Density Evolution and Functional Threshold. for the Noisy Min-Sum Decoder

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1 Dnsity Evolution and Functional Thrshold 1 for th Noisy Min-Sum Dcodr (Extndd Vrsion arxiv: v1 [cs.it] 26 May 214 Christian L. Kamni Ngassa,#, Valntin Savin, Elsa Dupraz #, David Dclrcq # CEA-LETI, Minatc Campus, Grnobl, Franc {christian.kamningassa, valntin.savin}@ca.fr # ETIS, ENSEA / CNRS UMR-851 / Univ. Crgy-Pontois, Franc {lsa.dupraz, dclrcq}@nsa.fr Abstract This papr invstigats th bhavior of th Min-Sum dcodr running on noisy dvics. Th aim is to valuat th robustnss of th dcodr in th prsnc of computation nois,.g. du to faulty logic in th procssing units, which rprsnts a nw sourc of rrors that may occur during th dcoding procss. To this nd, w first introduc probabilistic modls for th arithmtic and logic units of th th finit-prcision Min-Sum dcodr, and thn carry out th dnsity volution analysis of th noisy Min-Sum dcodr. W show that in som particular cass, th nois introducd by th dvic can hlp th Min-Sum dcodr to scap from fixd points attractors, and may actually rsult in an incrasd corrction capacity with rspct to th noislss dcodr. W also rval th xistnc of a spcific thrshold phnomnon, rfrrd to as functional thrshold. Th bhavior of th noisy dcodr is dmonstratd in th asymptotic limit of th cod-lngth by using noisy dnsity volution quations and it is also vrifid in th finit-lngth cas by Mont-Carlo simulation. I. INTRODUCTION In traditional modls of communication or storag systms with rror corrction coding, it is assumd that th oprations of an rror corrction ncodr and dcodr ar dtrministic and that th randomnss This work was supportd by th Svnth Framwork Programm of th Europan Union, undr Grant Agrmnt numbr (i-risc projct. This papr is an xtndd vrsion of th papr with sam titl, submittd to IEEE Transactions on Communications

2 2 xists only in th transmission or storag channl. Howvr, with th advnt of nanolctronics, th rliability of th forthcoming circuits and computation dvics is bcoming qustionabl. It is thn bcoming crucial to dsign and analyz rror corrcting dcodrs abl to provid rliabl rror corrction vn if thy ar mad of unrliabl componnts. Excpt th pionrd works by Taylor and Kuzntsov on rliabl mmoris [1] [3], latr gnralizd in [4], [5] to th cas of hard-dcision dcodrs, this nw paradigm of noisy dcodrs has mrly not bn addrssd until rcntly in th coding litratur. Howvr, ovr th last yars, th study of rror corrcting dcodrs, spcially Low-Dnsity Parity-Chck (LDPC dcodrs, running on noisy hardwar attractd mor and mor intrst in th coding community. In [6] and [7] hardwar rdundancy is usd to dvlop fault-compnsation tchniqus, abl to protct th dcodr against th rrors inducd by th noisy componnts of th circuit. In [8], a class of modifid Turbo and LDPC dcodrs has bn proposd, abl to dal with th nois inducd by th failurs of a low-powr buffring mmory that stors th input soft bits of th dcodr. Vry rcntly, th charactrization of th ffct of noisy procssing on mssag-passing itrativ LDPC dcodrs has bn proposd. In [9], th concntration and convrgnc proprtis wr provd for th asymptotic prformanc of noisy mssag-passing dcodrs, and dnsity volution quations wr drivd for th noisy Gallagr-A and Blif-Propagation dcodrs. In [1] [12], th authors invstigatd th asymptotic bhavior of th noisy Gallagr-B dcodr dfind ovr binary and non-binary alphabts. Th Min-Sum dcoding undr unrliabl mssag storag has bn invstigatd in [13], [14]. Howvr, all ths paprs dal with vry simpl rror modls, which mulat th noisy implmntation of th dcodr, by passing ach of th xchangd mssags through a noisy channl. In this work w focus on th Min-Sum dcodr, which is widly implmntd in ral communication systms. In ordr to mulat th noisy implmntation of th dcodr, probabilistic rror modls ar proposd for its arithmtic componnts (addrs and comparators. Th proposd probabilistic componnts ar usd to build th noisy finit-prcision dcodrs. W furthr analyz th asymptotic prformanc of th noisy Min-Sum dcodr, and provid usful rgions and targt-ber-thrsholds [9] for a wid rang of paramtrs of th proposd rror modls. W also highlight a wid varity of mor or lss convntional bhaviors and rval th xistnc of a spcific thrshold phnomnon, which is rfrrd to as functional thrshold. Finally, th asymptotic rsults ar also corroboratd through finit lngth simulations. Th rmaindr of th papr is organizd as follows. Sction II givs a brif introduction to LDPC cods and itrativ dcoding. Sction III prsnts th rror modls for th arithmtic componnts. Th dnsity volution quations and asymptotic analysis of th noisy finit-prcision Min-Sum dcoding ar prsntd in Sction V and Sction VI rspctivly. Sction VII provids th finit-lngth prformanc

3 3 and Sction VIII concluds th papr. II. LDPC CODES AND THE MIN-SUM ALGORITHM A. LDPC Cods LDPC cods [15] ar linar block cods dfind by spars parity-chck matrics. Thy can b advantagously rprsntd by bipartit (Tannr graphs [16] and dcodd by mssag-passing (MP itrativ algorithms. Th Tannr graph of an LDPC cod is a bipartit graph H, whos adjacncy matrix is th parity-chck matrix H of th cod. Accordingly, H contains two typs of nods: variabl-nods, corrsponding to th columns of H, or quivalntly to th codword bits, and chck-nods, corrsponding to th rows of H, or quivalntly to th parity quations th codword bits ar chckd by. W considr an LDPC cod dfind by a Tannr graph H, with N variabl-nods and M chck-nods. Variabl-nods ar dnotd by n {1,2,...,N}, and chck-nods by m {1,2,...,M}. W dnot by H(n and H(m th st of nighbor nods of th variabl-nod n and of th chck-nod m, rspctivly. Th numbr of lmnts of H(n (or H(m is rfrrd to as th nod-dgr. Th Tannr graph rprsntation allows rformulating th probabilistic dcoding initially proposd by Gallagr [15] in trms of Blif-Propagation 1 (BP: an MP algorithm proposd by J. Parl in 1982 [17] to prform Baysian infrnc on trs, but also succssfully usd on gnral graphical modls [18]. Th BP dcoding is known to b optimal on cycl-fr graphs (in th sns that it outputs th maximum a postriori stimats of th codd bits, but can also b succssfully applid to dcod linar cods dfind by graphs with cycls, which is actually th cas of all practical cods. Howvr, in practical applications, th BP algorithm might b disadvantagd by its computational complxity and its snsitivity to th channl nois dnsity stimation (inaccurat stimation of th channl noisy dnsity may caus significant dgradation of th BP prformanc. B. Min-Sum Dcoding On way to dal with complxity and numrical instability issus is to simplify th computation of mssags xchangd within th BP dcoding. Th most complx stp of th BP dcoding is th computation of chck-to-variabl mssags, which maks us of computationally intnsiv hyprbolic tangnt functions. Th Min-Sum (MS algorithm is aimd at rducing th computational complxity of 1 Also rfrrd to as Sum-Product (SP

