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

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. Outline. Introduction and Motivation

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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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