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IET Communications

Doubly multistage decoding of low-density parity-check codes over ℤ 2 m

Haifeng Yuan, M.A. Armand

Published Oct 16, 2012• 0 citations
Publisher

Abstract

A doubly multistage decoder (DMD) for low-density parity-check codes over ℤ2m, m>1, which fuses the multistage decoding approaches of Armand et al. and Varnica et al., is proposed. Two variants of the DMD are considered. The first (respectively, second) performs belief-propagation (BP) [respectively, offset min-sum (OMS)] decoding in each decoding stage and is referred to as DMD - BP (respectively, DMD - OMS). For the moderate-length codes considered in this study, the computer simulations show the DMD - BP (respectively, DMD - OMS) achieving coding gains of up to 0.43 dB (respectively, 0.67 dB) over standard BP decoding at a bit error rate of 10−6 on an additive-white-Gaussian-noise channel, while requiring significantly less computational power. Remarkably, DMD - OMS outperforms DMD - BP, yet has lower computational complexity than DMD - BP. Snapshots of the log-likelihood ratio densities of the decoded bits midway through the decoding process explain the superiority of the DMD over standard BP decoding.

DOI
10.1049/iet-com.2011.0552

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A doubly multistage decoder (DMD) for low-density parity-check codes over ℤ2m, m>1, which fuses the multistage decoding approaches of Armand et al. and Varnica et al., is proposed. Two variants of the DMD are considered. The first (respectively, second) performs belief-propagation (BP) [respectively, offset min-sum (OMS)] decoding in each decoding stage and is referred to as DMD - BP (respectively, DMD - OMS). For the moderate-length codes considered in this study, the computer simulations show the DMD - BP (respectively, DMD - OMS) achieving coding gains of up to 0.43 dB (respectively, 0.67 dB) over standard BP decoding at a bit error rate of 10−6 on an additive-white-Gaussian-noise channel, while requiring significantly less computational power. Remarkably, DMD - OMS outperforms DMD - BP, yet has lower computational complexity than DMD - BP. Snapshots of the log-likelihood ratio densities of the decoded bits midway through the decoding process explain the superiority of the DMD over standard BP decoding.
A novel representation, using soft-bit messages, of the belief propagation (BP) decoding algorithm for low-density parity-check codes is derived as an alternative to the log-likelihood-ratio (LLR)-based BP and min-sum decoding algorithms. A simple approximation is also presented. Simulation results demonstrate the functionality of the soft-bit decoding algorithm. Floating-point soft-bit and LLR BP decoding show equivalent performance; the approximation incurs 0.5-dB loss, comparable to min-sum performance loss over BP. Fixed-point results show similar performance to LLR BP decoding; the approximation converges to floating-point results with one less bit of precision.
Many classes of high-performance Low-density parity check codes are based on parity check matrices composed of permutation sub matrices. The emulation-simulation framework further allows the algorithm and implementation to be iteratively redefined to improve the error floor performance of message passing decoder. Log-Like hood-Ratio (LLR) based Belief-Propagation (BP) algorithm is presented for Low Density Parity Check codes. Numerically accurate representation of check node update computation used in LLR-BP decoding is described. The implementation of Sum-Product algorithm (SPA) within Low Density Parity Check Code (LDPC) decoder is described in this paper and the correction term is used to improve the decoding performance of min-sum algorithm (MSA). Quantization and log-tanh function approximation in sum-product decoder strongly affect which absorbing set dominates in error floor region. For LDPC decoder, bit error rate (BER) decreases with increase in the signal to noise ratio.
This paper proposes a new finite precision iterative decoder for low-density parity-check (LDPC) codes. The proposed decoder, named Sign-Preserving Min-Sum (SP-MS), significantly improves the decoding performance compared to the classical Offset Min-Sum (OMS) decoder when messages are quantized onq=2q=2q=2, 3, or 4 bits. The particularity of the SP-MS decoder is that messages cannot take the 0 value, and can fully benefit from theqqqbits of precision. The optimization of the SP-MS decoder is investigated in the asymptotic limit of the code length using density evolution (DE). Our study shows that 3-bit SP-MS decoders can achieve the same error-correcting performance as 5-bit OMS decoders, and 2-bit SP-MS decoders outperform 3-bit OMS decoders. The finite-length simulations confirm the conclusions of the DE analysis for several LDPC codes. Our SP-MS decoder shows a signal-to-noise ratio (SNR) gain up to 0.43 dB, with a memory/wire reduction of up to 40%, compared to the OMS decoder. Moreover, the SP-MS decoder converges faster and uses fewer iterations than the OMS decoder, with an improvement of up to 83.3% of the average decoding throughput. On an FPGA, the SP-MS decoder reduces resource utilization by up to 56% compared to the OMS decoder.
Various log-likelihood-ratio-based belief-propagation (LLR-BP) decoding algorithms and their reduced-complexity derivatives for low-density parity-check (LDPC) codes are presented. Numerically accurate representations of the check-node update computation used in LLR-BP decoding are described. Furthermore, approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check-node update, or symbol-node update, or both. In particular, two main approaches for simplified check-node updates are presented that are based on the so-called min-sum approximation coupled with either a normalization term or an additive offset term. Density evolution is used to analyze the performance of these decoding algorithms, to determine the optimum values of the key parameters, and to evaluate finite quantization effects. Simulation results show that these reduced-complexity decoding algorithms for LDPC codes achieve a performance very close to that of the BP algorithm. The unified treatment of decoding techniques for LDPC codes presented here provides flexibility in selecting the appropriate scheme from performance, latency, computational-complexity, and memory-requirement perspectives.
Two-dimensional (2-D) correction schemes are proposed to improve the performance of conventional min-sum (MS) decoding of irregular low density parity check codes. An iterative procedure based on parallel differential optimization is presented to obtain the optimal 2-D factors. Both density evolution analysis and simulation show that the proposed method provides a comparable performance as belief propagation (BP) decoding while requiring less complexity. Interestingly, the new method exhibits a lower error floor than that of BP decoding. With respect to conventional MS and 1-D normalized MS decodings, the 2-D normalized MS offers a better performance. The 2-D offset MS decoding exhibits a similar behavior.