saturates the analog-to-digital converter, generating a noise burst in the readback signal.
The first work on -ary LDPC ( -LDPC) codes appeared in [6] and [7]. Similar to B-LDPC codes, a -LDPC code can be described by a low-density parity-check matrix . Each
element of is now an element from GF .
A row vector xxx of length is a codeword if
(1)
Similar to B-LDPC codes, a -LDPC code can be regarded as a collection of subcodes, which are simply parity-check codes [3]. For regular -LDPC codes, column weight and row
Manuscript received March 15, 2002; revised December 1, 2002.
The authors are with the School of Electrical and Computer Engineering, The University of Oklahoma, Norman, OK 73019 USA.
Digital Object Identifier 10.1109/TMAG.2003.808600
weight can be defined as the number of nonzero GF ele- ments in each column and each row in .
It was shown in [6]–[8] that -LDPC codes of rates to outperform B-LDPC codes on the AWGN channel. It is reasonable to expect that this might hold for any code rate, and that -LDPC codes might perform better than B-LDPC codes on MRCs with AWGN. On channels with noise bursts and/or erasures, the consecutive bits in the burst window are grouped into fewer symbols and, therefore, it is also reasonable to expect that -LDPC codes may have an advantage over binary codes.
These conjectures motivated our experimental investigation of -LDPC codes.
In Section II, a -LDPC code design for burst channels is briefly introduced. In Section III, a reduced-complexity decoding algorithm for -LDPC codes is presented. In Sec- tion IV, magnetic recording systems with -LDPC codes are investigated and simulation results comparing them with Reed–Solomon (RS) coded systems are provided. Concluding remarks are given in Section V.
II. -LDPC CODEDESIGN
In a -LDPC code over GF , each code symbol con- tains bits. In principle, -LDPC codes can be generated from B-LDPC codes. By substituting each element one in the parity check matrix for a B-LDPC code with a nonzero element randomly chosen from GF , a -LDPC parity check matrix is obtained [6], [7]. It is shown in [8] that the GF ele- ments replacing the ones in cannot be all the same, otherwise the resultant -LDPC code is simply composed of -disjointed (also interleaved) B-LDPC codes.
Conceptually, any B-LDPC code (random or algebraic, reg- ular or irregular) parity check matrix can be used to gen- erate a -LDPC code parity check matrix . However, since irregular LDPC codes have larger decoding complexity than regular LDPC codes, only random regular -LDPC codes are considered.
For a low-density matrix, the minimum space distance (MSD) is defined as the minimum length of runs of zeros in all rows, and denoted as . It is shown in [9] that a B-LDPC code with MSD is guaranteed to recover a burst erasure of length bits. The key observation is that a burst erasure of length up to
bits causes at most one unknown bit in each parity check equation. A -bit-symbol -LDPC code with MSD is guar- anteed to recover a burst erasure of symbols, or bits. Burst erasures shorter than symbols can be recov- ered in one LDPC iteration. Since one of the reasons for con- sidering -LDPC codes is to improve the error correction capa-
0018-9464/03$17.00 © 2003 IEEE
bility under long bursts, the MSD should be maximized. For a matrix with columns and row weight , clearly . To obtain a parity check matrix with large MSD, the fol- lowing method is used. First, a reasonable value is chosen. Then, starting from the first column, locations are randomly chosen and filled with ones. For each latter column, both cycle-four and MSD constraints are checked, and priority is given to the row locations with the smallest current row weight (row weight of all previous columns). So, the generated matrix will have uniform but not necessarily uniform row weights, but typically the row weights do not vary much.
Considering the complexity (see Section III), GF(16) is prob- ably the largest field of practical interest for -LDPC codes, and only codes with are considered. For sector-size codes, i.e., 4096 bits, three -LDPC codes designed on GF(16) are considered and their parameters are summarized in Table I. No- tice that Code 1 has rate , while the maximum code rate for a , LDPC code is 0.9267 [2], showing that the MSD rule does not hinder the design of high-rate codes.
Consider the AWGN channel and model the worst case era- sures as the received channel value being zero. As shown in Table I, Code 2 has . On a binary erasure channel, where only a single burst erasure can occur per codeword, this code is guaranteed to recover erasures of 31 four-bit symbols. In the worst case, a 118-bit erasure can result in a 31-symbol erasure, with the first and the last bit in the sequence the only erased bit in the corresponding erased symbols. Therefore, Code 2 is guar- anteed to recover single burst erasures of length 118 bits.
Since we assume that only a single burst erasure can occur in a codeword, a length -bit burst erasure can occur at
different locations. Examination of the cases shows that Code 2 is able to recover all single-burst erasures of length up to 344 bits. Notice that an interleaved RS code on GF with the same length (in bit) and code rate would have 64 eight-bit-symbol redundancy and, therefore, would able to recover up to 512-bit burst erasures.
