AbstractâTurbo codes and low-density parity check (LDPC) codes with iterative decoding have received significant research at- tention because of their ...
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Iterative Decoding for Partial Response (PR), Equalized, Magneto-Optical (MO) Data Storage Channels Hongwei Song, Student Member, IEEE, B. V. K. Vijaya Kumar, Senior Member, IEEE, Erozan Kurtas, Yifei Yuan, Laura L. McPheters, and Steven W. McLaughlin, Senior Member, IEEE
Abstract—Turbo codes and low-density parity check (LDPC) codes with iterative decoding have received significant research attention because of their remarkable near-capacity performance for additive white Gaussian noise (AWGN) channels. Recently, turbo code and LDPC code variants are being investigated as potential candidates for high-density magnetic recording channels suffering from low signal-to-noise ratios (SNR). In this paper, we address the application of turbo codes and LDPC codes to magneto-optical (MO) recording channels. Our results focus on a variety of practical MO storage channel aspects, including storage density, partial response targets, the type of precoder used, and mark edge jitter. Instead of focusing just on bit error rates (BER), we also study the block error statistics. Our results for MO storage channels indicate that turbo codes of rate 16/17 can achieve coding gains of 3–5 dB over partial response maximum likelihood (PRML) methods for a 10 4 target BER. Simulations also show that the performance of LDPC codes for MO channels is comparable to that of turbo codes, while requiring less computational complexity. Both LDPC codes and turbo codes with iterative decoding are seen to be robust to mark edge jitter. Index Terms—Iterative decoding, low-density parity-check (LDPC) codes, magneto-optical (MO) recording, partial response maximum likelihood (PRML).
I. INTRODUCTION
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URBO codes, or parallel concatenated codes, originally proposed by Berrou et al. in 1993 [1], were shown to achieve remarkable, near-Shannon capacity performance for additive, white Gaussian noise (AWGN) channels. Shortly after, serial concatenated codes with iterative decoding were shown to provide similar performance gains [2]. In the wake of the excellent performance of turbo codes, another class of codes called low-density parity-check (LDPC) or Gallager codes [3], which exhibit similar characteristics and performance, was rediscovered [4]. Recently, these codes with iterative decoding have been investigated [5]–[8] as potential candidates for low signal-to-noise ratio (SNR), high-density magnetic recording
Manuscript received March 1, 2000; revised July 22, 2000. This work was supported by the National Science Foundation, under Grant ECD-8907068. H. Song, B. V. K. V. Kumar, and Y. Yuan are with the Data Storage Systems Center, Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 USA. E. Kurtas is with Seagate Technology, Seagate Research Pittsburgh, Pittsburgh, PA 15203 USA. L. L. McPheters and S. W. McLaughlin are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. Publisher Item Identifier S 0733-8716(01)01749-8.
channels. Simulations show that for high rates (e.g., 8/9, 16/17, etc.), iterative decoding requires about 3 to 5 dB less SNR compared with the uncoded partial response maximum likelihood (PRML) method. Relatively less attention has been paid to the application of iterative decoding methods for optical data storage channels [9]–[11]. Information is represented in optical storage systems by modulating either the magnetic (as in magneto-optical (MO) media) or optical (e.g., in phase change media) properties of the recording medium. We will focus our attention in this paper on binary MO recording where the resulting magnetized regions are called marks and spaces. Kerr effect (the property that the magnetization of the medium affects the polarization of the transmitted/reflected light) is commonly employed [12] for the MO readback. A significant feature of the MO readout process is that it responds to the DC component. Thus, the transition response (i.e., the readback signal induced by a change from mark to space) looks more like a smoothened step function than a pulse function. In contrast, the transition response longitudinal magnetic recording is best modeled as a smoothened pulse. Because of this difference in channel transition response, partial response (PR) targets appropriate (unlike for MO data storage [13], [14] are of the form for magnetic recording channel), where denotes a 1-bit delay and is usually chosen to be 1, 2, or 3. Thus, iterative decoding may perform differently when applied to MO data storage channels, and in this paper, we address the application of turbo codes and LDPC codes with iterative decoding to MO PR channels. In the recording system we consider for this paper, the channel detector is a maximum a posteriori (MAP) detector instead of a Viterbi detector; and Turbo/serial/LDPC codes being investigated are applied in between the ECC module and the write module. Although optical storage systems usually run-length limited (RLL) codes with , the employ effect of RLL codes is not considered in this paper, because the combination of RLL coding with iterative decoding is not trivial, and we hope to discuss related RLL coding issues in a later paper. Related studies [5]–[9] have mostly assumed ideal equalization and AWGN and, thus, cannot adequately capture the effect of increasing the storage density. In this paper, we will discuss the following aspects of applying iterative decoding to MO recording channels. • Iterative decoding is investigated using an MO recording channel model [13], [14] employing a finite impulse re-
