Analytical Solution of Covariance Evolution for Regular LDPC Codes

Analytical Solution of Co v a riance Ev olution for Re gular LDPC Codes T akayuk i Nozaki ∗ , Kenta Kasai ∗ , an d Kohichi Saka niwa ∗ ∗ Dept. of Communication s and Integrated Systems, T okyo Institute of T echnology Email: { nozak i, kenta, sakaniwa } @comm.ss.titech.ac.jp Abstract —The co variance e volution is a system of differential equations with respect to the co variance of th e number of edges connecting to the nodes of each residual degree. Solving the cov ariance e volution, we can derive di stributions of the number of check nodes of residual degree 1, which helps us to estimate the b lock error probability for finite-length LDPC code. Amraoui et a l. r esorted to numerical co mputations to solve the cov ariance ev olution. In this paper , we give the analytical solution of the cov ariance ev olution. I . I N T R O D U C T I O N Gallager in vented low-density parity-check (LDPC) codes [1] in 19 63. LDPC codes are lin ear codes defin ed b y sparse bipartite graphs. Lu by et al. introd uced the peeling algorithm (P A) [2], [4] fo r th e b inary erasure channel ( BEC). P A is an iterativ e algorithm which is defined on T anner graphs. P A and brief p ropagatio n (BP) deco der have the same decoding result. As P A procee ds, edges and nodes ar e progressiv ely removed. The residual grap hs co nsist o f nodes and ed ges tha t are still unknown at each iteration. Th e decoding successfu lly ha lts if the graph v anishes. Amraoui [3] showed that distrib utions of the n umber of check nodes of degree one in the residual graph conv ergences weakly to a Gau ssian as blo cklength tends to in finity . Amraoui also showed that block and bit error prob ability of finite-length LDPC co des are d eriv ed by th e av erage and the variance o f the number of check nodes of degree one in th e resid ual graph. Th e average number o f ch eck no des of degree one in the re sidual grap h is determined from a system of differential equations, which was d erived and solved b y L uby et al. [2]. The v ariance o f the number of check nodes of degree o ne in the residual g raph is also deter mined fro m a system of differential equations called covariance evolution , which was derived by Amraou i et al. [3]. Since analytical solution of covariance e volution has not been kn own so far, we had to resor t to numer ical computations to solve the covariance ev olution. An alternative way to determine the variance of the num ber of check nodes of degree on e was pr oposed in [3]. Th e variance of the nu mber o f check nodes of degree one in the residual graph can b e co mputed by deter mining the variance of the number of erased message s of BP for BEC with parameter ǫ ∗ where ǫ ∗ is th e thre shold of the ensemble u nder BP