4 4 th BP, by using max-log approximations of th parity-chck to codd-bit mssags [19] [21]. Th only oprations rquird by th MS dcoding ar additions, comparisons, and sign (±1 products, which solvs th complxity and numrical instability problms. Th prformanc of th MS dcoding is also known to b indpndnt of th channl nois dnsity stimation, for most of th usual channl modls. For th sak of simplicity, w only considr transmissions ovr binary-input mmorylss noisy channls, and assum that th channl input alphabt is { 1, +1}, with th usual convntion that +1 corrsponds to th -bit, and 1 corrsponds to th 1-bit. W furthr considr a codword x = (x 1,...,x N { 1,+1} N and dnot by y = (y 1,...,y N th rcivd word. Th following notation will b usd throughout th papr, with rspct to mssag passing dcodrs: γ n is th log-liklihood ratio (LLR valu of x n according to th rcivd y n valu; it is also rfrrd to as th a priori information of th dcodr concrning th variabl-nod n; γ n is th a postriori information (LLR valu of th dcodr concrning th variabl-nod n; α m,n is th variabl-to-chck mssag snt from variabl-nod n to chck-nod m; β m,n is th chck-to-variabl mssag snt from chck-nod m to variabl-nod n. Th (infinit prcision MS dcoding is dscribd in Algorithm 1. It consists of an initialization stp (in which variabl-to-chck mssags ar initializd according to th a priori information of th dcodr, followd by an itration loop, whr ach itration compriss thr main stps as follows: CN-procssing (chck-nod procssing stp: computs th chck-to-variabl mssags β m,n ; VN-procssing (variabl-nod procssing stp: computs th variabl-to-chck mssags α m,n ; AP-updat (a postriori information updat stp: computs th a postriori information γ n. Morovr, ach itration also compriss a hard dcision stp, in which ach transmittd bit is stimatd according to th sign of th a postriori information, and a syndrom chck stp, in which th syndrom of th stimatd word is computd. Th MS dcoding stops whn whthr th syndrom is +1 (th stimatd word is a codword or a maximum numbr of itrations is rachd. Th a priori information (LLR of th dcodr is dfind by γ n = log Pr(x n = +1 y n, and for th Pr(x n = 1 y n two following channl modls (prdominantly usd in this work, it can b computd as follows: For th Binary Symmtric Channl (BSC, y { 1,+1} N is obtaind by flipping ach ntry of x with som probability ε, rfrrd to as th channl s crossovr probability. Consquntly: ( 1 ε γ n = log y n (1 ε For th Binary-Input Additiv Whit Gaussian Nois (BI-AWGN channl, y R N is obtaind by

5 5 Algorithm 1 Min-Sum (MS dcoding Input: y = (y 1,...,y N Y N (Y is th channl output alphabt Output: ˆx = (ˆx 1,...,ˆx N { 1,+1} N Initialization for all n = 1,...,N do γ n = log Pr(x n = +1 y n Pr(x n = 1 y n ; rcivd word stimatd codword for all n = 1,...,N and m H(n do α m,n = γ n ; Itration Loop for all m = 1,...,M and n H(m do β m,n = ( sgn(α m,n min α m,n ; n H(m\n n H(m\n for all n = 1,...,N and m H(n do α m,n = γ n + β m,n; m H(n\m for all n = 1,...,N do γ n = γ n + β m,n ; m H(n for all {v n } n=1,...,n do ˆx n = sgn( γ n ; if ˆx is a codword thn xit th itration loop CN-procssing VN-procssing AP-updat hard dcision syndrom chck End Itration Loop y n = x n +z n, whr z n is th whit Gaussian nois with varianc σ 2. It follows that: γ n = 2 σ 2y n (2 Rmark: It can b asily sn that if th a priori information vctor γ = (γ 1,...,γ N is multiplid by a constant valu, this valu will factor out from all th procssing stps in Algorithm 1 (throughout th dcoding itrations, and thrfor it will not affct in any way th dcoding procss. It follows that for both th BSC and BI-AWGN channl modls, on can simply dfin th a priori information of th dcodr by γ n = y n, n = 1,...,N. III. ERROR INJECTION AND PROBABILISTIC MODELS FOR NOISY COMPUTING A. Noisy Mssag-Passing dcodrs Th modl for noisy MP dcodrs proposd in [9] incorporats two diffrnt sourcs of nois: computation nois du to noisy logic in th procssing units, and mssag-passing nois du to noisy wirs

6 6 (or noisy mmoris usd to xchang mssags btwn nighbor nods. Th computation nois is modld as a random variabl, which th variabl-nod or th chcknod procssing dpnds on. Put diffrntly, an outgoing mssag from a (variabl or chck nod dpnds not only on th incoming mssags to that nod (including th a priori information for th variabl-nod procssing, but also on th ralization of a random variabl which is assumd to b indpndnt of th incoming mssags. Th mssag-passing nois is simply modld as a noisy channl. Hnc, transmitting a mssag ovr a noisy wir is mulatd by passing that mssag through th corrsponding noisy channl. Howvr, in [9] it has bn notd that thr is no ssntial loss of gnrality by combining computation nois and mssag-passing nois into a singl form of nois (s also [22, Lmma 3.1]. Consquntly, th approach adoptd has bn to mrg noisy computation into mssag-passing nois, and to mulat noisy dcodrs by passing th xchangd mssags through diffrnt noisy channl modls. Thus, th noisy Gallagr-A dcodr has bn mulatd by passing th xchangd mssags ovr indpndnt and idntical BSC wirs, whil th noisy BP dcodr has bn mulatd by corrupting th xchangd mssags with boundd and symmtrically distributd additiv nois (.g. uniform nois or truncatd Gaussian nois. Th approach w follow in this work diffrs from th on in [9] in that th computation nois is modld at th lowr lvl of arithmtic and logic oprations that compos th variabl-nod and chcknod procssing units. This finr-graind nois modling is aimd at dtrmining th lvl of nois that can b tolratd in ach typ of opration. As th main focus of this work is on computation nois, w shall considr that mssags ar xchangd btwn nighbor nods through rror-fr wirs (or mmoris. Howvr, w not that this work can radily b xtndd to includ diffrnt rror modls for th mssag-passing nois (as dfind in [9]. Altrnativly, w may assum that th mssag-passing nois is mrgd into th computation nois, in th sns that adding nois in wirs would modify th probabilistic modl of th noisy logic or arithmtic oprations. B. Error Injction Modls W only considr th cas of finit-prcision oprations, maning that th inputs (oprands and th output of th oprator ar assumd to b boundd intgr numbrs. W simulat a noisy oprator by injcting rrors into th output of th noislss on. In th following, V Z dnots a finit st consisting of all th possibl outputs of th noislss oprator.

