Parallel Concatenated Codes Research Articles

Generalized Spatially-Coupled Parallel Concatenated Codes With Partial Repetition

A new class of spatially-coupled turbo-like codes (SC-TCs), dubbed generalized spatially coupled parallel concatenated codes (GSC-PCCs), is introduced. These codes are constructed by applying spatial coupling on parallel concatenated codes (PCCs) with a fraction of information bits repeated <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> times. GSC-PCCs can be seen as a generalization of the original spatially-coupled parallel concatenated codes proposed by Moloudi <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> , 2017. To characterize the asymptotic performance of GSC-PCCs, we derive the corresponding density evolution equations and compute their decoding thresholds. The threshold saturation effect is observed and proven. Most importantly, we rigorously prove that the rate- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> GSC-PCC ensemble with 2-state convolutional component codes achieves at least a fraction <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1-\frac {R}{R+q}$ </tex-math></inline-formula> of the capacity of the binary erasure channel (BEC) for repetition factor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q\geq 2$ </tex-math></inline-formula> and this multiplicative gap vanishes as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> tends to infinity. To the best of our knowledge, this is the first class of SC-TCs that are proven to be capacity-achieving. Further, the connection between the strength of the component codes, the decoding thresholds of GSC-PCCs, and the repetition factor is established. The superiority of the proposed codes with finite blocklength is exemplified by comparing their error performance with that of existing SC-TCs via computer simulations.

Spatially Coupled Turbo-Like Codes

In this paper, we introduce the concept of spatially coupled turbo-like codes (SC-TCs) as the spatial coupling of a number of turbo-like code ensembles. In particular, we consider the spatial coupling of Berrou et al. parallel concatenated codes (PCCs) and Benedetto et al. serially concatenated codes (SCCs). Furthermore, we propose two extensions of braided convolutional codes (BCCs), a class of turbo-like codes which have an inherent spatially coupled structure, to higher coupling memories, and show that these yield improved belief propagation (BP) thresholds as compared to the original BCC ensemble. We derive the exact density evolution (DE) equations for SC-TCs and analyze their asymptotic behavior on the binary erasure channel. We also consider the construction of families of rate-compatible SC-TC ensembles. Our numerical results show that threshold saturation of the belief propagation (BP) decoding threshold to the maximum a-posteriori threshold of the underlying uncoupled ensembles occurs for large enough coupling memory. The improvement of the BP threshold is especially significant for SCCs and BCCs, whose uncoupled ensembles suffer from a poor BP threshold. For a wide range of code rates, SC-TCs show close-to-capacity performance as the coupling memory increases. We further give a proof of threshold saturation for SC-TC ensembles with identical component encoders. In particular, we show that the DE of SC-TC ensembles with identical component encoders can be properly rewritten as a scalar recursion. This allows us to define potential functions and prove threshold saturation using the proof technique recently introduced by Yedla et al..

Parallel Concatenated Tree Hadamard Codes

In this letter, a new class of low-rate error-correcting codes named parallel concatenated tree Hadamard codes (CTrHC) are designed. Tree Hadamard codes consist of Hadamard segments which are connected by carefully chosen positions from a Hadamard codeword. With the same encoding and decoding complexity as the parallel concatenated zigzag Hadamard codes, CTrHC with short code length can achieve much better error floor performance illustrated by simulation results.

Design of Optimized Convolutional and Serially Concatenated Convolutional Codes in the Presence of A-priori Information

In this paper, we focus on the design of optimized binary convolutional codes (CCs) and serially concatenated convolutional codes (SCCCs) in the presence of a-priori information (API) at the receiver. For large signal-to-noise ratios (SNRs), we first propose a CC design criterion based on the minimization of a union bound on the bit error probability (BEP). In this case, relevant performance gains, with respect to previously proposed CCs, are obtained. These gains persist even in the presence of estimation errors on the API. Then, we apply the same union bound-based design criterion to SCCCs. Since the BEP of SCCCs is characterized by a typical waterfall shape, the proposed union bound-based design criterion is accurate only at large SNR, to estimate the BEP floor. In order to complement this analysis, we propose a density evolution-based approach to optimize the SCCC design in terms of minimization of the SNR of the knee of the BEP curve. The obtained simulation results show substantial gains with respect to previously proposed parallel concatenated convolutional coding (PCCCing) schemes optimized under the assumption of no API at the decoder. Moreover, in the presence of strong API the proposed SCCCs allow to approach the Shannon limit (SL) more than any previously proposed turbo coding scheme.