decodin g. This method is a valid ap proxim ation f or the erasure prob ability close to ǫ ∗ . Mo reover , E zri et al. e xtende d to this method to m ore general cha nnels [5]. However , if we solve th e cov ariance ev olution analytically , we can d erive the variance of the nu mber o f check nodes of degree one in the residual gr aph fo r all ǫ where ǫ is the channel p arameter for the BEC. In th is pape r , we show an analytical solution o f th e covari- ance e volution fo r regular LDPC code ensembles. I I . C O V A R I A N C E E V O L U T I O N [ 3 ] In this section, we br iefly revie w the covariance evolution and initial covariance in [3]. W e consider the tr ansmission ov er the BEC with channel erasure probab ility ǫ using LDPC co des in a ( b , d )-regular LDPC code ensemble. Let t denote the iteration round and ξ be th e total number of edges in the original gr aph. W e define that τ := t ξ . (1) Define a par ameter y such that d y / d τ = − 1 / ( ǫy b − 1 ) and y = 1 when τ = 0 . Let l b,t denote a rando m variable correspo nding to the number of ed ges connecting to variable nodes of degree b in th e resid ual g raph at the iter ation r ound t . Let r k,t denote a rando m variable correspo nding to the number of edg es con necting to check nodes o f d egree k in the r esidual graph at the iteratio n round t . Th ose random variables depend s on th e choice of the gra ph from ( b, d )-regular L DPC cod e ensemble, the chann el outputs and the random cho ices made by P A. W e define D t := { l b,t , r 1 ,t , r 2 ,t , . . . , r d − 1 ,t } . T o simp lify the notation, we drop the subscript t . F or i ∈ D t , we define ¯ i ( y ) by ¯ i ( y ) := E [ i ] ξ . W e also define δ ( i,j ) ( y ) by the cov ariance of i and j ( i, j ∈ D t ) divided by the total numb er of edge s in the orig inal grap h i.e. δ ( i,j ) ( y ) := Cov[ i, j ] ξ . In [3], Amrao ui showed these parameters satisfy the following system of dif ferential equations in the limit of the block length. This system is referred to as covariance ev olution. d δ ( i,j ) ( y ) d y = − e ( y ) y  X k ∈D  ∂ ˆ f ( i ) ∂ ¯ k δ ( j,k ) + ∂ ˆ f ( j ) ∂ ¯ k δ ( i,k )  + ˆ f ( i,j ) ( y )  , (2) where ∂ ˆ f ( l b ) ∂ ¯ l b = 0 , ∂ ˆ f ( l b ) ∂ ¯ r j = 0 , ∂ ˆ f ( r j ) ∂ ¯ l b = − j ( b − 1 ) ¯ r j +1 − ¯ r j ¯ l 2 b , ∂ ˆ f ( r j ) ∂ ¯ r k = j b − 1 ¯ l b  I { k = j +1 } − I { k = j }  , ∂ ˆ f ( r d − 1 ) ∂ ¯ l b = ( d − 1)( b − 1 ) ¯ l b + ¯ r d − 1 − ¯ r d ¯ l 2 b , ∂ ˆ f ( r d − 1 ) ∂ ¯ r j = − (1 + I { j = d − 1 } ) ( d − 1)( b − 1) ¯ l b , ˆ f ( l b l b ) = 0 , ˆ f ( l b r k ) = 0 , ˆ f ( r k r j ) = k j b − 1 ¯ l b n − ( ¯ r k +1 − ¯ r k )( ¯ r j +1 − ¯ r j ) ¯ l b + I { k = j } ( ¯ r j +1 + ¯ r j ) − I { k = j +1 } ¯ r k − I { j = k +1 } ¯ r j o , where x := ǫ y b − 1 , ˜ x := 1 − x , ˜ ǫ := 1 − ǫ , e ( y ) = ¯ l b = xy , ¯ r j =  d − 1 j − 1  x j ˜ x d − j , ¯ r 1 = x ( y − 1 + ˜ x d − 1 ) and I { k = s } is the ind icator function which equals to 1 if k = s and 0 otherwise. The in itial con ditions of the covariance ev olution a re given by initial covariances. The initial covariances are the covari- ances o f th e nu mber o f e dges o f ea ch d egree at th e start o f th e decodin g d i vided by the total n umber of edges in the orig inal graph. F or j, k ∈ { 1 , 2 , . . . , d } , initial cov ariances are derived in [3], as fo llows. δ ( l b ,l b ) (1) = bǫ ˜ ǫ, δ ( l b ,r j ) (1) = − b  d − 1 j − 1  ǫ j ˜ ǫ d − j ( dǫ − j ) , δ ( r j ,r k ) (1) = I { k = j } j  d − 1 j − 1  ǫ j ˜ ǫ d − j − d  d − 1 j − 1  d − 1 k − 1  ǫ j + k ˜ ǫ 2 d − j − k + ( b − 1)  d − 1 j − 1  d − 1 k − 1  ǫ j + k − 1 ˜ ǫ 2 d − j − k − 1 · ( dǫ − j )( dǫ − k ) . I I I . A N A LY T I C A L S O L U T I O N O F C O V A R I A N C E E V O L U T I O N W e sho w in the f ollowing theor em the analytical solu tion of the covariance ev olution, f or a ( b , d )-r egular LDPC code ensemble. The proof 1 is gi ven in Section III-C. 1 T a king the deriv ati v e of both sides of (3), (4) and (5 ) with respect to y , we can check that those equati ons fulfill (2). Theorem 1. Let τ be th e norma lized iter ation rou nd of P A as defined in (1). For a ( b , d )-regular LDPC code en semble and j, k ∈ { 1 , 2 , . . . , d − 1 } , in the limit of t he code length , we obtain the following. δ ( l b ,l b ) = bǫ ˜ ǫ, (3) δ ( l b ,r j ) = − G j { ǫ ˜ ǫ ( b − 1) y − 1 + ˜ ǫx } + I { j =1 } bǫ ˜ ǫ, (4) δ ( r k ,r j ) = b − 1 b G k G j { ǫ ˜ ǫ ( b − 1) y − 2 − ( ǫ − ˜ ǫ ) xy − 1 + x 2 } − d  d − 1 k − 1  d − 1 j − 1  x k + j ˜ x 2 d − k − j + I { k = j }  d − 1 k − 1  k x k ˜ x d − k + I { k =1 ,j =1 } ( bǫ ˜ ǫ − x ˜ x ) −  I { k =1 } G j + I { j =1 } G k  { ǫ ˜ ǫ ( b − 1) y − 1 − ǫx + x 2 } , (5) where G j :=  d − 1 j − 1  x j − 1 ˜ x d − j − 1 ( dx − j ) + I { j =1 } and y is defined by d y/ d τ = − 1 / ( ǫy b − 1 ) with y = 1 when τ = 0 . A. scaling parameter α In [3], scaling parameter α is gi ven by α = − p δ ( r 1 ,r 1 ) ( ǫ ∗ , y ∗ ) p ξ /n ∂ ¯ r 1 ∂ ǫ   ǫ ∗ ; y ∗ , (6) where ǫ ∗ is th e thr eshold of the en semble u nder BP decod ing , y ∗ is th e non- zero solution of ¯ r 1 ( y ) at th e threshold , n is the blockleng th and ξ is the total num ber of edges in the original graph. W e define x ∗ := ǫ ∗ ( y ∗ ) b − 1 and ˜ x ∗ := 1 − x ∗ . Since ¯ r 1 ( ǫ ∗ , y ∗ ) = 0 a nd ∂ ¯ r 1 ∂ y | ǫ ∗ ; y ∗ = 0 , we see that y ∗ = 1 − ( ˜ x ∗ ) d − 1 and y ∗ = ( b − 1)( d − 1) x ∗ ( ˜ x ∗ ) d − 2 . Using tho se equation s, w e have from (5) δ ( r 1 ,r 1 ) ( ǫ ∗ , y ∗ ) = x ∗ y ∗ b − 1 ( y ∗ − x ∗ ) . (Note that G 1 = b b − 1 y ∗ ). Recall that ¯ r 1 ( ǫ, y ) = x ( y − 1 + ˜ x d − 1 ) . W e see that ∂ ¯ r 1 ∂ ǫ   y ∗ ; ǫ ∗ = − x ∗ y ∗ ǫ ∗ ( b − 1) . From (6), w e can obtain α = ǫ ∗ s b − 1 b  1 x ∗ − 1 y ∗  . This is the same r esult as in [3] for regu lar L DPC code ensembles. B. Example of Solution of Covariance Evolution Figure 1 shows the solu tion of the c ov ariance ev olution δ ( r j ,r j ) ( ǫ, y ) , j ∈ { 1 , 2 , . . . , 5 } , as a function of y for ( 3,6)- regular LDPC code ensemb le. Figure 2 shows the solution of the covariance ev olution δ ( r j ,r j ) ( ǫ, y ) , j ∈ { 1 , 2 , 3 } , as a function of y fo r (2 ,4)-r egular LDPC code ensemble. C. Ou tline of pr oo f 1) Pr oof fo r δ ( l b ,l b ) : From (2), w e get d δ ( l b ,l b ) d y ( y ) = 0 . From initial cov ariance, we h av e δ ( l b ,l b ) = bǫ ˜ ǫ . This leads to (3). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P S f r a g r e p l a c e m e n t s y δ ( r 1 ,r 1 ) δ ( r 2 ,r 2 ) δ ( r 3 ,r 3 ) δ ( r 4 ,r 4 ) δ ( r 5 ,r 5 ) Fig. 1. The solution of the cov arianc e ev olution δ ( r j ,r j ) , j ∈ { 1 , 2 , . . . , 5 } , as a function of the parameter y for the (3,6)-regular LDPC code ensemble. The channel parameter is ǫ = 0 . 42943 98 ≈ ǫ ∗ . 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P S f r a g r e p l a c e m e n t s y δ ( r 1 ,r 1 ) δ ( r 2 ,r 2 ) δ ( r 3 ,r 3 ) δ ( r 4 , r 4 ) δ ( r 5 , r 5 ) Fig. 2. The solution of the cov ariance ev olution δ ( r j ,r j ) , j ∈ { 1 , 2 , 3 } , as a func tion of the paramete r y for the (2,4)-re gular LDPC code ensemble. The channe l parameter is ǫ = 0 . 333333 ≈ ǫ ∗ . 2) Pr oof for δ ( l b ,r j ) : In orde r to solve δ ( l b ,r j ) , we de- fine A ( l b , Σ) := P d − 1 j =1 δ ( l b ,r j ) , wh ich gi ves d A ( l b , Σ) d y = P d − 1 j =1 d δ ( l b ,r j ) d y . From (2), we see tha t d A ( l b , Σ) d y = b − 1 y  dA ( l b , Σ) + D ( l b , Σ)  , (7) where D ( l b , Σ) := d ( x d − 1 y − 1 − 1) δ ( l d ,l d ) . This equ ation is a first order linear dif ferential equation and the solution is gi ven by A ( l b , Σ) = y d ( b − 1) n Z ( b − 1) D ( l b , Σ) y d ( b − 1)+1 d y + C l b , Σ o = G d ǫ ˜ ǫ ( b − 1) y − 1 + bǫ ˜ ǫ + C l b , Σ y d ( b − 1) , with a con stant C l b , Σ determined fro m th e initial covariance where G j :=  d − 1 j − 1  x j − 1 ˜ x d − j − 1 ( dx − j ) + I { j =1 } . Note th at A ( l b , Σ) (1) = P d − 1 j =1 δ ( l b ,r j ) (1) = bǫ ˜ ǫ − bdǫ d ˜ ǫ . W e see th at C l b , Σ = − dǫ d ˜ ǫ . W e have A ( l b , Σ) = G d { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } + bǫ ˜ ǫ. (8) From (2) and (8), we get for j ∈ { 1 , . . . , d − 1 } d δ ( l b ,r j ) d y = b − 1 y { j δ ( l b ,r j ) + D ( l b ,r j ) } , where D ( l b ,r j ) := j { ¯ r j +1 − ¯ r j ¯ l b bǫ ˜ ǫ − δ ( l b ,r j +1 ) I { j 6 = d − 1 } + ( A ( l b , Σ) − bǫ ˜ ǫ ) I { j = d − 1 } } . Those eq uations ar e fir st order line ar d ifferential equations. The solutions are given by δ ( l b ,r j ) = y j ( b − 1) n Z ( b − 1) D ( l b ,r j ) y j ( b − 1)+1 d y + C l b ,r j o , with constants C l b ,r j determined from the initial covariances. Those equations can be solved by m athmatical in duction for j ∈ { 2 , 3 , . . . , d − 1 } . W e sho w that δ ( l b ,r d − 1 ) fulfill (4). From (8), we can write D ( l b ,r d − 1 ) =( d − 1)˜ ǫxG d + ǫ ˜ ǫy − 1 { bG d − 1 + ( b − 1)( d − 1) G d } . Using the same way in the induction step, we can obtain δ ( l b ,r d − 1 ) = − G d − 1 { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } . W e show that if δ ( l b ,r j +1 ) fulfill (4), then also δ ( l b ,r j ) fulfill (4) . Assume δ ( l b ,r j +1 ) = − G j +1 { ǫ ˜ ǫ ( b − 1) + ˜ ǫx } . Using the induction hypothesis, we can write D ( l b ,r j ) = j ˜ ǫxG j +1 + ǫ ˜ ǫy − 1 { bG j + ( b − 1) j G j +1 } . (9) Using ˜ x k = P k s =0  k s  ( − x ) s , we see that y j ( b − 1) Z ( b − 1) j ˜ ǫxG j +1 y j ( b − 1)+1 d y = − ˜ ǫxG j + d d − j − 1  d − 1 j − 1  ˜ ǫ x j . (10) Similarly , we hav e y j ( b − 1) Z ( b − 1) ǫ ˜ ǫy − 1 bG j y j ( b − 1)+1 d y =( b − 1) bǫ ˜ ǫ  d − 1 j − 1  x j d − j X s =0  d − j s  ( j + s )( − ǫ ) s − 1 K s − 1 , (11) y j ( b − 1) Z ( b − 1) 2 ǫ ˜ ǫy − 1 j G j +1 y j ( b − 1)+1 d y = − ( b − 1) 2 ǫ ˜ ǫ  d − 1 j − 1  x j d − j X s =0  d − j s  s ( j + s )( − ǫ ) s − 1 K s − 1 , (12) where K s := y s ( b − 1) − 1 s ( b − 1) − 1 I { s ( b − 1) 6 =1 } + log y I { s ( b − 1)=1 } . Note that { s ( b − 1) − b } K s − 1 = y ( s − 1)( b − 1) − 1 − I { ( s − 1)( b − 1)=1 } . From (11) and ( 12), we ha ve y j ( b − 1) Z ( b − 1) ǫ ˜ ǫy − 1 { bG j + ( b − 1) j G j +1 } y j ( b − 1)+1 d y = − ( b − 1) ǫ ˜ ǫ  d − 1 j − 1  x j · d − j X s =0  d − j s  ( j + s ) { s ( b − 1) − b } K s − 1 = − ( b − 1) ǫ ˜ ǫy − 1 G j + ( b − 1) ǫ ˜ ǫx j P j , (13) where P j :=  d − 1 j − 1  d − j X s =0  d − j s  ( j + s )( − ǫ ) s − 1 I { ( s − 1)( b − 1)=1 } . From (9),(10) and (13), we have δ ( l b ,r j ) = − G j { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } + d d − j − 1  d − 1 j − 1  ˜ ǫ x j + ( b − 1) ǫ ˜ ǫx j P j + C l b ,r j y j ( b − 1) . From initial cov ariance, we have C l b ,r j = − d d − j − 1  d − 1 j − 1  ˜ ǫ ǫ j − ( b − 1) ǫ ˜ ǫǫ j P j . Hence we obtain δ ( l b ,r j ) = − G j { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } . This lads to ( 4) for j ∈ { 2 , 3 , . . . , d − 1 } . Note th at δ ( l b ,r 1 ) = A ( l b , Σ) − P d − 1 j =2 δ ( l b ,r j ) and that − G 1 = P d j =2 G j . W e ha ve δ ( l b ,r 1 ) = − G 1 { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } + bǫ ˜ ǫ. Hence we obtain (4). 