7 7 Dfinition 1: An rror injction modl on V, dnotd by (E,p E,ı V, is givn by: A finit rror st E Z togthr with a probability mass function p E : E [,1], rfrrd to as th rror distribution; A function ı : V E V, rfrrd to as th rror injction function. For a givn st of inputs, th output of th noisy oprator is th random variabl dfind by ı(v,, whr v V is th corrsponding output of th noislss oprator, and is drawn randomly from E according to th probability distribution p E. Th rror injction probability is dfind by p = 1 δ ı(v, v V p E(, (3 v whr δ v ı(v, = if v = ı(v,, and δ v ı(v, = 1 if v ı(v,. In othr word, p = Pr(v ı(v,, assuming that v is drawn uniformly from V and is drawn from E according to p E. Th abov dfinition maks som implicit assumptions which ar discussd blow. Th st of possibl outputs of th noisy oprator is th sam as th st of possibl outputs of th noislss oprator (V. This is justifid by th fact that, in most common cass, V is th st of all (signd or unsignd intgrs that can b rprsntd by a givn numbr of bits. Thus, rror injction will usually altr th bit valus, but not th numbr of bits. Th injctd rror dos not dpnd on th output valu of th noislss oprator and, consquntly, nithr on th givn st of inputs. In othr words, th injctd rror is indpndnt on th data procssd by th noislss oprator. Th validity of this assumption dos actually dpnd on th siz of th circuit implmnting th oprator. Indd, this assumption tnds to hold fairly wll for larg circuits [23], but bcoms mor tnuous as th circuit siz dcrass. Obviously, it would b possibl to dfin mor gnral rror injction modls, in which th injctd rror would dpnd on th data (currntly and/or prviously procssd by th oprator. Such an rror injction modl would crtainly b mor ralistic, but it would also mak it vry difficult to analytically charactriz th bhavior of noisy MP dcodrs. As a sid ffct, th dcoding rror probability would b dpndnt on th transmittd codword, which would prvnt th us of th dnsity volution tchniqu for th analysis of th asymptotic dcoding prformanc (sinc th dnsity volution tchniqu rlis on th all-zro codword assumption. Howvr, th fact that th rror injction modl is data indpndnt dos not guarant that th dcoding rror probability is indpndnt of th transmittd codword. In ordr for this to happn, th rror injction modl must also satisfy a symmtry condition that can b statd as follows.

8 8 Dfinition 2: An rror injction modl (E,p E,ı V is said to b symmtric if V is symmtric around th origin (maning that v V v V, but dos not ncssarily blong to V, and th following quality holds φ (ı V { ı(v,=w} p E ( = { ı( v,= w} p E (, v,w V (4 Th maning of th symmtry condition is as follows. Lt V b a random variabl on V. Lt φ (ı V and dnot th probability mass functions of th random variabls obtaind by injcting rrors in th output of V and V, rspctivly. Thn th abov symmtry condition is satisfid if and only if for any V th following quality holds φ (ı V (w = φ(ı V ( w, w V (5 A particular cas in which th symmtry condition is fulfilld is whn ı( v, = ı(v,, for all v V and E. In this cas, th rror injction modl is said to b highly symmtric. Mssags xchangd within mssag-passing dcodrs ar gnrally in blif-format, maning that th sign of th mssag indicats th bit stimat and th magnitud of th mssag th confidnc lvl. As a consqunc, rrors occurring on th sign of th xchangd mssags ar xpctd to b mor harmful than thos occurring on thir magnitud. This motivats th following dfinition, which will b usd in th following sction (s also th discussion in Sction IV-C. Dfinition 3: An rror injction modl (E,p E,ı V is said to b sign-prsrving if for any v V and E, v and ı(v, ar ithr both non-ngativ ( or both non-positiv (. C. Bitwis-XOR Error Injction W focus now on th two main symmtric rror injction modls that will b usd in this work. Both modls ar basd on a bitwis XOR opration btwn th noislss output v and th rror. Th two modls diffr in th dfinition of th rror st E, which is chosn such that th bitwis XOR opration may or may not affct th sign of th noislss output. In th first cas, th bitwis XOR rror injction modl is said to b full-dpth, whil in th scond it is said to b sign-prsrving. Ths rror injction modls ar rigorously dfind blow. In th following, w fix θ 2 and st V = { Θ,..., 1,,+1,...,+Θ}, whr Θ = 2 θ W also fix a signd numbr binary rprsntation, which can b any of th sign-magnitud, on s complmnt, or two s complmnt rprsntation. Thr ar xactly 2 θ signd numbrs that can b

9 9 rprsntd by θ bits in any of th abov formats, on of which dos not blong to V (not that V contains only 2Θ+1 = 2 θ 1 lmnts for symmtry rasons!. W dnot this lmnt by ζ. Hnc: In sign-magnitud format, ζ =, with binary rprsntation 1 ; In on s complmnt format, ζ =, with binary rprsntation 11 1; In two s complmnt format, ζ = (Θ+1, with binary rprsntation 1. For any u,v V, w dnot by u v th bitwis XOR opration btwn u and v. From th abov discussion, it follows that u v V {ζ}. 1 Full-dpth rror injction: For this rror modl, th rror st is E = V. Th rror injction probability is dnotd by p, and all th possibl rror valus ar assumd to occur with th sam probability (for symmtry rasons. It follows that th rror distribution function is givn byp E ( = 1 p and p E ( = p 2Θ,. Finally, th rror injction function is dfind by: v, if v V ı(v, =, if v = ζ (6 2 Sign-prsrving rror injction: For this rror modl, th rror st is E = {,+1,...,+Θ}. Th rror injction probability is dnotd by p, and all th possibl rror valus ar assumd to occur with th sam probability (for symmtry rasons. It follows that th rror distribution function is givn by p E ( = 1 p and p E ( = p Θ,. Finally, th rror injction function is dfind by: v, if v and v V ı(v, = ±, if v =, if v = ζ In th abov dfinition, ı(, is randomly st to ithr or +, with qual probability (this is du onc again to symmtry rasons. Not also that th last two conditions, namly v = and v = ζ, cannot hold simultanously (sinc ζ. Finally, w not that both of th abov modls ar highly symmtric, if on of th sign-magnitud or th on s complmnt rprsntation is usd. In cas that th two s complmnt rprsntation is usd, thy ar both symmtric, but not highly symmtric. An xampl of sign-prsrving bitwis-xor rror injction is givn in Tabl I. Th numbr of bits is θ = 5 and two s complmnt binary rprsntation is usd. Th sign bit of th rror is not displayd, as it is qual to zro for any E. Th positions of 1 s in th binary rprsntation of corrspond to th positions of th rronous bits in th noisy output. (7

