Iterative decoding for partial response (PR), equalized,magneto-optical (MO) data storage channels

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. Previously, 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). 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     

Performance analysis and code optimization of low densityparity-check codes on Rayleigh fading channels

A numerical method has been presented to determine the noise thresholds of low density parity-check (LDPC) codes that employ the message passing decoding algorithm on the additive white Gaussian noise (AWGN) channel. In this paper, we apply the technique to the uncorrelated flat Rayleigh fading channel. Using a nonlinear code optimization technique, we optimize irregular LDPC codes for such a channel. The thresholds of the optimized irregular LDPC codes are very close to the Shannon limit for this channel. For example, at rate one-half, the optimized irregular LDPC code has a threshold only 0.07 dB away from the capacity of the channel. Furthermore, we compare simulated performance of the optimized irregular LDPC codes and turbo codes on a land mobile channel, and the results indicate that at a block size of 3072, irregular LDPC codes can outperform turbo codes over a wide range of mobile speeds     

Adaptive Transmission with Variable-Rate Turbo Bit-Interleaved Coded Modulation

We study an adaptive transmission scheme based on variable-rate turbo bit-interleaved coded modulation (VR- Turbo-BICM). The proposed coding scheme employs punctured turbo codes. A continuously varying transmission rate can be obtained by changing the code rate through both puncturing of the coded bits and adapting of the modulation constellation size. The main results are elaborated in two parts. First, we derive a closed-form expression for a set of achievable rate bounds (called rate thresholds) for VR-Turbo-BICM by employing recent results on the parallel channel performance of turbo code ensembles and the BICM parallel channel analysis model. The derived rate threshold is expressed as a fraction of the capacity of BICM with Gray mapping, where this fraction is a turbo code weight spectrum parameter. Simulation results illustrate that introduced rate thresholds predict well the rate versus SNR performance of VR-Turbo-BICM for a wide range of codeword error probabilities and codeword lengths. Next, based on a simplified rate threshold, we derive a power, puncturing rate, and modulation constellation size assignment policy for a slow fading channel.     

Capacity Achieving LDPC Codes Through Puncturing

The performance of punctured low-definition parity-check (LDPC) codes under maximum-likelihood (ML) decoding is studied in this correspondence via deriving and analyzing their average weight distributions (AWDs) and the corresponding asymptotic growth rate of the AWDs. In particular, it is proved that capacity-achieving codes of any rate and for any memoryless binary-input output-symmetric (MBIOS) channel under ML decoding can be constructed by puncturing some original LDPC code with small enough rate. Moreover, it is shown that the gap to capacity of all the punctured codes can be the same as the original code with a small enough rate. Conditions under which puncturing results in no rate loss with asymptotically high probability are also given in the process. These results show high potential for puncturing to be used in designing capacity-achieving codes, and in rate-compatible coding under any MBIOS channel.      

Product accumulate codes: a class of codes with near-capacity performance and low decoding complexity

We propose a novel class of provably good codes which are a serial concatenation of a single-parity-check (SPC)-based product code, an interleaver, and a rate-1 recursive convolutional code. The proposed codes, termed product accumulate (PA) codes, are linear time encodable and linear time decodable. We show that the product code by itself does not have a positive threshold, but a PA code can provide arbitrarily low bit-error rate (BER) under both maximum-likelihood (ML) decoding and iterative decoding. Two message-passing decoding algorithms are proposed and it is shown that a particular update schedule for these message-passing algorithms is equivalent to conventional turbo decoding of the serial concatenated code, but with significantly lower complexity. Tight upper bounds on the ML performance using Divsalar's (1999) simple bound and thresholds under density evolution (DE) show that these codes are capable of performance within a few tenths of a decibel away from the Shannon limit. Simulation results confirm these claims and show that these codes provide performance similar to turbo codes but with significantly less decoding complexity and with a lower error floor. Hence, we propose PA codes as a class of prospective codes with good performance, low decoding complexity, regular structure, and flexible rate adaptivity for all rates above 1/2.     

Scalable image and video transmission using irregular repeat-accumulate codes with fast algorithm for optimal unequal error protection

This paper considers designing and applying punctured irregular repeat-accumulate (IRA) codes for scalable image and video transmission over binary symmetric channels. IRA codes of different rates are obtained by puncturing the parity bits of a mother IRA code, which uses a systematic encoder. One of the main ideas presented here is the design of the mother code such that the entire set of higher rate codes obtained by puncturing are good. To find a good unequal error protection for embedded bit streams, we employ the fast joint source-channel coding algorithm in Hamzaoui et al. to minimize the expected end-to-end distortion. We test with two scalable image coders (SPIHT and JPEG-2000) and two scalable video coders (3-D SPIHT and H.26L-based PFGS). Simulations show better results with IRA codes than those reported in Banister et al. with JPEG-2000 and turbo codes. The IRA codes proposed here also have lower decoding complexity than the turbo codes used by Banister et al.     