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sponse (FIR) filter so that the density effects, the equalizer length effects, and the noise coloring caused by PR equalizer are captured. Also, the effect of mark edge jitter on the BER performance is studied. • PR targets suitable for MO recording are investigated and the least mean squares (LMS) algorithm is used to train the PR equalizer coefficients to understand the effects of a limited training set on the turbo code performance. • We investigate the BER for different architectures of concatenated codes and LDPC codes when applied to the MO recording channels. • We study the block error statistics of turbo codes and LDPC codes rather than just the BERs. Such statistics will determine the demands on outer codes in architectures employing concatenated codes. The remainder of this paper is organized as follows. The MO channel model will be discussed in Section II. Two different channel models are discussed. The first model does not take into account mark edge jitter, whereas the second one does. In Section III, we will briefly review both parallel and serial concatenated turbo code architectures under consideration. We will also show how the PRML MO channel naturally fits into a serial concatenated structure, in the same way a PRML magnetic recording channel does [5]–[8]. LDPC codes have simpler decoding algorithms compared with concatenated codes and, thus, may be more compatible with the high data-rate requirement of storage channels. LDPC codes will be briefly reviewed in Section III, and we will discuss how to apply LDPC to MO channels. BER simulation results will be presented in Section IV. Conclusions will be provided in Section V.
II. MAGNETO-OPTICAL (MO) CHANNEL MODELING A. Linear Superposition Model is modeled in terms of the isolated The readback signal , the response to an isolated recorded transistep response tion from space to a mark. In the MO medium, this transition corresponds to a change in the direction of magnetization just as in a longitudinal magnetic recording medium. On the other hand, if the medium was a phase change (PC) medium, the transition usually refers to a change in an optical property (e.g., re, produced by a transition that flectivity). A good model for moves past the focused spot at a linear velocity , is an error function [15]; i.e.,
FWHM
(1)
where (2) NA is the full width at half-maximum and FWHM (FWHM) spot size at the disk, which is governed by the wavelength and the numerical aperture (NA) of the optical system.
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Assuming that linear superposition holds [16], modeled as
can be
(3) denotes additive Gaussian noise with two-sided where , is the bit interval, and power spectral density denotes the absence or presence of transitions for a transition from mark to space and for a ( transition from space to mark). can also be modeled in the following The readback signal alternative form: (4) is the channel pulse rewhere is the recorded binary sequence represponse, . senting spaces and marks, and By decreasing , we increase the recording density and a useful measure of achieved density is the normalized density defined as follows: FWHM
(5)
Larger means more closely spaced transitions and, hence, more intersymbol interference (ISI). B. Channel Model with Mark Edge Jitter A significant problem in high-density MO storage and, for that matter, in most other optical storage methods is that transitions (also called mark edges) do not necessarily appear where we intend them to. The mark formation process usually involves the interaction between the thermal content of the illuminating laser spot (affected by the optics) and the properties of the moving optical recording medium. Although careful writing strategies are employed to achieve marks of specific sizes, there is bound to be some error in the transition positions. This error can be either deterministic or random or can contain both types. The deterministic mark size change (caused by laser power being too high or too low or the medium being too sensitive or not sensitive enough) is known as bloom. Residual random shifts in transition locations (caused by a variety of sources, including random media sensitivity or laser noise) are termed mark edge jitter [13]. Both lead to BER degradation, as the transitions are not where they are supposed to be in the absence of such mark edge jitter. Let us denote the shift of the transition in the th bit cell by , modeled as a zero-mean, truncated Gaussian random variis uncorrelated able with variance . We also assume that for different and statistically independent of the additive noise . Thus, the readback signal with mark edge jitter is modeled as follows: (6)
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Fig. 1. Schematic of the optical recording channel model.
