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Reliability-Based Schedule for Decoding Low-Density Parity-Check Codes

This study, conducted by Ahmed Nouh and Amir H. Banihashemi from Carleton University, focuses on enhancing the performance of Low-Density Parity-Check (LDPC) codes using a reliability-based decoding schedule. The proposed framework adjusts the participation timing of nodes based on the reliability of their information, significantly improving decoding speed and performance, especially for short block lengths. The research includes simulation results demonstrating the advantages of the reliability-based schedule over traditional flooding methods.

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Reliability-Based Schedule for Decoding Low-Density Parity-Check Codes

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  1. Reliability-Based Schedule for Decoding Low-Density Parity-Check Codes by Ahmed Nouh and Amir H. Banihashemi Department of Systems and Computer Engineering Carleton University Ottawa, Ontario, Canada

  2. Outline • Introduction and Motivation • Reliability-Based Schedule (RBS): General Framework and Optimization • Simulation Results: Performance Speed of Convergence • Concluding Remarks

  3. Introduction and Motivation • Improving the performance of iterative decoding of LDPC codes at short block lengths is of practical importance • Reliability-based decoding, Fossorier, 2001 • Probabilistic scheduling, Mao and Banihashemi, 2001 • Graph-based schedules, Xiao and Banihashemi, 2002 • Generalized belief propagation, Yedidia, Freeman and Weiss, 2003 • Normalized and offset belief propagation, Yazdani, Hemati and Banihashemi, 2003 • Fixing the Tanner graph (TG) of the code and the decoding algorithm, is it possible to improve the performance? • Graph-based schedules • Reliability-based schedule

  4. Reliability-Based Schedule • Main idea: The participation timing of each node in iterative decoding is adjusted by the reliability of its information • Can even be applied to iterative algorithms with binary messages (cost: 2-bit representation of initial messages) • Improves performance and decoding speed significantly!

  5. General Framework • Iterative decoding of LDPC codes over BI-AWGN channel • Identifying unreliable bit nodes: • Reliability threshold vectors: (α(ℓ )), ℓ = 1, 2, 3, … • Reliability measure for the j th bit (can be the estimate of LLR): Rjℓ • If Rjℓ<αj(ℓ) , then bit j is ``unreliable,” otherwise it is ``reliable.” • At each iteration, only reliable bit nodes and reliable check nodes pass messages • At the beginning of each iteration, reliable messages coming from check nodes and the channel message are used to compute

  6. Optimization • Vectors (α(ℓ )), ℓ = 1, 2, 3, …can be optimized to achieve minimum error rate (for a given code, decoding algorithm and Eb/N0) • Optimization is very complex • For simplification: • α(ℓ) = 0, ℓ ≥ 2 (flooding after the 2nd iteration) • αj(1) = α (independent of j) • Rj1 = |rj| • Threshold value α is optimized by simulation

  7. Simulation Results • Algorithms: Belief Propagation (BP), Gallager’s Algorithm A (GA), Sipser-Spielman’s algorithm (SS) • Codes: optimized (1000,500) irregular (C1), (273,191) regular (C2), (273,191) projective geometry (C3) • Optimal α for each code and decoding algorithm is a function of Eb/N0 (α/σ however is independent of Eb/N0) • Optimal values of α/σ : 0.3, 0.7, 0.4.

  8. Simulation Results (Performance) • RBS provides significant improvement over flooding • RBS-GA and RBS-SS outperform GA and SS by about 1 dB and 0.6 dB at BER=10-5, respectively • RBS algorithms close a large part of the gap between bit-flipping (BF) and weighted BF (WBF) algorithms • At large SNR, RBS-GA is only about 0.2 dB inferior to WBF for BER • For WER, RBS-GA even outperforms WBF at high SNR • RBS-SS is inferior to WBF by only about 0.25 and 0.1 dB for BER and MER, respectively • RBS-BP performs very close to BP • In all cases, graph-based schedules perform the same as flooding

  9. Simulation Results (Speed of Convergence)

  10. Simulation Results (Speed of Convergence) • RBS on average converges faster than flooding, in some cases by more than a factor of 2. • Compared to WBF, RBS-BF algorithms converge up to more than 3 times faster on average • For RBS-BP only the standard deviation of number of iterations is reduced compared to BP

  11. Concluding Remarks • A reliability-based schedule for iterative decoding of LDPC codes was proposed. • A simplified version of the proposed schedule appears to be particularly effective for bit-flipping algorithms. • Significant improvements in performance and convergence speed are obtained at the very low cost of calculating 1-bit reliability information per coded bit for initial messages. • In many cases where graph-based schedules fail to provide any improvement over flooding, RBS can provide a significantly better performance/decoding-time tradeoff compared to flooding. • For soft-decision algorithms, the simplified version of RBS does not provide much improvement.

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