Block-error performance of root-ldpc codes. Author(s): Andriyanova, Iryna; Boutros, Joseph J.; Biglieri, Ezio; Declercq, David

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1 Research Collecton Conference Paper Bloc-error perforance of root-ldpc codes Authors: Andryanova, Iryna; Boutros, Joseph J.; Bgler, Ezo; Declercq, Davd Publcaton Date: 00 Peranent Ln: Rghts / Lcense: In Copyrght - Non-Coercal Use Pertted Ths page was generated autoatcally upon download fro the ETH Zurch Research Collecton. For ore nforaton please consult the Ters of use. ETH Lbrary

2 Int. Zurch Senar on Councatons IZS, March 3-5, 00 Bloc-Error Perforance of Root-LDPC Codes Iryna Andryanova ETIS group ENSEA/UCP/CNRS-UMR Cergy-Pontose, France Joseph J. Boutros Elec. Eng. Departent Texas A&M Unversty at Qatar 3874, Doha, Qatar Ezo Bgler WISER S.r.l. Va Fue Lvorno, Italy e.bgler@eee.org Davd Declercq ETIS group ENSEA/UCP/CNRS-UMR Cergy-Pontose, France declercq@ensea.fr Abstract Ths paper nvestgates the error rate of root-ldpc RLDPC codes. These codes were ntroduced n [], as a class of codes achevng full dversty D over a nonergodc blocfadng transsson channel, and hence wth an error probablty decreasng as SNR D at hgh sgnal-to-nose ratos. As for ther structure, root-ldpc codes can be vewed as a specal case of ultedge-type LDPC codes []. However, RLDPC code optzaton for nonergodc channels does not follow the sae crtera as those appled for standard ergodc erasure or Gaussan channels. Whle prevous analyses of RLDPC codes were based on ther asyptotc bt threshold for nforaton varables under teratve decodng, n ths wor we nvestgate asyptotc bloc threshold. A stablty condton s frst derved for a gven fadng channel realzaton. Then, n a slar way as for unstructured LDPC codes [3], wth the help of Bhattacharyya paraeter, we state a suffcent condton for a vanshng bloc-error probablty wth the nuber of decodng teratons. I. INTRODUCTION AND MOTIVATION OF OUR WORK When a bloc of encoded data s sent, after beng splt nto F subblocs, through F ndependent slow-fadng channels, the approprate channel odel s nonergodc. Ths odel ay correspond to a parallel MIMO systes or to a sequental HARQ protocols data-transsson schee. It turns out that specal desgn crtera are needed for codes to be used wth such a odel n partcular, full transt dversty s sought, whch guarantees that, at large sgnal-tonose ratos SNR, the error probablty of the transsson schee scales as /SNR D, wth D the axu dversty order achevable. It has been shown n [] that standard sparsegraph code ensebles allow one to obtan error probabltes decreasng only as /SNR, and hence they are not fulldversty ensebles. Even nfnte-length rando code ensebles cannot acheve full dversty, as shown va a dversty populaton evoluton technque n [4]. The ey dea for codes achevng full dversty s to ensure that each nforaton node s recevng ultple essages affected by ndependent fadng coeffcents. Ths dea has been pleented n RLDPC codes [] desgned for bloc-fadng channels wth F = by ntroducng the concept of root checnodes. A root checnode protects a essage receved fro the second subchannel when the varable node s receved fro the frst subchannel. RLDPC codes are full-dversty codes thus, they are also Maxu Dstance Separable n the Ths wor was supported by the European FP7 ICT-STREP DAVINCI proect under the contract No. INFSO-ICT-603. Sngleton-bound sense and can be devsed for any dversty order. In ths paper we focus on rate-/, dversty- RLDPC codes, and study ther stablty under teratve decodng. We also derve a suffcent condton for vanshng bloc-error probablty. As