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

2. Team LDPC, SoC Lab. CS Dept. National Taiwan University 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

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

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

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

6. Team LDPC, SoC Lab. CS Dept. National Taiwan University 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})

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

8. Team LDPC, SoC Lab. CS Dept. National Taiwan University Trellis & Tanner graph x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 Every path is valid configuration

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

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

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

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

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

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

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

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

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

18. Team LDPC, SoC Lab. CS Dept. National Taiwan University Min-Sum Algorithm Final cost function (μ s, μ E )

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

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

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

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

23. Team LDPC, SoC Lab. CS Dept. National Taiwan University Min-Sum Algorithm (3 of 8) 00111001

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

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

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

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

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

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

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

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

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

33. Team LDPC, SoC Lab. CS Dept. National Taiwan University Sum-product Algorithm Final cost function (μ s, μ E )

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

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