Performance of Parallel Concatenated Coding Schemes

In this paper, ensembles of parallel concatenated codes are studied and rigorous results on their asymptotic performance, under the assumption of maximum-likelihood (ML) decoding, are presented. In particular, it is proven that in any parallel concatenation scheme with k branches where all k encoders are recursive and the Bhattacharyya parameter of the channel is sufficiently small, the bit-error rate (BER) and the word-error rate go to 0 exactly like N 1- k and N 2- k, respectively. Different types of ensembles by changing the subgroup of permutations used to interconnect the various encoders, are considered.

A simple remedy for the exaggerated extrinsic information produced by the SOVA algorithm

In this paper, we propose a novel and simple approach for dealing with the exaggerated extrinsic information produced by the soft-output Viterbi algorithm (SOVA). The proposed remedy is based on mathematical analysis and it involves using two attenuators, one applied to the immediate output of the SOVA and another applied to the extrinsic information before it is passed to the other decoder (assuming iterative decoding). The use of these attenuators aims at reducing the inherent strong correlation between the intrinsic information (input to the SOVA) and extrinsic information (output of the SOVA). We examine the modified SOVA (MSOVA) on additive white Gaussian noise (AWGN) and flat fading channels for parallel concatenated codes (PCCs) and serial concatenated codes (SCCs). We show that the MSOVA provides substantial performance improvements over both channels. For example, it provides improvements of about 0.8 to 1.0 dB at P/sub b/ = 10/sup -5/ in AWGN, and about 1.4 to 2.0 dB at P/sub b/ = 10/sup -5/ on fading channels. We also show that there are cases where the MSOVA is superior to the a posteriori probability (APP) algorithm. With this motivation, we extend the proposed modification to the APP algorithm with favorable results. We demonstrate that the modified APP (MAPP) provides performance improvements between 0.3 to 0.6 dB at P/sub b/ = 10/sup -5/ relative to the APP. We lastly mention that the proposed modifications, while they provide considerable performance improvements, keep the complexity of these decoders almost the same, which is remarkable.

Improvements in SOVA-based decoding for turbo-coded storage channels

In this paper, we propose a novel and simple approach for dealing with the exaggerated extrinsic information produced by the soft-output Viterbi algorithm (SOVA). We first identify the reason behind these exaggerated values and then propose a simple remedy for it. We argue that what leads to this optimistic extrinsic information is the inherent strong correlation between the intrinsic information (input to the SOVA) and extrinsic information (output of the SOVA). Our proposed remedy is based on mathematical analysis, and it involves using two attenuators, one applied to the immediate output of the SOVA and another applied to the extrinsic information before it is passed to the other decoder (assuming iterative decoding). We examine the modified SOVA (MSOVA) on idealized partial response (PR) channels and the Lorentzian channel equalized to a PR target. We consider both parallel concatenated codes (PCCs) and serial concatenated codes (SCCs). We show that the MSOVA provides substantial performance improvements over both channels. For example, it provides improvements of about 0.8 to 1.6 dB at P/sub b/=10/sup -5/. Finally, we note that the proposed modifications, while they provide considerable performance improvements, introduce only two additional multipliers to the complexity of the SOVA algorithm, which is remarkable.

Convergence Analysis of Turbo Decoding of Serially Concatenated Block Codes and Product Codes

The geometric interpretation of turbo decoding has founded a framework, and provided tools for the analysis of parallel-concatenated codes decoding. In this paper, we extend this analytical basis for the decoding of serially concatenated codes, and focus on serially concatenated product codes (SCPC) (i.e., product codes with checks on checks). For this case, at least one of the component (i.e., rows/columns) decoders should calculate the extrinsic information not only for the information bits, but also for the check bits. We refer to such a component decoder as a serial decoding module (SDM). We extend the framework accordingly and derive the update equations for a general turbo decoder of SCPC and the expressions for the main analysis tools: the Jacobian and stability matrices. We explore the stability of the SDM. Specifically, for high SNR, we prove that the maximal eigenvalue of the SDM's stability matrix approaches d - 1, where d is the minimum Hamming distance of the component code. Hence, for practical codes, the SDM is unstable. Further, we analyze the two turbo decoding schemes, proposed by Benedetto and Pyndiah, by deriving the corresponding update equations and by demonstrating the structure of their stability matrices for the repetition code and an SCPC code with 2×2 information bits. Simulation results for the Hamming [(7, 4, 3)]2 and Golay [(24, 12, 8)]2 codes are presented, analyzed, and compared to the theoretical results and to simulations of turbo decoding of parallel concatenation of the same codes.