expected, snce root checnodes occupy a sngle edge n each nforaton varable, stablty and blocerror perforance of RLDPC codes depend on the fracton of varables wth degrees and 3. II. TRANSMISSION MODEL Under our assuptons, a bloc of encoded data a codeword s dvded nto two equal subblocs, each one beng transtted over an ndependent Raylegh fadng channel wth SNR= γ and fadng coeffcents α and α. Therefore, the observaton y correspondng to the bnary transtted sybol x = ± receved fro the -th channel s y = α x + z, α [0, +, and z N0,σ wth σ =/γ. III. RLDPC CODES: DEFINITION AND DENSITY EVOLUTION A. Defnton Gven an ntal λ, ρ LDPC enseble, one defnes a λ, ρ RLDPC enseble wth dversty through the ultnoals λ root μ,x and ρ root μ,x, wth μ μ,μ and x x,x,x 3,x 4,x 5,x 6 : λ root μ,x λ μ x + λ μ x + λ μ x 3 +λ μ x 4 + λ μ x 5 + λ μ x 6, ρ root μ,x ρ x fe x 4 g e x 5 +x 6 fe x 3ge x, the fractons f e and g e wll be defned n next subsecton. In words, the structure of the RLDPC enseble conssts of four types of varable nodes, p,, p, two sets of chec nodes c, c, and 6 dfferent edge classes see Fg.a. Perutatons of edges wthn edge classes are chosen unforly at rando. Varable nodes and p correspond to nforaton and redundancy bts, respectvely, n a codeword 94

3 Int. Zurch Senar on Councatons IZS, March 3-5, 00 sent through the frst fadng subchannel. Slarly, varable nodes and p correspond to bts sent through the second subchannel. Note that the nforaton varable nodes =, are connected to chec nodes of the sae type, c, through exactly one edge; all other edges are connected to chec nodes of the other type. Redundancy varable nodes are always connected to chec nodes of dfferent type. In -, μ and μ correspond to two fadng subchannels, and the varables x,x,...,x 6 to the followng edge classes: c, c, p c, p c, c, and c. We have thus obtaned a code enseble of rate /. As shown n [], such a constructon guarantees transt dversty, whch s the axu we can obtan wth two ndependent transsson subchannels. a b Fg.. Structure of a λ, ρ RLDPC code enseble of dversty. B. Densty Evoluton RLDPC codes are decoded, as standard LDPC codes, usng an teratve algorth. An asyptotc analyss of teratve decodng s provded n [], [4] and we shall suarze t here, after gvng soe notaton. We denote the probablty densty functons pdfs of channel LLR outputs fro the two transsson subchannels by μ x and μ x, respectvely. These are noral pdfs wth eans α/γ and α/γ and varances 4α/γ and 4α/γ, respectvely. Further, we denote by the operaton of convoluton of two pdfs. We also defne the followng operaton: Defnton : The R-convoluton of two pdfs αx and βx s α βx =fˆαx ˆβx, and ˆαx αth x βth x, ˆβx x = x fx =cosh ˆα ˆβx th ˆα ˆβx. Note that the R-convoluton of pdfs corresponds to the followng operaton over the correspondng rando varables A and B: th tha/ + thb/, whch s exactly the operaton perfored at the chec nodes. Let us denote the average pdfs for 6 edge sets by q x, f x, g x, g x,f x, and q x as shown n Fg.b. Then the evoluton of the pdfs at the teraton + can be descrbed by the followng recursons: q + x = μ x λ ρq x,f e f x+g e g x f + x = μ x λ ρq x,f e f x+g e g x ρf e f x+g e g x g + x = μ x λ ρq x,f e f x+g e g x g + x = μ x λ ρq x,f e f x+g e g x f + x = μ x λ ρq x,f e f x+g e g x ρf e f x+g e g x q + x = μ x λ ρq x,f e f x+g e g x we have borrowed fro [4] the followng notaton: λx ρx f e d b d b d c d c λ λ + = λx db Also, we defne ρq, x λ x ; ρ x ; λ x ; d c d c db / dc / d b d b ; g e f e ; ρx d c ρ q x 3. IV. STABILITY CONDITIONS λ /; ρ /; ρ x. We are nterested n defnng stablty condtons for RLDPC codes. The