Layered Belief Propagation Research Articles

Non-Gaussian Reconciliation for Continuous-Variable Quantum Key Distribution

Non-Gaussian modulation can improve the performance of continuous-variable quantum key distribution (CV QKD). For Gaussian-modulated coherent-state CV QKD, photon subtraction can realize non-Gaussian modulation, which can be equivalently implemented by non-Gaussian postselection. However, non-Gaussian reconciliation has not been deeply researched, which is one of the key technologies in CV QKD. In this paper, we propose a non-Gaussian reconciliation method to obtain identical keys from non-Gaussian data. Multidimensional reconciliation and multiedge-type low-density parity-check codes (MET LDPC) are used in a non-Gaussian reconciliation scheme, where the layered belief propagation decoding algorithm of MET LDPC codes is used to reduce the decoding complexity. Furthermore, we compare the error-correction performance of Gaussian data and non-Gaussian data. The results show that the error-correction performance of non-Gaussian data is better than Gaussian data, where the frame error rate can be reduced by 50 % for code rate 0.1 at SNR of 0.1554 and the average iteration number can be reduced by 25 %.

Performance Analysis of Different Flexible Decoding Algorithms for NR-LDPC Codes

Channel coding technique is a fundamental building block in any modern communication system to realize reliable, fast, and secure data transmission. At the same time, it is a challenging and crucial task, as the data transmission happens in a channel where noise, fading, and other impairments are present. The Low-Density Parity-Check (LDPC) codes give substantial results close to the Shannon limit when the complexity and processing delay time are unlimited. In this paper, the performance of the LDPC decoding with four algorithms was investigated. The investigated four algorithms were Belief Propagation (BP), Layered Belief Propagation (LBP), Normalized min-sum (NMS), and Offset min-sum (OMS). These algorithms were examined for code rates ranging from 1/3 to 9/10 and message block lengths (64, 512, 1024, and 5120) bits. The simulation results revealed the flexibility of these decoders in supporting these code rates and block lengths, which enables their usage in a wide range of applications and scenarios for fifth-generation (5G) wireless communication. In addition, the effect of the maximum number of decoding iterations on the error correction performance was investigated, and a gain of 5.6 dB can be obtained by using 32 decoding iterations at BER=2*10-3 instead of one decoding iteration. The results showed that the decoders performed better for longer message blocks than for short message blocks, and less power was required for transmitting longer messages. Finally, the comparison results of their performance in terms of bit error rate (BER) under the same conditions showed a gain of 0.8 dB using LBP at BER= 10-5 compared with the NMS decoding algorithm.

Generalized Mutual Information-Maximizing Quantized Decoding of LDPC Codes With Layered Scheduling

In this paper, we propose a framework of the mutual information-maximizing (MIM) quantized decoding for low-density parity-check (LDPC) codes by using simple mappings and fixed-point additions. Our decoding method is generic in the sense that it can be applied to LDPC codes with arbitrary degree distributions, and can be implemented based on either the belief propagation (BP) algorithm or the min-sum (MS) algorithm. In particular, we propose the MIM density evolution (MIM-DE) to construct the lookup tables (LUTs) for the node updates. The computational complexity and implementation complexity are discussed and compared to the LUT decoder variants. To accelerate the convergence speed of decoding quasi-cyclic LDPC codes, we consider the layered schedule, and develop the layered MIM-DE to design the LUTs based on MS algorithm, leading to the MIM layered quantized MS (MIM-LQMS) decoder. An optimization method is further introduced to reduce the memory requirement for storing the LUTs. Simulation results show that the MIM quantized decoders outperform the state-of-the-art LUT decoders in the waterfall region with both 3-bit and 4-bit precision. Moreover, the 4-bit MIM-LQMS decoder can approach the error performance of the floating-point layered BP decoder within 0:1 dB.

