Finite-Length Analysis of Irregular Expurgated LDPC Codes under - PowerPoint PPT Presentation

Finite-Length Analysis of Irregular Expurgated LDPC Codes under Finite Number of Iterations Ryuhei Mori Toshiyuki Tanaka Kenta Kasai Kohichi Sakaniwa ISIT2009

LDPC codes over the binary erasure channel The aim of our research To estimate the bit error probability P b ( n , ǫ , t ) of LDPC codes over the BEC under belief propagation decoding where n : blocklength ■ ǫ : erasure probability of BEC ■ t : the number of iterations ■ 2 / 15

Previous Results Analysis for the BEC Exact or Number of Computational Irregular Blocklength Method Asymptotic Iterations Complexity Ensembles ∞ O ( t ) � t Exact Density Evolution [1] O ( n 3 ) ∞ △ n Exact Stopping Sets [2] ∞ O (1) � n Asymptotic Scaling Law [3] × → � O ( t 3 ) This Research n t Asymptotic [1] Richardson and Urbanke 2001 [2] Di et al. 2002 [3] Amraoui et al. 2004 The Main Result of This Work Our result presented in ISIT2008 is generalized for irregular ensembles 3 / 15

Asymptotic Expansion Asymptotic Expansion w.r.t n while t is fixed � 1 � P b ( n , ǫ , t ) = P b ( ∞ , ǫ , t ) + α ( ǫ , t )1 n + O n 2 Coefficient of 1 / n α ( ǫ , t ) : = lim n →∞ n (P b ( n , ǫ , t ) − P b ( ∞ , ǫ , t )) Approximation P b ( n , ǫ , t ) ≈ P b ( ∞ , ǫ , t ) + α ( ǫ , t )1 n Our purpose is to derive α ( ǫ , t ) for irregular ensembles 4 / 15

Neighborhoods � P b ( n , ǫ , t ) = P n ( G )P b ( G , ǫ ) G ∈ the set of all neighborhoods of depth t ε(1−(1−ε) 2 ) 2 ε 2 (1−(1−ε) 2 ) ε 3 ε(1−(1−ε) 2 ) ε 2 (1+ε(1−ε)) ε P b ( G , ǫ ) (2n-6)(2n-8) 2(2n-6) 1 2 4(2n-6) 2 P n ( G ) (2n-1)(2n-5) (2n-1)(2n-5) (2n-1)(2n-5) (2n-1)(2n-5) (2n-1)(2n-5) (2n-1) -1 -2 -2 -1 -1 1 n n n n n Order of P n ( G ) G 5 / 15

Number of cycles The basic fact If G has k cycles P n ( G ) = Θ( n − k ). The large blocklength limit of the bit error probability � P b ( ∞ , ǫ , t ) = lim P n ( G )P b ( G , ǫ ) n →∞ G ∈ the set of all neighborhoods of depth t � = lim P n ( G )P b ( G , ǫ ) . n →∞ G ∈ the set of all cycle-free neighborhoods of depth t 6 / 15

Calculation of α ( ǫ , t ) α ( ǫ , t ) : = lim n →∞ n (P b ( n , ǫ , t ) − P b ( ∞ , ǫ , t ))   � = lim P n ( G )P b ( G , ǫ ) − P b ( ∞ , ǫ , t ) n →∞ n  G ∈ the set of all cycle-free neighborhoods of depth t � �� � β ( ǫ , t ) � + lim P n ( G )P b ( G , ǫ ) . n →∞ n G ∈ the set of all single-cycle neighborhoods of depth t � �� � γ ( ǫ , t ) In the previous work [Mori et al., ISIT2008], γ ( ǫ , t ) was obtained for irregular ensembles but β ( ǫ , t ) was obtained only for regular ensembles 7 / 15

Contribution of Cycle-Free Neighborhoods β ( ǫ , t ) =   1 i j � �  E t [ K ( K − 1) P ] − E t [ V i ( V i − 1) P ] − E t [ C j ( C j − 1) P ]  2 L ′ (1) λ i ρ j i j The expectations are taken on the tree ensemble of depth t P ∞ ( G ) := lim n →∞ P n ( G ) K : the number of edges in a tree neighborhood ■ V i : the number of variable nodes of degree i in a tree neighborhood ■ C j : the number of check nodes of degree j in a tree neighborhood ■ P : the erasure probability of the root node after BP decoding ■ on a tree neighborhood 8 / 15

