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Low-complexity LDPC decoding algorithms for ultra-high-order modulated signals

Hongjun Zhu, Meiyu Fu, Chen Hou, Guijun Hu

Published Nov 14, 2023• 2 citations
Publisher

Abstract

Low-complexity decoding algorithms and ultra-high-order modulation formats are necessary to meet data rate requirements in excess of 1 Tbps. The information bottleneck algorithm has been effectively applied to the LDPC decoding algorithm in recent years, and its performance is comparable to that of the double-precision information propagation technique. However, the application of information bottleneck decoding algorithms in ultra-high order modulation forms has received little attention. Furthermore, the number of table lookups required for a single decoding loop is square to the node degree, which is undesirable for optical communications. We present a low-complexity LDPC decoding technique for ultra-high-order modulated signals in this study. First, the algorithm employs multivariate covariates to build an information bottleneck framework, which introduces the processing required for applying the information bottleneck algorithm to 1024-QAM signals and the requirement of combining higher-order modulation formats with LDPC codes. The technique makes use of a bidirectional recursive network and the symmetry of quantized information to reuse the same set of tables, considerably reducing the number of table lookup operations required in the decoding process. Constructing a coherent optical communication system with 1024-QAM signals proves that the proposed algorithm can operate effectively. The performance sacrifice of 0.2 ∼ 0.3 dB reduces the number of table lookup operations required for decoding from square to linear magnitude, which greatly reduces the decoding time required in optical communication.

