Partial Response Channels Research Articles

A Unified, SNR-Aware SC-LDPC Code Design With Applications to Magnetic Recording

Spatially coupled (SC)-low-density parity-check (LDPC) codes are known to have outstanding error-correction performance and low decoding latency, which make them an excellent choice for high-density magnetic recording (MR) technologies. Whereas previous works on LDPC and SC-LDPC codes mostly take either an asymptotic or a finite-length design approach, we propose a unified framework for jointly optimizing the codes’ thresholds and cycle counts to address both regimes. We focus on circulant-based (CB) SC-LDPC code family as a representative, high-performance exemplar of structured SC-LDPC codes. The framework is based on efficient traversal and pruning of the code search space, building on the fact that the performance of a CB SC-LDPC code depends on some characteristics of the code’s partitioning matrix, which by itself is much smaller than the code’s full parity-check matrix. We then propose an algorithm that traverses all non-equivalent partitioning matrices and outputs a list of codes, each offering an attractive point on the trade-off between asymptotic and finite-length performance. Our simulations show that our framework results in SC-LDPC codes that outperform the state-of-the-art constructions, over both additive white Gaussian noise (AWGN) and partial response (PR) channel models, and that it offers the flexibility to choose low-signal-to-noise ratio (SNR), high-SNR, or in- between SNR region considering system requirements, e.g., that of the MR device.

Quantized Turbo Equalization for Non-binary LDPC Coded Partial-Response Channels

In this paper, we consider iterative detection and decoding (i.e., turbo equalization) for nonbinary low-density parity-check (LDPC) coded partial-response channels, where a quantizer is present to discretize the continuous received signal. We propose a turbo equalizer that uses the pre-computed quantized channel transition probabilities in the symbol-level BCJR channel detection algorithm, which significantly reduces the computational complexity by avoiding real-time floating-point multiplications. The proposed approach is further extended to nonbinary LDPC coded bit-patterned media recording (BPMR) channels. Simulation results show that with a small number of quantization bits, the proposed receiver approaches closely the performance of the conventional turbo equalizer operating on unquantized signals.

PR-NN: RNN-Based Detection for Coded Partial-Response Channels

In this paper, we investigate the use of recurrent neural network (RNN)-based detection of magnetic recording channels with inter-symbol interference (ISI). We refer to the proposed detection method, which is intended for recording channels with partial-response equalization, as Partial-Response Neural Network (PR-NN). We train bi-directional gated recurrent units (bi-GRUs) to recover the ISI channel inputs from noisy channel output sequences and evaluate the network performance when applied to continuous, streaming data. The computational complexity of PR-NN during the evaluation process is comparable to that of a Viterbi detector. The recording system on which the experiments were conducted uses a rate-2/3, (1,7) runlength-limited (RLL) code with an E <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> PR4 partial-response channel target. Experimental results with ideal PR signals show that the performance of PR-NN detection approaches that of Viterbi detection in additive white gaussian noise (AWGN). Moreover, the PR-NN detector outperforms Viterbi detection and achieves the performance of Noise-Predictive Maximum Likelihood (NPML) detection in additive colored noise (ACN) at different channel densities. A PR-NN detector trained with both AWGN and ACN maintains the performance observed under separate training. Similarly, when trained with ACN corresponding to two different channel densities, PR-NN maintains its performance at both densities. Experiments confirm that this robustness is consistent over a wide range of signal-to-noise ratios (SNRs). Finally, PR-NN displays robust performance when applied to a more realistic magnetic recording channel with MMSE-equalized Lorentzian signals.

A Channel-Aware Combinatorial Approach to Design High Performance Spatially-Coupled Codes

Because of their capacity-approaching performance and their complexity/latency advantages, spatially-coupled (SC) codes are among the most attractive error-correcting codes for use in modern dense data storage systems. SC codes are constructed by partitioning an underlying block code and coupling the partitioned components. Here, we focus on circulant-based SC codes. Recently, the optimal overlap (OO), circulant power optimizer (CPO) approach was introduced to construct high performance SC codes for additive white Gaussian noise (AWGN) and Flash channels. The OO stage operates on the protograph of the SC code to derive the optimal partitioning that minimizes the number of graphical objects that undermine the performance of SC codes under iterative decoding. Then, the CPO optimizes the circulant powers to further reduce this number. Since the nature of detrimental objects in the graph of a code critically depends on the characteristics of the channel of interest, extending the OO-CPO approach to construct SC codes for channels with intrinsic memory is not a straightforward task. In this paper, we tackle one relevant extension; we construct high performance SC codes for practical 1-D magnetic recording channels, i.e., partial-response (PR) channels. Via combinatorial techniques, we carefully build and solve the optimization problem of the OO partitioning, focusing on the objects of interest in the case of PR channels. Then, we customize the CPO to further reduce the number of these objects in the graph of the code. SC codes designed using the proposed OO-CPO approach for PR channels outperform prior state-of-the-art SC codes by up to around 3 orders of magnitude in frame error rate (FER) and 1.1 dB in signal-to-noise ratio (SNR). More intriguingly, our SC codes outperform structured block codes of the same length and rate by up to around 1.8 orders of magnitude in FER and 0.4 dB in SNR. The performance advantage of SC codes designed using the devised OO-CPO approach over block codes of the same parameters is not only pronounced in the error floor region, but also in the waterfall region.

Linear-Complexity ADMM Updates for Decoding LDPC Codes in Partial Response Channels

This letter investigates the application of alternating direction method of multipliers (ADMM) in decoding low-density parity-check (LDPC) codes over partial response (PR) channels. Unlike the ADMM decoding in memoryless channels, the ADMM decoding in PR channels becomes involved with the correlation among adjacent code symbols. As the update equations for code symbols constitute the most complicated part of the ADMM decoding in PR channels, this letter focuses on their derivation by making use of the iterative fashion of ADMM and incorporating the use of a penalty function. A notable advantage of the proposed ADMM penalized decoding algorithm is that its complexity increases linearly with the memory length of PR channels. Simulation results show that the proposed algorithm can achieve error performance comparable with that of Turbo equalization (TE) in the waterfall region. It can also outperform TE at high signal-to-noise ratios as iteration grows.

2-Correcting Lee Codes: (Quasi)-Perfect Spectral Conditions and Some Constructions

Let $p$ be an odd prime. Recently, Camarero and Martinez (in “Quasi-perfect Lee codes of radius 2 and arbitrarily large dimension”, IEEE Trans. Inform. Theory, vol. 62, no. 3, 2016) constructed some $p$ -ary 2-quasi-perfect Lee codes for $p\equiv \pm 5 \pmod {12}$ . In this paper, some infinite classes of $p$ -ary 2-quasi-perfect Lee codes for any odd prime $p$ with flexible length and dimension are presented. More specifically, we provide a new method for constructing quasi-perfect Lee codes. Our approach uses subsets derived from some quadratic curves over finite fields (in odd characteristic) to obtain two classes of 2-quasi-perfect Lee codes defined in the space $\pmb {\mathbb {Z}}_{p}^{n}$ for $n=\frac {p^{k}+1}{2}$ (with $p\equiv 1, -5 \pmod {12}$ and $k$ is any integer, or $p\equiv -1, 5 \pmod {12} $ and $k$ is an even integer) and $n=\frac {p^{k}-1}{2}$ (with $p\equiv -1, 5 \pmod {12}$ , $k$ is an odd integer and $p^{k}>12$ ). Our codes encompass the $p$ -ary ( $p\equiv \pm 5 \pmod {12}$ ) 2-quasi-perfect Lee codes constructed by Camarero and Martinez. Furthermore, we prove that the related Cayley graphs are Ramanujan or almost Ramanujan using Kloosterman sums. This generalizes the work of Bibak, Kapron, and Srinivasan (in “The Cayley graphs associated with some quasi-perfect Lee codes are Ramanujan graphs”, IEEE Trans. Inform. Theory, vol. 62, no. 11, 2016) from the case $p\equiv 3 \pmod {4}$ and $k=1$ to the case of any odd prime $p$ and positive integer $k$ . Finally, we derive some necessary conditions with the exponential sums of all 2-perfect codes and 2-quasi-perfect codes, and present a heuristic algorithm for constructing 2-perfect codes and 2-quasi-perfect codes. Our results show that, in general, the Cayley graphs associated with 2-perfect codes are Ramanujan. From the algorithm, some new 2-quasi-perfect Lee codes different from those constructed from quadratic curves are given. The Lee codes presented in this paper have applications in constrained and partial-response channels, flash memories, and decision diagrams.

