LOW-density parity-check (LDPC) codes first discovered

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1 Desgn of LDPC Codes Robust to Nosy Message-Passng Decodng Alla Targhat, Student Member, IEEE, Hamed Farhad, Member, IEEE and Farshad Lahout, Senor Member, IEEE arxv: v3 [cs.it] Jun 05 Abstract We address nosy message-passng decodng of lowdensty party-check (LDPC) codes over addtve whte Gaussan nose channels. Message-passng decoders n whch certan processng unts teratvely exchange messages are common for decodng LDPC codes. The exchanged messages are n general subject to nternal nose n hardware mplementaton of these decoders. We model the nternal decoder nose as addtve whte Gaussan nose (AWGN) degradng exchanged messages. Usng Gaussan approxmaton of the exchanged messages, we perform a two-dmensonal densty evoluton analyss for the nosy LDPC decoder. Ths makes t possble to track both the mean, and the varance of the exchanged message denstes, and hence, to quantfy the threshold of the LDPC code n the presence of nternal decoder nose. The numercal and smulaton results are presented that quantfy the performance loss due to the nternal decoder nose. To partally compensate ths performance loss, we propose a smple method, based on EXIT chart analyss, to desgn robust rregular LDPC codes. The smulaton results ndcate that the desgned codes can ndeed compensate part of the performance loss due to the nternal decoder nose. I. INTRODUCTION LOW-densty party-check (LDPC) codes frst dscovered by Gallager [] and redscovered by Spelman et al. [] and MacKay et al. [3], [4] due to ther outstandng performance have attracted much nterest and have been studed extensvely durng the recent years. They have also been ncluded n several wreless communcaton standards, e.g., DVB-S and IEEE 80.6e [5], [6]. Effectve decodng of LDPC codes can be accomplshed by usng teratve messagepassng schemes such as belef-propagaton algorthm or sum product algorthm [7]. Moreover, powerful analytcal tools ncludng EXIT chart analyss [8], [9] and densty evoluton analyss [0] are developed for desgnng LDPC codes and quantfyng ther performance lmts. The sum-product algorthm s based on elementary computatons usng sum-product modules []. These modules can be mplemented usng dgtal crcuts or analog crcuts. The dgtal mplementaton of sum-product modules for LDPC decodng, e.g., the one presented n [], s subject to nosy message passng due to the quantzaton of messages. The mpact of quantzaton error on LDPC decodng performance A. Targhat s wth the ACCESS Lnnaeus Centre, department of sgnal processng, KTH Royal Insttute of Technology, Sweden (e-mal: allat@kth.se). H. Farhad s wth the Department of Sgnals and Systems, Chalmers Unversty of Technology, Sweden (e-mal: farhad@chalmers.se). F. Lahout s wth the Center for Wreless Multmeda Communcatons, School of Electrcal and Computer Engneerng, College of Engneerng, Unversty of Tehran, Iran (e-mal: lahout@ut.ac.r). The materal n ths paper was presented n part at the 6th IEEE Int. Symp. Turbo Codes and Iteratve Inform. Processng (ISTC), Brest, France, 00. s nvestgated n [3], where the smulatons ndcate the resultng substantal performance degradaton. Quantzaton error s often modeled as an addtve nose and s assumed Gaussan under certan condtons [4]. In [5], Varshney consders the mpact of quantzaton error on message passng algorthm, and suggests a sgnal-ndependent addtve truncated Gaussan nose model. More recently, Leduc-Prmeau et al. studed the mpact of tmng devatons (faults) on dgtal LDPC decoder crcuts n [6]. They model the devatons as addtve nose to the messages passed n the decoder, and show that under certan condtons the nose can be assumed Gaussan. Loelger et al. n [] and Hagenauer and Wnklhofer n [7] ntroduced soft gates n order to mplement