an dffculty here les n the fact that not all essages need be recovered exactly or, n LDPC argon, not all pdfs converge to δ. It s not hard to prove that only the pdfs responsble for the convergence of nforaton essages,.e., f and f, need to converge for exact recovery of the nforaton bts ths condton s also suffcent. The an concept of the proof s that f and f are strctly better than q and q. In ths secton we derve the stablty condton for RLDPC codes based on the recovery of nforaton bts only. Before startng our dervaton, let us frst apply the tradtonal stablty condton [] to RLDPC codes, assung that all the code bts should be recovered. In such case the RLDPC codes are sply vewed as a ult-edge code enseble, A. RLDPCs as Mult-Edge Codes The stablty condton for ult-edge codes conssts n ensurng that the spectral radus of a atrx M s <, M BμΛP, wth Bμ the vector of Bhattacharyya paraeters for all transsson channels, the Λ atrx correspondng to the varable node sde of the graph, and P correspondng to 95

4 Int. Zurch Senar on Councatons IZS, March 3-5, 00 the chec node sde. Applyng the expressons derved n [], we fnd that Bμ = Bμ Bμ Bμ Bμ Bμ Bμ T I Λ= db λ d b λ db d λ λ b λ d b P = T P P P P 3 P 4 P 3 d b λ Fnally, the approxaton of f lnear n ε s obtaned as f = μ x λ + λ ρq x,g eg x + λ ε f ef q x,g eg x ρg eg x + ε f ef g eg x + const δ = μ x [ λ + λ ρq x,g eg x] ρg eg x + ε f e[ λ + λ ρq x,g eg x] F g eg x + λ ε f ef q x,g eg x ρg eg x + c δ wth P d c g e d c f e 0 P 0 ρ f e ρ g e 0 0 ρ P 3 ρ 0 0 ρ f e ρ g e 0 P 4 0 d c f e d c g e Note that two egenvalues of M are already 0. B. RLDPCs as Full-Dversty Codes wth By loong at RLDPC as at full-dversty codes, we only as C 0 x [ λ + λ ρq x,g e g x] ρg e g x for the convergence of f and f to δ. To derve a stablty C x [ λ + condton for ths case, assue that, at teraton, λ ρq x,g e g x] F g e g x C x λ F q x,g e g x ρg e g x f = ε δ 0 + ε δ, f = ε δ 0 + ε δ. Therefore, we have the followng relaton: and fnd an approxaton of essages f and f at the next f ε teraton whch s lnear n ε. f = a + f e A, ε To do ths, let us frst fnd a lnear approxaton of ρf e fx+g e gx: wth μ x C a 0 x μ ρf efx+g egx = ρf eεδ 0 + f e εδ + g egx = gx x C 0 x = and ρ ge gx + f eε fe ge gx μ x C x μ x C x A =0 μ x C x μ x C x + c δ = ρg egx + εf ef g egx + c δ, Denote now by qf the Bhattacharyya paraeter related to c s a constant, and ρg e gx denotes the frst ter the pdf f, n the su, whle F g e gx denotes the second one. Over Bf e x/ fxdx. the bnary erasure channel, ρg e gx and F g e gx can R be coputed explctly, whle, n the general case, the two B s closely related to the bt error probablty P functons should be coputed by runnng the densty evoluton b correspondng to fx, and t has been shown n [5] that P b teratons. Also note that one can bound the pdf of gx by the ntal pdf correspondng to the channel estate. If the 0 Bf 0. Knowng ths, and tang nto account the transsson channel s bad, the bound wll be qute tght. propertes of convoluton and of R-convoluton, we obtan that Next, ρq x,f efx+g egx = ρ ge q x gx ρ f eε fe 3 ge q x gx =0 + c δ = ρq x,g egx + εf ef q x,g egx + c δ. Further calculatons yeld λ ρq x,f e fx+g e gx = = λ ρq x,f e εδ 0 + f e εδ + g e gx = λ + λ ρq x,g e gx + εf e F q x,g e gx+c δ. = μ x C 0 x+ε f ec x+ε f ec x + c δ C 0 x [ λ + λ ρq x,g e g x] ρg e g x C x λ F q x,g e g x ρg e g x C x [ λ + λ ρq x,g e g x] F g e g x Slarly, f = μ x C0 x+ε f e C x+ε f e C x + c δ C = Bf Bf [C + f e BA] ε ε,. 0 Bμ λge ρ g e Bμ λge ρ g e 0 Note that we splfed the expressons by boundng Bg Bq and Bg Bq, and further boundng C 0 x and C 0 x. Defne next D C + f e BA. Then the followng recurrence relaton can be obtaned: Bf Bf Bf D Bf, 96

