Analysis of Non-binary Hybrid LDPC Codes

Transcription

1 Anayss of Non-bnary Hybrd LDPC Codes Luce Sassate and Davd Decercq ETIS ENSEA/UCP/CNRS UMR Cergy, FRANCE Abstract Ths paper s egbe for the student paper award. In ths paper, we anayse asymptotcay a new cass of LDPC codes caed Non-bnary Hybrd LDPC codes, whch has been recenty ntroduced n [7]. We use densty evouton technques to derve a stabty condton for hybrd LDPC codes, and prove ther threshod behavor. We study ths stabty condton to concude on asymptotc advantages of hybrd LDPC codes compared to ther non-hybrd counterparts. I. INTRODUCTION Le Turbo Codes, LDPC codes are pseudo-random codes whch are we-nown to be channe capacty-approachng. LDPC codes have been redscovered by MacKay under ther bnary form and soon after ther non-bnary counterpart have been studed by Davey []. Non-bnary LDPC codes have recenty receved a great attenton because they have better performance than bnary LDPC codes for short boc ength and/or hgh order moduatons [3], [], [4]. However, good short ength non-bnary LDPC codes tend to be utra-sparse, and have worse convergence threshod than bnary LDPC codes. Our man motvaton n ntroducng and studyng the new cass of hybrd LDPC codes s to combne the advantages of both fames of codes, bnary and non-bnary. Hybrd codes fames am at achevng ths trade-off by mxng dfferent order for the symbos n the same codeword. Our resutng codes are caed Non-bnary Hybrd LDPC Codes because of the mxture of dfferent symbo sets n the codeword. In [7], we have demonstrated the nterest of the Hybrd LDPC codes by desgnng codes that compare favoraby wth exstng codes for qute moderate code ength (a few thousands bts). Hybrd LDPC codes appear to be especay nterestng for ow rate codes, R.25. In ths paper, we study the asymptotc behavor and propertes of Hybrd LDPC codes under teratve beef propagaton (BP) decodng. The secton two of ths paper hghghts the generaty of our new codes structure, and expans why we have focused the asymptotc study on the partcuar subcass of near codes. In thrd and fourth sectons, we present the context of the study, and deta symmetry and near-nvarance propertes whch are usefu for the stabty condton.ths condton s then expressed and anayzed to show theoretc advantages of Hybrd LDPC codes. Ths wor was supported by the French Armament Procurement Agency (DGA). II. THE CLASS OF HYBRID CODES We defne a Non-bnary Hybrd LDPC code as LDPC code whose varabe nodes beong to fnte sets of dfferent orders. More precsey, ths cass of codes s not defned n a fnte fed, but n fnte groups. We w ony consder groups whose cardnaty q s a power of 2, that says groups of the type G(q ) = ( Z p 2Z) wth p = og 2 (q ). Thus to each eement of G(q ) corresponds a bnary map of p bts. Let us ca the mnmum order of codeword symbos q mn, and the maxmum order of codeword symbos q max. The cass of hybrd LDPC codes s defned on the product group ( Z pmn ( 2Z)... Z ) pmax. 2Z Let us notce that ths type of LDPC codes but on product groups has aready been proposed n the terature [2], but no optmzaton of the code structure has been proposed and ts appcaton was restrcted to the mappng of the codeword symbos to dfferent moduaton orders. Party chec codes defned on (G(q mn )... G(q max )) are partcuar snce they are near n Z 2Z, but coud be non-near n the product group. Athough t s a oss of generaty, we have decded to restrct ourseves to hybrd LDPC codes that are near n ther product group, n order to bypass the encodng probem. We w therefore ony consder upper-tranguar party chec matrces and a specfc sort of the symbo orders n the codeword, whch ensures the nearty of the hybrd codes. The structure of the codeword and the assocated party chec matrx s depcted n Fgure. We herarchcay sort the H = Redundancy. Informaton c = G(qmax)... G(q r+ ) G(qr )... G(q mn ) Fg.. G(q max ) G(q r+ ) Hybrd codeword and party-chec matrx. dfferent group orders n the rows of the party-chec matrx, and aso n the codeword, such that q mn <... < q <... < q max. To encode a redundancy symbo, we consder