The performance of -LDPC Code 2 is shown in Fig. 1 (labeled as Q) on the AWGN channel with and without a single burst erasure of length 144 bits. Also shown is the performance of a B-LDPC code (labeled as B) of column weight four, which has the same code length (in bit) and code rate as Code 2. In addition, the performance of a binary irregular code (labeled as Irr B) is also shown, which has 4352 information bits and code rate 0.9 [9]. It can be seen that although the -LDPC code performance is very similar to the weight-four B-LDPC code in AWGN, it performs 0.2 dB better at a bit-error rate (BER) of
Fig. 1. Performance ofQ-LDPC and B-LDPC codes on the AWGN channel.
BER in the presence of 144-bit erasures. The irregular B-LDPC code does not perform as well as the other two codes in both cases.
It is well known that Reed–Solomon codes achieve maximum Hamming distance and perform better with random errors (era- sures) than LDPC codes if decoding is bound by the half Ham- ming distance. However, -LDPC codes offer a way of com- bining soft iterative decoding with nonbinary codes, a powerful synergy when the channel has bursty impairments.
III. -LDPC DECODING
Any decoding method for B-LDPC codes can be extended to -LDPC codes by using the proper field operations. However, the efficient implementation of the belief propagation (BP) al- gorithm for B-LDPC codes using log-likelihood-ratios (LLRs) cannot be done for -LDPC codes. This fact increases the de- coding complexity of -LDPC codes.
A. BP Decoding for -LDPC Codes
Given the probability mass function pmf , ,
where can be any GF , BP
decoding for -LDPC codes is done in exactly the same two steps as for B-LDPC codes: a row step and a column step [2]
subcode is satisfied
pmf for (2)
(3)
where and ;
and .
In the row step, the subcodes are decoded. Let us simplify the notation for subcode constraint to , in which only the bits participating in the subcode are included. If we define the state at stage as , then this subcode can be represented by a trellis with states and radix- . An example of a trellis section for is shown in Fig. 2.
The well-known Bahl–Cocke–Jelinek–Raviv (BCJR) algo- rithm can be used for maximum a posteriori (MAP) decoding
Fig. 2. Diagram of a trellis section ofQ-LDPC subcodes.
[14] and involves three steps: forward recursion, backward re- cursion, and a combination step. The forward recursion is
(4) where is the inverse of in GF .
In the column step, message nodes are updated with the inde- pendence assumption
(5)
and the posterior probabilities are computed as
(6)
The hard decision is made as .
B. Fast Implementation of the BP Algorithm Using Fast Fourier Transforms
The computation complexity of the algorithm described above is , but it can be reduced. The idea of using a fast Fourier transform (FFT) in the BP decoding was proposed in [8] and [11].
Notice that in the row step, decoding of the subcodes consists in finding pmf with known pmf , and pmf
is the same as the convolution of all pmf , which can be ef- ficiently computed using the FFT
pmf IFFT FFT pmf (7)
where IFFT is the inverse FFT. It is worth noting that for B-LDPC codes, (7) is actually the same as the difference BP in [2].
Since the function pmf is defined on GF , FFT pmf is not a -point FFT but a -dimension two-point FFT, where is the number of bits in a GF field with . An example for is illustrated in Fig. 3.
The field elements are represented in polynomial form. In the first layer, the FFT computes the sum and difference of the probabilities of two field elements differing from each other by only one bit location.
Fig. 3. FFT of pmf(x ) for q = 8.
Using the FFT, the forward recursion (4) becomes
FFT pmf FFT pmf FFT pmf (8)
The column step remains the same as in (5). We refer to this algorithm as the FFT-BP algorithm.
C. Logarithm Domain Implementation of the FFT-BP Algorithm
In a practical implementation of the decoder, it is highly de- sirable to eliminate the need for real-valued multiplications. In the following, a technique is described to meet this requirement.
In the FFT-BP algorithm, real-valued multiplication occurs in both the row step and the column step. In the column step, the multiplicands are pmf . Intuitively, one should define new variables as the logarithm of these multiplicands. Let be a probability, and define
(9) Then, in the column step, only additions are needed.
In the row step, as in (8), the multiplicands are FFT pmf . Since FFT pmf may have negative values, the definition of the logarithm domain variables is complicated. Define
by
(10) where is the field of reals. The inverse
is
(11)
Then, for , and
, where , define the operations , , and such that
(12) where stands for any of the four operations. It is straightfor- ward to show that
1)
(13)
(15) where is determined as
if or or if or or
(16)
and is calculated in two cases:
a)
(17) b)
(18) 4)
(19) where and can be determined similarly to (16)–(18).
