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arXiv:2107.11479 · IEEE Communications Letters

Error Floor Analysis of LDPC Column Layered Decoders

Ali Farsiabi, Amir H. Banihashemi

Published Dec 21, 2021• 3 citations
arXivHTMLPDF

Abstract

In this letter, we analyze the error floor of column layered decoders, also known as shuffled decoders, for low-density parity-check (LDPC) codes under saturating sum-product algorithm (SPA). To estimate the error floor, we evaluate the failure rate of different trapping sets (TSs) that contribute to the frame error rate in the error floor region. For each such TS, we model the dynamics of SPA in the vicinity of the TS by a linear state-space model that incorporates the information of the layered message-passing schedule. Consequently, the model parameters and the failure rate of the TS change as a result of the change in the order by which the messages of different layers are updated. This, in turn, causes the error floor of the code to change as a function of scheduling. Based on the proposed analysis, we then devise an efficient search algorithm to find a schedule that minimizes the error floor. Simulation results are presented to verify the accuracy of the proposed error floor estimation technique.

DOI
10.1109/lcomm.2021.3137203

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Error Floor Analysis of LDPC Row Layered Decoders

Ali Farsiabi, Amir H. Banihashemi

In this paper, we analyze the error floor of quasi-cyclic (QC) low-density parity-check (LDPC) codes decoded by the sum-product algorithm (SPA) with row layered message-passing scheduling. For this, we develop a linear state-space model of trapping sets (TSs) which incorporates the layered nature of scheduling. We demonstrate that the contribution of each TS to the error floor is not only a function of the topology of the TS, but also depends on the row layers in which different check nodes of the TS are located. This information, referred to as TS layer profile (TSLP), plays an important role in the harmfulness of a TS. As a result, the harmfulness of a TS in particular, and the error floor of the code in general, can significantly change by changing the order in which the information of different layers, corresponding to different row blocks of the parity-check matrix, is updated. We also study the problem of finding a layer ordering that minimizes the error floor, and obtain row layered decoders with error floor significantly lower than that of their flooding counterparts. As part of our analysis, we make connections between the parameters of the state-space model for a row layered schedule and those of the flooding schedule. Simulation results are presented to show the accuracy of analytical error floor estimates.

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Code-Independent Error Floor Estimation Techniques for Flooding and Layered Decoders of LDPC Codes

Ali Farsiabi

Finite-length low-density parity-check (LDPC) codes under iterative message passing algorithms suffer from error floor, i.e., as the channel quality improves, at some point, the error rate does not decrease as fast as its initial rate of decrease.The literature on the error floor analysis/estimation of LDPC codes can be, in general, partitioned into two main categories.The first category requires the full knowledge of the parity-check matrix or Tanner graph of the code to estimate the error floor.The second category, on the other hand, is code-independent, in that, rather than the full knowledge of the code's Tanner graph, these techniques only require the multiplicity and topology of harmful substructures of the Tanner graph, referred to as trapping sets (TSs), and possibly the degree distributions of the graph, to estimate the error floor.The proposed techniques in this thesis fit within the second category.The linear state-space model is a well-known code-independent method to 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groups of C 2 for the two schedules (1, 2, 3, 4, 5, 6) and (4, 1, 5, 2, 3, 6) (saturation level 15.75). .86 4.14 Simulation and estimation results of C 2 for the two schedules (1, 2, 3, 4, 5, 6) and (4, 1, 5, 2, 3, 6) (saturation level 15.75,I max = 30). .87 5.1 A labelled (5,3) LETS of the Tanner (155, 64) code in which the flow of the state variables are shown.The VNs, mis-satisfied CNs and unsatisfied CNs are shown by black circles, white squares and gray squares, respectively.The directed edges related to the layers number 1, 2 and 3 are represented by red, blue and purple colors, respectively.93 xiii 5.2 The VNs v 1 and v 4 related to Fig. 5.1 with the mis-satisfied CN c j connecting them.Figures (a) and (b) are related to virtual VNs, v k and v k , that are used during the update of VNs from layers L 1 and L 3 , respectively.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 5.3 Base graph of Example 16: The solid (red), dashed (green), dotted (blue) and dashed-dotted 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6.2 Simulation and estimation results for C 1 decoded by SPA over the AWGN channel (I max = 200). . . . . . . . . . . . . . . . . . . . . . .134 6.3 Simulation and extrapolated estimation results for C 1 decoded by SPA over the AWGN channel (I max = 200).The extrapolated curves are obtained based on the results at E b /N 0 = 4.5 dB . . . . . . . . . . . .136 6.4 Simulation and estimation results for C 2 decoded by SPA over the AWGN channel (I max = 200, C th = 25).The extrapolation in DSLM E is based on the results at E b /N 0 = 2.75 dB. . . . . . . . . . . . . . . .136 6.5 Simulation and estimation results for C 3 decoded by SPA over the AWGN channel (I max = 200).The extrapolation in DSLM E is based on the results at E b /N 0 = 4.5 dB. . . . . . . .

