1. Wireless Mobile Communication and Transmission Lab. Theory and Technology of Error Control Coding Chapter 7 Low Density Parity Check Codes

2. Wireless Mobile Communication and Transmission Lab. 2/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

3. Wireless Mobile Communication and Transmission Lab. 3/42 Introduction of LDPC codes 1960 1970 1980199020002004 Gallager Zyablov PinskerTanner MacKay Neal Wiberg Davey MacKay Yu Kou Shu Lin Fossorier SY Chung Urbanke Richardson Burshtein Miller McEliece Luby Mitzenmacher Spielman...... Some important research of LDPC codes since 1962

4. Wireless Mobile Communication and Transmission Lab. 4/42 Introduction of LDPC codes  Regular LDPC code(6,4)  parity check matrix H  Two classes of nodes in a Tanner  graph (variable nodes and check nodes)  Check node j is connected to variable  node i whenever element in H is 1  Bold line constructs a cycle  of length 6 in a Tanner Graph

5. Wireless Mobile Communication and Transmission Lab. 5/42 Introduction of LDPC codes

6. Wireless Mobile Communication and Transmission Lab. 6/42 Introduction of LDPC codes  rate=1/4, AWGN Channel, Thesis of M. C. Davey

7. Wireless Mobile Communication and Transmission Lab. 7/42 Introduction of LDPC codes  Local girth distribution histogram of variable nodes  Block length approaching infinity, the assumption of cycle freeness is asymptotically fulfilled  The relationship of girth, minimum distance and performance

8. Wireless Mobile Communication and Transmission Lab. 8/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

9. Wireless Mobile Communication and Transmission Lab. 9/42 Encoding of LDPC codes  H=[P|I]  G ＝ [I|P’]  C=M*G

10. Wireless Mobile Communication and Transmission Lab. 10/42 Encoding of LDPC codes

11. Wireless Mobile Communication and Transmission Lab. 11/42 Encoding of LDPC codes

12. Wireless Mobile Communication and Transmission Lab. 12/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

13. Wireless Mobile Communication and Transmission Lab. 13/42 Construction of parity check matrix  Random construction methods  Structured construction methods

14. Wireless Mobile Communication and Transmission Lab. 14/42 Construction of parity check matrix  Gallager method 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0

15. Wireless Mobile Communication and Transmission Lab. 15/42 Construction of parity check matrix  Mackay methods

16. Wireless Mobile Communication and Transmission Lab. 16/42 Construction of parity check matrix  Bit-filling

17. Wireless Mobile Communication and Transmission Lab. 17/42 Construction of parity check matrix  Extended Bit-filling

18. Wireless Mobile Communication and Transmission Lab. 18/42 Construction of parity check matrix  Hesuristic girth distribution

19. Wireless Mobile Communication and Transmission Lab. 19/42 Construction of parity check matrix  Progressive edge growth (PEG)

20. Wireless Mobile Communication and Transmission Lab. 20/42 Construction of parity check matrix  Random construction methods  Structured construction methods

21. Wireless Mobile Communication and Transmission Lab. 21/42 Construction of parity check matrix  FG-LDPC:EG-LDPC and PG-LDPC  n points and J lines : n*J incidense matrix H  Each line is composed of p points  There is one and only one line between two points  Each point lies on q lines  Any pare of lines has only one common point or no common point

22. Wireless Mobile Communication and Transmission Lab. 22/42 Construction of parity check matrix  Partial geometry LDPC Steiner 2-design; Net or transversal design (TD); Generalized quadrangle (GQ); Proper PG

23. Wireless Mobile Communication and Transmission Lab. 23/42 Construction of parity check matrix  BIBD-LDPC

24. Wireless Mobile Communication and Transmission Lab. 24/42 Construction of parity check matrix  Block-LDPC

25. Wireless Mobile Communication and Transmission Lab. 25/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

26. Wireless Mobile Communication and Transmission Lab. 26/42 Decoding of LDPC codes  Bit flipping method  Belief propagation and related methods  Weighted bit flipping methods

27. Wireless Mobile Communication and Transmission Lab. 27/42 Decoding of LDPC codes  Bit flipping method =0 =1 Connected to two unsatisfied check nodes: flipped

28. Wireless Mobile Communication and Transmission Lab. 28/42 Decoding of LDPC codes  Bit flipping method  Belief propagation and related methods  Weighted bit flipping methods

29. Wireless Mobile Communication and Transmission Lab. 29/42 Decoding of LDPC codes  Belief propagation method All the effective decoding strategies for LDPC codes are message passing algorithms The best algorithm known is the Belief Propagation algorithm (1) Complicated calculations are distributed among simple node processors (2) After several iterations, the solution of the global problem is available (3) BP algorithm is the optimal if there are no cycles or ignore cycles

30. Wireless Mobile Communication and Transmission Lab. 30/42 Decoding of LDPC codes  Belief propagation method (log domain) Probability information transmitting among connected codes through the edge Two types of message: The probability that one bit is 1 or 0, obtained via the connected checks nodes other than the check node that received the probability. The conditional probability of that one check node is satisfied if one connected bit is 1 or 0

31. Wireless Mobile Communication and Transmission Lab. 31/42 Decoding of LDPC codes  Belief propagation method: message passing in two steps

32. Wireless Mobile Communication and Transmission Lab. 32/42 Decoding of LDPC codes  UMP-BP based (min sum)

33. Wireless Mobile Communication and Transmission Lab. 33/42 Decoding of LDPC codes  Normalized UMP-BP based Reduce the complexity of horizontal step: The function value is greatly decided by the variable with minimum absolute value, L2 is greater than L1, Normalized factor is used to compensate the performance loss

34. Wireless Mobile Communication and Transmission Lab. 34/42 Decoding of LDPC codes  Bit flipping method  Belief propagation and related methods  Weighted bit flipping methods

35. Wireless Mobile Communication and Transmission Lab. 35/42 BPSK Modulation: The smaller the absolute value, the fewer the reliability Output of the check node Flipping the variable node n with largest weight Decoding of LDPC codes  Weighted bit flipping methods

36. Wireless Mobile Communication and Transmission Lab. 36/42 Decoding of LDPC codes Some improvements of WBF algorithm Consider the reliability of the bit (MWBF): Modified check node output (IMWBF):  Weighted bit flipping methods

37. Wireless Mobile Communication and Transmission Lab. 37/42 Decoding of LDPC codes Some improvements of WBF algorithm Consider both of the maximum and minimum symbols (LP): Add a check weight factor (MLP): Consider the ratio (RRWBF):  Weighted bit flipping methods

38. Wireless Mobile Communication and Transmission Lab. 38/42 Decoding of LDPC codes Developed from IMWBF which is a counterpart to Normalized BP Based algorithm Consider all the symbol in each check with the constraint of extrinsic information: Linear combination  Weighted bit flipping methods

39. Wireless Mobile Communication and Transmission Lab. 39/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

40. Wireless Mobile Communication and Transmission Lab. 40/42 Density Evolution  Messages passed in the factor graph are random variables. The calculations performed under the SPA are functions of random variables.  Messages passed through the graph are conditionally independent  Symmetry Condition

41. Wireless Mobile Communication and Transmission Lab. 41/42 EXIT VNDCND AWGN channel output Iterative Decoding of LDPC Decision

42. Wireless Mobile Communication and Transmission Lab. 42/42 EXIT