In (17), can be obtained by table
lookup. Similarly, in (18), can also
be obtained by table lookup. Therefore, neglecting binary op- erations, the computations needed for (15) are one comparison, one addition, and one table lookup. The above algorithm is re- ferred to as the Log-FFT-BP.
To summarize, (15) and (19) are used in the FFT; (13) and (14) are used in the forward–backward recursion. Also, calcu- lating for all can be efficiently implemented by first calculating , then subtracting each (similar idea cannot be applied to for all because of the “divide by zero”
problem).
In Table II, the decoding complexity of B-LDPC and -LDPC codes is compared. The decoding complexity of B-LDPC codes is given in [12]. For , the Log-FFT-BP -LDPC decoding is 12 times more complex than the Log-BP B-LDPC decoding algorithm.
IV. -LDPC CODEDMAGNETICRECORDINGCHANNELS
Two -LDPC coded systems, one on an equalized extended PR4 (EPR4) channel with partial response and the other on an equalized modified extended EPR4 (ME PR4)
channel with partial response , are
investigated. These systems are simulated at signal-to-noise ra- tios (SNRs) lower than the actual operating points with AWGN only, and also simulated at somewhat higher SNRs with burst impairments. This gives some indication of the performance of an actual system with both AWGN and burst impairments.
A. -LDPC Coded Equalized EPR4 System
Shown in Fig. 4 is the diagram of a -LDPC coded system.
The Lorentzian channel model is assumed [13]. The channel is equalized to the EPR4 target. The rate 16/17 run-length-limited (RLL) code is not implemented in the simulation, but it is in- cluded in the diagram to indicate that we are taking into account the coding penalty present in the actual system.
The system is simulated with Code 1 and Code 2, respec- tively, and is compared with the uncoded system, at user density . The BP decoder is set to perform at most 50 iter- ations. Turbo equalization is not implemented. Plotted in Fig. 5 are the BER and the symbol-error rate (SymER) performance.
These two codes perform very similarly, and both provide more than 3.5-dB gain over the uncoded system at BER . At BER , less than three iterations are executed on average.
For comparison, the performance of the weight-four B-LDPC code and the irregular B-LDPC code are also shown. It can be seen that the weight-four B-LDPC code performs marginally better than the two -LDPC codes, which is consistent with the performance comparison on AWGN in Fig. 1. Again, the irreg- ular B-LDPC code does not perform as well as the other codes.
This system with Code 2 is also simulated at SNR dB with full erasures, half erasures, and thermal asperities of dif- ferent lengths, and the performance is shown in Table III. The sector error rate is in the form of sectors in error per number of sectors simulated. Roughly, this system is able to correct full erasures of length up to 160 bits. Intuitively, the system should be able to correct longer partial erasures. It can be seen that 280-bit half erasures can be corrected, almost doubled the length for full erasures.
For simplicity, the TA is modeled as a rectangular window in which the readback signal equals the maximum signal level possible for the partial response target. Table III also shows the
Fig. 4. Q-LDPC coded EPR4-equalized magnetic recording system.
Fig. 5. Performance ofQ-LDPC coded EPR4-equalized magnetic recording channel.
TABLE III
SECTORERRORRATEWITHNOISEBURSTS,Q-LDPC CODE2
TABLE IV
SECTORERRORRATEWITHNOISEBURSTS, B-LDPC CODE
simulation results for the system with TAs. The maximal length of a correctable thermal asperity is 80 bits, which is not as good as in the case of erasures. The reason, intuitively, is as follows.
With erasures, there is no information from the channel detector, and it is essentially an erasure to the LDPC decoder. In contrast, with TAs, the channel detector provides some estimate of the bits involved, statistically half of which have the wrong polarity.
For comparison, the performance of the system with the weight-four B-LDPC code is shown in Table IV. This code can only correct erasures of about half the length of the -LDPC
Fig. 6. Model for an RS coded system.
Fig. 7. Q-LDPC coded system.
code. In addition, it performs poorly in the presence of thermal asperities. The irregular B-LDPC code was also simulated, and it can correct longer erasures and thermal asperities than the weight-four B-LDPC code, but it is not as effective as the -LDPC Code 2. In addition, as shown in Fig. 5, it performs significantly worse with AWGN only.
In practice, TAs may be detectable, in which case the channel values in the TA window can simply be zeroed out. The noise condition is therefore improved, and the system must perform better than in the presence of full erasures of the same length as the TA. Furthermore, one can perform channel detection ex- cluding the TA window, and set the LLR to zero in the thermal asperity window, as done in [9].
It is verified through the above simulations that -LDPC codes perform well on MRCs with burst impairments. Since the SNR is quite high in these simulations, the results reflect the error correction capability on erasure-dominated systems. For a practical system, it is necessary to know the performance of the system at lower SNR. An extensive simulation was carried out for the system shown in Fig. 4 at SNR dB with 80-bit full erasures. Out of 10 sectors simulated, only three sectors were in error, which corresponds to a sector error rate of ap- proximately 3 10 . This is probably acceptable for systems where burst erasures do not exceed 80 bits.
