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Codes and Decoding on General Graphs

Codes and Decoding on General Graphs. Speaker: Yi-hsin Jian. Outline. Introduction Equation and Tanner Graph Configuration and Behavior System and Check Structure Tanner and Trellis Graph Complexity Turbo Code Decoding Algorithms Min-sum Algorithm Sum-product Algorithm Conclusion.

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Codes and Decoding on General Graphs

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  1. Codes and Decoding on General Graphs Speaker: Yi-hsin Jian Team LDPC, SoC Lab. CS Dept. National Taiwan University

  2. Outline • Introduction • Equation and Tanner Graph • Configuration and Behavior • System and Check Structure • Tanner and Trellis Graph • Complexity • Turbo Code • Decoding Algorithms • Min-sum Algorithm • Sum-product Algorithm • Conclusion Team LDPC, SoC Lab. CS Dept. National Taiwan University

  3. Introduction • Algebraic Decoding • Probabilistic Decoding • Decoding Complexity Team LDPC, SoC Lab. CS Dept. National Taiwan University

  4. Equations & Tanner Graph Team LDPC, SoC Lab. CS Dept. National Taiwan University

  5. Configuration and Behavior • Configuration space • Valid configurations and Behavior • (Visible) Sites • Component of codeword in Tanner graph Team LDPC, SoC Lab. CS Dept. National Taiwan University

  6. System • System (N, W, B) • Set of sites (e.g. N={1,…,6}) • Configuration space (e.g. W=GF(2)6) • Valid configurations/Behaviors (e.g. B={000000,110001,…,101010}) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  7. Check Structure • Check Structure (Q) for (N,W,B) =Q is a collection of subset of N such that x belong to W satisfying xE belong to BE for all check sets. (e.g. Q={{1,2,3},{3,4,5},{5,6,1},{2,4,6}}) • BE Called local behavior (e.g. BE={000,110,101,011}) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  8. Trellis & Tanner graph Every path is valid configuration x1 x2 x3 x4 x5 x6 Team LDPC, SoC Lab. CS Dept. National Taiwan University

  9. Complexity • Local behavior • Site alphabets Team LDPC, SoC Lab. CS Dept. National Taiwan University

  10. Trivial realization Even parity check Team LDPC, SoC Lab. CS Dept. National Taiwan University

  11. Hidden site • Not a component of codewords • Indicate some state Team LDPC, SoC Lab. CS Dept. National Taiwan University

  12. Distinct value for each valid configuration, are unsuitable for decoding Example Team LDPC, SoC Lab. CS Dept. National Taiwan University

  13. Key module Turbo Codes Team LDPC, SoC Lab. CS Dept. National Taiwan University

  14. Share information sequence Produce many cycles Turbo Codes (Tanner) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  15. Decoding algorithms Team LDPC, SoC Lab. CS Dept. National Taiwan University

  16. Min-Sum Algorithm &Sum-Product Algorithm • Min-Sum algorithm is generalization of “Viterbi algorithm” • Sum-Product algorithm is generalization of “Forward and backward algorithm” Team LDPC, SoC Lab. CS Dept. National Taiwan University

  17. Initialized to zero for the first iteration In typical channel decoding, this value are set to zero Min-Sum Algorithm • Local cost function (γs, γE) • Intermediate cost function (μs,E, μE,s) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  18. Min-Sum Algorithm • Final cost function (μs, μE) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  19. Min-Sum Algorithm • Global cost function ( G(x) ) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  20. Case (weight 3) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  21. Min-Sum Algorithm (1 of 8) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  22. Min-Sum Algorithm (2 of 8) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  23. 00 11 10 01 Min-Sum Algorithm (3 of 8) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  24. Min-Sum Algorithm (4 of 8) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  25. Min-Sum Algorithm (5 of 8) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  26. Min-Sum Algorithm (6 of 8) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  27. Min-Sum Algorithm (7 of 8) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  28. Min-Sum Algorithm (8 of 8) Decision was made according to this cost info. Team LDPC, SoC Lab. CS Dept. National Taiwan University

  29. Binary Optimization • Only interested in the difference between “1” cost and the “0” cost. Team LDPC, SoC Lab. CS Dept. National Taiwan University

  30. Sum-product Algorithm • Global cost function( G(x) ) Maximizing this function rather than minimizing Team LDPC, SoC Lab. CS Dept. National Taiwan University

  31. Sum-product Algorithm • Local check cost (γE) • Local site cost (γs) • Global cost of a configuration x is strictly positive if x is valid. Team LDPC, SoC Lab. CS Dept. National Taiwan University

  32. Sum-product Algorithm • Intermediate cost function (μs,E, μE,s) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  33. Sum-product Algorithm • Final cost function (μs, μE) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  34. Case (weight 3) Team LDPC, SoC Lab. CS Dept. National Taiwan University

  35. Conclusion • These algorithms are optimal with cycle-free graph • Principle of decoding algorithms • Graph description • Find the right paper which bridges the gap Team LDPC, SoC Lab. CS Dept. National Taiwan University

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