sum-product modules usng analog transstor crcuts. They have also shown that by the varatons of a sngle basc crcut, the entre famly of sum-product modules can be mplemented. Usng these crcuts, any network of sum-product modules, n partcular, teratve decoder of LDPC codes can be drectly mplemented n analog very-large-scale ntegrated (VLSI) crcuts. In analog decoders, the exchanged messages are n general subject to addtve ntrnsc nose. The power of the nose depends n part on the chp temperature [8]. To capture ths phenomenon, n [8] Koch et al. consdered a channel that s subject to an addtve whte Gaussan nose. Ths channel s motvated by pont-to-pont communcaton between two termnals that are embedded n the same chp. Therefore, the nternal decoder nose may affect the communcaton of soft gates, and hence degrades the performance of the teratve analog decoder. Snce n practce dgtal or analog LDPC decoders are subject to nternal nose, the mpact of ths nose on the performance of teratve decodng needs to be nvestgated. Performance analyss of nosy LDPC decodng has attracted extensve nterest recently (see e.g. [5], [9] [3] and the references theren). The performance of a nosy bt-flppng LDPC decoder over a bnary symmetrc channel (BSC) s studed n [5]. In ths settng, the decoder messages are exchanged over bnary symmetrc nternal channels between the varable nodes and the check nodes. It has been shown that the performance degrades smoothly as the decoder nose probablty ncreases. Tabatabae et al. studed the performance lmts of LDPC decoder when t s subject to transent processor error [0], []. Ths research was further generalzed n [] by consderng both transent processor errors and permanent memory errors, usng densty evoluton analyss for regular LDPC codes. In ths paper, we analyze the performance of LDPC codes

2 v c u v v c c K u dv v u u 0 v dc u v v N (a) (b) Fg.. Tanner graph of a regular (3,6) LDPC code, where squares denote check nodes and crcles denote varable nodes. Fg.. (b). Message flow through a varable node (a), and through a check node transmtted over an AWGN communcaton channel, when a sum-product decodng algorthm s employed n whch exchanged messages are degraded by ndependent addtve whte Gaussan nose. We frst nvoke a densty evoluton (DE) analyss to track the probablty dstrbuton of exchanged messages durng decodng, and quantfy the performance degradaton due to the decoder nose. We compute the densty evoluton equatons for both regular and rregular LDPC codes. Also, we ntroduce an algorthm to fnd the EXIT curves of a nosy decoder. Fnally we propose an algorthm for the desgn of robust rregular LDPC codes usng EXIT chart for nosy decoders to partally compensate the performance loss due to the nternal decoder nose. Ths paper s organzed as follows. In Secton II, we present the defntons and model for nosy message-passng decoder. In Secton III, we derve the densty evoluton equatons for the nosy message-passng decoder. Next, numercal results of the densty evoluton analyss and smulaton results of fntelength codes are presented. In Secton IV, EXIT chart analyss of the nosy decoder s presented. Usng the EXIT charts, a method for desgnng robust codes for the nosy decoder s presented n Secton V. Fnally, Secton VI concludes the paper. II. LDPC CODES AND NOISY MESSAGE-PASSING DECODING PRINCIPLES Consder a regular bnary (d v,d c ) LDPC code wth length N. The code can be represented by a K N party check matrx H wth bnary elements, where the weght of each column and row of the matrx are d v and d c, respectvely. There s a Tanner graph correspondng to the party check matrx wth N varable nodes and K N dv d c check nodes. Every varable node n the graph s connected to d v check nodes and every check node s connected to d c varable nodes. Correspondng to the