Non-Binary LDPC Codes for Magnetic Recording Channels: Error Floor Analysis and Optimized Code Design

In this paper, we provide a comprehensive analysis of the error floor along with code optimization guidelines for structured and regular non-binary low-density parity-check (NB-LDPC) codes in magnetic recording (MR) applications. While the topic of the error floor performance of binary LDPC codes over additive white Gaussian noise (AWGN) channels has recently received considerable attention, very little is known about the error floor performance of NB-LDPC codes over other types of channels, despite the early results demonstrating superior characteristics of NB-LDPC codes relative to their binary counterparts. We first show that, due to the outer looping between the detector and the decoder in the receiver, the error profile of NB-LDPC codes over partial-response (PR) channels is qualitatively different from the error profile over AWGN channels—this observation motivates us to introduce new combinatorial objects aimed at capturing decoding errors that dominate the PR channel error floor region. We call these objects balanced absorbing sets (BASs), which are viewed as a special subclass of previously introduced absorbing sets (ASs). Aided by these new objects (BASs), we develop a method that combines analytical equations and biased simulations to predict the error floor performance of NB-LDPC codes over PR channels without the need to execute extensive Monte Carlo (MC) simulations. We show that explicitly incorporating the inter-symbol interference of MR channels into our prediction method makes the accuracy of the error floor estimate within 0.2 of an order of magnitude from the traditional MC simulation. In addition, we prove that, due to the more restrictive definition of BASs (relative to the more general class of ASs), an additional degree of freedom can be exploited in the code design for PR channels. We then demonstrate that the proposed code optimization aimed at removing dominant BASs offers performance improvements in the frame error rate in the error floor region by up to 2.5 orders of magnitude over the unoptimized designs. Our code optimization technique carefully, yet provably, removes BASs from the code while preserving its overall structure (node degree, quasi-cyclic property, regularity, and so forth). The resulting codes outperform the existing binary and NB-LDPC solutions for PR channels by about 2.5 and 1.25 orders of magnitude, respectively.

Performance Analysis of Gray Code based Structured Regular Column-Weight Two LDPC Codes

The Low-Density Parity-Check (LDPC) code is a linear block code specified by a parity-check matrix H. The construction process of LDPC codes considers the parameters such as row weight, column weight, density of H-matrix, code rate, code length and girth. A novel flexible method for constructing structured regular LDPC codes considering all these parameters, using Gray-code representations, was reported in our earlier work. LDPC codes with column-weight two have low computational complexity and are promising for data storage and partial response channels. This paper presents the performance analysis of column-weight two LDPC codes constructed using the method proposed in our previous work. The BER performance of the obtained codes is comparable to that of the standard Gallagers random codes and QC-LDPC codes.

Iterative detection and decoding for non‐binary LDPC coded partial‐response channels with written‐in errors

In this study, the authors investigate the performance of iterative detection and decoding for non-binary low-density parity-check (LDPC) coded partial-response (PR) channels, where written-in errors are present to introduce bit-flipping to LDPC code bits prior to the transmission over PR channels. From a probabilistic perspective, the written-in errors are modelled as the output of a binary symmetrical channel (BSC) with crossover probability equal to the write error probability. Several iterative receivers are presented for the considered system. Specifically, the authors propose a symbol-level Bahl-Cocke-Jelinek-Raviv (BCJR) channel detector based on a sectionalised representation of the joint BSC and PR trellis to produce directly the soft information of non-binary LDPC code symbols, thus avoiding suboptimal symbol/bit log-likelihood ratio conversions as required in bit-level detection schemes. Simulation results show that the proposed symbol-level iterative receiver provides much better bit-error-rate performance over bit-level schemes. Moreover, the proposed receiver enables non-binary LDPC codes to achieve a notable coding gain over their binary counterparts for channels affected with written-in errors.

The Design of Protograph LDPC Codes for 2-D Magnetic Recording Channels

This paper investigates the performance of the protograph-based low-density parity-check (LDPC) codes for 2-D intersymbol interference (ISI) channels with magnetic recording density. The protograph LDPC codes have been shown to possess simple hardware implementation and excellent error performance over additive white Gaussian noise channels and over partial response channels. We propose a design scheme and construct two new protograph LDPC codes for 2-D ISI channel, with the help of extrinsic information transfer chart analysis and asymptotic ensemble weight distribution analysis. The theoretical analyses and simulated results show that the proposed codes outperform the previously optimized irregular LDPC code in both low- and high-SNR regions. Thus, the proposed codes are good alternatives compared with other error-correction codes for the use in magnetic recording systems.