Reliable channel regions for good binary codes transmitted over parallel channels

We study the average error probability performance of binary linear code ensembles when each codeword is divided into J subcodewords with each being transmitted over one of J parallel channels. This model is widely accepted for a number of important practical channels and signaling schemes including block-fading channels, incremental redundancy retransmission schemes, and multicarrier communication techniques for frequency-selective channels. Our focus is on ensembles of good codes whose performance in a single channel model is characterized by a threshold behavior, e.g., turbo and low-density parity-check (LDPC) codes. For a given good code ensemble, we investigate reliable channel regions which ensure reliable communications over parallel channels under maximum-likelihood (ML) decoding. To construct reliable regions, we study a modifed 1961 Gallager bound for parallel channels. By allowing codeword bits to be randomly assigned to each component channel, the average parallel-channel Gallager bound is simplified to be a function of code weight enumerators and channel assignment rates. Special cases of this bound, average union-Bhattacharyya (UB), Shulman-Feder (SF), simplified-sphere (SS), and modified Shulman-Feder (MSF) parallel-channel bounds, allow for describing reliable channel regions using simple functions of channel and code spectrum parameters. Parameters describing the channel are the average parallel-channel Bhattacharyya noise parameter, the average channel mutual information, and parallel Gaussian channel signal-to-noise ratios (SNRs). Code parameters include the union-Bhattacharyya noise threshold and the weight spectrum distance to the random binary code ensemble. Reliable channel regions of repeat-accumulate (RA) codes for parallel binary erasure channels (BECs) and of turbo codes for parallel additive white Gaussian noise (AWGN) channels are numerically computed and compared with simulation results based on iterative decoding. In addition, an examp     

Serial concatenated TCM with an inner accumulate code - part II: density-evolution analysis

In a companion paper, we showed the existence of decoding thresholds for maximum-likelihood (ML) decoding of a serial concatenated trellis-coded modulation (SCTCM) system with one or more inner accumulate codes. In this paper, we compute the decoding thresholds for an iterative, non-ML decoder by density evolution (DE), assuming infinite blocklengths. We also derive a stability condition for the particular case of an outer parity-check code and a single inner accumulate code. We show that, for equiprobable signaling, the bit-wise log-likelihood ratio densities for higher order constellations are symmetric. Furthermore, when used in DE, these densities can be averaged without significantly affecting the resulting threshold values. For an outer single parity-check code, the lowest decoding thresholds are achieved with two inner accumulate codes. For an outer repeat code, a single inner accumulate code gives the lowest thresholds. At code rates r/sub c/>2/3, the decoding thresholds for the SCTCM system are within 1 dB of the constellation-constrained channel capacity for additive white Gaussian noise channels, and within 1.5 dB for independent, identically distributed Rayleigh channels. Simulation results verify the computed thresholds.     

Accumulate–Repeat–Accumulate Codes: Capacity-Achieving Ensembles of Systematic Codes for the Erasure Channel With Bounded Complexity

This paper introduces ensembles of systematic accumulate-repeat-accumulate (ARA) codes which asymptotically achieve capacity on the binary erasure channel (BEC) with bounded complexity, per information bit, of encoding and decoding. It also introduces symmetry properties which play a central role in the construction of new capacity-achieving ensembles for the BEC. The results here improve on the tradeoff between performance and complexity provided by previous constructions of capacity-achieving code ensembles defined on graphs. The superiority of ARA codes with moderate to large block length is exemplified by computer simulations which compare their performance with those of previously reported capacity-achieving ensembles of low-density parity-check (LDPC) and irregular repeat-accumulate (IRA) codes. ARA codes also have the advantage of being systematic.     

The Development of Turbo and LDPC Codes for Deep-Space Applications

The development of error-correcting codes has been closely coupled with deep-space exploration since the early days of both. Since the discovery of turbo codes in 1993, the research community has invested a great deal of work on modern iteratively decoded codes, and naturally NASA's Jet Propulsion Laboratory (JPL) has been very much involved. This paper describes the research, design, implementation, and standardization work that has taken place at JPL for both turbo and low-density parity-check (LDPC) codes. Turbo code development proceeded from theoretical analyses of polynomial selection, weight distributions imposed by interleaver designs, decoder error floors, and iterative decoding thresholds. A family of turbo codes was standardized and implemented and is currently in use by several spacecraft. JPL's LDPC codes are built from protographs and circulants, selected by analyses of decoding thresholds and methods to avoid loops in the code graph. LDPC encoders and decoders have been implemented in hardware for planned spacecraft, and standardization is under way.     

Recent Advances in Turbo Code Design and Theory

The discovery of turbo codes and the subsequent rediscovery of low-density parity-check (LDPC) codes represent major milestones in the field of channel coding. Recent advances in the design and theory of turbo codes and their relationship to LDPC codes are discussed. Several new interleaver designs for turbo codes are presented which illustrate the important role that the interleaver plays in these codes. The relationship between turbo codes and LDPC codes is explored via an explicit formulation of the parity-check matrix of a turbo code, and simulation results are given for sum product decoding of a turbo code.     

On the performance of high-rate TPC/SPC codes and LDPC codes overpartial response channels

This paper evaluates two-dimensional turbo product codes based on single-parity check codes (TPC/SPC) and low-density parity check (LDPC) codes for use in digital magnetic recording systems. It is first shown that the combination of a TPC/SPC code and a precoded partial response (PR) channel results in a good distance spectrum due to the interleaving gain. Then, density evolution is used to compute the thresholds for TPC/SPC codes and LDPC codes over PR channels. Analysis shows that TPC/SPC codes have a performance close to that of LDPC codes for large codeword lengths. Simulation results for practical block lengths show that TPC/SPC codes perform as well as LDPC codes in terms of bit error rate, but possess better burst error statistics which is important in the presence of an outer Reed-Solomon code. Further, the encoding complexity of TPC/SPC codes is only linear in the codeword length and the generator matrix does not have to be stored explicitly. Based on. the results in the paper and these advantages, TPC/SPC codes seem like a viable alternative to LDPC codes     

Coding for Parallel Channels: Gallager Bounds and Applications to Turbo-Like Codes

The transmission of coded communication systems is widely modeled to take place over a set of parallel channels. This model is used for transmission over block-fading channels, rate-compatible puncturing of turbo-like codes, multicarrier signaling, multilevel coding, etc. New upper bounds on the maximum-likelihood (ML) decoding error probability are derived in the parallel-channel setting. We focus on the generalization of the Gallager-type bounds and discuss the connections between some versions of these bounds. The tightness of these bounds for parallel channels is exemplified for structured ensembles of turbo codes, repeat-accumulate (RA) codes, and some of their recent variations (e.g., punctured accumulate-repeat-accumulate codes). The bounds on the decoding error probability of an ML decoder are compared to computer simulations of iterative decoding. The new bounds show a remarkable improvement over the union bound and some other previously reported bounds for independent parallel channels. This improvement is exemplified for relatively short block lengths, and it is pronounced when the block length is increased. In the asymptotic case, where we let the block length tend to infinity, inner bounds on the attainable channel regions of modern coding techniques under ML decoding are obtained, based solely on the asymptotic growth rates of the average distance spectra of these code ensembles.     

Parity-Check Density Versus Performance of Binary Linear Block Codes: New Bounds and Applications

The moderate complexity of low-density parity-check (LDPC) codes under iterative decoding is attributed to the sparseness of their parity-check matrices. It is therefore of interest to consider how sparse parity-check matrices of binary linear block codes can be a function of the gap between their achievable rates and the channel capacity. This issue was addressed by Sason and Urbanke, and it is revisited in this paper. The remarkable performance of LDPC codes under practical and suboptimal decoding algorithms motivates the assessment of the inherent loss in performance which is attributed to the structure of the code or ensemble under maximum-likelihood (ML) decoding, and the additional loss which is imposed by the suboptimality of the decoder. These issues are addressed by obtaining upper bounds on the achievable rates of binary linear block codes, and lower bounds on the asymptotic density of their parity-check matrices as a function of the gap between their achievable rates and the channel capacity; these bounds are valid under ML decoding, and hence, they are valid for any suboptimal decoding algorithm. The new bounds improve on previously reported results by Burshtein and by Sason and Urbanke, and they hold for the case where the transmission takes place over an arbitrary memoryless binary-input output-symmetric (MBIOS) channel. The significance of these information-theoretic bounds is in assessing the tradeoff between the asymptotic performance of LDPC codes and their decoding complexity (per iteration) under message-passing decoding. They are also helpful in studying the potential achievable rates of ensembles of LDPC codes under optimal decoding; by comparing these thresholds with those calculated by the density evolution technique, one obtains a measure for the asymptotic suboptimality of iterative decoding algorithms     

On decoding of low-density parity-check codes over the binary erasure channel

This paper investigates decoding of low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximum-likelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graph-theoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finite-length codes.     