Fig. 2. Receiver structure for an iterative decoded channel.
C. Front-End Low-Pass Filter and Finite Impulse Response (FIR) Equalizer Fig. 1 shows a block diagram of an MO recording channel containing a front-end low-pass filter (LPF) and an FIR equalizer. The role of the analog front-end LPF, placed before the sampling, is to reduce out-of-band noise. In some optical storage systems (e.g., DVD), this filter is designed to provide some equalization also by boosting frequency components near the edge of the pass band. The phase response of this LPF must be linear so that pulse symmetry is preserved after this filtering. A commonly employed equiripple type LPF with seven poles and two zeros [17] is used in our channel. It has two parameters that define the cutoff frequency and boost. The function of the FIR equalizer is to shape the overall channel response (combined effect of the channel pulse re, the front-end LPF, and the FIR equalizer) to be sponse close to the selected PR target response. Because of the nonzero DC response of the MO channels, PR targets of interest here factor. We used an FIR equalizer filter do not contain a with 11 coefficients that are determined by LMS training, and a training sequence of 4000 samples. Because the number of FIR filter coefficients is finite and the length of the training sequence is also limited, the overall channel response after the FIR equalizer will not perfectly match the ideal PR target response. In addition, the FIR filter colors the input white noise so that the noise entering the channel detector is not white, as is usually assumed in many iterative decoding investigations. III. REVIEW OF CODING METHODS In the wake of the great interest in turbo codes and related structures, various variations of concatenations and interleavers have been studied and shown to offer performance similar to the original parallel concatenated code structure. These codes use iterative soft decoding algorithms. Fig. 2 illustrates the receiver structure for the iterative decoded channel. In this section, we will briefly review the three code categories of interest, namely, parallel concatenated codes (turbo codes), serial concatenated codes, and LDPC codes. A. Parallel Concatenated Codes Turbo codes, or parallel concatenated codes, consist of two constituent encoders, separated by a pseudorandom interleaver
of length . The encoding is straightforward and described by the standard turbo encoding [1]. Some of the parity bits are omitted to obtain a desired code rate. The channel detector is based on the PR target response and computes the log-likelihood ratios (LLRs) of the channel input based on the received noisy samples. The LLRs are demultiplexed and deinterleaved as systematic and parity samples, which are used by the channel decoder. The main principle in the parallel concatenated scheme is that with the parallel encoding, we decode the two encoded streams with an iterative process using two soft-in, soft-out decoders, one corresponding to each of the encoders. A turbo code tutorial with several simplifications for the decoder for this type of code is given in [18]. In the decoding architecture of Fig. 2, if we do not pass any information from the channel decoder to the channel detector, we call it a partial turbo decoder [6], [10]. Simulation results using this partial turbo decoder will be denoted in our figures by the label “Part.” As pointed out elsewhere [6], [10], we may be able to achieve better performance by allowing exchange of soft information between the channel detector and the channel decoders. A receiver structure that allows such iteration is known as the full turbo decoder [6], [10]. We will denote the results from the full turbo decoder by the label “Full” in our figures. B. Serial Concatenation with PR Equalized Recording Channel In magnetic recording, several researchers realized that a precoded PRML channel could be viewed as a simple rate 1 inner code [5], [6]. This observation leads to a configuration that offers significant complexity reduction over the full and the partial turbo decoding structures; i.e., only one convolutional encoder is used instead of the two parallel encoders, and the iterative turbo decoder is replaced by one decoder. The information exchange occurs between the channel detector and the channel decoder. Such computational simplicity is essential for storage applications, as they demand very high data rates. In this paper, a serial concatenated system with the MO channel model is investigated for different partial response targets. We will denote the results from this architecture by the label “Seri” in our plots. We will denote the results from the PRML channel without the outer encoder and the corresponding decoder by label “PRML.” C. Low-Density Parity Check Codes LDPC codes first introduced by Gallager [3] are error-correcting codes based on sparse parity check matrices. After the introduction of turbo codes, there was a renewed interest in LDPC codes because of their computational simplicity and their corresponding iterative decoding algorithms [19]. It is shown that properly designed LDPC codes can provide turbo-like performance in AWGN channels. The major difference between turbo and LDPC codes is in the simplicity of the LDPC decoding algorithm compared with the decoding algorithms used for turbo codes. It is possible to associate a factor graph [20] with the parity check matrix used in the construction of the LDPC codes. Then, a simple neural network–inspired algorithm known as the belief propagation algorithm can be used to obtain the estimates of the recorded symbols. Details of the belief propagation (or message-passing)
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Fig. 3. BER versus SNR for PR channels employing no outer decoding and no precoding: circles, S 1:0, (1 + D) PR target; diamonds, S = 1:0, (1 + D ) PR target; squares, S = 2:0, (1 + D ) PR target; triangles, S = 2:0, (1 + D ) PR target.