3) Pr oof for B ( · , Σ) : In order to solve δ ( r j ,r k ) , we d efine B ( r j , Σ) := P d − 1 k =1 δ ( r j ,r k ) and B (Σ , Σ) := P d − 1 j =1 B ( r j , Σ) . From (2), we get for j ∈ { 1 , 2 , . . . , d − 1 } d B (Σ , Σ) d y = b − 1 y  D (Σ , Σ) + 2 dB (Σ , Σ)  , (14) d B ( r j , Σ) d y = b − 1 y  D ( r j , Σ) + ( d + j ) B ( r j , Σ)  , (15) where D (Σ , Σ) := d ¯ r d − ¯ l b ¯ l b  d ¯ r d + 2 A ( l b , Σ)  , D ( r j , Σ) := j ¯ r j +1 − ¯ r j ¯ l b ( d ¯ r d + A ( l b , Σ) ) + d ¯ r d − ¯ l b ¯ l b δ ( l b ,r j ) − j B ( r j +1 , Σ) I { j 6 = d − 1 } + ( d − 1)( B (Σ , Σ) − dr d − A ( l b , Σ) ) I { j = d − 1 } . The solution of (14 ) is given b y B (Σ , Σ) = y 2 d ( b − 1) n Z ( b − 1) D (Σ , Σ) y 2 d ( b − 1)+1 d y + C Σ , Σ o = b − 1 b G 2 d { ǫ ˜ ǫ ( b − 1) y − 2 − ( ǫ − ˜ ǫ ) xy − 1 } + 2 G d { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } + dx d + bǫ ˜ ǫ + C Σ , Σ y 2 d ( b − 1) , with a con stant C Σ , Σ which dete rmined f rom the inital covari- ance. From the in itial cov ariance, we get B (Σ , Σ) (1) = bǫ ˜ ǫ − 2 bdǫ d ˜ ǫ + dǫ d − dǫ 2 d + ( b − 1) d 2 ǫ 2 d − 1 ˜ ǫ. W e see that C Σ , Σ = b − 1 b d 2 ǫ 2 d − dǫ 2 d . Hence we ha ve B (Σ , Σ) = b − 1 b G 2 d { ( b − 1) ǫ ˜ ǫy − 2 − ( ǫ − ˜ ǫ ) xy − 1 + x 2 } + 2 G d { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } + dx d − dx 2 d + bǫ ˜ ǫ. (16) From (15), we get B ( r j , Σ) = y ( d + j )( b − 1) n Z ( b − 1) D ( r j , Σ) y ( d + j )( b − 1)+1 d y + C r j , Σ o , with co nstants C r j , Σ . For j ∈ { 2 , 3 , . . . , d − 1 } , tho se equ ation are solved by mathmatical indu ction as the proof for δ ( l b ,r j ) . From the initial covariances, no te that B ( r j , Σ) (1) = d ( b − 1)  d − 1 j − 1  ǫ d + j − 1 ˜ ǫ d − j ( dǫ − j ) + d  d − 1 j − 1  ǫ d + j ˜ ǫ d − j − b  d − 1 j − 1  ǫ j ˜ ǫ d − j ( dǫ − j ) . W e ha ve B ( r j , Σ) = − b − 1 b G d G j { ( b − 1) ǫ ˜ ǫy − 2 − ( ǫ − ˜ ǫ ) xy − 1 + x 2 } − G j { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } +  d − 1 j − 1  dx d + j ˜ x d − j . (17) Using B ( r 1 , Σ) = B (Σ , Σ) − P d − 1 j =2 B ( r j , Σ) , we ha ve B ( r 1 , Σ) = − b − 1 b G d G 1 { ( b − 1) ǫ ˜ ǫy − 2 − ( ǫ − ˜ ǫ ) xy − 1 + x 2 } − G 1 { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } + dx d +1 ˜ x d − 1 + G d { ( b − 1) ǫ ˜ ǫy − 1 + ˜ ǫx } + dx d ˜ x + bǫ ˜ ǫ. (18) 4) Pr oof fo r δ ( r k ,r j ) : From (2), we g et for k , j ∈ { 1 , 2 , . . . , d − 1 } d δ ( r k ,r j ) d y = b − 1 y { ( k + j ) δ ( r k ,r j ) + D ( r k ,r j ) } , where D ( r k ,r j ) := H k,j + H j,k − ¯ l b b − 1 ˆ f ( r k ,r j ) , H k,j := k ¯ r k +1 − ¯ r k ¯ l b δ ( l b ,r j ) − k δ ( r k +1 ,r j ) I { k 6 = d − 1 } − ( d − 1)( δ ( l b ,r j ) − B ( r j , Σ) ) I { k = d − 1 } . The solutions of th ose differential e quations are gi ven by δ ( r k ,r j ) = y ( k + j )( b − 1) n Z ( b − 1) D ( r k ,r j ) y ( k + j )( b − 1)+1 d y + C r k ,r j o , with con stants C r k ,r j . Using (17), we can solve those equ a- tions by mathmatical induction for j, k ∈ { 2 , 3 , . . . , d − 1 } . W e have δ ( r k ,r j ) = b − 1 b G k G j { ǫ ˜ ǫ ( b − 1) y − 2 − ( ǫ − ˜ ǫ ) xy − 1 + x 2 } − d  d − 1 k − 1  d − 1 j − 1  x k + j ˜ x 2 d − k − j + I { k = j }  d − 1 j − 1  j x j ˜ x d − j . Note that δ ( r k ,r 1 ) = B ( r k , Σ) − P d − 1 j =2 δ ( r k ,r j ) . W e have for k ∈ { 2 , . . . , d − 1 } δ ( r k ,r 1 ) = b − 1 b G k G 1 { ǫ ˜ ǫ ( b − 1) y − 2 − ( ǫ − ˜ ǫ ) xy − 1 + x 2 } − d  d − 1 k − 1  x k +1 ˜ x 2 d − k − 1 − G k { ( b − 1) ǫ ˜ ǫy − 1 − ǫx + x 2 } . Since δ ( r 1 ,r 1 ) = B ( r 1 , Σ) − P d − 1 j =2 δ ( r 1 ,r j ) , we ha ve δ ( r 1 ,r 1 ) = b − 1 b G 2 1 { ǫ ˜ ǫ ( b − 1) y − 2 − ( ǫ − ˜ ǫ ) xy − 1 + x 2 } − dx 2 ˜ x 2 d − 2 + x ˜ x d − 1 − 2 G 1 { ( b − 1) ǫ ˜ ǫy − 1 − ǫx + x 2 } + ( bǫ ˜ ǫ − x ˜ x ) . Thus, we can obtain ( 5). I V . R E L AT I O N S H I P T O S TA B I L I T Y C O N D I T I O N In this section, we consider the relationship between the stability condition [6], [4] an d lim y → 0 δ ( l b ,r 1 ) ( ǫ, y ) . For a ( b, d ) -regular LDPC code ensemble ( b ≥ 3 ), we see from (4) and ( 5) that lim y → 0 δ ( l b ,r j ) ( ǫ, y ) = bǫ ˜ ǫI { j =1 } , lim y → 0 δ ( r j ,r j ) ( ǫ, y ) = bǫ ˜ ǫI { j =1 } . For a (2 , d ) -regular LDPC code ensemble, we see from (4) and (5) that lim y → 0 δ ( l b ,r j ) ( ǫ, y ) =      2 ǫ ˜ ǫ { 1 − ( d − 1) ǫ } , if j = 1 2 ǫ ˜ ǫ ( d − 1) ǫ, if j = 2 0 , otherwise , lim y → 0 δ ( r j ,r k ) ( ǫ, y ) =          2 ǫ ˜ ǫ { 1 − ( d − 1) ǫ } 2 , if j = k = 1 2 ǫ 2 ˜ ǫ ( d − 1) { 1 − ( d − 1) ǫ } , if ( j, k ) = (1 , 2) , (2 , 1) 2 ǫ ˜ ǫ ( d − 1) 2 ǫ 2 , if j = k = 2 0 , otherwise. If we define th e correlation coef ficient for i, j ∈ D by ρ i,j ( ǫ ) := lim y → 0 δ ( i,j ) ( ǫ, y ) p δ ( i,i ) ( ǫ, y ) δ ( j,j ) ( ǫ, y ) , -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 P S f r a g r e p l a c e m e n t s y δ ( r 1 ,r 1 ) δ ( r 2 , r 2 ) δ ( r 3 , r 3 ) δ ( r 4 , r 4 ) δ ( r 5 , r 5 ) ǫ δ ( l 2 ,r 1 ) δ ( r 2 ,r 1 ) Fig. 3. Solution of cov ari ance ev oluti on lim y → 0 δ ( j,r 1 ) ( ǫ, y ) , j ∈ { l 2 , r 1 , r 2 } , as a function of the ch annel paramete r ǫ for the (2,4)-re gular LDPC code ensemble. we obtain ρ l b ,r 1 ( ǫ ) = ( 1 , if I { b =2 } ( d − 1) ǫ ≤ 1 − 1 , if I { b =2 } ( d − 1) ǫ > 1 . (19) Note th at I { b =2 } ( d − 1 ) ǫ ≤ 1 agree with the stability co ndition for regular LDPC code ensembles. Figure 3 shows the solution o f covariance ev olution lim y → 0 δ ( j,r 1 ) ( ǫ, y ) , j ∈ { l 2 , r 1 , r 2 } , as a fu nction of the ch an- nel par ameter ǫ fo r the (2,4)-r egular LDPC code en semble. From Figur e 3, we see that δ ( r 2 ,r 1 ) > 0 and δ ( l 2 ,r 1 ) > 0 when ( d − 1) ǫ < 1 . Also we see that δ ( r 2 ,r 1 ) < 0 and δ ( l 2 ,r 1 ) < 0 when ( d − 1) ǫ > 1 . V . C O N C L U S I O N A N D F U T U R E W O R K In this pap er , we ha ve solved an alytically th e covariance ev olution f or regular L DPC code ensemb les. 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