B. -LDPC Versus RS Systems on an Equalized ME PR4 Channel
Shown in Fig. 6 is a magnetic recording system diagram sim- plified for simulation purposes. The random data at the input of the MRC are assumed to be RS codewords, and pseudo-RS de- coding is performed. The overall code rate is 0.8425.
The -LDPC system studied is shown in Fig. 7. These two systems have similar code rates. The -LDPC code is Code 3 in Table I with rate 0.8890 and MSD . The overall code rate is 0.8298, close to the RS system.
The two systems were simulated at on
Lorentzian–Gaussian channels [15] with purely AWGN, and also on channels with 90% jitter noise power [16]. At most, 50 LDPC iterations were allowed.
Shown in Fig. 8 is the performance of the -LDPC and RS coded systems under purely AWGN, with both sector and byte (8-bit) error rates shown. At a sector error rate (SecER) of 10 ,
Fig. 8. Performance on magnetic recording channels with purely AWGN.
Fig. 9. Performance on magnetic recording channels with 90% jitter noise power.
TABLE V
ERASUREPERFORMANCE OFQ-LDPCON ANME PR4 EQUALIZEDSYSTEM
the -LDPC coded system outperforms current RS systems by 2.2 dB. Shown in Fig. 9 are similar results for the -LDPC and RS coded systems under 90% jitter noise power. At a sector error rate of 10 , the investigated -LDPC coded system outper- forms current RS systems by 1.4 dB.
The performance of the -LDPC system with burst erasures is shown in Table V. With purely AWGN, the raw BER at SNR dB is approximately ; and with 90% jitter noise power, the raw BER at SNR dB is also around . In both cases, the -LDPC system cannot correct 80-bit burst erasures.
erage LLR magnitude inside the erasure window was obtained through simulation, as well as the average LLR magnitude out- side the erasure window. The ratio of the former to the latter is found to be 0.41 for the ME PR4-equalized channel and 0.23 for the equalized EPR4 channel. The large LLR magnitude in the erasure window indicates higher noise. This may explain the difference in performance.
If full erasures or thermal asperities are detected, then by ze- roing the channel BCJR detector output LLRs in the impairment window, the 80-bit bursts are correctable.
V. CONCLUSION
A reduced-complexity decoding algorithm for -LDPC codes was presented, which brings the complexity of LDPC codes over GF(16) to about 12 times that of comparable binary codes. This reduced-complexity algorithm makes -LDPC codes attractive for magnetic recording. We have investigated the performance of these codes on Lorentzian–Gaussian magnetic recording channel models equalized to high-order PR targets. -LDPC codes were shown to be a good alternative to B-LDPC or RS codes for magnetic recording, because they perform well with AWGN and outperform B-LDPC codes when burst impairments are present. We further conclude that future hard disk drive systems could use a single sector-size LDPC code over GF(16) without the need for an outer RS code.
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Hongxin Song (S’00–M’03) was born in Jiangsu, China. He received the B.S.
and M.S. degrees from Southeast University, Nanjing, China, in 1991 and 1994, respectively, and the Ph.D. degree from the University of Oklahoma, Norman, in 2002, all in electrical engineering.
He is now with Marvell Semiconductor, Inc., working on signal processing for communications.
J. R. Cruz (S’75–M’79–SM’85–F’01) received the B.S. degree from the Uni- versity of Porto, Portugal, and the M.S. and Ph.D. degrees from the University of Houston, Houston, TX, in 1974, 1977 and 1980, respectively, all in electrical engineering.
From 1980 to 1981, he was with Computer Sciences Corporation at the NASA Johnson Space Center, Houston, TX, and from 1981 to 1982, he was with Mo- torola, Inc., as a Project Engineer in the Research Department of the Mobile Products Division, Ft. Worth, TX. In 1982, he joined the University of Ok- lahoma, Norman, OK, where he currently holds the Tilley Chair in Electrical Engineering. His current research interests include applications of signal pro- cessing to wireless communications and magnetic storage.
Prof. Cruz is a member of Eta Kappa Nu, Sigma Xi, Phi Kappa Phi, and the American Association for the Advancement of Science, and a Fellow of the Radio Club of America. He is the Past President of the IEEE Vehicular Technology Society, a former Editor-in-Chief of the IEEE TRANSACTIONS ON VEHICULARTECHNOLOGY, and served on the Board of Editors for the Inter- national Journal of Wireless Information Networks, ACM/Baltzer Journal on Wireless Networks, and Wireless Personal Communications. He is the recipient of the 1995 Outstanding Service Award from the IEEE Vehicular Technology Society and the IEEE Third Millennium Medal.
