Iterative detection/decoding of majority-logic decodable nonbinary LDPC codes over partial response channels

Towards the goal of achieving better error correction performance in data storage systems, iterative soft decoding of low density parity check (LDPC) codes and soft-decision decoding of Reed-Solomon (RS) codes have started receiving increasing research attention. However, even with increased computing power, complexities of soft-decision decoding algorithms are still too high for real products which require high throughput and small hardware area. Another problem is that the performance gains of those approaches are smaller for magnetic recording channels than they are for memoryless additive white Gaussian noise (AWGN) channels. We propose a new soft-decision decoding algorithm (based on the Chase algorithm), which takes advantage of pattern reliability instead of symbol reliability or bit reliability. We also present a modified Viterbi algorithm that provides probable error patterns with corresponding reliabilities. Simulation results of the proposed algorithms over the partial response (PR) channel show attractive performance gains. The proposed algorithm dramatically reduces the number of iterations compared to the conventional Chase2 algorithm over the PR channel.

We present a systematic way to construct multilevel concatenated codes for partial response (PR) channels using: (1) a structured set partition (SSP) of multiple channel output sets and (2) a set of conventional block codes with different error correcting capabilities. A lower bound on the minimum squared Euclidean distance of the constructed codes is given. This bound is based on the interset minimal Euclidean distances of the SSP and the minimum Hamming distances of the used block codes. An example of SSP for the extended class 4 partial response channel (EPR4) is presented. Iterative suboptimal decoding, which combines Viterbi detection on the trellis of the PR channel with algebraic error detection/correction, can be applied to the constructed concatenated codes. Truncated versions of the iterative decoding scheme are simulated and compared with each other.

A number of papers have recently been published on the concatenation of an outer code with a partial response (PR) channel, where the outer code is a turbo code, a convolutional code, or a low density parity check code. This paper deals with the second case assuming EPR4 and modified extended EPR4 (MEEPR4) partial response targets. The goal in this work is the joint optimization of interleaver and precoder for a fixed outer convolutional code and PR target. We introduce mathematical and algorithmic tools for accomplishing this goal and present simulation results which support our approach. Among the positive results is the ability to lower the well-known error rate floor of these concatenated schemes for arbitrary partial response channels.

Summary form only given. The ultimate performance limit or threshold of low-density parity-check (LDPC) codes is close to capacity over various memoryless channels (S.-Y. Chung, IEEE Trans. Info. Theory, vol. 47, pp. 657-670, 2001). Since the capacity of binary-input partial response (PR) channels is unknown, we estimate the capacity by calculating the threshold of capacity-approaching LDPC codes over these channels. We assume a BCJR equalizer for the channel and a sum-product message-passing decoder for the LDPC code. We also assume that the log-likelihood ratios (LLRs) in the channel BCJR and the LDPC decoder are Gaussian. We use our results to guide the design of decoder schedules that minimize the number of computationally expensive BCJR steps. The threshold calculation method can be used to optimize the left and right degree sequences of the LDPC code factor graph, using fast linear programming methods, to provide better thresholds than regular codes.

Combination of turbo codes with partial response (PR) channels was comprehensively studied in magnetic recording systems. In this paper, we investigate a decoding scheme called 'iterative tri-stage decoding' for the turbo code as the outer code and the precoded PR channel as the inner code. This decoding scheme reduces the computational complexities by reducing the number of iterations without degrading the coding gain.

The dependence of MSE and marginalized BER on amplitude asymmetry for two partial response channels is reported. To eliminate the impact of variations in other head parameters, we varied the bias point of a given head using an external magnetic field. We find that the MSE degrades continuously as the magnitude of the amplitude asymmetry is increased; the marginalized BER is at most weakly dependant on amplitude asymmetry for magnitudes less than 15%. At larger magnitudes the channel performance is more severely impacted, Finally, we were able to simulate these results using a simplified model of the read channel.

Recording media written-in transition jitter noise can limit magnetic recording systems. Read channel intersymbol interference (ISI) modulates media jitter in a head/disk/channel, attenuating jitter at low linear density due to classical ISI but amplifying at high density. This paper assesses whether jitter modulation is significant in typical disk drive peak detect (PD) and partial response (PR) channel design, where maximum linear bit density is sought at a specified bit error rate (BER). Experimental isolated pulse data, transition noise, and electronic noise measurements from a 65 mm head/disk recording design are used as inputs to a Matlab simulation of PD and PR Lorentzian channel low pass and boost filters, PD differentiator filter, PR digital filter, and PD/PR bit detectors. The goal of such models is optimizing the designable PD and PR channel parameters for realistic maximum linear density prediction. However their complexity obscures the jitter modulation contribution, which is demonstrated here by the shift in the EER versus density curves, when the jitter modulation is mathematically turned off. Jitter modulation is thereby shown to be significant but not dominant in (1,7) PD density limits. About an order of magnitude BER loss due to jitter modulation can occur at typical PD linear channel bit densities P/sub 50//T/spl ap/2. Jitter amplification in PD channels can occur at channel density 3.8 (above usable PD limits). PR channels only amplify media jitter, degrading EER by about an order of magnitude at typical (0, k) PR4 densities, and degrading timing channel accuracy twice as fast as channel density increases. Lorentzian isolated pulse (ISOP) modeling is compared to recording physics mathematical pulses, and simulation statistical validity checks are made.