10 1 Tabl I EXAMPLE OF SIGN-PRESERVING BITWISE-XOR ERROR INJECTION intgr 2 s complmnt binary rprsntation noislss output: v rror: noisy output: ı(v, bit position θ = Rmark: It is also possibl to dfin a variabl dpth rror injction modl, in which rrors ar injctd in only th λ last significant bits, with λ θ. Hnc, λ = θ corrsponds to th abov full-dpth modl, whil λ = θ 1 corrsponds to th sign-prsrving modl. Howvr, for λ < θ 1 such a modl is not symmtric, if th th two s complmnt rprsntation is usd. D. Output-Switching Error Injction A particular cas is rprsntd by rror injction on binary output. Assuming that V = {,1}, th bit-flipping rror injction modl is dfind as follows. Th rror st is E = {,1}, with rror distribution function givn by p E ( = 1 p and p E (1 = p, whr p is th rror injction probability, and th rror injction function is givn by ı(v, = v. Put diffrntly, th rror injction modl flips th valu of a bit in V with probability p. Clarly, th abov rror injction modl can b applid on any st V with two lmnts, by switching on valu to anothr with probability p. In this cas, w shall rfr to this rror injction modl as output-switching, rathr than bit-flipping. Morovr, if on taks V = { 1,+1} (with th usual,1 to ±1 convrsion, it can b asily vrifid that this rror injction modl is highly symmtric. E. Probabilistic modls for noisy addrs, comparators and XOR-gats In this sction w dscrib th probabilistic modls for noisy addrs, comparators and xor-gats, built upon th abov rror injction modls. Ths probabilistic modls will b usd in th nxt sction, in ordr to mulat th noisy implmntation of th quantizd (finit-prcision MS dcodr. 1 Noisy addr modl: W considr a θ-bit addr, with θ 2. Th inputs and th output of th addr ar assumd to b in V = { Θ,..., 1,,+1,...,+Θ}, whr Θ = 2 θ 1 1.

11 11 W dnot by s V : Z V, th θ-bit saturation map, dfind by: Θ, if v < Θ s V (v = v, if v V +Θ, if v > +Θ For inputs (x,y V, th output of th noislss addr is dfind as s V (x +y. Hnc, for a givn rror injction modl (E,p E,ı V, th output of th noisy addr is givn by: a pr (x,y = ı(s V (x+y,, (9 whr is drawn randomly from E according to th probability distribution p E. Th rror probability of th noisy addr, assuming uniformly distributd inputs, is qual to th rror injction probability (paramtr p dfind in (3, and will b dnotd in th squl by p a. 2 Noisy comparator modl: Lt lt dnot th noislss lss than oprator, dfind by lt(x,y = 1 if x < y, and lt(x,y = othrwis. Th noisy lss than oprator, dnotd by lt pr, is dfind by injcting rrors on th output of th noislss on, according to th bit-flipping modl dfind in Sction III-D. In othr words, th output of th noislss lt oprator is flippd with som probability valu, which will b dnotd in th squl by p c. Finally, th noisy minimum oprator is dfind by: x, if lt pr (x,y = 1 m pr (x,y = y, if lt pr (x,y = 3 Noisy XOR modl: Th noisy XOR oprator, dnotd by x pr is dfind by flipping th output of th noislss oprator with som probability valu, which will b dnotd in th squl by p x (according to th bit-flipping rror injction modl in Sction III-D. It follows that: x y, with probability 1 p x x pr (x,y = (11 x y, with probability p x Assumption: W furthr assum that th inputs and th output of th XOR oprator may tak valus in ithr {,1} or { 1,+1} (using th usual,1 to ±1 convrsion. This assumption will b implicitly mad throughout th papr. Rmark: As a gnral rul, w shall rfr to a noisy oprator according to its undrlying rror injction modl. For instanc, a sign-prsrving (rsp. full-dpth or sign-prsrving bitwis-xord noisy addr, is a noisy addr whos undrlying rror injction modl is sign-prsrving (rsp. on of th bitwis- XOR rror injction modls dfind in Sction III-C. W shall also say that a noisy oprator is (highly symmtric if its undrlying rror injction modl is so. (8 (1

12 12 F. Nstd Oprators As it can b obsrvd from Algorithm 1, svral arithmtic/logic oprations must b nstd 2 in ordr to comput th xchangd mssags. Sinc all ths oprations (additions, comparisons, XOR ar commutativ, th way thy ar nstd dos not hav any impact on th infinit-prcision MS dcoding. Howvr, this is no longr tru for finit-prcision dcoding, spcially in cas of noisy oprations. Thrfor, on nds an assumption about how oprators xtnd from two to mor inputs. Our assumption is th following. For n 2 inputs, w randomly pick any two inputs and apply th oprator on this pair. Thn w rplac th pair by th obtaind output, and rpat th abov procdur until thr is only on output (and no mor inputs lft. Th formal dfinition gos as follows. Lt Ω Z and ω : Ω Ω Ω b a noislss or noisy oprator with two oprands. Lt {x i } i=1:n Ω b an unordrd st of n oprands. W dfin: ω({x i } i=1:n = ω( (ω(x π(1,x π(2,,x π(n, whr π is a random prmutation of 1,...,n. IV. NOISY MIN-SUM DECODING A. Finit-Prcision Min-Sum Dcodr W considr a finit-prcision MS dcodr, in which th a priori information (γ n and th xchangd mssags (α m,n and β m,n ar quantizd on q bits. Th a postriori information ( γ n is quantizd on q bits, with q > q (usually q = q +1, or q = q +2. W furthr dnot: M = { Q,..., 1,,+1,...,Q}, whrq = 2 q 1 1, th alphabt of both th a priori information and th xchangd mssags; M = { Q,..., 1,,+1,..., Q}, whr Q = 2 q 1 1, th alphabt of th a postriori information; q : Y M, a quantization map, whr Y dnots th channl output alphabt; s M : Z M, th q-bit saturation map (dfind in a similar mannr as in (8; s M : Z M, th q-bit saturation map Rmark: Th quantization map q dtrmins th q-bit quantization of th dcodr soft input. Sinc q is dfind on th channl input (i.. y n valus, it must also ncompass th computation of th corrsponding LLR valus, whnvr is ncssary (s also th Rmark at th nd of Sction II-B. 2 For instanc, (d n 1 additions whr d n dnots th dgr of th variabl-nod n ar rquird in ordr to comput ach α m,n mssag. Similarly, ach β m,n mssag rquirs (d m 2 XOR oprations and (d m 2 comparisons.

