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IEEE International Symposium on Information Theory (ISIT)

Decoding LDPC codes with mutual information-maximizing lookup tables

Francisco Javier Cuadros Romero, Brian M. Kurkoski

Published Jun 1, 2015• 35 citations
Publisher

Abstract

A recent result has shown connections between statistical learning theory and channel quantization. In this paper, we present a practical application of this result to the implementation of LDPC decoders. In particular, we describe a technique for designing the message-passing decoder mappings (or lookup tables) based on the ideas of channel quantization. This technique is not derived from sum-product algorithm or any other LDPC decoding algorithm. Instead, the proposed algorithm is based on an optimal quantizer in the sense of maximization of mutual information, which is inserted in the density evolution algorithm to generate the lookup tables. This algorithm has low complexity since it only employs 3-bit messages and lookup tables, which can be easily implemented in hardware. Two quantized versions of the min-sum decoding algorithm are used for comparison. Simulation results for a binary-input AWGN channel show 0.3 dB and 1.2 dB gains versus the two quantized min-sum algorithms. On the binary symmetric channel also a gain is seen.

DOI
10.1109/isit.2015.7282490

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A recent result has shown connections between statistical learning theory and channel quantization. In this paper, we present a practical application of this result to the implementation of LDPC decoders. In particular, we describe a technique for designing the message-passing decoder mappings (or lookup tables) based on the ideas of channel quantization. This technique is not derived from sum-product algorithm or any other LDPC decoding algorithm. Instead, the proposed algorithm is based on an optimal quantizer in the sense of maximization of mutual information, which is inserted in the density evolution algorithm to generate the lookup tables. This algorithm has low complexity since it only employs 3-bit messages and lookup tables, which can be easily implemented in hardware. Two quantized versions of the min-sum decoding algorithm are used for comparison. Simulation results for a binary-input AWGN channel show 0.3 dB and 1.2 dB gains versus the two quantized min-sum algorithms. On the binary symmetric channel also a gain is seen.
In this paper, we propose a general framework of the mutual infomration-maximizing (MIM) quantized decoding for low-density parity-check (LDPC) codes, which can outperform the state-of-the-art lookup table (LUT) decoder by using simple mappings and fixed-point additions for the node updates. Our decoding method is generic in the sense that it can be applied to LDPC codes with arbitrary degree distributions, and it can be implemented based on either the belief propagation (BP) algorithm or the min-sum (MS) algorithm, leading to the MIM quantized BP (MIM-QBP) decoder and the MIM quantized MS (MIM-QMS) decoder, respectively. In particular, we approximate the check node (CN) update of the MIM-QBP decoder by a max-product operation and obtain the MIM-QMS decoder, which simplifies the decoder design and requires less resource consumption. To avoid the precision degradation, we introduce a dynamic reconstruction method to optimize the variable node update for different iterations. Some practical aspects of the proposed decoders such as the design and decoding complexity are also discussed. Simulation results show that the MIM-QBP decoder outperforms the LUT decoders in the waterfall region with both 3-bit and 4-bit precision. Moreover, the 4-bit MIM-QMS decoder can even surpass the floating-point BP decoder in the error-floor region.
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 memory requirements are discussed and compared to the LUT decoder variants. For applications with low-latency requirement, we consider the layered schedule to accelerate the convergence speed of decoding quasi-cyclic LDPC codes. In particular, we develop the layered MIM-DE to design the LUTs based on the 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 over the additive white Gaussian noise channels. For small decoding iterations, the MIM quantized decoders also achieve a faster convergence speed compared to the benchmarks. Moreover, the 4-bit MIM-LQMS decoder can approach the error performance of the floating-point layered BP decoder within 0.3 dB in the moderate-to-high SNR regions, over both the AWGN channels and the fast fading channels.
A severe problem for mutual information-maximizing lookup table (MIM-LUT) decoding of low-density parity-check (LDPC) code is the high memory cost for using large tables, while decomposing large tables to small tables deteriorates decoding error performance. In this paper, we propose a systematic method, called mutual information- maximizing quantized belief propagation (MIM-QBP) decoding, to remove the lookup tables used for MIM-LUT decoding. Our method leads to a very practical decoder, namely the MIM-QBP decoder, which can be implemented based only on simple mappings and additions. Simulation results show that the proposed MIM-QBP decoder can outperform the state-of-the-art MIM-LUT decoder. Moreover, the MIM-QBP decoder with only 3 bits per message can outperform the floating-point belief propagation (BP) decoder at high signal-to-noise ratio (SNR) regions with a maximum of 10 iterations.
In this paper, we propose a finite alphabet message passing algorithm for LDPC codes that replaces the standard min-sum variable node update rule by a mapping based on generic look-up tables. This mapping is designed in a way that maximizes the mutual information between the decoder messages and the codeword bits. We show that our decoder can deliver the same error rate performance as the conventional decoder with a much smaller message bit-width. Finally, we use the proposed algorithm to design a fully unrolled LDPC decoder hardware architecture.
Abstract—The quantization of the output of a binary-input discrete memoryless channel to a smaller number of levels is considered. The optimal quantizer, in the sense of maximizing mutual information between the channel input and the quantizer output, may be found by an algorithm with complexity which is quadratic in the number of channel outputs. This is a concave optimization problem, and results from the field of concave optimization are invoked. The quantizer design algorithm is a realization of a dynamic program. Then, this algorithm is applied to the design of message-passing decoders for low-density parity-check codes, over arbitrary discrete memoryless channels. A general, systematic method to find message-passing decoding maps which maximize mutual information at each iteration is given. This may contrasted with existing quantized message-passing algorithms which are heuristically derived. The method finds message-passing decoding maps similar to those given by Richardson and Urbanke’s Algorithm E. Using four bits per message, noise thresholds similar to belief-propagation decoding are obtained. Index Terms—discrete memoryless channel, channel quantiza-tion, mutual information maximization, LDPC decoding I.
We propose a quantized decoding algorithm for low- density parity-check codes where the variable node update rule of the standard min-sum algorithm is replaced with a look-up table (LUT) that is designed using an information-theoretic criterion. We show that even with message resolutions as low as 3 bits, the proposed algorithm can achieve better error rates than a floating-point min-sum decoder. Moreover, we study in detail the effect of different decoder design parameters, like the design SNR and the LUT tree structure on the performance of our decoder, and we propose some complexity reduction techniques, such as LUT re-use and message alphabet downsizing.