ones n the columns and the rows of H, the varable nodes and the check nodes are connected to each other n the Tanner graph. Fg. exemplfes a Tanner graph for a regular (3,6) LDPC code wth length N. Varable node v and check node c j are known as neghbors, f they are connected to each other. Message-passng decodng algorthm can be represented as teratve exchange of messages between check nodes and varable nodes of the Tanner graph. Specfcally, every check node receves messages from ts d c neghbor varable nodes and sends the computed messages back. Smlarly, each varable node exchanges messages wth ts d v neghbor check nodes. We consder the output messages of the varable and check nodes as log-lkelhood rato (LLR) values, where the sgn of a varable node message specfes the bt estmate and ts magntude ndcates the relablty of the estmaton. Accordng to the sum-product decodng algorthm, the message at teraton l from a varable node to a check node, denoted by v (l), s v (l) = d v =0 u (l ), () where u (l ), =,...,d v, are ncomng LLRs from varable node neghbors at teraton l, except the check node that s to receve the output message v (l), and u 0 s the ncomng LLR message from the communcaton channel. The message u (l) from a check node to a varable node at teraton l can be obtaned as follows tanh u(l) dc = tanh v(l) j. () j= Fg. shows the schematcs of message-passng for a varable node and a check node. For a nosy decoder, the output messages of the varable and check nodes are subject to addtve whte Gaussan nose. The conventonal model shown n Fg. can be extended to the one n Fg. 3, where n and ν j denote the addtve whte Gaussan nose affectng the output messages of check nodes and varable nodes, respectvely. Hence, γ j and µ are nosy versons of v j and u, respectvely. Therefore, the ncomng messages to the varable nodes and the check nodes are µ (l) = u (l) +n, (3) γ (l) j = v (l) j +ν j, (4) wheren andν j are assumed to be ndependent and dentcally dstrbuted (..d.),.e., n,ν j N(0,σd ). Accordng to the sum-product algorthm, at teraton l the decodng s performed based on the followng updatng equatons d v v (l) = u 0 + = µ (l ), (5)

3 3 u dv n dv... n µ dv µ µ (a) u v u n u 0 Fg. 3. The model of (a) a varable node (b) a check node n nosy belefpropagaton decoder. tanh u(l) j= v dc ν dc... ν γ dc v γ u (b) γ v ν dc = tanh γ(l) j. (6) In order to generalze these equatons to rregular case one can follow the same steps as the one descrbed n [7]. In the next secton, we propose an approach for the performance analyss of ths nosy message-passng decodng algorthm. III. DENSITY EVOLUTION ANALYSIS OF NOISY DECODER In ths secton, usng Gaussan approxmaton, we wll fnd the densty evoluton equatons for the nosy decoder wth regular varable and check degrees. We further generalze the results to the rregular case, and fnally use the derved densty evoluton equatons to evaluate the performance of nosy decoders. shft keyng (BPSK) modulaton. Thus, the LLR values receved over an AWGN communcaton channel are Gaussan dstrbuted. The mean and the varance of the receved LLR and σ0 = 4 σ, n s the channel nose varance [7]. We assume that the values are respectvely equal to m 0 = σn whereσn random varables u, v, u and v j are all Gaussan dstrbuted. Frst, we check whether the messages of a nosy sum-product LDPC decoder are consstent. To ths end, we consder the expected values of both sdes of (3) and (5) and obtan v = m 0 +(d v )m (l ) u, (7) where v and m u (l ) denote the means of output messages of varable nodes and check nodes, respectvely. The ndex s omtted snce u, =,...,d v, are..d. Next, by computng the varances of both sdes of (3) we have σ µ (l) = σ u (l) +σ d. (8) Usng (5), we obtan the varance of the varable node output ( dv ) ( d v ) σv(l) = σ 0 +var +cov u 0,, = µ (l ) = µ (l ) (9) where var(x) denotes the varance of random varable X, and cov(x,y) s the covarance of random varables X and Y. Snce µ, =,...,d v, are..d. Gaussan random varables, we have var ( dv = µ (l ) ) = d v = ( ) var µ (l ) = (d v )σµ(l ). (0) A. Gaussan Approxmaton and Consstency The densty evoluton analyss s an analytcal method for trackng the denstes of messages n teratve decoders. Ths can be used to predct the performance lmts of an LDPC code measured by code s threshold [4]. The code s threshold s the smallest (largest) communcaton channel SNR (nose varance) for whch an arbtrarly small decodng bt-error probablty can be acheved by suffcently long codewords. For an AWGN communcaton channel and an LDPC sumproduct decoder, the denstes of the exchanged messages between the check nodes and the varable nodes can be approxmated as Gaussan [7], [5]. Hence, these denstes may be characterzed only wth ther mean and varance. A Gaussan random varable whose varance s twce ts mean s sad to be consstent [5]. The consstency assumpton smplfes densty evoluton as a one-dmensonal recursve equaton based on the mean (or the varance) of the messages. In [7], ths assumpton s used for the DE analyss of a noseless LDPC decoder, and subsequently, quantfyng the threshold of the code. The key assumpton n the densty evoluton analyss of nose-free decoders s that the code block length s suffcently large, based on whch t may be assumed that the Tanner graph of the LDPC code s cycle-free. Snce the code s lnear and the communcaton channel s symmetrc, we consder the transmsson of an all-one codeword usng a bnary phase The last term n (9) s zero, as the Tanner graph of the code s assumed to be cycle-free and u 0 s ndependent of the nosy messages. Therefore, the varance of a varable node output message at teraton l can be smplfed as follows σ v(l) = σ 0 +(d v )σ u(l ) +(dv )σ d. () To verfy the consstency of the nosy decoder, we plug n σv (l) = v and σu (l ) = m u (l ) nto () and compare t wth (7). It s clear that as long as σd s nonzero, the two quanttes are not equal and hence the nosy LDPC decoder s not consstent. Therefore, t does not suffce to track only the mean values of the nodes output messages. Instead, t s requred to track both the mean and the varance of nodes output messages. A smlar stuaton has been shown n the smulaton results of [6], when there s an ncorrect estmaton of the communcaton channel SNR at a (noseless) LDPC decoder. However, snce the code s lnear and the communcaton channel s symmetrc, sendng all-one codeword s suffcent for statstcal performance evaluaton of the code [6]. For an rregular LDPC code, the degree dstrbuton of varable nodes s λ(x) = D v = λ x and that of the check nodes s ρ(x) = D c = ρ x, where λ and ρ are the percentage of edges that are connected to varable nodes and check nodes of degree, respectvely. D v s the maxmum degree of varable nodes and D c s the maxmum degree of

4 4 TABLE I RELATION BETWEEN THE THRESHOLD AND DECODER NOISE VARIANCE σd SNR threshold SNR th (σ n ) th 0.63 db db db db check nodes. In ths case, by smlar steps as the ones for the regular LDPC codes, t can be shown that for the mean and the varance of a varable node of degree at teraton l v, = m 0 +( )m (l ) u, () σv, (l) = σ 0 +( )σu (l ) +( )σ d, (3) where v, and σ v,(l) are the mean and the varance of a varable node of degree at teraton l, respectvely. From () and (3) t can be nferred that the consstency s not vald for the rregular case. B. Densty Evoluton wth Gaussan Approxmaton for Nosy Message-Passng Decoder In the case of a nosy LDPC decoder, we have shown that consstency does not hold and we should track both the mean and the varance of nodes output messages. In order to do ths, we use the key equatons (3)-(8) and (). By computng the expected