This paper addresses a new scheme by Vasic and Pedagani to combine low-density parity-check (LDPC) codes with run length limited (RLL) constraints. In this method, the RLL constraints are embedded into the LDPC codewords by deliberately flipping the bits of LDPC codewords that violate the RLL constraints. It is important to keep the number of flipped bits small in order to not overburden the LDPC decoder. In this paper, we introduce a method to control the number of flipped bits by using pseudorandom sequences. We present a new low-complexity iterative decoding and detection scheme to correct both the flipped bits and channel errors in a partial response channel. Analyses and simulation results show that the proposed method has good performance and reasonable complexity for (0,k) RLL constraints.

Transmitter and receiver filter design plays a key role in communication systems with partial response channels. For a partial response system, a nonmaximally decimated multirate filter bank can be used as a precoder. Based on the MMSE criterion, a closed-form solution to the joint transmitter-receiver optimization problem for noisy partial response channels is presented in this paper. It is found that the redundancy introduced by the nonmaximally decimated filter bank precoder can compensate for spectral nulls in partial response channels that can impair several known joint optimization methods. Simulation results corroborate the analysis that joint transmitter-receiver optimization can achieve significant performance improvement over inverse filter receiver and receiver optimization schemes.

In this paper, we investigate the soft iterative decoding technique for ternary recoding systems with run-length-limited (RLL) constraints. We employ a simple binary-to-ternary RLL encoder following the LDPC (low density parity check) encoder. In the decoder, the iteratively passing of soft information between the LDPC decoder and a detector is used, where the detector is constructed for a combination of the RLL encoder, PLM (pulse length modulation) precoder and the partial response channel. We provide two different decoding algorithms. For one of the decoding algorithm, we are able to obtain bit-error-rate performance which is inferior to the comparable system without considering the RLL constraint for the high sign-to-noise ratio (SNR) regime and is better for the low-to-moderate SNR regime.

The partial response channel can be viewed as a rate-1 encoder in which the output alphabet differs from the input alphabet. In serially concatenated coding schemes, the partial response channel can serve as the inner encoder. Recent works on the application of turbo decoding techniques to partial response channels have used channel models to investigate system performance. While the models provided some insight into the performance of turbo decoding techniques, performance in a real system, with electronic and signal-dependent media noise, remained unknown. This paper will demonstrate the performance of turbo decoding techniques using real and modeled spinstand data. A simple serial concatenation system will be presented that uses as its outer code a single convolutional code and as its inner code the partial response channel. Significant gains will be shown over QMTR coded systems.

In this paper, the combined normalized and offset min-sum algorithm is proposed for efficient decoding of low-density parity-check (LDPC) codes in optical and magnetic storage systems. The proposed algorithm can be considered as a general case of the normalized and offset algorithms, and offers better BER performance compared to them. The performance improvement is presented for a partial response magnetic recording channel. Our simulations are based on an LDPC code from the class of finite-geometry LDPC codes. It will be shown that the proposed method has a performance very close to that of the belief propagation algorithm which has a much higher complexity.

Partial-response (PR) signaling is used to model communications channels with intersymbol interference (ISI) such as the magnetic recording channel and the copper-wire channel for digital subscriber lines. Coding for improving noise immunity in higher order partial-response channels, such as the extended class-4 channels denoted EPR4, E/sup 2/PR4, E/sup 3/PR4, has become an important subject as the linear densities in magnetic recording approach those at which these partial-response channels are the best models of real channels. In this paper, we consider partial-response channels for which ISI is so severe that the channels fail to achieve the matched-filter bound (MFB) for symbol error rate, assuming maximum-likelihood decoding. We show that their performance can be improved to the MFB by high-rate codes based on constrained systems, some of which may even simplify the Viterbi (1979) detectors relative to the uncoded channels. We present several examples of high-rate constrained codes for E/sup 2/PR4 and E/sup 3/PR4 channels and evaluate their performance by simulation.

The partial response channel can be viewed as a rate-1 encoder in which the output alphabet differs from the input alphabet. In serially concatenated coding schemes, the partial response channel can serve as the inner encoder. Recent work has focused on convolutional codes and parity-check codes as the outer code in such serial concatenation schemes. This work focuses instead on the use of product codes as the outer code. Systems using both hard and soft decoding, with both iterative and non-iterative decoding, are presented. Simulation results are presented comparing performance of product codes which use Reed-Solomon and BCH codes as the constituent codes.

In this paper, a novel decoding algorithm for low-density parity-check (LDPC) codes based on convex optimization is presented. The decoding algorithm, called interior point decoding, is designed for linear vector channels. The linear vector channels include many practically important channels such as inter symbol interference channels, partial response channels and MIMO channels. It is shown that the maximum likelihood decoding (MLD) rule for a linear vector channel can be relaxed to a convex optimization problem, which is called a relaxed MLD problem. Approximate variations of gradient descent and Newton methods are used to solve the convex optimization problem.