13 13 Saturation maps s M and s M dfin th finit-prcision saturation of th xchangd mssags and of th a postriori information, rspctivly. B. Noisy Min-Sum Dcodr Th noisy (finit-prcision MS dcoding is prsntd in Algorithm 2. W assum that q-bit addrs ar usd to comput both α m,n mssags in th VN-procssing stp, and γ n valus in th AP-updat procssing stp. This is usually th cas in practical implmntations 3, and allows us to us th sam typ of addr in both procssing stps. This assumption xplains as wll th q-bit saturation of α m,n mssags in th VN-procssing stp. Not also that th saturation of γ n valus is actually don within th addr (s Equation (9. Finally, w not that th hard dcision and th syndrom chck stps in Algorithm 2 ar assumd to b noislss. W not howvr that th syndrom chck stp is optional, and if missing, th dcodr stops whn th maximum numbr of itrations is rachd. C. Sign-Prsrving Proprtis Lt U dnot any of th VN-procssing or CN-procssing units of th noislss MS dcodr. W dnot by U pr th corrsponding unit of th noisy MS dcodr. W say that U pr is sign-prsrving if for any incoming mssags and any nois ralization, th outgoing mssag is of th sam sign as th mssag obtaind whn th sam incoming mssags ar supplid to U. Clarly, CN pr is sign-prsrving if and only if th XOR-oprator is noislss (p x =. In cas that th noisy XOR-oprator svrly dgrads th dcodr prformanc, it is possibl to incras its rliability by using classical fault-tolrant tchniqus (as for instanc modular rdundancy, or multi-voltag dsign by incrasing th supply voltag of th corrsponding XOR-gat. Th pric to pay, whn compard to th siz or th nrgy consumption of th whol circuit, would b rasonabl. Concrning th VN-procssing, it is worth noting that th VN pr is not sign-prsrving, vn if th noisy addr is. This is du to th fact that multipl addrs must b nstd in ordr to complt th VNprocssing. Howvr, a sign-prsrving addr might hav svral bnfits. First, th rror probability of th sign of variabl-nod mssags would b lowrd, which would crtainly hlp th dcodr. Scond, if th noisy addr is sign-prsrving and all th variabl-nod incoming mssags hav th sam sign, thn 3 In practical implmntation, th γ n is computd first, and thn α m,n is obtaind from γ n by subtracting th incoming β m,n mssag

14 14 Algorithm 2 Noisy Min-Sum (Noisy-MS dcoding Input: y = (y 1,...,y N Y N (Y is th channl output alphabt Output: ˆx = (ˆx 1,...,ˆx N { 1,+1} N rcivd word stimatd codword Initialization for all n = 1,...,N do γ n = q(y n ; for all n = 1,...,N and m H(n do α m,n = γ n ; Itration Loop for all m = 1,...,M and n H(m do CN-procssing β m,n = x pr ( {sgn(αm,n } n H(m\n mpr ( { αm,n } n H(m\n ; for all n = 1,...,N and m H(n do α m,n = a pr ( {γn } {β m,n} m H(n\m ; VN-procssing α m,n = s M (α m,n ; for all n = 1,...,N do AP-updat γ n = a pr ( {γn } {β m,n } m H(n ; for all {v n } n=1,...,n do ˆx n = sgn( γ n ; if ˆx is a codword thn xit th itration loop hard dcision syndrom chck End Itration Loop th VN pr dos prsrv th sign of th outgoing mssag. Put diffrntly, in cas that all th incoming mssags agr on th sam hard dcision, th noisy VN-procssing may chang th confidnc lvl, but cannot chang th dcision. This may b particularly usful, spcially during th last dcoding itrations. Finally, th motivation bhind th sign-prsrving noisy addr modl is to invstigat its possibl bnfits on th dcodr prformanc. If th bnfits ar worth it (.g. on can nsur a targt prformanc of th dcodr, th sign-bit of th addr could b protctd by using classical fault-tolrant solutions. V. DENSITY EVOLUTION A. Concntration and Convrgnc Proprtis First, w not that our dfinition of symmtry is slightly mor gnral than th on usd in [9]. Indd, vn if all th rror injction modls usd within th noisy MS dcodr ar symmtric, th noisy MS dcodr dos not ncssarily vrify th symmtry proprty from [9]. Howvr, this proprty is vrifid

15 15 in cas of highly symmtric fault injction 4. Nvrthlss, th concntration and convrgnc proprtis provd in [9] for symmtric noisy mssag-passing dcodrs, can asily b gnralizd to our dfinition of symmtry. W summariz blow th most important rsults; th proof rlis ssntially on th sam argumnts as in [9]. W considr an nsmbl of LDPC cods, with lngth N and fixd dgr distribution polynomials [24]. W choos a random cod C from this nsmbl and assum that a random codword x { 1,+1} N is snt ovr a binary-input mmorylss symmtric channl. W fix som numbr of dcoding itrations l >, and dnot by E (l C (x th xpctd fraction of incorrct mssags5 at itration l. Thorm 1: Assum that all th rror injction modls usd within th MS dcodr ar symmtric. Thn, th following proprtis hold: 1 [Conditional Indpndnc of Error] For any dcoding itration l >, th xpctd fraction of incorrct mssags E (l C (x dos not dpnd on x. Thrfor, w may dfin E(l C := E(l C (x. 2 [Cycl-Fr Cas] If th graph of C contains no cycls of lngth 2l or lss, th xpctd fraction of incorrct mssags E (l C dos not dpnd on th cod C or th cod-lngth N, but only on th dgr distribution polynomials; in this cas, it will b furthr dnotd by E (l (x. 3 [Concntration Around th Cycl-Fr Cas] For any δ >, th probability that E (l C lis outsid ( th intrval E (x δ,e (l (l (x+δ convrgs to zro xponntially fast in N. B. Dnsity Evolution Equations In this sction w driv dnsity volution quations for th noisy finit-prcision MS dcoding for a rgular (d v,d c LDPC cod. Th study can b asily gnralizd to irrgular LDPC cods, simply by avraging according to th dgr distribution polynomials. Th objctiv of th dnsity volution tchniqu is to rcursivly comput th probability mass functions of xchangd mssags, through th itrativ dcoding procss. This is don undr th indpndnc assumption of xchangd mssags, holding in th asymptotic limit of th cod lngth, in which cas th dcoding prformanc convrgs to th cycl-fr cas. Du to th symmtry of th dcodr, th analysis 4 According to th probabilistic modls introducd in Sction III-E, th noisy comparator and th noisy XOR-oprator ar highly symmtric, but th noisy addr dos not ncssarily b so! 5 Hr, mssags may hav any on of th thr following manings: variabl-nod mssags, or chck-nod mssags, or a postriori information valus.

16 16 can b furthr simplifid by assuming that th all-zro codword is transmittd through th channl. W not that our analysis applis to any mmorylss symmtric channl. Lt l > dnot th dcoding itration. Suprscript (l will b usd to indicat th mssags and th a postriori information computd at itration l. To indicat th valu of a mssag on a randomly slctd dg, w drop th variabl and chck nod indxs from th notation (and w procd in a similar mannr for th a priori and a postriori information. Th corrsponding probability mass functions ar dnotd as follows. C(z = Pr(γ = z, z M C (l ( z = Pr( γ (l = z, z M A (l (z = Pr ( α (l = z, z M B (l (z = Pr ( β (l = z, z M 1 Exprssion of th input probability mass function C: Th probability mass function C dpnds only on th channl and th quantization map q : Y M, whr Y dnots th channl output alphabt (Sction IV-A. W also not that for l =, w hav A ( = C. W giv blow th xprssion of C for th BSC and th BI-AWGN channl modls (s Sction II-B. For th BSC, th channl output alphabt is Y = { 1,+1}, whil for th BI-AWGN channl, Y = R. Lt µ b a positiv numbr, such that µ Q. Th quantization map q µ is dfind as follows: q µ : Y M, q µ (y = s M ([µ y], (12 whr [µ y] dnots th narst intgr to µ y, and s M is th saturation map (Sction IV-A. For th BSC, w will furthr assum that µ is an intgr. It follows that q µ (y = µ y, y Y = { 1,+1}. Considring th all-zro (+1 codword assumption, th probability mass function C can b computd as follows. For th BSC with crossovr probability ε: C(z = 1 ε, if z = µ ε, if z = µ, othrwis (13 For th BI-AWGN channl with nois varianc σ 2 : ( 1 q Q+.5 µ µσ, if z = Q ( ( C(z = q z.5 µ µσ q z+.5 µ µσ q, if z = +Q ( Q.5 µ µσ, if Q < z < +Q (14