DOI
10.1364/oe.507292

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Low-complexity decoding algorithms and ultra-high-order modulation formats are necessary to meet data rate requirements in excess of 1 Tbps. The information bottleneck algorithm has been effectively applied to the LDPC decoding algorithm in recent years, and its performance is comparable to that of the double-precision information propagation technique. However, the application of information bottleneck decoding algorithms in ultra-high order modulation forms has received little attention. Furthermore, the number of table lookups required for a single decoding loop is square to the node degree, which is undesirable for optical communications. We present a low-complexity LDPC decoding technique for ultra-high-order modulated signals in this study. First, the algorithm employs multivariate covariates to build an information bottleneck framework, which introduces the processing required for applying the information bottleneck algorithm to 1024-QAM signals and the requirement of combining higher-order modulation formats with LDPC codes. The technique makes use of a bidirectional recursive network and the symmetry of quantized information to reuse the same set of tables, considerably reducing the number of table lookup operations required in the decoding process. Constructing a coherent optical communication system with 1024-QAM signals proves that the proposed algorithm can operate effectively. The performance sacrifice of 0.2 ∼ 0.3 dB reduces the number of table lookup operations required for decoding from square to linear magnitude, which greatly reduces the decoding time required in optical communication.
In high-throughput applications, low-complexity and low-latency channel decoders are inevitable. Hence, for low-density parity-check (LDPC) codes, message passing decoding has to be implemented with coarse quantization—that is, the exchanged beliefs are quantized with a small number of bits. This can result in a significant performance degradation with respect to decoding with high-precision messages. Recently, so-called information-bottleneck decoders were proposed which leverage a machine learning framework (i.e., the information bottleneck method) to design coarse-precision decoders with error-correction performance close to high-precision belief-propagation decoding. In these decoders, all conventional arithmetic operations are replaced by look-up operations. Irregular LDPC codes for next-generation fiber optical communication systems are characterized by high code rates and large maximum node degrees. Consequently, the implementation complexity is mainly influenced by the memory required to store the look-up tables. In this paper, we show that the complexity of information-bottleneck decoders remains manageable for irregular LDPC codes if our proposed construction approach is deployed. Furthermore, we reveal that in order to design information bottleneck decoders for arbitrary degree distributions, an intermediate construction step which we call message alignment has to be included. Exemplary numerical simulations show that incorporating message alignment in the construction yields a 4-bit information bottleneck decoder which performs only 0.15 dB worse than a double-precision belief propagation decoder and outperforms a min-sum decoder.
In recent years coarsely quantized LDPC decoding using a flooding schedule has been extensively studied. However, there exist few works addressing coarse quantization for a layered schedule, which enables improved convergence speed of the message passing algorithm. The layered schedule can especially be beneficial for high throughput applications like fiber optical systems. This paper presents innovative layered decoding approaches, where the information bottleneck method is used for the design of different coarsely quantized decoder architectures. The varieties of investigated node implementations include lookup tables, computational domain techniques as well as reduced complexity approximations. All structures are designed offline using a layered discrete density evolution method. The performance of multiple node architectures is investigated in terms of evolution of mutual information in the design phase and in terms of error rates. We focus in this paper on regular quasi-cyclic codes. Our simulations running on GPUs also allow insights into the error floor behavior.
A fast quantized belief propagation algorithm is proposed for decoding low-density parity-check(LDPC) codes. With operations performed by addressing tables, the proposed algorithm takes as operands the indices of labels rather than the quantization levels. The intermediate calculations have been contained in operation tables with much higher precision. The required storage can be significantly reduced by applying the symmetry property while keeping low time complexity. Without the increase in time complexity, it is convenient to realize the decoding with uniform quantization or any nonuniform quantizations that is designed for improving the performance. The time-varying decoder also becomes possible. This algorithm makes it possible to apply LDPC codes to practical systems or to realize fast simulation systems. Some simulation results are also given. The 6?bit logarithmic quantization results in gains of about 0.2?dB relative to the uniform one, approaching to the continuous decoding at low signal-to-noise ratios (SNRs) and yielding a loss of about 0.2?dB at high SNRs. Proper quantization of high order can achieve the performance of continuous decoding, and it may be superior to continuous decoding at high SNRs.
Recently, a novel lookup table based decoding method for binary low-density parity-check codes has attracted considerable attention. In this approach, mutual-information-maximizing lookup tables replace the conventional operations of the variable nodes and the check nodes in message passing decoding. Moreover, the exchanged messages are represented by integers with very small bit width. A machine learning framework termed the information bottleneck method is used to design the corresponding lookup tables. In this paper, we extend this decoding principle from binary to non-binary codes. This is not a straightforward extension but requires a more sophisticated lookup table design to cope with the arithmetic in higher order Galois fields. Provided bit error rate simulations show that our proposed scheme outperforms the log-max decoding algorithm and operates close to sum-product decoding.
Recently, a novel lookup table based decoding method for binary low-density parity-check codes has attracted considerable attention. In this approach, mutual-information-maximizing lookup tables replace the conventional operations of the variable nodes and the check nodes in message passing decoding. Moreover, the exchanged messages are represented by integers with very small bit width. A machine learning framework termed the information bottleneck method is used to design the corresponding lookup tables. In this paper, we extend this decoding principle from binary to non-binary codes. This is not a straightforward extension but requires a more sophisticated lookup table design to cope with the arithmetic in higher order Galois fields. Provided bit error rate simulations show that our proposed scheme outperforms the log-max decoding algorithm and operates close to sum-product decoding.
The Information Bottleneck method is a powerful and generic tool from the field of machine learning. It compresses an observation to a quantized variable while attempting to preserve the mutual information shared with a relevant random variable. This paper describes a new application of the Information Bottleneck method in communications. It explains in detail, how the Information Bottleneck method can be applied to construct discrete message passing decoders for regular low-density parity-check codes. The obtained decoders process only unsigned integers and use only simple lookup tables as node operations. As a consequence, the decoders can be implemented using only unsigned integer arithmetic which makes them significantly simpler and faster than the state-of-the-art decoders which process real valued log-likelihood ratios. Anyway, included results show that the considered discrete message passing decoders perform surprisingly close to optimum message passing decoders and even outperform state-of-the-art decoders, such as the min-sum decoder. We aim to take the reader on a journey from the theoretical idea of the Information Bottleneck method to a complete design framework for the considered discrete decoders. Several included figures and examples illustrate the decoder construction process and its analysis.