2 each symbo that partcpates n the party chec as an eement of the hghest group, whch s ony possbe f the groups are sorted as n Fgure. Ths ceary shows that encodng s feasbe n near tme by bacward computaton of the chec symbos. In order to expan the decodng agorthm for hybrd LDPC codes, t s usefu to nterpret a party chec of a hybrd code as a speca case of a party chec but on the hghest order group of the symbos of the row, denoted G(q ) and have a oo at the bnary mage of the equvaent code []. For codes defned over Gaos feds, the nonzero vaues of H correspond to the companon matrces of the fnte fed eements and are typcay rotaton matrces (because of the cycc property of the Gaos feds). In the case of hybrd LDPC codes, a nonzero vaue s a functon that connects a row n G(q ) and a coumn n G(q ),.e., that maps the q symbos of G(q ) nto a subset of q symbos that beongs to G(q ). Such appcaton s not necessary near, but n the case t s, ts equvaent bnary representaton s a matrx of dmenson (p p ). Note that, wth the above mentoned constrants, we have necessary p p. It s possbe to generaze the Beef propagaton decoder to hybrd codes, and t has been shown that even for those very specfc structures, t s possbe to derve a fast verson of the decoder usng FFTs [5]. In ths wor, we consder ony maps that are near appcatons, and hence that have a bnary representaton, n order to be abe to appy a nown resuts on near codes. We ca the message passng step through h j (cf. fgure 2) extenson when t s from G(q ) to G(q ) and truncaton when t s from G(q ) to G(q ). c G(q ) c 2 G(q 2 ) c 3 G(q 3 ) 2 4 h (c ) h 2 (c 2 ) h 3 (c 3 ) party-chec n G(q 3 ) h (c ) + h 2 (c 2 ) + h 3 (c 3 ) =, h j (c j ) G(q 3 ) defnes a component code n the group G = G(q ) G(q 2 ) G(q 3 ) Fg. 2. Party-chec of an hybrd LDPC code. q q 2 q 3 III. PROPERTIES OF LINEAR HYBRID LDPC CODES A. The Extenson and Truncaton Operatons We frst carfy the nature of the non-zero eements of the party-chec matrx of a hybrd LDPC code. We consder an eement A of the set of near extensons from G(q ) to G(q ). Im(A) denotes the mage of A. A beongs to the set of near appcatons from G(2) p to G(2) p whch are fu-ran (that s njectve snce dm(im(a))=ran(a)=p ). A : G(2) p G(2) p j denotes the bnary map of n G(q ) n G(2) p, wth p = og 2 (q ). That s, each ndex s taen to mean the th eement of G(q ), gven some enumeraton of the fed. x s the th eement of vector x. The extenson y, of the probabty vector x by A, s denoted by x A and defned by: for a =,..., f / Im(A), y = f Im(A), y = x j wth j such that = Aj A s caed extenson, and the nverse functon A truncaton from Im(A) to G(q ). The truncaton s defned by A : Im(A) G(2) p j wth j such that j = A The truncaton x of the probabty vector y by A s denoted by y A and defned by =,..., q, x = y j wth j such that j = A Gven a probabty-vector x of sze q, the components of the ogarthmc densty rato ( (LDR) ) vector w assocated wth x x are defned as w = og x, =,..., q. At channe output, LDR messages are actuay ogarthmc ehood rato (LLR) vectors. B. Parameterzaton of Hybrd LDPC famy An edge of the Tanner graph of an Hybrd LDPC code has four parameters (, q, j, q ). A hybrd LDPC code s then represented by π(, j,, ) whch s the proporton of edges connectng varabe nodes of degree n G(q ), to chec nodes of degree j n G(q ). Thus, hybrd LDPC codes have a very rch parameterzaton snce the parameter space has four dmensons. C. Symmetry defnton for densty evouton approach Let W be a LLR vector computed at the output of a dscrete memoryess channe, and v the component of the codeword sent, correspondng to the requred vaue for the data node the edge wth message W s connected to. W a denotes the cycc-permutaton of W. c denotes the vaue of the symbo ned to the edge wth the message W, and y the avaabe nformaton on a other edges of the graph. W( a component of W a and s defned by W a = og P (a c= y) P (a c= y) where denotes the mutpcaton n G(q). Le n [4], W a, for a a G(q) s defned by W a = W a+ W a, =... q A channe s cycc f output LLR vector W fufs P (W a v = ) = P (W v = a) s the th ),