17 whr q(x = 1 + xp 2π (also known as th Q-function. x ( u2 2 Exprssion of B (l as a function of A (l 1 : 2 du is th tail probability of th standard normal distribution In th squl, w mak th convntion that Pr(sgn( = 1 = Pr(sgn( = 1 = 1/2. Th following notation will b usd: y A [x,y] = A(z, for x y M z=x A [+,y] = 1 y 2 A(+ A(z, for y M, y > z=1 A [x, ] = A(+ A(z, for x M, x < z=x For th sak of simplicity, w drop th itration indx, thus B := B (l and A := A (l 1. W procd by rcursion on i = 2,...,d c 1, whr d c dnots th chck-nod dgr. Lt β 1 := α 1, and for i = 2,...,d c 1 dfin: β i = x pr (sgn(β i 1, sgn(α i m pr ( β i 1, α i Lt also B i 1 and B i dnot th probability mass functions of β i 1 and β i, rspctivly (hnc, B 1 = A. First of all, for z =, w hav: B i ( = Pr(β i = = A(B i 1 (+[B i 1 ((1 A(+A((1 B i 1 (](1 p c. 17 For z, w procd in svral stps as follows: For z > : F i df (z = Pr(β i z p x = [ ] = B i 1 [ +,z 1] A [z,q 1] +A [+,z 1]B i 1 [z,q 1] [ ] + B i 1 [1 z, ] A [ Q, z]+a [1 z, ]B i 1 [ Q, z] + B i 1 [z,q 1] A [z,q 1]+B i 1 [ Q, z] A [ Q, z] p c p c For z < : G df i (z = Pr(β i z p x = [ ] = B i 1 [ +, z 1] A [ Q,z]+A [+, z 1]B i 1 [ Q,z] p c [ ] + B i 1 [ z,q 1] A [z+1, ] +A [ z,q 1] B i 1 [z+1, ] + B i 1[ z,q 1] A [ Q,z] +A [ z,q 1] B i 1[ Q,z] p c F i (z = df Pr(β i z = (1 p x.f i (z+p x.g i ( z B i (z = Pr(β i = z = F i (z F i (z +1 Finally, w hav that B = B dc 1. G i (z = df Pr(β i z = (1 p x.g i (z+p x.f i ( z B i (z = Pr(β i = z = G i (z G i (z +1 3 Exprssion of A (l as a function of B (l and C: W driv at th sam tim th xprssion of as a function of B (l and C. C (l For simplicity, w drop th itration indx, so A := A (l, B := B (l, and C := C (l. W dnot ( by E,p E,ı M th rror injction modl usd to dfin th noisy addr. W dcompos ach noisy

18 18 addition into thr stps (noislss infinit-prcision addition, saturation, and rror injction, and procd by rcursion on i =,1,...,d v, whr d v dnots th variabl-nod dgr: For i = : df Ω = γ M M, C ( z = df Pr(Ω = z = For i = 1,...,d v : C( z, if z M, if z M\M df ω i = Ω i 1 +β mi,n Z, c i (w = df Pr(ω i = w = C u i 1 (ub(w u, w Z df ω i = s M (ω i M, c i ( w = df Pr( ω i = w = df Ω i = ı( ω i, M, Ci ( z = df Pr(Ω i = z = ω c i ( w, if w M\{± Q} w Q c i(w, if w = Q w + Q c i(w, if w = + Q δ z ı( ω, p E( c i ( ω, z M whr δ y x = 1 if x = y, and δ y x = if x y. Not that in th dfinition of Ω i abov, dnots an rror drown from th rror st E according to th rror probability distribution p E. Finally, w hav: A = s M ( Cdv 1 C = C dv In th first quation abov, applying th saturation oprator s M on th probability mass function C dv 1 mans that all th probability wights corrsponding to valus w outsid M must b accumulatd to th probability of th corrsponding boundary valu of M (that is, ithr Q or +Q, according to whthr w < Q or w < +Q. Rmark: If th noisy addr is dfind by on of th bitwis-xor rror injction modls (Sction III-C, thn th third quation from th abov rcursion (xprssion of C i as a function of c i may b rwrittn as follows: Sign-prsrving bitwis-xord noisy addr C i ( z = (1 p a c i ( z+ 1 Qp a ( ci[ ] c i (z, if z < (1 p a c i (+ 1 Q p a(1 c i (, if z = (15 (1 p a c i ( z+ 1 Qp a ( ci[ + ] c i (z, if z >

19 19 whr c i[ ] = ω< c i( ω+ 2 c 1 i(, and c i[ + ] = 1 2 c i(+ ω> c i( ω. Full-dpth bitwis-xord noisy addr C i ( z = (1 p a c i ( z+ 1 2 Q p a(1 c i ( z (16 Finally, w not that th dnsity volution quations for th noislss finit-prcision MS dcodr can b obtaind by stting p a = p c = p x =. C. Error Probability and Usful and Targt-BER rgions 1 Dcoding Error Probability: Th rror probability at dcoding itration l, is dfind by: P (l = 1 z= Q C (l ( z+ C (l ( 2 Proposition 1: Th rror probability at dcoding itration l is lowr-boundd as follows: (a For th sign-prsrving bitwis-xord noisy addr: P (l 1 2 Q p a. (b For th full-dpth bitwis-xord noisy addr: P (l 1 2 p a Q p a. Proof. (a Using C = C dv and quations (17 and (15, it follows that P (l 1 2 Q p a( 1 2 cdv [ ] + pa ( cdv [ ] 1 2 c d v ( (1 p a c dv [ ] Q p a( 1 2 cdv [ ] sinc th function (1 p a x+ 1 2 Q p a(1 2x is an incrasing function of x [,1]. (b Equations (17 and (16 imply thatp (l = 1 2 p a+(1 p a c dv [ ]+ 1 4 Q p a( 1 2 cdv [ ] (17 = (1 p a c dv [ ] + 1 p 2 Q a, 1 2 p a+ 1 4 Q p a Not that th abov lowr bounds ar actually infrrd from th rror injction in th last (th d v -th addition prformd whn computing th a postriori information valu. Thrfor, ths lowr bounds ar not xpctd to b tight. Howvr, if th channl rror probability is small nough, th sign-prsrving lowr bound provs to b tight in th asymptotic limit of l (this will b discussd in mor dtails in Sction VI. Not also that by protcting th sign of th noisy addr, th bound is lowrd by a factor of roughly Q, which rprsnts an xponntial improvmnt with rspct to th numbr of bits of th addr. In th asymptotic limit of th cod-lngth, P (l givs th probability of th hard bit stimats bing in rror at dcoding itration l. For th (noislss, infinit-prcision BP dcodr, th rror probability is usually a dcrasing function of l. This is no longr tru for th noislss, infinit-prcision MS dcodr, for which th rror probability may incras with l. Howvr, both dcodrs xhibit a thrshold