Low-Complexity Large MIMO Detection via Layered Belief Propagation in Beam Domain

In large multi-user multi-input multi-output systems, the computational cost and circuit scale of base stations (BSs) are effectively reduced using two-stage signal processing consisting of a slow varying outer beamformer (OBF) based on long-term channel statistics and group-specific multi-user detection for instantaneous channel variations. However, the dimensionality reduction of the group-specific beam-domain channel based on the OBF causes significant performance degradation in the subsequent spatial-filtering detection. To compensate for this drawback, this paper introduces a novel layered belief propagation (BP) detector with a concatenated structure of beam- and antenna-domain BP layers for post-stage OBF processing. The proposed detector is designed for improving the convergence of iterative detection by suppressing intra- and inter-group interference in stages. The layered structure provides the advantages of both beam and antenna domains while maintaining low signal-processing complexity. Numerical results show the validity of our proposed method in terms of the bit error rate performance in both the uncoded and coded cases and the computational complexity.

Neural Layered Decoding of 5G LDPC Codes

In this letter, we propose various low-complexity neural layered min-sum (NLMS) algorithms to improve the decoding performance of the fifth generation (5G) new radio (NR) low-density parity-check (LDPC) codes. Our proposals are based on the computationally-efficient normalized offset min-sum (NOMS) approach for the layered belief propagation (LBP). The main novelty of our proposals is a deep neural network (DNN) that implements the layered mode decoding and that additionally learns the normalization and the offset parameters of the NOMS scheme. The schemes that we propose use the DNN as well as adaptation and filtering for estimating the optimum values of these parameters. We show by simulation that our proposed decoders achieve a significant computational benefit compared to the standard (non-neural) LBP decoder without an appreciable performance penalty.

Two Effective Scheduling Schemes for Layered Belief Propagation of 5G LDPC Codes

Low-density parity-check (LDPC) codes have been chosen for the fifth-generation (5G) new radio (NR) enhanced Mobile Broad Band (eMBB) data channel. Since NR LDPC codes puncture the first two block columns of the parity-check matrix, the decoding complexity is reduced, but it slows down the convergence rate. In this letter, we propose a fixed schedule that decodes the layers with the least-punctured edges and those with the highest-degree (LPHD), which has much faster convergence speed than conventional schemes. The fixed scheduling schemes cannot take full advantage of the dynamic changes of decoding messages. As a result, the residual-based layered belief propagation (RB-LBP) is proposed to rearrange the layers dynamically among different iterations. Compared with some informed dynamic scheduling (IDS) strategies, RB-LBP reduces the decoding complexity with slight performance loss.

Layered Decoding Algorithm and Two-Level Quasi-Cyclic Matrix Construction for Rate Compatible Modulation

Rate-compatible modulation (RCM) achieves seamless rate adaptation under time-varying channels by varying the number of transmitted symbols at a small granularity. The implementation of effective decoders based on the belief propagation (BP) algorithm suffers from the varying number of symbols and highly complex symbol nodes. Considering these RCM-specific characteristics, we present a top–down design methodology, including decoding algorithm, mapping matrix construction, and hardware implementation. The layered BP algorithm provides the potential to implement the decoders that fine-tune the decoding throughput according to the number of symbols. To mitigate the impact of inter-layer data dependency on decoding throughput and facilitate the implementation of low parallelism decoders, a construction and optimization algorithm is proposed to obtain the two-level quasi-cyclic matrices with small sub-matrix size. The proposed algorithm and matrix are applied to a partially parallel RCM decoder for a ${5544 \times 5544}$ matrix on a field-programmable gate-array (FPGA) device. A memory access structure that requires only one barrel shifter is used to improve area efficiency. The post-route results show that the decoder achieves a throughput of 377 M ~ 1100 Mbit/s at six iterations with a clock frequency of 218 MHz.

Check‐node lazy scheduling approach for layered belief propagation decoding algorithm

A novel check-node lazy scheduling (CLS) approach is presented for the layered belief propagation (LBP) decoding algorithm in low-density parity-check (LDPC) applications. In the cases where the check nodes are highly reliable, this proposed CLS approach skips the check nodes in the belief propagation processes, and thus it clearly reduces the iteration times of LDPC decoding. Simulations show that, when this CLS approach is integrated into the classical LBP decoding algorithm, the needed maximum iterations is improved by nearly 30%.