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Many classes of high-performance Low-density parity check codes are based on parity check matrices composed of permutation sub matrices. The emulation-simulation framework further allows the algorithm and implementation to be iteratively redefined to improve the error floor performance of message passing decoder. Log-Like hood-Ratio (LLR) based Belief-Propagation (BP) algorithm is presented for Low Density Parity Check codes. Numerically accurate representation of check node update computation used in LLR-BP decoding is described. The implementation of Sum-Product algorithm (SPA) within Low Density Parity Check Code (LDPC) decoder is described in this paper and the correction term is used to improve the decoding performance of min-sum algorithm (MSA). Quantization and log-tanh function approximation in sum-product decoder strongly affect which absorbing set dominates in error floor region. For LDPC decoder, bit error rate (BER) decreases with increase in the signal to noise ratio.

International Conference on Communication Systems and Network Technologies•Apr 1, 2015•1 citation
In this letter, we analyze the error floor of column layered decoders, also known as shuffled decoders, for low-density parity-check (LDPC) codes under saturating sum-product algorithm (SPA). To estimate the error floor, we evaluate the failure rate of different trapping sets (TSs) that contribute to the frame error rate in the error floor region. For each such TS, we model the dynamics of SPA in the vicinity of the TS by a linear state-space model that incorporates the information of the layered message-passing schedule. Consequently, the model parameters and the failure rate of the TS change as a result of the change in the order by which the messages of different layers are updated. This, in turn, causes the error floor of the code to change as a function of scheduling. Based on the proposed analysis, we then devise an efficient search algorithm to find a schedule that minimizes the error floor. Simulation results are presented to verify the accuracy of the proposed error floor estimation technique.
In this paper, we analyze the error floor of quasi-cyclic (QC) low-density parity-check (LDPC) codes decoded by the sum-product algorithm (SPA) with row layered message-passing scheduling. For this, we develop a linear state-space model of trapping sets (TSs) which incorporates the layered nature of scheduling. We demonstrate that the contribution of each TS to the error floor is not only a function of the topology of the TS, but also depends on the row layers in which different check nodes of the TS are located. This information, referred to as TS layer profile (TSLP), plays an important role in the harmfulness of a TS. As a result, the harmfulness of a TS in particular, and the error floor of the code in general, can significantly change by changing the order in which the information of different layers, corresponding to different row blocks of the parity-check matrix, is updated. We also study the problem of finding a layer ordering that minimizes the error floor, and obtain row layered decoders with error floor significantly lower than that of their flooding counterparts. As part of our analysis, we make connections between the parameters of the state-space model for a row layered schedule and those of the flooding schedule. Simulation results are presented to show the accuracy of analytical error floor estimates.
Finite-length low-density parity-check (LDPC) codes under iterative message passing algorithms suffer from error floor, i.e., as the channel quality improves, at some point, the error rate does not decrease as fast as its initial rate of decrease.The literature on the error floor analysis/estimation of LDPC codes can be, in general, partitioned into two main categories.The first category requires the full knowledge of the parity-check matrix or Tanner graph of the code to estimate the error floor.The second category, on the other hand, is code-independent, in that, rather than the full knowledge of the code's Tanner graph, these techniques only require the multiplicity and topology of harmful substructures of the Tanner graph, referred to as trapping sets (TSs), and possibly the degree distributions of the graph, to estimate the error floor.The proposed techniques in this thesis fit within the second category.The linear state-space model is a well-known code-independent method to estimate the contribution of a trapping set structure to the error floor of low-density paritycheck codes.In this thesis we first provide an in-depth analysis of this method by incorporating a more accurate model for the incoming messages to the TS structure that takes into account the randomness and the correlation among such messages.Based on this analysis, we demonstrate that both randomness and correlation result in the over-estimation of the failure probability of the TS.We then propose an alternate code-independent technique for the error floor estimation of iterative LDPC decoders that can accurately estimate the contribution of different TS structures in the error floor.Compared to the linear state-space model, the proposed method is not only more accurate, but also more general, in that, it is applicable to any saturating iterative message-passing decoder, symmetrically quantized or unquantized, over any memoryless binary-input output-symmetric channel.To the best of our knowledge, all the existing work on the theoretical analysis of error floor is limited to two-phase message passing algorithms, also known as flooding or parallel schedule.There are however a variety of message passing schedules which iii 4.5 The steps of constructing the digraph D l (S) of the (5, 3) LETS for a layered decoder from the flooding digraph D f (S).