3 Defnton On a cycc channe, a LDR message s symmetrc, f the foowng expresson hods a G(q), P (W = w v = a) = e wa P (W = w v = ) Most practca channes are cycc, and thus, n ths wor, we assume transmsson on arbtrary memoryess cyccsymmetrc channes. The generazaton of the resuts n ths paper to non-symmetrc channes can be done thans to the coset approach as n [4]. The symmetry property s the essenta condton for any asymptotc study snce t ensures that the error probabty s ndependent of the codeword sent. Lemma : If W s a symmetrc LDR-vector random varabe, then ts extenson W A, by any near extenson A wth fu ran, s aso symmetrc. The same emma hods for truncaton. The data pass and the chec pass of beef propagaton have aready been shown to preserve symmetry. Thus, emma ensures that the hybrd decoder preserves the symmetry property f the nput messages are symmetrc. Lemma 2: The error-probabty of a code n a hybrd famy, used on a cycc-symmetrc channe, s ndependent on the codeword sent. For ac of space, we do not gve the proof of ths emma, whch s a drect generazaton of [6]. D. Lnear Appcaton-Invarance Now we ntroduce a property that s specfc to the hybrd codes fames. Bennatan et a. n [4] used permutatonnvarance to derve a stabty condton for non-bnary LDPC codes, and to approxmate the denstes of graph messages usng one-dmensona functonas, for extrnsc nformaton transfert (EXIT) charts anayss. The dfference between nonbnary and Hybrd LDPC codes hods n the non-zeros eements of the party-chec matrx. Indeed, they do not correspond anymore to cycc permutatons, but to near extensons or truncatons, that we denote by near appcatons. The goa s to prove that near appcaton-nvarance (shortened by LAnvarance) of messages s nduced by choosng unformy the near extensons whch are the non-zero eements of the hybrd party-chec matrx. In partcuar, LA-nvarance aows to characterze message denstes wth ony one scaar parameter [7]. We wor wth probabty vector random varabes, but a the defntons and proofs gven n the remanng aso appy to LDR-vector random varabes. We denote by E the set of near extensons from G(q ) to G(q 2 ), and by T the set of nverse functons of E, what we ca the set of near truncatons from G(q 2 ) to G(q ) (see prevous secton on near extensons). Defnton 2: Y s LA-nvarant f and ony f for a (A, B ) T T, the probabty-vector random varabes Y A and Y B are dentcay dstrbuted. Lemma 3: If a probabty-vector random varabe Y of sze q 2 s LA-nvarant, then for a (, j) G(q 2 ) G(q 2 ), the random varabes Y and Y j are dentcay dstrbuted. Defnton 3: Let X be a q -szed probabty-vector random varabe, we defne the random-extenson of sze q 2 of X, denoted X, as the probabty-vector random varabe X A, where A s unformy chosen n E and ndependent on X. Lemma 4: A probabty-vector random varabe Y s LAnvarant f and ony f there exsts a probabty-vector random varabe X such that Y = X. For ac of space reason, we w deta the proof of ths emma, whch s easy, n a future pubcaton. Thans to emma 4, the chec node ncomng messages are LA-nvarant n the code famy made of a the possbe cyce-free ntereavers and unformy chosen near extensons (and hence correspondng truncatons). Moreover, random-truncatons, at chec node output, ensures LA-nvarance of varabe node ncomng messages. Thus, as shown n [7] under Gaussan approxmaton, the denstes of vector messages are characterzed by ony one parameter. IV. THE STABILITY CONDITION FOR HYBRID LDPC CODES The stabty condton, ntroduced n [6], s a necessary and suffcent condton for the error probabty to approach arbtrary cose to zero, assumng t has aready dropped beow some vaue at some teraton. In ths paragraph, we generaze the stabty condton to hybrd LDPC codes. Gven a hybrd famy defned by