Shuffled Belief Propagation Decoding Juntan Zhang and Marc - PowerPoint PPT Presentation

Shuffled Belief Propagation Decoding Juntan Zhang and Marc Fossorier Department of Electrical Engineering University of Hawaii at Manoa Honolulu, HI 96816

Outline � Review of LDPC Codes � Standard Belief Propagation Algorithm � Shuffled Belief Propagation Algorithm � Optimality and Convergence � Parallel Shuffled Belief Propagation � A Small Example � Simulation Results � Conclusion

Low-Density Parity Check (LDPC) Codes First proposed by R. G. Gallager in 1960 ’ s, and resurrected � recently [Gallager-IRE62, MacKay-IT99] . Can achieve near Shannon limit performance with belief � propagation (BP) or sum-product algorithm [Richardson- Urbanke-IT01] . Advantages over turbo codes: � better distance properties; parallel decoding structure for high speed decoders. Disadvantages: � encoding complexity is high; decoder complexity is high for full parallel implementation.

Representations of LDPC Codes bipartite graph parity check matrix   1 0 1 0 0 1 0  L L L L    0 0 0 0 1 0 0 O       1 0 0 0 0 0 1 O    = H 0 0 0 0 0 0 0 M  O      M M M O M M M M     M M M L L L L M M M M      1 0 0 0 0 0 1 L L L L    Check nodes Bit(variable) N nodes

Regular and Irregular LDPC Codes An LDPC code is regular if H has constant row weight and column � weight, or equivalently, the check nodes have constant degree d c and variable nodes have constant degree d v ; An LDPC code is irregular if row and column weights are not � constants. Irregular LDPC codes are defined by degree distributions � Long irregular LDPC codes have better performance than regular � LDPC codes, and can beat turbo codes [Richardson-Urbanke-IT01] .

Geometric LDPC Codes Originally studied for majority logic decoding decades ago, and � constructed based on finite geometries (Euclidean and projective geometries) [Weldon-Bell66, Rudolph-IT67]; BP algorithm can be applied to the decoding of this family of � codes [Lucas-Fossorier-Kou-Lin-COM00, Kou-Lin-Fossorier-IT01]; Encoding can be easily implemented with shift registers since � they are cyclic codes; They have very good minimum distance properties; � Decoding complexity is high. �

An example: (7, 3) DSC code:   1 0 1 1 0 0 0   0 1 0 1 1 0 0     0 0 1 0 1 1 0   = H 0 0 0 1 0 1 1     1 0 0 0 1 0 1   1 1 0 0 0 1 0     0 1 1 0 0 0 1   � Equal number of bit nodes � Parity matrix is Squared ! and check nodes. � Not full rank. � Node degrees are larger.

Some one-step majority decodable codes: PG-LDPC codes (DSC) EG-LDPC codes rate ( N , K ) d min rate ( N , K ) d min (7, 3) 0.429 4 (15, 7) 0.467 5 (21, 11) 0.524 6 (63, 37) 0.587 9 (73, 45) 0.616 10 (255, 175) 0.686 17 (273, 191) 0.700 18 (1023, 781) 0.763 33 (1057, 813) 0.769 34 (4095, 3367) 0.822 65 (4161, 3431) 0.825 66

Processing in check nodes: Principles: incoming messages + constraints ⇒ outgoing messages z z mn mn 1 1 L z mn mn 1 2 L L Check Node mn mn 2 2 m L mn   3 z z ( )  ∏   L − = 1 L 2 tanh tanh z 2 mn mn 3 3 mn  ′ 4 mn m n   ′ ∈ n N ( m ) \ n z z mn mn Bit Nodes 4 4 N(m)

Processing in bit nodes: ∑ L = + z F L m n 1 ′ mn n m n ′ ∈ L m M ( n ) \ m z m n 2 m n z 1 Bit Node m n 2 n z L ∑ m n = + 3 m n z F L , 3 z for hard decision n n mn m n 4 ∈ m M ( n ) F n L m n 4 Check Nodes M(n)

Standard IDBP Initialization � Step1: Update the Belief Matrix � Horizontal Step: Update the whole Check-to-Bit Matrix � Vertical Step: Update the whole Bit-to-Check Matrix � Step2: Hard decision and stopping test � Step3: Output the decoded codeword �

Standard BP Algorithm; Step 1: ∏ ( i ) Horizontal Step : − + ( i 1 ) 1 tanh( z / 2 ) mn ' ∈ n ' N ( m ) \ n ε = ( i ) log ∏ mn − − ( i 1 ) 1 tanh( z / 2 ) mn ' ∈ n ' N ( m ) \ n ( ii )Vertical Step: ∑ = + ε ( i ) ( i ) z F mn n m ' n ∈ m ' M ( n ) \ m ∑ = + ε ( i ) ( i ) z F n n mn ∈ m M ( n )