(The colors (edge types) red (dotted), blue (dashed) and purple (dash-dotted) represent L 1 , L 2 and L 3 , respectively.) . . . . . . . . . . . . . . . . . . . . . .69 4.6 QC-LDPC codes used for simulations.The entries of the matrices, that are not equal to -1, represent the right circular shift of the identity matrix to create the corresponding block of the parity-check matrix.The -1 entries represent zero blocks. . . . . . . . . . . . . . . . . . .80 4.7 Simulation and estimation results of C 1 for different saturation levels.The maximum number of iterations I max = 30. . . . . . . . . . . . .81 4.8 The (5, 5) LETS structure of C 1 in which the row layers for different CNs are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81 4.9 The approximate estimate of the failure probability of the (5, 5) LETS of C 1 for various row layered schedules at E b /N 0 = 6 dB and saturation level 31.75.The schedules are sorted based on r. . . . . . . . . . . .83 4.10 The simulation and estimation results of C 1 for different row schedules.(The saturation level is 31.75, and the maximum number of iterations for layered schedules and the flooding schedule are set to 30 and 60, respectively.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83 4.11 The effect of various row layered schedules on the failure probability of different classes of LETSs in C 2 (E b /N 0 = 6 dB, saturation level 15.75).85 4.12 The effect of various row layered schedules on the total FER of C 2 (E b /N 0 = 6 dB, saturation level 15.75). . . . . . . . . . . . . . . . . .85 4.13 The error estimation of the ten (7, 1) LETS groups of C 2 for the two schedules (1, 2, 3, 4, 5, 6) and (4, 1, 5, 2, 3, 6) (saturation level 15.75). .86 4.14 Simulation and estimation results of C 2 for the two schedules (1, 2, 3, 4, 5, 6) and (4, 1, 5, 2, 3, 6) (saturation level 15.75,I max = 30). .87 5.1 A labelled (5,3) LETS of the Tanner (155, 64) code in which the flow of the state variables are shown.The VNs, mis-satisfied CNs and unsatisfied CNs are shown by black circles, white squares and gray squares, respectively.The directed edges related to the layers number 1, 2 and 3 are represented by red, blue and purple colors, respectively.93 xiii 5.2 The VNs v 1 and v 4 related to Fig. 5.1 with the mis-satisfied CN c j connecting them.Figures (a) and (b) are related to virtual VNs, v k and v k , that are used during the update of VNs from layers L 1 and L 3 , respectively.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .99 5.3 Base graph of Example 16: The solid (red), dashed (green), dotted (blue) and dashed-dotted (purple) arrows, respectively, illustrate the updating order which the column layered decoder perform, accordingly.109 5.4 The error estimation related to (5, 5) LETS of C 1 for various column layered schedules at E b /N 0 = 6 dB.All the schedules result in the same dominant eigenvalue of the layered transition matrix, r = 16.9536.The saturation level is 15.75. . . . . . . . . . . . . . . . . . . . . . . . . .111 5.5 The simulation and estimation results related to code C 1 showing the effect of different schedules.The saturation level is 15.75 and the maximum number of iterations is 30. . . . . . . . . . . . . . . . . . .111 5.6 The simulation and estimation results related to code C 2 showing the effect of different schedules.The saturation level is 15.75.The maximum number of iterations is 30. . . . . . . . . . . . . . . . . . . . .113 6.1 The histogram of the conditional error indicator function of a (5, 5) ETS of (640, 192) QC-LDPC code [3] at different iterations. . . . . .128 6.2 Simulation and estimation results for C 1 decoded by SPA over the AWGN channel (I max = 200). . . . . . . . . . . . . . . . . . . . . . .134 6.3 Simulation and extrapolated estimation results for C 1 decoded by SPA over the AWGN channel (I max = 200).The extrapolated curves are obtained based on the results at E b /N 0 = 4.5 dB . . . . . . . . . . . .136 6.4 Simulation and estimation results for C 2 decoded by SPA over the AWGN channel (I max = 200, C th = 25).The extrapolation in DSLM E is based on the results at E b /N 0 = 2.75 dB. . . . . . . . . . . . . . . .136 6.5 Simulation and estimation results for C 3 decoded by SPA over the AWGN channel (I max = 200).The extrapolation in DSLM E is based on the results at E b /N 0 = 4.5 dB. . . . . . . .