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Fig. 4. BER versus SNR for PR channels with a normalized density of S = 2:0 and no precoding: diamonds, (1 + D ) PR target, no turbo decoding; asterisks, (1 + D ) PR target, serial decoding; triangles, (1 + D ) PR target, serial decoding; plusses, (1 + D ) PR target, serial decoding.
algorithm are beyond the scope of this paper and can be found elsewhere [20], [21]. The application of LDPC codes to magnetic recording channels was studied in [8], and it was shown that LDPC codes provide turbo-like performance for ideal and Lorentzian equalized PR targets. IV. SIMULATION RESULTS In this section, we show BER as a function of SNR for a selected set of PR targets, precoder and decoding architecture choices. The SNR is defined as SNR
(7)
is the normalized energy of the partial response where is the code rate, and is the two-sided power target, spectral density of the white noise being added to the readback signal before inputting it into the equalizer. Recall that transition noise (i.e., mark edge jitter) is modeled as a truncated, zero-mean, Gaussian-distributed random variable, with in our second model. The percentage of transition variance %. noise for the recording channel is For the results shown in this paper, the iterative turbo decoding employed 15 iterations, although we show only the results of the last iteration. We show the results for turbo codes codes and interleaving block size of using rate . Both the convolutional codes employed use same code polynomials (31, 23) . To ensure a fair comparison between LDPC codes and turbo codes, we use the same code rate (16/17) and the same block length for LDPC codes. The column weight of the regular LDPC code parity check matrix we used is as the target BER unless stated otherwise. 3. We consider A. Simulation Results for Turbo Codes Fig. 3 shows the BER versus SNR results for the uncoded PRML (i.e., no outer encoding and decoding is used) for two and and for normalized densities and PR targets. For , the target is
Fig. 5. BER versus SNR for the (1 + D ) PR channel with no precoder, recording density S = 2:0: plusses, uncoded; circles, partial turbo decoding; diamonds, full turbo decoding; triangles, serial concatenated decoding.
better than the target by about 1 dB (for a target BER ). For , the target is better than the of target by about 3.0 dB. Comparing the best BERs at for and the each density [i.e., the target for ], we see that the uncoded PRML channels target BER require a 2-dB SNR increase to achieve the [with target] to as we increase density from [with target]. We show in Fig. 4 the BER versus SNR for the turbo-coded and no precoding. At this higher recording channel with density, using the serial concatenated decoding scheme, the PR target is worse than the other two PR targets by about 1.5 PR target is similar to that dB. The performance of the target, but the detector for the target of the is significantly more complex than the detector for the target. PR target Based on Figs. 3 and 4, we conclude that the and PR is the suitable for normalized density
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Fig. 6. BER versus SNR for selected PR channels using serial concatenated 1:0, (1 + D) PR target and no precoding: diamonds, decoding: circles, S S = 1:0, (1 + D) PR target with 1=(1 D) precoding; asterisks, S = 2:0, (1 + D ) PR target with no precoding; squares, S = 2:0, (1 + D ) PR target with (1 D ) precoding.
=
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Fig. 7. BER versus SNR for selected PR channels using serial concatenated codes and LDPC codes: circles, serial concatenated codes, S = 1:0, (1 + D ) PR target with 1=(1 D ) precoding; squares, LDPC codes, S = 1:0, (1 + D ) PR target with no precoding; diamonds, serial concatenated codes, S = 2:0, D ) precoding; triangles, S = 2:0, LDPC (1 + D ) PR target with 1=(1 codes, (1 + D ) PR target with no precoding.
