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arXiv:1504.06025

XJ-BP: Express Journey Belief Propagation Decoding for Polar Codes

Jingwei Xu, Tiben Che, Gwan Choi

Published Apr 23, 2015• 2 citations
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Abstract

This paper presents a novel propagation (BP) based decoding algorithm for polar codes. The proposed algorithm facilitates belief propagation by utilizing the specific constituent codes that exist in the factor graph, which results in an express journey (XJ) for belief information to propagate in each decoding iteration. In addition, this XJ-BP decoder employs a novel round-trip message passing scheduling method for the increased efficiency. The proposed method simplifies min-sum (MS) BP decoder by 40.6%. Along with the round-trip scheduling, the XJ-BP algorithm reduces the computational complexity of MS BP decoding by 90.4%; this enables an energy-efficient hardware implementation of BP decoding in practice.

DOI
10.48550/arxiv.1504.06025

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XJ-BP: Express Journey Belief Propagation Decoding for Polar Codes

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This paper presents a novel propagation (BP) based decoding algorithm for polar codes. The proposed algorithm facilitates belief propagation by utilizing the specific constituent codes that exist in the factor graph, which results in an express journey (XJ) for belief information to propagate in each decoding iteration. In addition, this XJ-BP decoder employs a novel round-trip message passing scheduling method for the increased efficiency. The proposed method simplifies min-sum (MS) BP decoder by 40.6%. Along with the round-trip scheduling, the XJ-BP algorithm reduces the computational complexity of MS BP decoding by 90.4%; this enables an energy-efficient hardware implementation of BP decoding in practice.
This paper presents a novel belief propagation (BP) based decoding algorithm for polar codes. The proposed algorithm facilitates belief propagation by utilizing the specific constituent codes that exist in the factor graph, which results in an express journey (XJ) for belief information to propagate in each decoding iteration. In addition, this XJ-BP decoder employs a novel round-trip message passing scheduling method for the increased efficiency. The proposed method simplifies min-sum (MS) BP decoder by 40.6%. Along with the round-trip scheduling, the XJ-BP algorithm reduces the computational complexity of MS BP decoding by 90.4%; this enables an energy-efficient hardware implementation of BP decoding in practice.
This paper presents a novel belief propagation (BP) based decoding algorithm for polar codes. The proposed algorithm facilitates belief propagation by utilizing the specific constituent codes that exist in the factor graph, which results in an express journey (XJ) for belief information to propagate in each decoding iteration. In addition, this XJ-BP decoder employs a novel round-trip message passing scheduling method for the increased efficiency. The proposed method simplifies min-sum (MS) BP decoder by 40.6%. Along with the round-trip scheduling, the XJ-BP algorithm reduces the computational complexity of MS BP decoding by 90.4%; this enables an energy-efficient hardware implementation of BP decoding in practice.
The belief propagation (BP) decoding algorithm not only is an alternative to the successive cancelation (SC) decoders of polar codes, but also provides soft outputs that are necessary for joint detection and decoding. The BP decoders with the flooding schedule achieve high throughput with excessive hardware cost especially when the block length is large. The soft-cancelation (SCAN) decoders for polar codes have reduced memory complexity compared to the BP decoders based on the flooding schedule. The simplified SC aided reduced complexity soft-cancelation (S-RCSC) decoders further reduce the computational and memory complexity of the SCAN decoders at the cost of negligible error performance degradation. Both the SCAN and S-RCSC decoders have limited throughput due to their serial decoding schedules. In this paper, we first propose an improved S-RCSC (IS-RCSC) decoding algorithm and then present a high throughput decoder architecture based on our IS-RCSC algorithm. Our IS-RCSC decoding algorithm performs the message passing on a binary tree representation of a polar code. Compared to the S-RCSC decoding algorithm, our IS-RCSC decoding algorithm accelerates the computing of the returned soft messages when certain types of nodes are activated. The corresponding hardware architecture of our IS-RCSC decoder is also proposed. In terms of area efficiency, the hardware implementation results demonstrate that our IS-RCSC decoders are 19% to 43% better than decoders in the literature.
We describe a novel approach to interpret a polar code as a low-density parity-check (LDPC)-like code with an underlying sparse decoding graph. This sparse graph is based on the encoding factor graph of polar codes and is suitable for conventional belief propagation (BP) decoding. We discuss several pruning techniques based on the check node decoder (CND) and variable node decoder (VND) update equations, significantly reducing the size (i.e., decoding complexity) of the parity-check matrix. As a result, iterative polar decoding can then be conducted on a sparse graph, akin to the traditional well-established LDPC decoding, e.g., using a fully parallel sum-product algorithm (SPA). This facilitates the systematic analysis and design of polar codes using the well-established tools known from analyzing LDPC codes. We show that the proposed iterative polar decoder has a negligible performance loss for short-to-intermediate codelengths compared to Arikan's original BP decoder. Finally, the proposed decoder is shown to benefit from both reduced complexity and reduced memory requirements and, thus, is more suitable for hardware implementations.
The parallel nature of the belief propagation (BP) decoding algorithm for polar codes opens up a real possibility of high throughput and low decoding latency during hardware implementation. To address the problem that the BP decoding algorithm introduces high-complexity non-linear operations in the iterative messages update process, this paper proposes to simplify these operations and develops two novel low complexity BP decoding algorithms, namely, exponential BP (Exp-BP) decoding algorithm and quantization function BP (QF-BP) decoding algorithm. The proposed algorithms simplify the compound hyperbolic tangent function by using probability distribution fitting techniques. Specifically, the Exp-BP algorithm simplifies two types of non-linear operations into single non-linear operation using the piece-wise exponential model function, which can approximate the hyperbolic tangent function in the updating formula. The QF-BP algorithm eliminates non-linear operations using the non-uniform quantization in the updating formula, which is effective in reducing computational complexity. According to the simulation results, the proposed algorithms can reduce the computational complexity up to 50% in each iteration with a loss of less than 0.1 dB compared with the BP decoding algorithm, which can facilitate the hardware implementation.
In this paper, we propose a simplified belief propagation(SBP) decoder for the polar codes, where the update calculation for the messages of the so-called frozen nodes is replaced by setting these nodes' messages by prior probabilities during the BP procedure. Numerical results show that the derived SBP yields almost the same performance as the BP decoders in the literature, while the computational complexity is reduced significantly, which make our SBP more suitable for practical application as compared to the existing BP decoders.