π(, j,, ), we defne the foowng famy parameter: Ω = j,, π( = 2,, j, ) q (j ) Aso for a gven memoryess symmetrc output channe wth transton probabtes p(y x) and a mappng δ( ), we defne the foowng channe parameter: = π(, ), E.g., for BI-AWGN channe, we have = π(, ), p(y δ())p(y δ())dy exp( 2σ 2 n ) where n s the number of ones n the bnary map of G(q ). Theorem: Let assume (π, δ) gven for a hybrd LDPC set. Let P denotes the probabty dstrbuton functon of nta messages () for a. Let Pe t = P e (R t ) denotes the average error probabty at teraton t under densty evouton. If Ω, then there exsts a postve constant ξ = ξ(π, P ) such that Pe t > ξ for a teratons t. If Ω <, then there exsts a postve constant ξ = ξ(π, P ) such that f Pe t < ξ at some teraton t, then Pe t approaches zero as t approaches nfnty. For ac of space reason, we gve there ony a setch of the proof. Proof We frst gve the genera nes of the proof of the necessary condton. Let t+n denotes the varabe node

4 outcomng messages n G(q ) at teraton t + n, where n =,,.... Snce we consder ony cycc-symmetrc channes, we can appy emma 4 from [4]. It ensures that there exsts an erasurzed channe such that the cyccsymmetrc channe s a degraded verson of t, and hence provdes a ower bound on the error probabty. Let ˆ t+n, n =,,..., denote the respectve messages of the erasurzed channe, and ˆɛ the erasure probabty. In the remander of the proof, we swtch to og-densty representaton of messages. Let ˆR () t+n denote the LDR-vector () representaton of ˆR t+n, n =,,.... Q () n (w) denotes the dstrbuton of ˆR () t+n. P () denotes the dstrbuton of the nta message R () of the cycc-symmetrc channe. The overne notaton X apped to vector X represents the vector resutng from random extenson foowed by random truncaton of X. Provded that random extenson and truncaton are such that X and X are of same sze, we can show that the error probabtes are equa. Thus, f Q () n s the dstrbuton of ˆR () t+n, we have P e (Q n ) = π()p e (Q () n ) = π()p e (Q () n ) Therefore P e (Q n ) s ower bounded by a constant strcty greater than zero f and ony f there exsts such that P e (Q () n ) s ower bounded by a constant strcty greater than zero. Defnng and Ω = j 2, π( = 2, j, ) q (j ) P = () π()p, we show that P e (Q () n ) 2 2(q max ) 2 ˆɛ() Ω Ω n n P We prove that P s symmetrc n the bnary sense, and as n [4], we obtan ( ( m n n og P (W ) n = og E )) 2 R where R s the shortened notaton for the frst component of the mxture of decoder nput LLR-vector random varabes R (). We have ( E ) ( ( )) R 2 R A B = E A,B E A, B and fnay obtan ( E ) 2 R = π(, ), and E ( R R ) R A B E ( R = p(y δ()p(y δ()))dy R ) () Hence, we fnd E ( 2 ) R =. Ths ast equaton combned wth equaton () eads to the concuson that P e (Q () n ) s ower bouded by a strcty postve constant, as n tends to nfnty, as soon as Ω. Ths condton s the same for a. Thus, the necessary condton for stabty s Ω <. We gve now the man steps for the proof of the suffcency of the condton. X () denotes a probabty-vector random varabe of sze. We defne D n and D a : D n (X () ) = E X() X () = E = π( ) q D a (X () ) = E j= X() j X () X() X () E X() X () To shorten the notatons we can omt the ndex of teraton t. The data pass s transated by = R ()() n= L () We obtan v 2 v Bu D n(r t) = t R() C A = X 3 u π() t L() A5 L () Frst, we are gong to prove the recursve nequaty (2) We show the three foowng equatons. E L() = D n (L () ) D n (L () ) = L () π( ) q D a(l () ) D a (L () ) j π(j )( D a(r () )) j +O(D a (R () ) 2 ) Connectng D a (R () ) to D n (R t ) ends up wth the proof of equaton 2: D a (R () ) = q π( ) E R() D n (R t ) = q π() π( ) E we obtan D n (R t ) = π()d a (R () ) R()