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target for . For the rest of the simulations, we will use these targets for corresponding normalized densities. Fig. 5 shows the BER versus SNR for the three turbo-decoding architectures as well as for the uncoded PRML case. No precoding is employed, and the PR target for a normalized recording density of used was . Over the investigated range of SNRs, the full turbo decoding appears to perform about 0.5 dB better than the partial turbo decoding and about 3 dB better than the uncoded case. Surprisingly, the simpler serial decoder outperforms the partial PR target over turbo and the full turbo decoder for the the BER range investigated. For larger SNR values, the serial decoder exhibits an error floor, whereas the other two schemes exhibit a significantly faster decrease in BER with increasing SNR.
Fig. 8. Effect of transition noise on the BER performance of uncoded PRML channel and serial turbo coded channel with (1 + D ) PR target, no precoding, and normalized density S = 2:0. The percentage of transition noise is defined as a percentage of =T .
Fig. 9. Effect of transition noise on the BER performance of uncoded PRML channel and LDPC coded channel with (1 + D ) PR target, no precoding, and normalized density S = 2:0. The percentage of transition noise is defined as a percentage of =T .
Although not shown here, similar performances were obwith served for normalized recording density PR target and with PR target. Thus, we can conclude that the full turbo decoder performs about 0.5 dB better than does the partial turbo, which is contributed by the additional soft information exchange between the channel detector and the turbo decoder. Also, the simpler serial turbo decoding outperforms the full turbo decoder at low SNRs. Because low SNR is more likely the case at higher densities, it is probably safe to say that the simpler serial concatenated code is a good candidate for high-density, low-SNR MO storage channels. But the serial turbo performance curves flatten out or lower BER. The “error floor” is believed to be a at consequence of the relatively low free distance of the code and can be lowered by either increasing the size of the interleaver or increasing the free distance of the code [22]. We show in Fig. 6 the effect of precoders when used with PR target, use of the serial concatenated codes. For the
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Fig. 10.
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Number of blocks as a function of the number of bit errors for serial concatenated decoder with (1 + D) target, no precoding and S = 2:0.
precoder seems to improve the BER performance when the BER values are better (i.e., smaller) than certain BER ). Similarly, use of precoder seems threshold ( PR target when to be beneficial in the case of the . We studied many other PR the BER is better than targets with different precoders and have observed that serial concatenated codes with no precoders have better performance at low SNRs, but exhibit earlier error floors than do codes with precoders. Precoders seem to affect the minimum code weight distribution in a favorable manner, thus, pushing the error floor further down. Thus, the need for a precoder might very much depend on what kind of BERs we need from turbo codes. B. Comparison of LDPC Codes with Serial Concatenated Codes In Fig. 7, we compare the simulation results of serial conand catenated codes and LDPC codes for PR targets , at density 1.0 and 2.0, respectively. For the PR target suitable for , LDPC codes perform more or less the same as serial concatenated codes when the BER values . For the are smaller than some threshold target suitable for density , the performance of LDPC codes is only about 0.3 dB worse than serial turbo decoding. These results suggest that LDPC codes are comparable to turbo codes in terms of BER performance while offering the potential for complexity reduction. C. Effect of Mark Edge Jitter on BER Performance We show in Fig. 8 the effect of transition noise (modeling the mark edge jitter found in many MO recording systems) on the BER performance for the uncoded PRML case and serial PR target, no precoder, concatenated decoding with . For the uncoded PRML chanand recording density nels, 5% transition noise causes little SNR degradation, and it BER in the requires a 2.5-dB SNR increase to achieve
presence of 10% transition noise. In contrast, for serial turbo decoding schemes, 10% transition noise only causes 0.5-dB loss in SNR. We also see that the uncoded channel performs poorly in the presence of 15% transition noise, but the serial concatenated scheme only experiences about a 1.0-dB SNR loss. Fig. 9 shows similar results of the effect of transition noise on the BER performance, for the uncoded PRML case and LDPC codes with PR target, recording density . For LDPC-coded channels, 5% transition noise has a small effect (0.2-dB degradation) on BER performance, and it requires about BER if 10% transition a 0.8-dB SNR increase to achieve noise is introduced. Again, the LDPC codes can accommodate 15% transition noise, with less than a 2.0-dB SNR loss. Based on Figs. 8 and 9, we can conclude that both LDPC codes and serial concatenated codes with iterative decoding are robust to transition noise, showing only a modest (about 1.0 2.0 dB) SNR loss, even for 15% transition noise. D. Block Failure Rates Although BER comparisons are important, they do not reveal if any correlations exist in the error patterns. To facilitate this investigation, our simulation also produces the number of blocks with a certain of number of errors in that block. As one illustration of the importance of observing the block failure statistics, we show in Fig. 10 the number of blocks as a function of the number of errors for the serial concatenated scheme with PR target, no precoder, and recording density . Recall that each block has 4096 bits. A large number of blocks are error free, and the total number of blocks with no errors is not being shown in this figure for the sake of clarity in the subplots. In Fig. 10, we plot the block error statistics for four , , , and BER levels (namely, ), the top one corresponds to the waterfall region of the BER versus SNR plot, and the other three correspond to the “error floor” region. We observe that the most common errors are the ones with two bit errors per block for all of the cases.
