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Theory and Technology of Error Control Coding

Chapter 7 Low Density Parity Check Codes. Theory and Technology of Error Control Coding. Outline. Introduction of LDPC codes Encoding of LDPC codes Construction of parity check matrix Decoding of LDPC codes Density evolution and EXIT. Y. u. K. o. u. S. h. u. L. i. n. F. o. s.

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Theory and Technology of Error Control Coding

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  1. Chapter 7 Low Density Parity Check Codes Theory and Technology ofError Control Coding

  2. Outline • Introduction of LDPC codes • Encoding of LDPC codes • Construction of parity check matrix • Decoding of LDPC codes • Density evolution and EXIT

  3. Y u K o u S h u L i n F o s s o r i e r S Y C h u n g S o m e i m p o r t a n t r e s e a r c h U r b a n k e R i c h a r d s o n B u r s h t e i n o f L D P C c o d e s s i n c e 1 9 6 2 M i l l e r W i b e r g M c E l i e c e L u b y D a v e y M i t z e n m a c h e r M a c K a y S p i e l m a n M a c K a y Z y a b l o v . . . . . . N e a l G a l l a g e r P i n s k e r T a n n e r 1 9 8 0 1 9 9 0 2 0 0 0 2 0 0 4 1 9 7 0 1 9 6 0 Introduction of LDPC codes

  4. 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. Introduction of LDPC codes

  6. Introduction of LDPC codes • rate=1/4, AWGN Channel, Thesis of M. C. Davey

  7. 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. Outline • Introduction of LDPC codes • Encoding of LDPC codes • Construction of parity check matrix • Decoding of LDPC codes • Density evolution and EXIT

  9. Encoding of LDPC codes • H=[P|I] • G＝[I|P’] • C=M*G

  10. Encoding of LDPC codes

  11. Encoding of LDPC codes

  12. Outline • Introduction of LDPC codes • Encoding of LDPC codes • Construction of parity check matrix • Decoding of LDPC codes • Density evolution and EXIT

  13. Construction of parity check matrix • Random construction methods • Structured construction methods

  14. 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 Construction of parity check matrix • Gallager method

  15. Construction of parity check matrix • Mackay methods

  16. Construction of parity check matrix • Bit-filling

  17. Construction of parity check matrix • Extended Bit-filling

  18. Construction of parity check matrix • Hesuristic girth distribution

  19. Construction of parity check matrix • Progressive edge growth (PEG)

  20. Construction of parity check matrix • Random construction methods • Structured construction methods

  21. 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. Construction of parity check matrix • Partial geometry LDPC Steiner 2-design; Net or transversal design (TD); Generalized quadrangle (GQ); Proper PG

  23. Construction of parity check matrix • BIBD-LDPC

  24. Construction of parity check matrix • Block-LDPC

  25. Outline • Introduction of LDPC codes • Encoding of LDPC codes • Construction of parity check matrix • Decoding of LDPC codes • Density evolution and EXIT

  26. Decoding of LDPC codes • Bit flipping method • Belief propagation and related methods • Weighted bit flipping methods

  27. Decoding of LDPC codes • Bit flipping method =0 =1 Connected to two unsatisfied check nodes: flipped

  28. Decoding of LDPC codes • Bit flipping method • Belief propagation and related methods • Weighted bit flipping methods

  29. 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. 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. Decoding of LDPC codes • Belief propagation method: message passing in two steps

  32. Decoding of LDPC codes • UMP-BP based (min sum)

  33. 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. Decoding of LDPC codes • Bit flipping method • Belief propagation and related methods • Weighted bit flipping methods

  35. Decoding of LDPC codes • Weighted bit flipping methods • BPSK Modulation: The smaller the absolute value, the fewer the reliability • Output of the check node • Flipping the variable node n with largest weight

  36. Decoding of LDPC codes • Weighted bit flipping methods • Some improvements of WBF algorithm • Consider the reliability of the bit (MWBF): • Modified check node output (IMWBF):

  37. Decoding of LDPC codes • Weighted bit flipping methods • Some improvements of WBF algorithm • Consider both of the maximum and minimum symbols (LP): • Add a check weight factor (MLP): • Consider the ratio (RRWBF):

  38. Decoding of LDPC codes • Weighted bit flipping methods • 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

  39. Outline • Introduction of LDPC codes • Encoding of LDPC codes • Construction of parity check matrix • Decoding of LDPC codes • Density evolution and EXIT

  40. 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. Decision AWGN channel output VND CND Iterative Decoding of LDPC EXIT

  42. EXIT

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