value of both sdes of equaton (6), and notng that γ (l) j, j =,...,d c are..d., we have ] [ dc E [tanh u(l) ] = E tanh γ(l) j j= ( ]) (4) dc = E [tanh γ(l), where γ (l) s defned n (4) and has the followng dstrbuton γ (l) N( v, σ v(l) +σ d ). (5) Next, by computng the expected value of squared tanh rule, we obtan the second major equaton as follows ( )] ( ( )]) u E [tanh (l) γ = E [tanh (l) dc. (6) The densty evoluton can be obtaned by smultaneously solvng equatons (4) and (6). Specfcally, representng γ (l) usng (5) and v, σv(l) from (7) and (), we obtan the DE equatons for check nodes as follows ( ( ( f g ) u,σ u(l) = ( ) m u (l),σu(l) = f ( g )) dc v, σ v(l) +σ d, ( )) v, σv (l) dc (7) +σ d. The auxlary functons f(m,σ ) and g(m,σ ) are defned as follows [ f(m,σ ) E g(m,σ ) E ( )] X tanh [ ( X tanh, )], (8) where X N(m,σ ). These equatons can be used to track u and σu(l) n the decodng teratons of a regular (d v,d c ) LDPC for gven values of communcaton channel nose varance and nternal decoder nose varance. Because of non-lnearty of the equatons n (7), t s not easy to fnd a closed form expresson for the mean and the varance at each teraton. As such, we resort to Monte-Carlo smulatons to solve (7) and adopt the sem-gaussan approxmaton method used n [7] and [9]. Smlarly, for rregular LDPC codes, the message dstrbutons are approxmated by Gaussan mxture [7]. For each check node of degree at teraton l we have ( ) D v ( ) f u,,σ u,(l) = λ j f v,j, σ (l) v,j +σ d, ( ) g u,,σ u,(l) = j= D v j= ( ) λ j g v,j, σ (l) v,j +σ d, (9) and from Gaussan mxture equatons, the densty of check node n teraton l has the followng mean and varance values = u = D c = σu (l) D c ) ] = ρ [σ u, (l) + ( ρ u,, (0) u, ( u ). () Therefore, the DE can be found by solvng (9) for a check node wth degree and the dstrbuton of a check node s then found usng (0) and (), teratvely. C. Numercal Results of the Densty Evoluton Analyss We solve the densty evoluton equatons teratvely for a (3,6) regular LDPC code consderngm (0) u = 0 and σu(0) = 0 as ntal condtons. Ths provdes the mean and the varance of check nodes outputs and allows for the computaton of the threshold for the gven varances of nternal decoder nose and communcaton channel nose. Table I shows the relaton between the threshold and the nternal decoder nose varance σd resultng from (7). It can be observed that the SNR threshold SNR th ( E b N 0 ) th ncreases as the nternal decoder nose varance ncreases. Ths s n lne wth a smlar observaton n [5] on the performance of btflppng LDPC decodng n the presence of nosy messagepassng over BSC channels, where the performance deterorates as the cross-over probablty of the nternal decoder nose ncreases.

5 5 Probablty of Error σ d = 0 σ d = σ d = σ d = SNR (db) Fg. 4. Error probablty performance of (3, 6) regular fnte length code wth decoder nose varance σd. Smulaton results confrm that our analytcal results accurately predct the performance of fnte-length codes as well. Fg. 4 depcts the smulaton results for the performance of a fnte-length (3,6) regular code wth length N = 008. It s evdent that the threshold of ths fnte-length code s farly the same as our analytcal threshold for varous values of the nternal decoder nose varance σd. One can use (0) and () to also track the densty of rregular LDPC codes for nosy decoder. In general, the presented analyss could be used to desgn rregular LDPC codes for nosy decoders. Snce the problem of desgnng rregular LDPC codes by densty evoluton s not a convex problem, fndng a good degree dstrbuton requres complex computatons and extensve search [7]. Therefore, n the remanng sectons after nvestgatng the performance lmts of the nosy decoder by means of EXIT chart, we wll ntroduce a smple and effectve method to desgn robust LDPC codes for the