A Survey on Protograph LDPC Codes and Their Applications

Low-density parity-check (LDPC) codes have attracted much attention over the past two decades since they can asymptotically approach the Shannon capacity in a variety of data transmission and storage scenarios. As a type of promising structured LDPC codes, the protograph LDPC codes not only inherit the advantage of conventional LDPC codes, i.e., excellent error performance, but also possess simple representations to realize fast encoding and efficient decoding. This paper provides a comprehensive survey on the state-of-the-art in protograph LDPC code design and analysis for different channel conditions, including the additive white Gaussian noise (AWGN) channels, fading channels, partial response (PR) channels, and Poisson pulse-position modulation (PPM) channels. Moreover, the applications of protograph LDPC codes to joint source-and-channel coding (JSCC) and joint channel-and-physical-layer-network coding (JCPNC) are reviewed and studied. In particular, we focus our attention on the encoding design and assume the decoder is implemented by the belief propagation (BP) algorithm. Hopefully, this survey may facilitate research in this area.

EXIT Chart Analysis of Nonbinary Protograph LDPC Codes for Partial Response Channels

Low-density parity-check (LDPC) codes over finite fields GF(q) provide an error correction in the noisy partial response (PR) channels. The extrinsic information transfer (EXIT) chart can be used to predict the threshold decoding of protograph LDPC codes, however, previous works only consider the binary protograph LDPC codes in the PR channels. In this paper, we propose to perform the EXIT chart analysis on the nonbinary protograph LDPC codes for the PR channels. Unlike prior works, the actual extrinsic information of the channel detector is measured, then the extrinsic information to the variable nodes is generated with the measured statistics. Moreover, since the mutual information of the variable nodes depends on GF(q), we use the Monte Carlo method to approximate the mutual information. The analysis on the regular (2,4) code, regular (3,6) code, RA code, and AR3A code on the PR channels reveal that, for the PR! channel, the RA code outperforms the others for q = 2, 4, and 8, but the regular (2,4) code is the best for q = 16 and 32. For the PR2 channel, the RA code is the best code for q = 2 and 4, but the regular (2,4) code is the best code for q > 4. The simulation of the codes for q = 4 and 16 are then used to confirm the theoretical results.

An Improved TA Suppression Method for Coded PR Channels

Thermal asperity (TA) resulting from the collision between the slider and the asperity on a magnetic medium during read process can deteriorate the performance of hard disk drives (HDDs). Without TA detection and correction algorithms, the system performance can be unacceptable, depending on how severe the TA is. This paper presents an improved TA suppression method for coded partial response (PR) channels, which consists of two channels running in parallel. Specifically, one channel is matched to the target H(D), while the other is matched to the target H(D)G(D), where G(D) = 1 – D2 is a bandpass filter and D is a delay operator. The soft-output Viterbi algorithm (SOVA) detector in the H(D) channel yields the high-quality soft information in absence of the TA, while that in the G(D)H(D) channel produces the high-quality soft information in presence of the TA. Then, the overall soft information chosen from these two detectors, depending on if a TA is detected or not, is sent to the decoder according to the turbo equalization principle. Experimental results show that the proposed method performs better than the conventional and the previously proposed ones, when operating at high signal-to-noise ratio (SNR) region where a practical HDD works.

Cycle Slip Detection and Correction Through Classification of Modulation Code Failures

In high-density recording systems, synchronization might be lost due to increased noise and/or inaccurate sampling/filtering. In such, long bursts of errors occur due to bits either erased from or inserted into the bit stream. At low signal-to-noise ratios, such bit slips (also known as cycle slips) lead to excessive post-error correction code (post-ECC) failures. In order to reduce the rate of the post-ECC failures, cycle slips should be detected and corrected. In the past, the majority of the research was concentrated on iterative joint timing recovery and error decoding methodologies. In this study, we propose to detect and correct cycle slips using a decoding failure analysis of maximum transition run (MTR) codeword sequences in conjunction with a general-purpose classification algorithm. First, noncyclic properties of constrained coding are identified by realizing that MTR codewords render invalid more often if bit shifts happen. Later, we present a method to use such invalid codewords to help detect and correct cycle slips. Finally, some numerical results are presented to show the effectiveness of the proposed detection/correction scheme using MTR codes and the support vector machines. It is demonstrated that such a combination has the potential for BER and post-ECC failure rate performance improvements by eliminating cycle slips for the generalized partial response channels.