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Fig. 11.
Number of blocks as a function of the number of bit errors for serial concatenated decoder with (1 + D) target, 1=(1
Fig. 12.
Number of blocks as a function of the number of bit errors for LDPC codes with (1 + D ) target, S = 2:0.
For the higher BER, we see that there are some blocks with more than 100 errors. Although for the other three BERs corresponding to “error floor,” the number of bit errors per block (given that the block is in error) is typically contained within a few bits. So it is safe to say that the “error floor” is contributed by large number of blocks with a few error bits, in particular, two bit errors, which supports an earlier [22] conclusion that the “error floor” is a consequence of the relatively small free distance of the code. The key point here is that if an outer error correction code such a Reed–Solomon code is used, then the errors introduced in the “error floor” region can be corrected with high probability. We show in Fig. 11 similar block error statistics as in Fig. 10, with the only difference being that a precoder is employed. Two differences between Figs. 10 and 11 are apparent. First, the number of blocks with small bit errors no longer dominates. In fact, blocks with just a few errors seem to
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D
) precoding and S = 2:0.
rarely happen. Second, the inclusion of the precoder seems to lead to blocks with an even larger number of errors. Also, the distribution of the number of error blocks as a function of the number of errors appears to be more uniform with the precoder than without the precoder. We show in Fig. 12 similar block error statistics, but for LDPC codes. For LDPC codes, most blocks with errors have between 20 to 120 error bits (depending on the SNR), and few blocks with a small number of bit errors were observed. Also, the maximum number of errors in a block for LDPC codes is comparable to that in Fig. 11 (serial turbo schemes with precoding) rather than that in Fig. 10 (without precoding). It is clear that turbo codes and LDPC codes lead to different types of error patterns. This will affect the specifications of an outer ECC, such as a Reed–Solomon code. Not surprisingly, the block error statistics plots are very much consistent with the BER versus SNR plots. In the low SNR re-
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gion, the precoder introduces more errors, but it functions like a recursive convolutional code, which can change the weight distribution after the outer convolutional code. As a result, the overall codeword has less number of low weight codewords, which can lower the error floor. As Gallager proved [3], the LDPC codes have good weight distribution with high probability; so the performance of LDPC codes looks more like serial turbo codes with precoding rather than serial turbo codes without precoding, both in terms of BER versus SNR plot and block error distribution. V. CONCLUSION In this paper, we have discussed how turbo codes and LDPC codes can be applied to MO recording channels. Based on our simulations, there are a few conclusions we can draw regarding the applicability of iterative decoding to MO data storage channels. • Of the three concatenation schemes considered, the simpler serial turbo decoding seems to perform the best at low SNRs. However, serial turbo decoding seems to reach the error floor more quickly. • Turbo codes yield a 4–5-dB SNR benefit over the uncoded . LDPC codes are comparable PRML channels for to turbo codes in terms of BER performance, although their main advantages lie in their reduced complexity. • Use of precoding does not appear to improve BER at low SNR. However, precoding seems to benefit when the SNR is high and seems to delay the onset of the error floor. Inclusion of precoder changes the block error distributions for serial concatenated schemes. • Both LDPC codes and concatenated codes with iterative decoding are robust to mark edge jitter found in optical recording, showing only a modest SNR (about 1 dB) loss even for 15% transition noise. • For serial turbo decoding, the number of blocks with small number of bit errors dominates if no precoder is employed. Inclusion of precoder leads to small number of error blocks, but larger number of error bits if a block is in error. LDPC codes exhibit block error distributions more like serial turbo codes with precoding rather than that without precoding.
