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Interlaced Column-Row Message-Passing Schedule for Decoding LDPC Codes

Saleh Usman, Mohammad M. Mansour, Ali Chehab

Published Dec 1, 2016• 5 citations
Publisher

Abstract

This paper investigates efficient decoding algorithms for LDPC codes. Alternating column-row message-passing (ACRMP) and Interlaced column-row message-passing (I-CRMP) schedules for decoding of LDPC codes are proposed and investigated in this work. Existing serial scheduling schemes for LDPC decoding are based either on column message-passing (MP) or row MP, and roughly converge twice as fast as Gallager's flooding-based MP schedule at high signal-to-noise ratio (SNR). To further accelerate the convergence speed of serial decoders, hybrid column-row MP schedules that perform multiple message passes between check and variable nodes within or between iterations are proposed. Our proposed I-CRMP schedule converges in less than half the number of iterations compared to the best existing serial decoding schedules. Compared to column MP, the added complexity of this scheme is proportional only to check-node-degree times more additions at variable nodes. This increase in complexity is moderate compared to the convergence acceleration factor that the scheme achieves. Superior performance of the proposed I-CRMP scheme is confirmed by decoding randomly generated as well as IEEE 802.11n/ac LDPC codes.

DOI
10.1109/glocom.2016.7841913

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This paper investigates efficient decoding algorithms for LDPC codes. Alternating column-row message-passing (ACRMP) and Interlaced column-row message-passing (I-CRMP) schedules for decoding of LDPC codes are proposed and investigated in this work. Existing serial scheduling schemes for LDPC decoding are based either on column message-passing (MP) or row MP, and roughly converge twice as fast as Gallager's flooding-based MP schedule at high signal-to-noise ratio (SNR). To further accelerate the convergence speed of serial decoders, hybrid column-row MP schedules that perform multiple message passes between check and variable nodes within or between iterations are proposed. Our proposed I-CRMP schedule converges in less than half the number of iterations compared to the best existing serial decoding schedules. Compared to column MP, the added complexity of this scheme is proportional only to check-node-degree times more additions at variable nodes. This increase in complexity is moderate compared to the convergence acceleration factor that the scheme achieves. Superior performance of the proposed I-CRMP scheme is confirmed by decoding randomly generated as well as IEEE 802.11n/ac LDPC codes.
Conventionally, in each low-density parity-check (LDPC) decoding iteration all the variable nodes and subsequently all the check nodes send messages to their neighbors (flooding schedule). An alternative, more efficient, approach is to update the nodes' messages serially (serial schedule). A theoretical analysis of serial message passing decoding schedules is presented. In particular, the evolution of the computation tree under serial scheduling is analyzed. It shows that the tree grows twice as fast in comparison to the flooding schedule's one, indicating that the serial schedule propagates information twice as fast in the code's underlying graph. Furthermore, an asymptotic analysis of the serial schedule's convergence rate is done using the density evolution (DE) algorithm. Applied to various ensembles of LDPC codes, it shows that for long codes the serial schedule is expected to converge in half the number of iterations compared to the standard flooding schedule, when working near the ensemble's threshold. This observation is generally proved for the binary erasure channel (BEC) under some natural assumptions. Finally, an accompanying concentration theorem is proved.
An efficient decoding schedule for low-density parity-check (LDPC) codes that outperforms the conventional approach, in terms of both complexity and performance, is presented. Conventionally, in each iteration, all symbol nodes and, subsequently, all the check nodes, send messages to their neighbors ("flooding schedule"). In contrast, in the proposed method, the updating of nodes is performed according to a serial schedule which propagates the information twice as fast. A density evolution (DE) algorithm for asymptotic analysis of the new schedule is derived, showing that, when working near the code's capacity, the decoder converges in approximately half the number of iterations. In addition, a concentration theorem is proved, showing that, for a randomly chosen serial schedule, code graph, and decoder input, the decoder's performance approaches its expected one as predicted by the DE algorithm, when the code length increases.
The major drawback of the LDPC codes versus the turbo-codes is their comparative low convergence speed: 25-30 iterations vs. 8-10 iterations for turbo-codes. Recently, Hocevar showed by simulations that the convergence rate of the LDPC decoder can be accelerated by exploiting a `turbo-scheduling' applied on the bit-node messages (rows of the parity check matrix). In this paper, we show analytically that the convergence rate for this type of scheduling is about two times increased for most of the regular LDPC codes. Second we prove that `turbo-scheduling' applied on the rows of the parity check matrix is identical belief propagation algorithm as standard message passing algorithm. Furthermore, we propose two new message passing schedules: 1) a turbo-scheduling is applied on the check-node messages (columns of the parity check matrix); and 2) a hybrid version of both previous schedules where the turbo-effect is applied on both check-nodes and bit-nodes. Frame error rate simulations validate the effectiveness of the proposed schedules
Serial decoding schedules for low-density parity-check (LDPC) codes are described and analyzed. Conventionally, in each iteration all the variable nodes and subsequently all the check nodes send messages to their neighbors (“flooding schedule”). In contrast, in the considered methods, the updating of the nodes is implemented according to a serial schedule. The evolution of the decoding algorithm’s computation tree under serial scheduling is analyzed. The analysis shows that it grows twice as fast in comparison to the flooding schedule’s computation tree, indicating that the serial schedule propagates information twice as fast in the code’s underlying graph. Furthermore, asymptotic analysis of the serial schedule’s convergence rate is done using the Density Evolution (DE) algorithm. Applied to various ensembles of LDPC codes, it shows that when working near the ensemble’s threshold, for long enough codes the serial schedule is expected to converge in half the number of iterations compared to the standard flooding schedule. This observation is generally proved for the Binary Erasure Channel (BEC) under some likely assumptions. Finally, an accompanying concentration theorem is proved, justifying the asymptotic DE analysis assumptions. 1
In this letter, a novel scheduling scheme for decoding irregular low-density parity-check (LDPC) code, based on the column weight of variable nodes in the code graph, is introduced. In this scheme, the irregular LDPC code is decoded using the shuffled belief-propagation (BP) algorithm by selecting the variable nodes in descending order of their column weight. Via numerical simulation, it is shown that the proposed high-to-low column-weight based decoding schedule can noticeably increase the convergence speed at medium to high signal-to-noise ratio (SNR) over AWGN and Rayleigh fading channels without introducing additional complexity or error rate degradation. Furthermore, it is found that the improvement in decoding convergence is proportional to the maximum column-weight in the code graph.