20 2 phnomnon, sparating th rgion whr rror probability gos to zro (as th numbr of dcoding itrations gos to infinity, from that whr it is boundd abov zro [24]. Things gt mor complicatd for th noisy (finit-prcision MS dcodr. First, th rror probability hav a mor unprdictabl bhavior. It dos not always convrg and it may bcom priodic 6 whn th numbr of itrations gos to infinity. Scond, th rror probability is always boundd abov zro (Proposition 1, sinc thr is a non-zro probability of fault injction at any dcoding itration. Hnc, a dcoding thrshold, similar to th noislss cas, cannot longr b dfind. Following [9], w dfin blow th notions of usful dcodr and targt rror rat thrshold. W considr a channl modl dpnding on a channl paramtr χ, such that th channl is dgradd by incrasing χ (for xampl, th crossovr probability for th BSC, or th nois varianc for th BI-AWGN channl. W will us subscript χ to indicat a quantity that dpnds on χ. Hnc, in ordr to account that P (l dpnds also on th valu of th channl paramtr, it will b dnotd in th following by P (l,χ. 2 Usful Rgion: Th first stp is to valuat th channl and hardwar paramtrs yilding a final probability of rror (in th asymptotic limit of th numbr of itrations lss than th input rror probability. Th lattr probability is givn by P,χ ( = 1 z= Q C(z+ 1 2C(, whr C is th probability mass function of th quantizd a priori information of th dcodr (s Sction V-B1. ( Following [9], th dcodr is said to b usful if is convrgnt, and: P (,χ P (l,χ l> df = lim P,χ (l < P,χ ( (18 l Th nsmbl of th paramtrs that satisfy this condition constituts th usful rgion of th dcodr. 3 Targt Error Rat Thrshold: For noislss-dcodrs, th dcoding thrshold is dfind as th suprmum channl nois, such that th rror probability convrgs to zro as th numbr of dcoding itrations gos to infinity. Howvr, for noisy dcodrs this rror probability dos not convrg to zro, and an altrnativ dfinition of th dcoding thrshold has bn introducd in [9]. Accordingly, for a targt bit-rror rat η, th η-thrshold is dfind 7 by: { } χ (η = sup χ P (,χ xists and P(,χ < η, χ [,χ] ( 6 In fact, for both BSC and BI-AWGN channls, th only cass w obsrvd, in which th squnc P (l l> (19 dos not convrg, ar thos cass in which this squnc bcoms priodic for l larg nough. { } 7 In [9], th η-thrshold is dfind by χ (η = sup χ P,χ ( xists and P,χ ( < η, and consquntly, thr might xist a channl paramtr valu χ < χ (η, for which P (,χ dos not xist. In ordr to avoid this happning, our dfinition is slightly diffrnt from th on in [9].

21 21 D. Functional Thrshold Although th η-thrshold dfinition allows dtrmining th maximum channl nois for which th bit rror probability can b rducd blow a targt valu, thr is not significant chang in th bhavior of th dcodr whn th channl nois paramtr λ incrass byond th valu of χ (η. In this sction, a nw thrshold dfinition is introducd in ordr to idntify th channl and hardwar paramtrs yilding to a sharp chang in th dcodr bhavior, similar to th chang that occurs around th thrshold of th noislss dcodr. This thrshold will b rfrrd to as th functional thrshold. Th aim is to dtct a sharp incras (.g. discontinuity in th rror probability of th noisy dcodr, whn λ gos byond this functional thrshold valu. Th thrshold dfinition w propos mak us of th Lipschitz constant of th function χ P ( (χ in ordr to dtct a sharp chang of P ( (χ with rspct to χ. Th dfinition of th Lipschitz constant is first rstatd for th sak of clarity. Dfinition 4: Lt f : I R b a function dfind on an intrval I R. Th Lipschitz constant of f in I is dfind as f(x f(y L(f,I = sup R + {+ } (2 x y I x y For a I and δ >, lt I a (δ = I (a δ,a + δ. Th (local Lipschitz constant of f in a I is dfind by: L(f,a = inf δ> L(f,I a(δ R + {+ } (21 Not that if a is a discontinuity point of f, thn L(f,a = +. On th opposit, if f is diffrntiabl in a, thn th Lipschitz constant in a corrsponds to th absolut valu of th drivativ. Furthrmor, if L(f,I < +, thn f is uniformly continuous on I and almost vrywhr diffrntiabl. In this cas, f is said to b Lipschitz continuous on I. Th functional thrshold is thn dfind as follows. Dfinition 5: For givn hardwar paramtrs and a channl paramtr χ, th dcodr is said to b functional if (a (b (c Th function x P ( (x is dfind on [,χ] P ( is Lipschitz continuous on [, χ] ( L P (,x is an incrasing function of x [,χ] Thn, th functional thrshold χ is dfind as: χ = sup{χ conditions (a,(b and (c ar satisfid} (22

22 22 Th us of th Lipschitz constant allows a rigorous dfinition of th functional thrshold, whil avoiding th us of th drivativ (which would rquir P ( (λ to b a picwis diffrntiabl function of λ. As it will b furthr illustratd in Sction VI, th functional thrshold corrsponds to a transition btwn two mods. Th first mod corrsponds to th channl paramtrs lading to a low lvl of rror probability, i.., for which th dcodr can corrct most of th rrors from th channl. In th scond mod, th ( channl paramtrs lad to a much highr rror probability lvl. If L P (, χ = +, thn χ is a ( discontinuity point of P ( and th transition btwn th two lvls is sharp. If L P (, χ < +, thn χ is an inflction point of P ( and th transition is smooth. With th Lipschitz constant, on can charactriz th transition in both cass. Howvr, th scond cas corrsponds to a dgnratd on, in which th hardwar nois is too high and lads to a non-standard asymptotic bhavior of th dcodr. That is why a st of admissibl hardwar nois paramtrs is dfind as follows. Dfinition 6: Th st of admissibl hardwar paramtrs is th st of hardwar nois paramtrs ( (p a,p c,p x for which L, χ = +. P ( In th following, as ach thrshold dfinition hlps at illustrating diffrnt ffcts, on or th othr dfinition will b usd, dpnding on th contxt. VI. ASYMPTOTIC ANALYSIS OF THE NOISY MIN-SUM DECODER In this sction, th dnsity volution quations drivd prviously ar usd to analyz th asymptotic prformanc (i.. in th asymptotic limit of both th cod lngth and numbr of itrations of th noisy MS dcodr. Unlss spcifid othrwis, th following paramtrs ar usd throughout this sction: Cod paramtrs: W considr th nsmbl of rgular LDPC cods with variabl-nod dgr d v = 3 and chck-nod dgr d c = 6 Quantization paramtrs: Th a priori information and xchangd mssags ar quantizd on q = 4 bits; hnc, Q = 7 and M = { 7,...,+7}. Th a postriori information is quantizd on q = 5 bits; hnc, Q = 15 and M = { 15,...,+15}. W analyz th dcoding prformanc dpnding on: Th quantization map q µ : Y M, dfind in Equation (12. Th factor µ will b rfrrd to as th channl-output scal factor, or simply th channl scal factor.

