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Minimum-Integer Computation Finite Alphabet Message Passing Decoder: From Theory to Decoder Implementations towards 1 Tb/s

Tobias Monsees, Oliver Griebel, Matthias Herrmann, Dirk Wübben, Armin Dekorsy, Norbert Wehn

Published Oct 12, 2022• 16 citations
Publisher

Abstract

In Message Passing (MP) decoding of Low-Density Parity Check (LDPC) codes, extrinsic information is exchanged between Check Nodes (CNs) and Variable Nodes (VNs). In a practical implementation, this information exchange is limited by quantization using only a small number of bits. In recent investigations, a novel class of Finite Alphabet Message Passing (FA-MP) decoders are designed to maximize the Mutual Information (MI) using only a small number of bits per message (e.g., 3 or 4 bits) with a communication performance close to high-precision Belief Propagation (BP) decoding. In contrast to the conventional BP decoder, operations are given as discrete-input discrete-output mappings which can be described by multidimensional LUTs (mLUTs). A common approach to avoid exponential increases in the size of mLUTs with the node degree is given by the sequential LUT (sLUT) design approach, i.e., by using a sequence of two-dimensional Lookup-Tables (LUTs) for the design, leading to a slight performance degradation. Recently, approaches such as Reconstruction-Computation-Quantization (RCQ) and Mutual Information-Maximizing Quantized Belief Propagation (MIM-QBP) have been proposed to avoid the complexity drawback of using mLUTs by using pre-designed functions that require calculations over a computational domain. It has been shown that these calculations are able to represent the mLUT mapping exactly by executing computations with infinite precision over real numbers. Based on the framework of MIM-QBP and RCQ, the Minimum-Integer Computation (MIC) decoder design generates low-bit integer computations that are derived from the Log-Likelihood Ratio (LLR) separation property of the information maximizing quantizer to replace the mLUT mappings either exactly or approximately. We derive a novel criterion for the bit resolution that is required to represent the mLUT mappings exactly. Furthermore, we show that our MIC decoder has exactly the communication performance of the corresponding mLUT decoder, but with much lower implementation complexity. We also perform an objective comparison between the state-of-the-art Min-Sum (MS) and the FA-MP decoder implementations for throughput towards 1 Tb/s in a state-of-the-art 28 nm Fully-Depleted Silicon-on-Insulator (FD-SOI) technology. Furthermore, we demonstrate that our new MIC decoder implementation outperforms previous FA-MP decoders and MS decoders in terms of reduced routing complexity, area efficiency and energy efficiency.

DOI
10.3390/e24101452

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In Message Passing (MP) decoding of Low-Density Parity Check (LDPC) codes, extrinsic information is exchanged between Check Nodes (CNs) and Variable Nodes (VNs). In a practical implementation, this information exchange is limited by quantization using only a small number of bits. In recent investigations, a novel class of Finite Alphabet Message Passing (FA-MP) decoders are designed to maximize the Mutual Information (MI) using only a small number of bits per message (e.g., 3 or 4 bits) with a communication performance close to high-precision Belief Propagation (BP) decoding. In contrast to the conventional BP decoder, operations are given as discrete-input discrete-output mappings which can be described by multidimensional LUTs (mLUTs). A common approach to avoid exponential increases in the size of mLUTs with the node degree is given by the sequential LUT (sLUT) design approach, i.e., by using a sequence of two-dimensional Lookup-Tables (LUTs) for the design, leading to a slight performance degradation. Recently, approaches such as Reconstruction-Computation-Quantization (RCQ) and Mutual Information-Maximizing Quantized Belief Propagation (MIM-QBP) have been proposed to avoid the complexity drawback of using mLUTs by using pre-designed functions that require calculations over a computational domain. It has been shown that these calculations are able to represent the mLUT mapping exactly by executing computations with infinite precision over real numbers. Based on the framework of MIM-QBP and RCQ, the Minimum-Integer Computation (MIC) decoder design generates low-bit integer computations that are derived from the Log-Likelihood Ratio (LLR) separation property of the information maximizing quantizer to replace the mLUT mappings either exactly or approximately. We derive a novel criterion for the bit resolution that is required to represent the mLUT mappings exactly. Furthermore, we show that our MIC decoder has exactly the communication performance of the corresponding mLUT decoder, but with much lower implementation complexity. We also perform an objective comparison between the state-of-the-art Min-Sum (MS) and the FA-MP decoder implementations for throughput towards 1 Tb/s in a state-of-the-art 28 nm Fully-Depleted Silicon-on-Insulator (FD-SOI) technology. Furthermore, we demonstrate that our new MIC decoder implementation outperforms previous FA-MP decoders and MS decoders in terms of reduced routing complexity, area efficiency and energy efficiency.
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This paper proposes a finite-precision decoding method for low-density parity-check (LDPC) codes that features the three steps of Reconstruction, Computation, and Quantization (RCQ). Unlike Mutual-Information-Maximization Quantized Belief Propagation (MIM-QBP), RCQ can approximate either belief propagation or Min-Sum decoding. MIM-QBP decoders do not work well when the fraction of degree-2 variable nodes is large. However, sometimes a large fraction of degree-2 variable nodes is used to facilitate a fast encoding structure, as seen in the IEEE 802.11 standard and the DVB-S2 standard. In contrast to MIM-QBP, the proposed RCQ decoder may be applied to any off-the-shelf LDPC code, including those with a large fraction of degree-2 variable nodes. Simulations show that a 4-bit Min-Sum RCQ decoder delivers frame error rate (FER) performance within 0.1 dB of floating point belief propagation (BP) for the IEEE 802.11 standard LDPC code in the low SNR region. The RCQ decoder actually outperforms floating point BP and Min-Sum in the high SNR region were FER less than 10-5. This paper also introduces Hierarchical Dynamic Quantization (HDQ) to design the time-varying non-uniform quantizers required by RCQ decoders. HDQ is a low-complexity design technique that is slightly sub-optimal. Simulation results comparing HDQ and optimal quantization on the symmetric binary-input memoryless additive white Gaussian noise channel show a mutual information loss of less than 10-6bits, which is negligible in practice.
In this paper, we propose a mutual information-maximizing quantized layered min-sum (MIM-QLMS) decoder for quasi-cyclic low-density parity-check (QC-LDPC) codes. Our proposed decoder operates similarly to a layered min-sum decoder with additional reconstruction and quantization operations by using single-input lookup tables (LUTs). In particular, we first develop the protograph-based MIM density evolution to design the LUTs, which may differ for each iteration and each edge in the protograph of the QC-LDPC codes. Furthermore, to minimize the memory requirement for storing the LUTs, we propose an optimization method to unify all LUTs into only four distinct LUTs, which can be used for all decoding iterations. To the best of our knowledge, the proposed MIM-QLMS decoders are the first class of layered finite alphabet iterative decoders (FAIDs) that are designed based on accurately tracking the probability distributions of the exchanged messages. Simulation results show that for 3-bit (resp. 4-bit) exchanged message precision, the proposed MIM-QLMS decoders can reasonably (resp. generally) outperform the state-of-the-art layered FAIDs and the layered normalized min-sum decoder, in terms of both the error rate performance and the average number of iterations.
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