Turbo product codes for partial response channels

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.

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.

We study a serially interleaved concatenated code construction, where the outer code is a standard convolutional code, and the inner code is a recursive convolutional code of rate 1. We focus on the ubiquitous inner differential encoder (used, in particular, to resolve phase ambiguities), double differential encoder (used to resolve both phase and frequency ambiguities), and another rate 1 recursive convolutional code of memory 2. We substantiate analytically the rather surprising result, that the error probabilities corresponding to a maximum-likelihood (ML) coherently detected antipodal modulation over the additive white Gaussian noise (AWGN) channel for this construction are advantageous as compared to the stand-alone outer convolutional code. This is in spite of the fact that the inner code is of rate 1. The analysis is based on the tangential sphere upper bound of an ML decoder, incorporating the ensemble weight distribution (WD) of the concatenated code, where the ensemble is generated by all random and uniform interleavers. This surprising result is attributed to the WD thinning observed for the concatenated scheme which shapes the WD of the outer convolutional code to resemble more closely the binomial distribution (typical of a fully random code of the same length and rate). This gain is maintained regardless of a rather dramatic decrease, as demonstrated here, in the minimum distance of the concatenated scheme as compared to the minimum distance of the outer stand-alone convolutional code. The advantage of the examined serially interleaved concatenated code, given in terms of bit and/or block error probability which is decoded by a practical suboptimal decoder, over the optimally decoded standard convolutional code is demonstrated by simulations, and some insights into the performance of the iterative decoding algorithm are also discussed. Though we have investigated only specific constructions of constituent inner (rate 1) and outer codes, we trust, hinging on the rational of the arguments here, that these results extend to many other constituent convolutional outer codes and rate 1 inner recursive convolutional codes.

We study the performance of turbo codes for magnetic recording channels using realistic channel models. The main objective is to provide a benchmark for the future studies on the subject. We examine a variety of schemes including the use of parallel concatenated turbo codes with and without turbo equalization and convolutional codes with an interleaver (can be regarded as a serial concatenation of convolutional codes with the partial response channel). We investigate the effect of the (turbo) code block length, the use of different code structures, and the effects of precoding and the media noise on the system performance. We assume that the partial response target is EPR4.

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.

Both polar code and Low-Density Parity-Check (LDPC) code have been adopted in 3GPP eMBB scenario. Some scholars have proposed a serial concatenation structure combining these two codes. However, the inner code needs to add redundant bits to protect the redundant bits of the outer code in the serial concatenation. In this paper, we propose a parallel concatenation code combining polar code and LDPC code on the AWGN channel, which does not distinguish the inner code from the outer code. In the proposed concatenated structure, the BP algorithm is used to decode polar code, and the output soft information of BP decoder is logarithmically processed before LDPC decoding. According to the simulation results, compared with the serial concatenated code with the same code rate and code length, the parallel concatenated code has better BER(Bit-Error-Rate) performance.

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.

In this paper, we propose a concatenated code by combining a Recursive Systematic Convolutional (RSC) code with a Low-Density Parity-Check (LDPC) code. The proposed concatenated code is encoded in parallel by an RSC encoder and an LDPC encoder without interleavers between them. When decoding the proposed concatenated code, the information bits are decoded by two decoding algorithms, and soft information is exchanged between the RSC decoder and the LDPC decoder. This allows for the elimination of wrong codewords output by each decoder. The use of an LDPC code allows us to halt the decoding process when the valid codeword is obtained. We evaluate the error rate performance of the proposed concatenated code by computer simulations. We show that the proposed concatenated code does not have a high error floor, and achieves better BLock Error Rate (BLER) performance than either conventional LDPC codes or conventional turbo codes in the high Eb/No region. We also show that since the proposed concatenated code can correct wrong codewords output by each decoder, it achieves better BLER than the regular LDPC code at the same Bit Error Rate (BER).

<div> <h2>Abstract</h2> <p><strong>Objective:</strong> In the code theory, parallel concatenation of codes becomes more popular after the introduction of turbo codes. In recent years, the Low Density Parity Check (LDPC) code has found remarkable advancement and has seen them outshine turbo codes in terms of performance especially in the error floor and higher code rate. The main objective of this paper is to address the various techniques of a parallel concatenation of LDPC code in code theory. <strong>Method/Finding:</strong> To reduce the complexity of encoding and decoding for longer codes various parallel concatenation of LDPC coding methods were introduced and the performance was compared with other work. <strong>Novelty:</strong> When a longer block length is used, the parallel LDPC decoder is suffered from the complexity of prohibitive implementation. To overcome this issue and to achieve the best performance for longer codes, the different methods for parallel concatenation of LDPC codes were introduced with reduced complexity. This will helps to break the long and complex LDPC code into less complex and smaller LDPC to distribute the decoding and encoding load. Also, this will provides scalability and scope for improving the performance of LDPC codes in practical delay-sensitive and energy-aware applications.</p> <p><strong>Keywords:</strong> Parallel concatenation; LowDensity ParityCheck (LDPC); turbo codes; Parallel Concatenated Gallager Codes (PCGC)</p> </div> <div></div> <div> </div>

Reed-Solomon convolutional concatenated (RSCC) code has been widely applied in wireless and space communications. However, iterative soft-decision decoding of the concatenated code is yet to be developed. This paper proposes a novel iterative soft decoding algorithm for the concatenated coding scheme. The maximum a posteriori (MAP) algorithm is used to decode the inner convolutional code. Its soft output will be deinterleaved and then passed to the soft-in-soft-out (SISO) decoding algorithm for the outer Reed-Solomon (RS) code. The outer SISO decoder integrates the adaptive belief propagation (ABP) algorithm and the Koetter-Vardy (KV) list decoding algorithm, attempting to find out the transmitted message. If it is found, the deterministic probabilities of the corresponding RS coded bits will be fed back. Otherwise, the extrinsic probabilities that are yielded by the ABP algorithm will be given as the feedback. With the proposed soft information exchange decoding mechanism, error-correction potential of the concatenated code can be better exploited. Our simulation results show that significant performance improvement can be achieved over the existing decoding algorithms.

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.

Serial concatenation scheme with a space-time trellis code as the outer code and recursive convolutional codes as the inner code are compared. A new encoding/decoding structure characterized by the inner encoding by a single recursive convolutional code and the combined decoding of simultaneous symbols from multiple antennas is proposed, which gives a large performance gain over the original multiple inner encoding/separate decoding scheme and enables a decoding algorithm much simpler than the multiple inner encoding/joint decoding scheme. Simulation results for a communication system with two transmit and a single receive antennas in quasi-static, flat Rayleigh fading channels are presented.

In this paper, we evaluate the performances of the Reed–Solomon (RS) and low density parity check (LDPC) codes in order to propose an efficient coding system for the outer code in high density optical recording channels. Hard decision decoding (HDD) RS code, iterative soft decision decoding (SDD) RS code or LDPC code is used as the outer code and LDPC code is used as the inner code. We compare the bit error rate (BER) performance of each coding system for various code rates, code lengths and burst errors. Simulation results show that the LDPC code has a better BER performance than the HDD RS and iterative SDD RS codes as the outer code in high density optical channels. Therefore, the LDPC code could be more attractive as the outer code than the RS code in high density optical channels.

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.

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.