23 23 Th paramtrs of th noisy addr, comparator, and XOR-oprator, dfind rspctivly in Equations (9, (1, and (11. A. Numrical rsults for th BSC For th BSC, th channl output alphabt is Y = { 1,+1} and th quantization map is dfind by q µ ( 1 = µ and q µ (+1 = +µ, with µ {1,...,Q}. Th infinit-prcision MS dcodr (Algorithm 1, is known to b indpndnt of th scal factor µ. This is bcaus µ factors out from all th procssing stps in Algorithm 1, and thrfor dos not affct in any way th dcoding procss. This is no longr tru for th finit prcision dcodr (du to saturation ffcts, and w will show in this sction that, vn in th noislss cas, th scal factor µ can significantly impact th prformanc of th finit prcision MS dcodr. W start by analyzing th prformanc of th MS dcodr with quantization map q 1, and thn w will analyz its prformanc with an optimizd quantization map q µ. 1 Min-Sum dcodr with quantization map q 1 : Th casµ = 1 lads to an unconvntional bhavior, as in som particular cass th nois introducd by th dvic can hlp th MS dcodr to scap from fixd points attractors, and may actually rsult in an incrasd corrction capacity with rspct to th noislss dcodr. This bhavior will b discussd in mor dtails in this sction. W start with th noislss dcodr cas. Figur 1 shows th asymptotic rror probability P ( a function of p. It can b sn that P ( valu p th =.39, whr P ( dcrass slightly with p, until p rachs a thrshold drops to zro. This is th classical thrshold phnomnon mntiond in Sction V-C: for p > p th, th dcoding rror probability is boundd far abov zro (P ( for p < p th, on has P ( =. as >.31, whil Now, w considr a p valu slightly gratr than th thrshold of th noislss dcodr, and invstigat th ffct of th noisy addr on th dcodr prformanc. Lt us fix p =.6. Figur 2(a shows th dcoding rror probability at itration l, for diffrnt paramtrs p a {1 3,1 15,1 5 } of th noisy addr. For ach p a valu, thr ar two suprimposd curvs, corrsponding to th full-dpth ( fd, solid curv and sign-prsrving ( sp, dashd curv rror modls of th noisy addr. Th rror probability of th noislss dcodr is also plottd (solid black curv: it can b sn that it incrass rapidly from th initial valu P ( = p and closly approachs th limit valu P ( =.323 aftr a fw numbr of itrations. Whn th addr is noisy, th rror probability incrass during th first dcoding itrations, and bhavs similarly as in th noislss cas. It may approach th limit valu from th noislss cas, but starts dcrasing aftr som numbr of dcoding itrations. Howvr, it rmains

24 (3,6 rgular LDPC; (4,5 quantization; noislss MS P ( as function of p P ( (dcoding rror probability p th = p (crossovr probability Figur 1. Asymptotic rror probability P ( of th noislss MS dcodr as a function of p.35 (3,6 rgular LDPC, (4,5 quantization, noisy addr 1 (3,6 rgular LDPC, (4,5 quantization, noisy addr P (l (dcoding rror probability at itration l p a = 1E 5 p a = 1E 15 p a = 1E 3 Noislss MS add pr [fd, p a =1 3] add pr [sp, p a =1 3] add pr [fd, p a =1 15] add pr [sp, p a =1 15] add pr [fd, p a =1 5] add pr [sp, p a =1 5] Itration numbr (l P (l (dcoding rror probability at itration l Noislss MS add pr [fd, p a =1 3] add pr [sp, p a =1 3] add pr [fd, p a =1 15] add pr [sp, p a =1 15] add pr [fd, p a =1 5] p a = 1E 5 p a = 1E 15 add [sp, p =1 5] pr a p a = 1E Itration numbr (l (a P (l plottd in linar scal (b P (l plottd in logarithmic scal Figur 2. Effct of th noisy addr on th asymptotic prformanc of th MS dcodr (p =.6 boundd abov zro, according to th lowr bounds from Proposition 1. This can b sn in Figur 2(b, whr P (l plottd in logarithmic scal. Th asymptotic valus P ( and th corrsponding lowr-bounds valus from Proposition 1 ar shown in Tabl II. It can b sn that ths bounds ar tight, spcially in th sign-prsrving cas. Th abov bhavior of th MS dcodr is xplaind by th fact that th nois prsnt in th addr hlps th MS dcodr to scap from fixd points attractors. Figur 3 illustrats th volution of th probability mass function C (l for th noislss dcodr. At itration l =, C( is supportd in ±1, with C ( ( 1 = p and C ( (+1 = 1 p. It volvs during th itrativ dcoding, and rachs a fixd point of th dnsity volution for l = 2. Not that sinc all variabl-nods ar of dgr d v = 3, it can

25 25 Tabl II ASYMPTOTIC ERROR PROBABILITY OF THE MS DECODING WITH NOISY ADDER (p =.6 full p a P ( dpth lowr-bound sign P ( prsrving lowr-bound (3,6 rgular LDPC, (4,5 quantization, noislss MS, itr =.4 (3,6 rgular LDPC, (4,5 quantization, noislss MS, itr = 5 (3,6 rgular LDPC, (4,5 quantization, noislss MS, itr = 2.4 A Postriori Information PDF ( C Alphabt ( M A Postriori Information PDF ( C Alphabt ( M A Postriori Information PDF ( C Alphabt ( M (a Itration l = (b Itration l = 5 (c Itration l = 2 Figur 3. Probability mass function of th a postriori information C (l (noislss MS dcodr (3,6 rgular LDPC, (4,5 quantization, noisy MS, itr = 2.4 (3,6 rgular LDPC, (4,5 quantization, noisy MS, itr = (3,6 rgular LDPC, (4,5 quantization, noisy MS, itr = 3.35 A Postriori Information PDF ( C Alphabt ( M A Postriori Information PDF ( C Alphabt ( M A Postriori Information PDF ( C Alphabt ( M (a Itration l = 2 (b Itration l = 23 (c Itration l = 3 Figur 4. Probability mass function of th a postriori information C (l (MS dcodr with full-dpth noisy addr, p a = 1 15 b asily sn that, for l 1, C (l is supportd only on vn valus. Ths gaps in th probability mass function sm lad to favorabl conditions for th occurrnc of dnsity-volution fixd-points. Figur 4 illustrats th volution of th probability mass function C (l whn th full-dpth noisy addr with p a = 1 15 is usd within th MS dcodr. At itration l = 2, C (l is virtually th sam as in th noislss cas. Howvr, th noisy addr allows th dcodr to scap from this fixd-point, as it can b sn for itrations l = 23 and l = 3. For l > 3, th C (l movs furthr on th right, until th corrsponding rror probability P (l rachs th limit valu P ( =