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[7] T. M. Duman and E. Kurtas, “Comprehensive performance investigation of turbo codes over high density magnetic recording channels,” in Proc. GLOBECOM 1999. [8] J. L. Fan, E. Kurtas, A. Friedmann, and S. W. McLaughlin, “LDPC codes for magnetic recording,” in Proc. 36th Allerton Conf., Monticello, IL, Sept. 1999, pp. 1314–1323. [9] L. McPheters and S. McLaughlin, “Concatenated codes and iterative (turbo) decoding for PRML optical recording channels,” in Proc. Optical Data Society Conf., 1999, pp. 342–343. [10] H. Song, B. V. K. Vijaya Kumar, E. Kurtas, and Y. Yuan, “Turbo decoding for optical data storage,” in Proc. Int. Conf. Commun. 2000, vol. 1, New Orleans, LA, pp. 104–108. [11] H. Song, B. V. K. Vijaya Kumar, and E. Kurtas, “Effect of transition noise on turbo decoding for optical data storage,” in Proc. SPIE Int. Symp. Optical Memory and Optical Data Storage 2000, pp. 283–288. [12] M. Mansuripur, The Physical Principles of Magneto-Optical Recording. Cambridge, U.K.: Cambridge Univ. Press, 1995. [13] R. Haeb and R. T. Lynch Jr., “Trellis codes for partial-response magnetooptical direct overwrite recording,” IEEE J. Select. Areas Commun., vol. 10, pp. 182–190, 1992. [14] I. Ozgunes, B. V. K. Vijaya Kumar, and M. Kryder, “Effect of transition noise on the signal-to-noise ratio of magneto-optic read channels,” IEEE Trans. Magn., vol. 32, pp. 3291–3304, 1996. [15] M. D. Levenson, R. T. Lynch, and S. M. Tan, “Edge detection for magneto-optical data storage,” Appl. Opt., vol. 30, pp. 232–252, 1991. [16] R. T. Lynch, “Modeling the performance of optical recording read channels,” Appl. Opt., vol. 27, pp. 723–727, 1988. [17] Silicon System, Specification of SSI 33p3725 Read Channel IC for DVD-ROM, 1996. [18] W. Ryan. A turbo code tutorial. [Online]. Available: http://www.ece.arizona.edu/~ryan/. [19] D. J. C. MacKay, “Good error correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, pp. 399–431, 1999. [20] B. Frey, Graphical Models for Machine Learning and Digital Communication. Cambridge, MA: MIT Press, 1998. [21] J. Pearl, Probabilistic Reasoning in Intelligent Systems, 2nd ed. San Francisco, CA: Morgan-Kaufman, 1988. [22] L. C. Perez, J. Seghers, and D. J. Costello, “A distance spectrum interpretation of turbo codes,” IEEE Trans. Inform. Theory, vol. 42, pp. 1698–1709, Nov. 1996.
Hongwei Song (S’01) was born in Zhejiang, China, in January 1975. He received the B.E. degree in optical engineering from Zhejiang University, Zhejiang, China, in 1996, and the M.S. degree in electrical engineering from Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai, China in 1999. He is currently pursuing the Ph.D. degree with the Department of Electrical and Computer Engineering, Carnegie Mellon University. His main research interests concern coding and signal processing for intersymbol interference channels as well as optical communication systems. He received the second place award of the International Disk Drive Equipment and Materials Association (IDEMA) 2000–2001 Fellowship.