Method of Generating Function � E t [ K ( K − 1) P ] = ∂ 2 E t [ x K P ] � � ∂ x 2 � x = 1 � E t [ V i ( V i − 1) P ] = ∂ 2 E t [ x V i P ] � � ∂ x 2 � x = 1 � E t [ C j ( C j − 1) P ] = ∂ 2 E t [ x C j P ] � � ∂ x 2 � x = 1 �� �� � E t [ x K P ] = 1 � y V k z C l � l P x E t � k � y k = x , z l = x for all k , l k l �� �� � � y V k z C l E t [ x V i P ] = E t � l P � k � y i = x , y k = 1, z l = 1 for all k � = i , l k l �� �� � � y V k z C l E t [ x C j P ] = E t � l P � k � z j = x , y k = 1, z l = 1 for all k , l � = j k l 9 / 15

The Mother Generating Function �� � � y V k z C l = ǫ L ( F ( t )), l P E t k k l where � 1, if t = 0 F ( t ) : = P ( g ( t )) − P ( G ( t )), otherwise, G ( t ) : = L ( f ( t − 1)) − ǫ L ( F ( t − 1)), � 1, if t = 0 f ( t ) : = P ( g ( t )), otherwise, g ( t ) : = L ( f ( t − 1)), and where � � � L i y i x i , λ i y i x i − 1 , ρ j z j x j − 1 . L ( x ) : = L ( x ) : = P ( x ) : = i i j 10 / 15

α ( ǫ , t ) for Optimized Irregular Ensemble 15 10 α ( ǫ , t ) 5 0 -5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ǫ λ ( x ) = 0.500 x + 0.153 x 2 + 0.112 x 3 + 0.055 x 4 + 0.180 x 8 ρ ( x ) = 0.492 x 2 + 0.508 x 3 R ≈ 0.192, ǫ BP ≈ 0.8, t = 1, 2, ... , 8, 50 11 / 15

Simulation Results 10 2 | α ( ǫ , t ) | n=360 n=720 n=5760 n | P b ( n , ǫ , t ) − P b ( ∞ , ǫ , t ) | 10 1 10 0 10 -1 10 -2 0.5 0.6 0.7 0.8 0.9 1 ǫ λ ( x ) = 0.500 x + 0.153 x 2 + 0.112 x 3 + 0.055 x 4 + 0.180 x 8 ρ ( x ) = 0.492 x 2 + 0.508 x 3 R ≈ 0.192, ǫ BP ≈ 0.8, t = 20 12 / 15

Ensembles with λ 2 = 0 10 1 n | P b ( n , ǫ , t ) − P b ( ∞ , ǫ , t ) | 10 0 10 -1 10 -2 | α ( ǫ , t ) | n=128 n=512 n=4096 10 -3 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 ǫ P b ( n , ǫ , ∞ ) = Θ(1 / n 2 ) for ǫ < ǫ BP (3, 6)-regular ensemble t = 5 For small ǫ , the small number of iteration is sufficient unless blocklength is sufficiently large 13 / 15

The Speed of Convergence For the irregular ensemble, λ ( x ) = 0.500 x + 0.153 x 2 + 0.112 x 3 + 0.055 x 4 + 0.180 x 8 ρ ( x ) = 0.492 x 2 + 0.508 x 3 when t = 20, n = 5760, α ( ǫ , t ) ≈ n (P b ( n , ǫ , t ) − P b ( ∞ , ǫ , t )) for any ǫ (Generally, λ 2 is larger and larger, the convergence is faster) α ( ǫ , t ) consists of contributions of cycle-free neighborhoods and single-cycle neighborhoods But the number of variable nodes in the smallest tree of depth 20 is 4194302 ≫ 5760 The probability of cycle-free and single-cycle neighborhoods is zero Open problem: Why is the speed of the convergence fast? 14 / 15

Conclusion and Open Problems Conclusion Using the generating function method, ■ β ( ǫ , t ) is obtained for irregular ensembles The speed of the convergence to α ( ǫ , t ) is fast ■ Open problems The fast convergence to α ( ǫ , t ) except for ■ ensembles with λ 2 = 0 and ǫ is small Minimization of P b ( n , ǫ , t ) + α ( ǫ , t ) / n on some conditions ■ Higher order terms i.e. coefficient of 1 / n 2 , 1 / n 3 ... ■ The limit parameter α ( ǫ , ∞ ) for irregular ensembles ■ Generalization to arbitrary binary memoryless symmetric channels ■ Asymptotic analysis of performance based on other limits e.g. n →∞ and ■ t →∞ simultaneously 15 / 15