Asymptotic performance comparison of concatenated (turbo) codes over GF(4)

AbstractIn this paper, we investigate and compare the asymptotic performance of concatenated convolutional coding schemes over GF(4) over additive white Gaussian noise (AWGN) channels. Both parallel concatenated codes (PCC) and serial concatenated codes (SCC) are considered. We construct such codes using optimal non‐binary convolutional codes where optimality is in the sense of achieving the largest minimum distance for a fixed number of encoder states. Code rates of the form k0/(k0 + 1) for k0=1, 8, and 64 are considered, which suite a wide spectrum of communications applications. For all of these code rates, we find the minimum distance and the corresponding multiplicity for both concatenated code systems. This is accomplished by feeding the encoder with all possible weight‐two and weight‐three input information patterns and monitoring, at the output of the encoder, the weight of the corresponding codewords and their multiplicity. Our analytical results indicate that the SCC codes considerably outperform their counterpart PCC codes at a much lower complexity. Inspired by the superiority of SCC codes, we also discuss a mathematical approach for analysing such codes, leading to a more comprehensive analysis and allowing for further improvement in performance by giving insights on designing a proper interleaver that is capable of eliminating the dominant error patterns. Copyright © 2004 John Wiley &amp; Sons, Ltd.

On Concatenated Zigzag Codes and Their Decoding Schemes

In this letter, we show that a concatenated zigzag code can be viewed as a low-density parity-check (LDPC) code. Based on the bipartite graph representation for such a parallel-concatenated code, various sum-product based decoding algorithms are introduced and compared. The results show that the improved versions of sum-product algorithm exhibit better convergence rate while maintaining the essential parallel form.

Combined concatenated coding and nonlinear equalization for magnetic recording channels

Summary form only given. Following the work done by Ryan (Proc. ICC Int. Conf. on Communications, 1998), considerable research has been reported on the application of concatenated codes to partial response (PR) channels. Both parallel concatenated codes (PCC) and serial concatenated codes (SCC) have been studied. In all of those works, a linear model for the channel was assumed, which is a valid assumption for low recording densities. However, as the recording density increases, the channel becomes nonlinear where nonlinearity arises from closely spaced recorded transitions. In this paper, we combine concatenated coding and nonlinear equalization for PR channels that suffer from nonlinear distortion (or intersymbol interference) according to the setup shown in Fig 1. We extend the RAM-search/Viterbi-detector concept to the a posteriori probability (APP) and soft-output Viterbi algorithm (SOVA) detectors that are used with turbo-coded partial response channels. The modified APP and SOVA are called RS-APP and RS-SOVA, respectively.

BER performance of turbo-coded PPM CDMA systems on optical fiber

We obtain upper bounds on the bit error rate (BER) for turbo-coded optical code-division multiple-access (CDMA) systems using pulse position modulation (PPM). We use transfer function bounding techniques to obtain these bounds, so our results correspond to the average bound over all interleavers of a given length. We consider parallel concatenated coding (PCC) schemes that use recursive convolutional codes as constituent codes. We consider systems using an avalanche photodiode (APD), and treat APD noise, thermal noise, and multi-user interference using a Gaussian approximation. We compare the performance of turbo-coded systems with that of BCH-coded systems with soft-decision decoding, and that of concatenated coding systems with outer Reed-Solomon (RS) code and inner convolutional code. We show that turbo-coded systems have better performance than BCH-coded systems. We also show that concatenated systems have better performance than turbo-coded systems when the block length is small and the received laser power is somewhat large.

Optimal code rates for concatenated codes on a PR4-equalized magnetic recording channel

At some nominal recording density, the read signal in digital magnetic recording resembles a Class IV partial response (PR4) signal and, hence, may be equalized to the PR4 shape with relatively little noise enhancement. When coding is added, for a fixed user density, the recording density must increase as a result of coding overhead, and the read signal will resemble PR4 to a lesser extent. Equalization to PR4 in this case will produce excessive noise enhancement. Thus, coding overhead (or rate) must be selected for optimum tradeoff between code strength and noise enhancement. Toward this end, we provide results for high-rate concatenated codes, assuming a Lorentzian recording channel model. In addition to examining optimal code rates, we compare parallel and serial concatenated code performance on the PR4 channel.