REFERENCES [1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo codes,” in Proc. 1993 Int. Conf. Commun., pp. 1064–1070. [2] S. Benedetto, G. Montorsi, D. Divsalar, and F. Pollara, “Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding,” TDA Progress Rep. 42-126, JPL, Aug. 15, 1996. [3] R. G. Gallager, Low-Density Parity Check Codes. Cambridge, MA: MIT Press, 1963. [4] D. J. C. Mackay and R. M. Neal, “Good codes based on very sparse matrices,” in Proc. Cryptography and Coding. 5th IMA Conf, 1995, [Online]. Available: http://wol.ra.phy.cam.ac.uk. [5] W. Ryan, L. McPheters, and S. McLaughlin, “Combined turbo coding and turbo equalization for PR4-equalized Lorentzian channels,” in Proc. Conf. Inform. Sci. Syst., Princeton, NJ, 1998, pp. 489–493. [6] T. Souvignier, A. Friedmann, M. Oberg, P. Siegel, R. E. Swanson, and J. K. Wolf, “Turbo codes for PR4: Parallel versus serial concatenation,” in Proc. 1999 Int. Conf. Commun..
B. V. K. Vijaya Kumar (S-’78–M’80–SM’90) received the B.Tech. and M.Tech. degrees in electrical engineering from the Indian Institute of Technology, Kanpur, and the Ph.D. degree in electrical engineering from Carnegie Mellon University (CMU), Pittsburgh, PA. Since 1982, he has been a faculty member in the Department of Electrical and Computer Engineering, CMU, where he is now a Professor. His research interests include image processing, pattern recognition, coding, and signal processing for data storage systems. He has authored or co-authored more than 250 technical papers in these areas in various conference proceedings and journals. Dr. Kumar served as a Topical Editor for the Information Processing Division of Applied Optics. He is a member of Sigma Xi, a Fellow of SPIE, and a Fellow of Optical Society of America. He is listed in Marquis’s Who’s Who in the World and in the American Men and Women of Sciences.
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Erozan Kurtas received the B.Sc. degree from the ECE Department, Bilkent University, Ankara Turkey, in 1991 and received the M.Sc. and Ph.D. degrees from the ECE Department, Northeastern University, Boston, MA, in 1993 and 1997, respectively. Dr. Kurtas’s research interests cover the general field of digital communication and information theory with special emphasis on coding and detection for inter-symbol interference channels. Dr. Kurtas is currently the head of the channels department at Seagate Technology, Seagate Research Pittsburgh.
Yifei Yuan was born in Shanghai, China, in October 1970. He received the B.S. and M.S. degrees in precision instruments from Tsinghua University, Beijing, China, in 1993 and 1996, respectively, and the Ph.D. degree in electrical and computer engineering from Carnegie Mellon University, Pittsburgh, in 2000. His research interests include signal processing for communications channels, probability decoding, and neural networks.
Laura L. McPheters, photograph and biography not available at the time of publication.
Steven W. McLaughlin (S’90–M’92–SM’99) received the B.S.E.E degree from Northwestern University, Chicago, IL, in 1985, the M.S.E. degree from Princeton University, Princeton, NJ, in 1986, and the Ph.D. degree from the University of Michigan, Ann Arbor, in 1992. He is an Associate Professor in the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta. He He has also held positions at the Rochester Institute of Technology, Booz, Allen and Hamilton, Eastman Kodak, and AT&T Bell Laboratories. He holds 18 patents in the area of coding for high-density magnetic and optical recording, and he has over 10 patents pending. His research interests include communication and information theory, including coding for constrained channels, signal processing and coding for high-density magnetic and optical recording channels, and source coding (quantization and compression). Dr. McLaughlin received the Presidential Early Career Award for Scientists and Engineers (PECASE) and the NSF CAREER award, in 1997, and he received the NSF Research Initiation award in 1993. He is member of the IEEE Information Theory Society Board of Governors. He served as the Publications Editor for the IEEE TRANSACTIONS ON INFORMATION THEORY from 1995 to 1999. He was also responsible (Ramesh Rao) for creating for IEEE TRANSACTIONS ON INFORMATION THEORY Digital Library, where the entire history of the TRANSACTIONS is now available online at galaxy.ucsd.edu/welcome.htm. He also co-edited Information Theory—50 Years of Discovery with Sergio Verdu (Piscataway, NJ: IEEE Press, 1999).