Effective lazy schedule for the layered belief propagation algorithm

This study presents an effective lazy schedule (LS) for the layered belief propagation (LS-LBP) algorithm; it significantly reduces the number of iterations and the error floors of decoding. This novel LS forces a check node into a sleep status when the reliabilities of its neighbour variable nodes exceed the threshold, and the check node will turn to awake as all the nodes are in sleep status. This LS reduces latency by ignoring useless check-node processes, and it also achieves a low error rate by overcoming some trapping sets. The simulation experiment shows that when compared to the classical layered belief propagation for world interoperability for microwave access (WiMAX), random codes constructed using the progressive edge-growth (PEG codes) algorithm, and digital video broadcasting-satellite-second generation (DVB-S2) codes, this LS increases the decoding convergence speed by up to 36% to achieve the bit error rate 1 × 10–6 and reduces the error floors to 20–45% for the PEG and DVB-S2 codes. Compared with other schedules for the layered belief propagation, the LS-LBP algorithm is more suitable for the codes with a long block length and has lower complexity.

An efficient dynamic schedule for layered belief-propagation decoding of LDPC codes

An efficient dynamic schedule for layered belief-propagation (LBP) decoding of low-density parity-check (LDPC) codes is presented, in which the check nodes connecting to a variable node with maximum relative message residual are chosen in each dynamic decoding iteration, and then LBP is performed for these check nodes. In this schedule we combined the features of lazy schedule and node-wise residual belief propagation (NWRBP) together while keeping the extra computational complexity very low. Simulation results show that the new schedule speeds up the convergence rate and greatly improves error performance at medium to high signal-to-noise ratio (SNR) when compared to the standard LBP decoding algorithm.

Performance Comparison between Turbo Code and Rate-Compatible LDPC Code for Evolved UTRA Downlink OFDM Radio Access

This paper compares the turbo code and rate-compatible low-density parity-check (LDPC) codes based on the block error rate (BLER) performance and decoding complexity in order to clarify which channel coding scheme is most appropriate for the channel coding scheme in the OFDM based Evolved UTRA (E-UTRA) downlink. Simulation results and the decoding complexity analysis show that although the Rate-Compatible/Quasi-Cyclic (RC/QC)-LDPC code employing an offset layered belief propagation (BP) method can reduce the computational complexity by approximately 30% for the channel coding rate of R ≥ 1/2, the required average received signal energy per bit-to-noise power spectrum density ratio (Eb/N0) is degraded by approximately 0.2-0.3dB for R=1/3, 1/2 and 3/4 compared to that for the turbo code. Moreover, the decoding complexity level of the RC/QC-LDPC code with the δ-min algorithm is almost the same or higher than that for the turbo code with a slight degradation in the required received Eb/N0. Although the decoding complexity level of the ZigZag code is lower than that of the turbo code, the code brings about a distinct loss in the required average received Eb/N0 of approximately 0.4dB. Finally, the turbo Single Parity Check (SPC) code improves the BLER performance compared to the ZigZag code, i.e., achieves almost the same BLER performance as that for the turbo code, at the cost of a two-fold increase in the decoding complexity. As a result, we conclude that the turbo code with a contention free interleaver is more promising than the LDPC codes for prioritizing the achievable performance over complexity and as the channel coding scheme for the shared data channel in the E-UTRA.

Layered BP Decoding for Rate-Compatible Punctured LDPC Codes

Rate-compatible punctured LDPC codes have shown to perform well over a wide variety of code rates, both theoretically and practically. However it has been reported that the belief propagation (BP) decoding for these codes converges slower than for unpunctured codes. Layered BP algorithm is a modified BP algorithm that accelerates the decoding convergence by means of sequential scheduling of check node updates. In this letter, we propose an efficient scheduling of check node updates for rate-compatible punctured LDPC codes that performs well. We show that the convergence speed of the proposed scheduling outperforms conventional (random) scheduling and conventional BP decoding. Performance improvements become more distinctive with the growing fraction of punctured bits