Update the belief matrix in i-th iteration with standard belief propagation decoding ε ε − ε ε − ε ε − − − − ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( i 1 ) ( ( i i ) 1 ) z z ( i ) z z z z 00 00 01 01 03 00 01 03 03 00 01 03 − − − − − − ε ε ( i ) ε ε ε ( i 1 ) ε ( i 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( i ) z z z z z z 11 11 12 14 11 12 14 14 12 14 11 12 − − − − − − ε ε ε ε ε ε ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) z z z z z z 22 23 23 25 25 22 22 23 23 25 25 22 − − − − − − ε ε ( i 1 ) ε ε ( i 1 ) ε ε ( i 1 ) ( i 1 ) ( i 1 ) ( i 1 ) ( i ) ( i ) ( i ) ( i ) ( i ) ( i ) z z z z z z 33 33 34 34 36 36 33 33 34 34 36 36 ε ε − ε − ε − − − − ( ( i i ) 1 ) ε ( ( i i ) 1 ) ε ( ( i i ) 1 ) ( i 1 ) ( ( i i ) 1 ) ( ( i i 1 ) ) ( i ) z z z z z z 40 40 44 44 45 45 40 40 44 44 45 45 ε ε − ε ε − ε ε − − − − ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( i 1 ) ( ( i i 1 ) ) ( ( i i ) 1 ) ( i ) z z z z z z 51 51 55 56 51 55 56 55 56 51 55 56 − − − − − − ε ε ε ε ε ( ( i i ) 1 ) ε ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) z z z z z z 60 60 62 66 66 60 60 62 62 66 66 62

Shuffled Belief Propagation � Initialization � Step1: Update the Belief Matrix, for n=0,..N-1 � Horizontal Step: Update the n-th column of Check-to-Bit Matrix � Vertical Step: Update the n-th column of Bit-to-Check Matrix � Step2: Hard decision and stopping test � Step3: Output the decoded codeword

Shuffled Belief Propagation; Step 1: ( i ) Horizontal Step: ∏ ∏ + − ( i ) ( i 1 ) 1 tanh( z / 2 ) tanh( z / 2 ) mn ' mn ' ∈ ∈ n ' N ( m ) \ n n ' N ( m ) \ n < > n ' n n ' n ε = ( i ) log ∏ ∏ mn − − ( i ) ( i 1 ) 1 tanh( z / 2 ) tanh( z / 2 ) mn ' mn ' ∈ ∈ n ' N ( m ) \ n n ' N ( m ) \ n < > n ' n n ' n ( ii )Vertical Step: ∑ = + ε ( i ) ( i ) z F mn n m ' n ∈ m ' M ( n ) \ m ∑ = + ε ( i ) ( i ) z F n n mn ∈ m M ( n )

Update the belief matrix in i-th iteration with shuffled belief propagation decoding ε ε − ε ε − ε ε − − − − ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i 1 ) ) ( i 1 ) ( ( i i ) 1 ) z z ( i ) z z z z 00 00 01 01 03 00 01 03 03 00 01 03 − − − − − − ε ε ( i ) ε ε ε ( i 1 ) ε ( i 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( i ) z z z z z z 11 11 12 14 11 12 14 12 14 11 12 14 − − − − − − ε ε ε ε ε ε ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) z z z z z z 22 23 23 25 25 22 22 23 23 25 25 22 − − − − − − ε ε ( i 1 ) ε ε ( i 1 ) ε ε ( i 1 ) ( i 1 ) ( i 1 ) ( i 1 ) ( i ) ( i ) ( i ) ( i ) ( i ) ( i ) z z z z z z 33 33 34 34 36 36 33 33 34 34 36 36 ε ε − ε − ε − − − − ( ( i i ) 1 ) ε ( ( i i ) 1 ) ε ( ( i i ) 1 ) ( i 1 ) ( ( i i ) 1 ) ( ( i i 1 ) ) ( i ) z z z z z z 40 40 44 44 45 45 40 40 44 44 45 45 ε ε − ε ε − ε ε − − − − ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( i 1 ) ( ( i i ) 1 ) ( ( i i ) 1 ) ( i ) z z z z z z 51 51 55 56 51 55 56 55 56 51 55 56 − − − − − − ε ε ε ε ε ( ( i i ) 1 ) ε ( ( i i ) 1 ) ( ( i i ) 1 ) ( ( i i 1 ) ) ( ( i i ) 1 ) ( ( i i ) 1 ) z z z z z z 60 60 62 66 66 60 60 62 62 66 66 62

Implementation of shuffled BP � Backward-forward implementation � Computation Complexity

Optimality and Convergence Property of Shuffled BP Given the Tanner Graph of the code is connected and acyclic. � Shuffled BP is optimal in the sense of MAP � Shuffled BP converges faster (or at least no slower) than Standard BP

Parallel Shuffled BP � Divide the N bits into G groups, each group contains bits. (regular N G partition) . � In each group, the updatings are processed in parallel. The processings of groups are in sequential.