5 Int. Zurch Senar on Councatons IZS, March 3-5, 00 and hence, f we perfor teratons of densty evoluton, we obtan that Bf Bf D Bf 0 Bf 0, we assue that the essages q and g for any teraton are bounded by q 0 and g 0. We are nterested n the case of Bf decreasng to 0. Tang all the above nto account, we have the followng suffcent stablty condton for full-dversty codes: Theore Suffcency part of the stablty condton: The bt error probablty P e for a full-dversty RLDPC enseble converges to 0 f all the absolute values of the egenvalues of D are <. Notce that the usual stablty condton entoned n Secton IV-A depends on λ, whle the stablty condton derved here depends on both λ and λ 3, hdden n λ and λ. V. BLOCK-ERROR RATE OF RLDPC CODES The an result of ths paper s the study of the bloc-error probablty P B of RLDPC codes. Usng the suffcent part of the stablty condton derved above, we can ln P B to the bt-error probablty P b, and show n whch cases P b 0 ples P B 0. Usng a unon bound at soe teraton, we obtan P B n 4 P l b+ n 4 P l b 4 ax M l 6+ε P l b+ 4 ax M l 6+ε P l b, n s the code length, and ax M ax M s the axu nuber of varable nodes n a coputaton tree of a varable node fro the set n the bpartte graph, after teratons. The second nequalty follows fro the sae reasonng used n [3, Secton II], to whch we refer the reader desrng a detaled proof. Now, to ensure that, as, PB decreases to 0 whle P b 0, one has to ensure that Pb decreases wth faster than the axu nuber of varable nodes n the coputaton tree. A. Case of λ = λ 3 =0 Let us consder the sple case of both λ and λ 3 beng 0. ths s slar to the case of standard LDPC codes wth λ =0. Repeatng the calculatons of [3, Secton VI.A], we obtan P B + v d ax c 6+ε+ [Bf 3/ + Bf 3/ ], 4 dax whch decreases to 0 as. B. General case Gven that for varable nodes and Boutput =Π Bnput Boutput Π Bnput for chec nodes, and snce Bq and Bg for any, q, and g are no greater than the correspondng Bμ, one can bound BC λ g e ρ g e Bμ ax{bμ,bμ } BC λ + λ ρg e g e Bμ B C λ + λ ρg e g e Bμ B C λ g e ρ g e Bμ ax{bμ,bμ } and obtan Bf Bμ w Bf +w Bf Bf Bμ w Bf +w Bf wth w f e λ ρ g e and w f e λ + λ ρg e + λ g e ρ g e. Thus, wth a lnear approxaton, Bf + Bμ w Bf +Bμ Bμ ww Bf Bf + Bμ w Bf +Bμ Bμ ww Bf. Consequently, the bloc error probablty P B + 4 dax v d ax c 6+ε+ [Bμ w Bf + Bμ w Bf +Bμ Bμ ww {Bf +Bf }], can be seen to decrease to 0, as, f the followng condtons are satsfed: Bμ w d ax v d ax c 3, Bμ w d ax v d ax c 3. VI. CONCLUSION In ths paper we have derved the condtons under whch the bloc-error rate of a RLDPC code enseble decreases to 0 as the bt-error rate does the sae. The nterest of our fndngs les n the fact that results exstng n the lterature deal wth errors related to all the of code bts, whle for RLDPC only errors affectng nforaton bts should be consdered. REFERENCES [] J. Boutros, A. G. Fabregas, E. Bgler, and G. Zeor, Low-densty party-chec codes for nonergodc bloc-fadng channels, 007, subtted to IEEE Trans. Infor. Theory. [] T. Rchardson and R. Urbane, Mult-edge LDPC codes, 004, subtted to IEEE Trans. Infor. Theory. [Onlne]. Avalable: vay/ultedge.pdf [3] H. Jn and T. Rchardson, Bloc error teratve decodng capacty for ldpc codes, n ISIT 05, Adelade, Australa, Septeber 005. [4] J. Boutros, Dversty and codng gan evoluton n graph codes, n ITA 09, San-Dego, USA, February 009. [5] T. Rchardson, A. Shorollah, and R. Urbane, Desgn of capactyapproachng rregular low-densty party-chec codes, IEEE Trans. Infor. Theory, vol. 47, no., pp , February