The linear state-space model is a well-known code-independent method to estimate the contribution of a trapping set (TS) structure to the error floor of low-density parity-check (LDPC) codes. In this paper, we provide an in-depth analysis of this method by incorporating a more accurate model for the incoming messages to the TS structure that takes into account the randomness and the correlation among such messages. Based on this analysis, we demonstrate that both randomness and correlation result in the over-estimation of the failure probability of the TS. We then propose an alternate code-independent technique for the error floor estimation of iterative LDPC decoders that can accurately estimate the contribution of different TS structures in the error floor. Compared to the linear state-space model, the proposed method is not only more accurate, but also more general, in that, it is applicable to any saturating iterative message-passing decoder, symmetrically quantized or unquantized, over any memoryless binary-input output-symmetric channel. The proposed technique can be viewed as the local application of importance sampling (IS) to the message-passing algorithm over the subgraph induced by the TS in the code's Tanner graph. In the message-passing process, to account for the effect of the rest of the Tanner graph, density evolution along with a simple correlation model is used to generate the messages coming into the TS from the rest of the Tanner graph. Extensive simulations demonstrate that the proposed technique can accurately estimate the error floor of LDPC codes over both additive white Gaussian noise (AWGN) channel and binary symmetric channel (BSC), for a variety of iterative decoding algorithms and quantization schemes.
This paper addresses the prediction of error floors of low-density parity-check codes transmitted over the additive white Gaussian noise channel. Using a linear state-space model to estimate the behavior of the sum-product algorithm (SPA) decoder in the vicinity of trapping sets (TSs), we study the performance of the SPA decoder in the log-likelihood ratio (LLR) domain as a function of the LLR saturation level. When applied to several widely studied codes, the model accurately predicts a significant decrease in the error floor as the saturation level is allowed to increase. For nonsaturating decoders, however, we find that the state-space model breaks down after a small number of iterations due to the strong correlation of LLR messages. We then revisit Richardson's importance-sampling methodology for estimating error floors due to TSs when those floors are too low for Monte Carlo simulation. We propose modifications that account for the behavior of a nonsaturating decoder and present the resulting error floor estimates for the Margulis code. These estimates are much lower, significantly steeper, and more sensitive to iteration count than those previously reported.
This paper addresses the prediction of error floors of variable-regular Low Density Parity Check (LDPC) codes in the Additive White Gaussian Noise (AWGN) channel. Specifically, we focus on the Sum-Product Algorithm (SPA) decoder in the log-domain at high SNRs. We hypothesize that several published error floor levels are due to numerical saturation within their decoders when handling high SNRs. We take care to develop a log-domain SPA decoder that does not saturate near-certain messages and find the error rates of our decoder to be lower by at least several orders of magnitude. We study the behavior of near-codewords / trapping sets that dominate the reported error floors. J. Sun, in his Ph.D. thesis, used a linear system model to show that error floors due to elementary trapping sets don't exist under certain conditions, assuming that the SPA decoder is non-saturating [1]. We develop a refined linear model which we find to be capable of predicting the error floors caused by elementary trapping sets for saturating decoders. Performance results of several codes at several levels of decoder saturation are presented.
Many classes of high-performance Low-density parity check codes are based on parity check matrices composed of permutation sub matrices. The emulation-simulation framework further allows the algorithm and implementation to be iteratively redefined to improve the error floor performance of message passing decoder. Log-Like hood-Ratio (LLR) based Belief-Propagation (BP) algorithm is presented for Low Density Parity Check codes. Numerically accurate representation of check node update computation used in LLR-BP decoding is described. The implementation of Sum-Product algorithm (SPA) within Low Density Parity Check Code (LDPC) decoder is described in this paper and the correction term is used to improve the decoding performance of min-sum algorithm (MSA). Quantization and log-tanh function approximation in sum-product decoder strongly affect which absorbing set dominates in error floor region. For LDPC decoder, bit error rate (BER) decreases with increase in the signal to noise ratio.