Joint channel-state estimation and decoding of low-density parity-check codes on the two-state noiseless/useless BSC block interference channel

We apply the density-evolution technique to determine the thresholds of low-density parity-check (LDPC) codes when the sum-product algorithm is employed to perform joint channel-state estimation and decoding. The channel considered is the two-state noiseless/useless binary symmetric channel (BSC) block interference channel, where a block of h consecutive symbols shares the same channel state, which is either a noiseless BSC (crossover probability 0) or a useless BSC (crossover probability 1/2). The channel state is selected independently and at random from block to block, according to a known prior distribution. The threshold of the joint channel-state estimation/decoding scheme when used over such a channel is shown to be greatly superior to that of a decoder that makes no attempt to estimate the channel state. These results are also confirmed by simulation. The maximum-likelihood (ML) performance of LDPC codes when used over this channel is investigated. Lower bounds on the error exponents of regular LDPC codes, when ML decoded, are shown to be close to the random coding channel error exponent when the LDPC variable node degree is high.     

Effect of precoding on the convergence of turbo equalization forpartial response channels

The effect of the precoder on the convergence of turbo equalization for precoded partial response channels is studied. The idea is to consider the turbo decoding algorithm as a one-parameter dynamical system and to study the effect of the precoder on the fixed points of the system. It is showed that precoding results in a loss in fidelity during the first iteration and that this loss depends on the precoder. Further, the rate at which the extrinsic information increases with iterations is also dependent on the precoder. The net result of these two effects is used to explain several existing results in the literature about the performance of different precoders. A design criteria based on the convergence is then proposed, and the impact of precoding on the design of the outer code is then studied. Finally, the design of precoders in the presence of an error correcting code, such as a Reed-Solomon code, is studied     

Turbo codes cascaded with high-rate block codes for (O,κ)-constrained channels

We consider several issues in the analysis and design of turbo coded systems for (O, κ) input-constrained channels. These constraints commonly arise in magnetic recording channels. This system is characterized by a high-rate turbo code driving a high-rate (n-1)/n, small-length (O, κ) block code. We discuss the properties of the (O, κ) code that affect its performance on both an additive white Gaussian noise (AWGN) and a precoded dicode channel. We address soft-in soft-out (SISO) decoding of linear and nonlinear (O, κ) codes and show that good (O, κ) codes exist even when d_min=1. For the (O, κ) constrained AWGN channel, we present several rate (n-1)/n block codes that optimally tradeoff bit-error-rate performance with κ. For the precoded dicode channel, we show that the systematic (O, n-1) modulation codes are superior to most other rate (n-1)/n block codes in terms of error-rate performance, and their attractiveness is increased by the fact that they do not contribute any significant complexity to the overall system     

Rice信道下LDPC码密度进化的研究

应用低密度奇偶校验(LDPC)码译码消息的密度进化可以得到码集的噪声门限,依此评价不同译码算法的性能,并可以用来优化非正则LDPC码的次数分布对。该文首先以Rice信道下正则LDPC码为例,讨论了不同量化阶数及步长时BP,BP-based 和offset BP-based 3种译码算法的DDE(Discrete Density Evolution)分析,接着在offset BP-based译码算法的DDE分析基础上,采用差分进化方法对Rice信道下非正则LDPC码的次数分布对进行了优化,得出了相应的噪声门限。最后,给出了Rice信道下码率为1/2的优化非正则LDPC码的概率聚集函数(PMF)进化曲线。     

Combinatorial Interleavers for Systematic Regular Repeat-Accumulate Codes [Transactions Letters]

This paper proposes novel interleaver and accumulator structures for systematic, regular repeat-accumulate (RA) codes. It is well known that such codes are amenable to iterative (sum-product) decoding on the Tanner graph of the code, yet are as readily encodable as turbo codes. In this paper, interleavers for RA codes are designed using combinatorial techniques as a basis for deterministic interleaver constructions, yielding RA codes whose Tanner graphs are free of 4-cycles. Further, a generalized RA code accumulator structure is proposed, leading to codes, termed w3RA codes, whose parity-check matrices have many fewer weight-2 columns than conventional RA codes. The w3RA codes retain the low-complexity encoding of conventional RA codes and exhibit improved error-floor performance.