Efficient detection and decoding of q-ary LDPC coded signals over partial response channels

Abstract In this study, we consider the ways for concatenating the intersymbol interference (ISI) detector with a q-ary low-density parity-check (LDPC) decoder for transmissions over partial response (PR) channels. LDPC codes allow achieving performance close to the channel capacity in additive white Gaussian noise channels, while designing receivers employing these codes for transmissions over channels affected by ISI is still an open issue. Turbo equalization schemes are considered with a novel joint message-passing-based receiver, which is derived from a recently proposed joint algorithm for binary LDPC codes. Simulation results provide performance evaluation of these systems over three different PR channels, together with an analysis of the trade-off between error-rate performance and complexity.

Design of Protograph LDPC Codes for Partial Response Channels

We investigate the performance of the protograph low-density parity-check (LDPC) codes, which have been shown to possess simple structures and outstanding error performance over additive white Gaussian noise (AWGN) channels, over partial response (PR) channels using the finite-length extrinsic information transfer (EXIT) algorithm. Due to the intersymbol interference (ISI) caused by the PR channels, we observe that the conventional protograph LDPC codes do not perform well in terms of error rates. We further propose a new design scheme and construct three new types of protograph LDPC codes. Unlike conventional protograph LDPC codes in which the highest-degree variable nodes are punctured, the new protograph LDPC codes have their lowest-degree variable nodes punctured. Moreover, some edge re-connections are made in one of the proposed codes. The EXIT-chart analysis, the convergence analysis and the bit-error-rate simulation have shown that all three new codes outperform the conventional protograph LDPC codes. Moreover, two of the proposed codes are superior to the regular column-weight-3 LDPC code and thus they are good alternatives compared to other error-correction codes for use in data storage systems.

Data-Dependent Noise-Predictive Symbol-Based Detector for Perpendicular Magnetic Recording Channels

This work introduces a new formulation of the well-known data-dependent noise-predictive (DDNP) approach to whiten noise in partial response channel detection. The new approach is tailored for whitening the branches in the recently introduced symbol-based detectors that are demonstrated to have optimal performance in concatenation with non-binary error correction codes, such as non-binary low density parity check (LDPC) codes. This paper describes a new approach in the context of the perpendicular magnetic channel, where a symbol-based BCJR is concatenated with a non-binary LDPC decoder. Several approximations based on conventional DDNP implementations are suggested to quantify the performance of the new algorithm.

Iterative Timing Recovery with the Split-Preamble Strategy for Coded Partial Response Channels

This paper proposes a modified per-survivor iterative timing recovery scheme, which exploits a new split-preamble strategy in conjunction with a per-survivor processing soft-output Viterbi algorithm (PSP-SOVA). The conventional split-preamble strategy places a preamble at the beginning of a data sector and uses it to run a phase-locked loop during acquisition to find an initial phase/frequency offset. However, the proposed scheme splits the preamble into two parts. The first part is placed at the beginning of the data sector, whereas the second part is divided into small clusters, each of which is then embedded uniformly within the data stream. This split preamble is utilized to adjust the branch metric calculation in PSP-SOVA to ensure that the survivor path occurs in a correct direction. Results indicate that the proposed scheme yields a better performance than a conventional receiver with separate timing recovery and turbo equalization, and the iterative timing recovery scheme proposed in [1],[2], especially when the timing jitter is large. In addition, we also show that the proposed scheme can automatically correct a cycle slip much more efficiently than the others.

Low Complexity Encoder of High Rate Irregular QC-LDPC Codes for Partial Response Channels

Low Complexity Encoder of High Rate Irregular QC-LDPC Codes for Partial Response Channels

Interior Point Decoding for Linear Vector Channels Based on Convex Optimization

In the present paper, a novel decoding algorithm for low-density parity-check (LDPC) codes based on convex optimization is presented. The decoding algorithm, which is referred to hereinafter as interior point decoding, is designed for linear vector channels. The linear vector channels include several practically important channels, such as inter-symbol interference channels and partial response (PR) 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. The proposed decoding algorithm is based on a numerical optimization technique known as the interior point method with barrier functions. Approximate variations of an interior point method based on the gradient descent and Newton methods are used to solve the relaxed MLD problem. Compared with a conventional joint message-passing decoder, from computer simulations, it is observed that the proposed decoding algorithm achieves better BER performance on PR channels with less decoding complexity in several cases. Furthermore, an extension of the proposed algorithm for high-order modulation formats, such as PAM and QAM, is presented.