nosy decoder. IV. EXIT CHART ANALYSIS OF NOISY DECODER The EXIT chart analyss, frst ntroduced n the poneerng work of ten Brnk [8], s a powerful tool for analyzng the performance of teratve turbo technques. It s manly based on keepng track of the mutual nformaton of channel nput bts and varable node and check node outputs. Let X be a bnary random varable denotng the BPSK modulated AWGN communcaton channel nput whch takes ± values wth equal probabltes. If f(y) s the probablty densty functon (pdf) of the communcaton channel soft output Y, then, the mutual nformaton of X and Y for a symmetrc channel [9], [30] s I(X;Y) = ( ) f(y x) f(y x)log dy. () f(y) x=± In order to fnd the EXIT functon of a noseless decoder, the varable node and the check node EXIT functons can u u dv.. n n dv VND u 0 NVND v v v dc. ν ν dc CND NCND Fg. 5. Modfcaton of decoder components (VND or CND) to nosy decoder components (NVND or NCND). Algorthm EXIT curve for NVND : Input: I A, d v, σ n, σ d, N : Output: I E 3: X =, [length N BPSK modulated codeword] 4: for k = : N do [compute decoder LLR nputs] Y(k) = σn (+n c ),n c N(0,σn ) 5: end for 6: for = : length(i A ) do where J(σ) σ A = J (I A ()), e (x σ /) σ πσ log (+e x )dx 7: for j = : N do [compute a pror LLRs] AP(j) = σ A +n, n N(0,σ A ) 8: end for 9: [compute nosy varable node outputs usng (5)] E = NVND ( d v, AP, Y,σd ) 0: [Compute extrnsc mutual nformaton usng ()] : end for I E () = I(X, E) be computed usng a J-functon [8], whch drectly results from the consstency assumpton of the decoder. Snce ths assumpton s volated n the case of a nosy decoder, to compute the a pror and extrnsc mutual nformaton, we compute these values accordng to the defnton of mutual nformaton. To obtan the EXIT curves for the nosy LDPC decoder, we use Algorthm and the emprcal dstrbutons computed by runnng smulatons for each degree of varable node and check node. Specfcally for each decoder component, we compute the extrnsc mutual nformaton I E correspondng to a pror mutual nformaton I A for the nosy varable (check) node decoder. To mantan the desred structure of the teratve decoder [8] for the nosy decoder, we modfed the varable nodes and check nodes as shown n Fg. 5. In fact, we have u

6 6 IE,V, IA,C 0.6 σ d = 0 Check node Varable node I A,V, I E,C IE,V, IA,C 0.6 σ d = Check node Varable node I A,V, I E,C IE,V, IA,C 0.6 σ d = Check node Varable node I A,V, I E,C IE,V, IA,C 0.6 σ d = 3 Check node Varable node I A,V, I E,C Fg. 6. EXIT charts of (3,6) regular LDPC code resultng from Algorthm for SNR= 3 db and dfferent decoder nose varances σ d. 0.5 f(y) y Fg. 7. nodes for I A = 0.5, σd 0.09 f(y) y Emprcal dstrbutons of outputs of check (left) and varable (rght) = and SNR = 3 db. consdered decoder nose as a part of varable (check) nodes and call them nosy varable (check) node decoders, NVND (NCND). In Algorthm, the proposed method for fndng the EXIT curve of NVND s descrbed. The procedure of fndng EXIT curves for the nosy check node s smlar to that of the nosy varable node; however, for the check node the communcaton channel output s not fed to the check node,.e., the only nput of the nosy check node s the vector of a pror LLRs from varable nodes (AP). The presented algorthm s an extenson of the algorthm 7.4 n [3] for Turbo codes. It s noteworthy that for a noseless decoder, nput messages to each decoder component (VND or CND) are modeled by Gaussan random varables of mean µ A and varance σ A = µ A. Then the mutual nformaton I A of each nput message and the channel nput X (a pror mutual nformaton) can be computed usng I A = J (σ A ). Ths relaton mples a one-to-one mappng from each I A to a σ A (and consequently µ A ). However, for a nosy decoder there s not such a one-to-one mappng and for each I A dfferent sets of (µ A,σA ) may be found. One way to tackle ths problem s to assume consstency for nput messages to the nosy components (NVND and NCND) as stated n step 6 of Algorthm, and calculate each a pror LLR as n step 7. Then, the nput messages to VND and CND are no longer consstent (snce they are corrupted wth decoder nose). Snce we smulate each decoder component separately, the assumpton made does not affect the computaton of EXIT curves for the other components, and as our experments valdate, the suggested approach provdes very accurate EXIT curves. Fg. 6 llustrates the evoluton of EXIT charts of (3, 6) regular LDPC code for dfferent values of decoder nose varance σd and for communcaton channel SNR 3 db. For σd = 0, there s an open tunnel between the curves, and ncreasng σd to one makes the tunnel tghter. However, for σd = and σ d = 3 the varable and the check EXIT curves cross each other. The EXIT charts n Fg. 6 llustrate that the SNR threshold of (3,6) regular LDPC code s less than 3 db when σd = 0 and σ d = and s greater than 3 db when σd = and σ d = 3. Ths s n lne wth the results obtaned from the densty evoluton analyss, as shown n Table I. Fg. 7 depcts the emprcal dstrbutons f(y) for the outputs of NVND of degree d v = 3 and NCND of degree d c = 6, used n () to fnd the extrnsc mutual nformaton I E. We observe that the dstrbuton of the nodes outputs are lght-taled for the check node and normal-taled for the varable node. Ths motvates the use of numercal ntegraton n () by ntegratng over a lmted ntegral doman. In the followng secton, we wll use the EXIT curves resultng from Algorthm to desgn robust LDPC codes for nosy decoders. V. DESIGN OF ROBUST IRREGULAR LDPC CODES FOR NOISY DECODER In ths secton, our goal s to desgn robust rregular LDPC codes for nosy decoder. The desgn procedure of LDPC codes s based on fttng EXIT curves correspondng to gven varable and check degree dstrbutons. In [8], for a noseless decoder, rght-regular LDPC codes are desgned for a fxed check node degree of d c, by fttng a weghted sum of the EXIT curves of varable nodes to the check node EXIT curve. In ths work, we desgn rregular LDPC codes and allow more than one degree for both varable nodes and check nodes. Our benchmark for comparsons are rregular LDPC codes wth the same rates desgned for a noseless decoder n [9]. We verfy the performance of the desgned codes by the smulatons of fnte-length LDPC codes. It s noteworthy that when the exchanged messages n the decoder are not consstent, the weghted sum of the EXIT chart curves of varable node and check node are not the exact curves of the rregular code. However, our smulaton results ndcate that ths approach s accurate enough for the case wth a nosy decoder. A. Code Desgn Algorthm Usng the EXIT curves obtaned from Algorthm, we propose a smple method for the desgn of rregular codes for the nosy decoder. Smlar to the code desgn approach

7 7 Algorthm Robust LDPC code desgn : Input: d c, D v, r, σ d,, S : Output: code C = (ρ(x),λ(x) ) 3: ntalze SNR th 4: σ n = (rsnr th) 5: for = : M + do 6: α = S() 7: [compute EXIT curves of check nodes wth degree 8: d c and degree d c usng Algorthm ] 9: [compute effectve EXIT curve I A,NCND usng ρ(x) 0: n (3)] : [compute EXIT curves of varable nodes wth degrees : d v = to d v = D v usng Algorthm ] 3: [check f there s a λ(x) such that C = (ρ(x),λ(x)) 4: satsfes the constrants n (5)] 5: f code C exsts then 6: SNR th = SNR th 7: C = C 8: go to step 4 9: end f 0: end for proposed n [9], we restrct our attenton to rregular codes wth two consecutve check node degrees as and varable node degrees as ρ(x) = αx dc +( α)x dc, (3) D v λ(x) = λ x. (4) = The effectve EXIT curve of a nosy rregular check (varable) node I A,NCND (I E,NVND ) s obtaned by averagng over the EXIT curves of check (varable) nodes wth consttuent