5 2 D n(r t+) X π(, ) 4 X, 3 π(, ) X π(j,, )( β td n(r t)) j A5 j 2 + O(D n(r t) 2 ) (2) We express D a (R () ) n terms of D n (R t ):.7 Evouton of nb 4 Evouton of Ω D a (R () ) mn D a (R () ) β t D n (R t ) as soon as β t s a functon of the teraton such that β t mn D a (R () ) D n (R t ). Thus, we obtan equaton (2). We then can prove that f Ω < then there exsts α such that f D n (R t ) < α at some teraton t, then m t D n (R t ) =. Moreover, f X () s a symmetrc probabty-vector random varabes of sze q, then D n (X () ) 2 P e (X () ) (q )D n (X () ) (3) q 2 Let us remnd that D n (R t ) = π()d n( t ), and that the sequence D n (R t ) t=t converges to zero. Thus for a, the sequences D n ( t ) t=t aso converge to zero. And hence P e ( t ) converges to zero. Ths s true for a, and snce we have P e (R t ) = π()p e( t ), P e (R t ) aso converges to zero. Ths proves the suffcency of the stabty condton. Thus, we have proved that, provded that a fxed pont of densty evouton exsts for hybrd codes, ths pont can be stabe under certan condton. Our hybrd codes have hence threshod behavor. V. ANALYSIS OF THE STABILITY CONDITION Now we are abe to compare the stabty condtons for hybrd LDPC codes whose hghest order group s G(q) and for non-bnary LDPC codes defned on the hghest fed GF (q). To ustrate advantages of hybrd codes over non-bnary codes concernng the stabty, we consder on fgure 3 a code rate of one haf, acheved by non-bnary codes on GF (q), wth q = , and hybrd codes of type G(2) G(q), hence wth graph rate varyng wth q. The nformaton part of hybrd LDPC codes s n G(2), and the redundancy n G(q). We assume reguar Tanner graphs for those codes, wth connecton degree of varabe nodes d v = 2. The connecton degree of chec nodes w be hence varyng whth the graph rate for hybrd codes. We consder BI-AWGN channe whose varance σ 2 s set to. We denote by Ω nb and Ω hyb the parameters of GF (q) LDPC codes and hybrd LDPC codes, respectvey. The same for nb and hyb. Remar : We note, on fgure 3, that Ω hyb Ω nb and hyb nb. Hence, wth those assumptons, a fxed pont of densty evouton s stabe at ower SNR for hybrd codes than for GF (q) codes. Remar 2: For a usua non-bnary GF (q) LDPC code, the hyb q Ω d v 2 Ω nb Ω hyb q Fg. 3. Channe and code parameters and Ω for hybrd and non-hybrd codes n terms of maxmum symbo order q. These fgures show that a hybrd code can be stabe when a non-bnary code s not. hybrd stabty condton reduces to non-hybrd stabty condton, gven by: Ω nb = ρ ()λ () nb = q exp( q 2σ 2 n ) wth n, the number of ones n the bnary map of G(q). Under ths form, we can prove that nb tends to zero as q goes to nfnty. On BI-AWGN channe, ths means that any fxed pont of densty evouton s stabe as q tends to nfnty for non-bnary LDPC codes, and for hybrd codes too (because of Remar ). Those resuts ndcate that optmzaton procedures w be more effcent snce there exst more stabe hybrd codes than non-hybrd LDPC codes for a gven set of channe and code parameters. The optmzaton and code desgn s reported n a future appcaton. REFERENCES [] M. Davey and D.J.C. MacKay, Low Densty Party Chec Codes over GF(q), IEEE Commun. Lett., vo. 2, pp , June 99. [2] D. Srdhara and T.E. Fuja, Low Densty party Chec Codes over Groups and Rngs, n the proc. of ITW 2, Bangadore, Inda, Oct. 22. [3] X.-Y. Hu and E. Eeftherou, Bnary Representaton of Cyce Tanner- Graph GF(2 q ) Codes, n the proc. of ICC 4, pp , Pars, France, June 24. [4] A. Bennatan and Davd Burshten, Desgn and Anayss of Nonbnary LDPC Codes for Arbtrary Dscrete-Memoryess Channes, IEEE Trans. on Inform. Theory, vo. 52, no. 2, pp , Feb. 26. [5] A. Goup, M. Coas, G. Gee and D. Decercq, FFT-based BP Decodng of Genera LDPC Codes over Abean Groups, to appear n the IEEE Trans. on Commun., 26. [6] T. Rchardson, A. Shoroah and R. Urbane, Desgn of Capacty- Approachng Irreguar LDPC Codes, IEEE Trans. on Inform. Theory, vo. 47, no. 2, pp , Feb. 2. [7] L. Sassate and D. Decercq, Non-bnary Hybrd LDPC Codes: Structure, Decodng and Optmzaton, n IEEE Inform. Theory Worshop, Chengdu, Chna, October 26, cache/cs/pdf/7/766v.pdf. [] C. Pouat, M. Fossorer and D. Decercq, Usng Bnary Image of Nonbnary LDPC Codes to Improve Overa Performance, n IEEE Intern. Symp. on Turbo Codes, Munch, Apr 26.