check (node) degrees [9]. In ths paper, we refer to a canddate code C = (ρ(x),λ(x)) of rate r as a successful code for a gven SNR, f t satsfes the followng constrants ) λ() =, ) ρ() =, 0 ) r = ρ(x)dx 0 λ(x)dx, v) I E,NVND I A,NCND, (5) where the last constrant mples that the effectve EXIT curve of ts varable node les above the effectve EXIT curve of ts check node. For a gven code rate r, we are lookng for the code C = (ρ(x),λ(x) ) correspondng to the threshold SNR, SNR th,.e., the mnmum SNR for whch there s a successful code. In Algorthm, we set the check node degree d c, the maxmum varable node degreed v, and desgn rate r. In order to speed up the desgn procedure, we let α take lmted values n S = [0 : /M : ]. Then, for a gven SNR th, by varyng α from zero to one, we check f there s a varable degree dstrbuton λ(x), as n (4), such that the constrants n (5) are satsfed. One way to do ths s for each α to fnd I A,NCND accordng to (3) and check f there s a vector (λ,...,λ Dv ) for whch the constrants n (5) are feasble. If so, we form the code C = (ρ(x),λ(x)). Subsequently, we reduce SNR th by and the algorthm does another teraton; otherwse, t termnates and the successful code from the prevous teraton s selected. The EXIT curves of a check node do not depend on SNR th ; By changng SNR th, the EXIT curves of varable nodes, and consequently I E,NVND are affected. Usng ths fact, the complexty of the proposed algorthm may be further reduced by pre-computng I A,NCND for each value of α once and storng them before runnng the algorthm. In the followng secton, we wll present some results llustratng the applcaton of the proposed algorthm n the desgn of robust LDPC codes for nosy decoders. B. Desgn Examples From Table of [9], for the maxmum varable degree of four, the ntroduced code of rate one-half has two types of check nodes wth degrees fve and sx. The threshold of ths code s db and has the followng degree dstrbutons λ(x) = 0.384x+0.04x x 3, ρ(x) = 0.4x x 5. However, when the decoder s not perfect, the threshold and also the performance of ths code degrade. For nstance, n nosy decoder the threshold has ncreased by about.7 db and.5 db for decoder nose varances σd = 0.5 and σ d =, respectvely. Consderng the same constrants n check degree dstrbuton and maxmum varable degree, we have desgned an rregular one-half rate LDPC code for the nosy decoder wth σd = 0.5. Usng Algorthm, we obtaned code A wth the followng degree dstrbutons λ(x) = 0.453x+0.547x 3, ρ(x) = 0.45x x 5. Fg. 8 shows the performance of Code A and the desgned code n [9] for dfferent decoder nose varances. When σd = 0.5 (sold curves), Code A provdes a better performance compared to the code n [9]. By ncreasng the decoder nose varance to σd = 0.7 (dashed curves) Code A stll outperforms the code n [9], whle by decreasng the decoder nose varance to σd = 0.3 (dotted curves) the codes show almost the same performance. It s evdent that Code A performs well when the decoder nose power vares n the proxmty of the desgned varance. It s noteworthy that n our smulatons the Tanner graph of codes are free from cycles of length four. The maxmum number of decoder teratons s set to 80, and for each SNR, the error probablty s computed after a total number of 50 block errors occur. As another example of code desgn, consderng the same constrants on degree dstrbutons, we have desgned an rregular code (Code B) for nosy decoder wth σd =. The

8 8 Probablty of error Code n [9], σ d = 0.7 Code A, σ d = 0.7 Code n [9], σ d = 0.5 Code A, σ d = 0.5 Code n [9], σ d = 0.3 Code A, σ d = SNR (db) Probablty of error Code n [9], σ d =. Code B, σd =. Code n [9], σd = Code B, σ d = Code n [9], σ d = 0.8 Code B, σ 0 4 d = SNR (db) Fg. 8. Error probablty performances of Code A and the code n [9] for dfferent decoder nose varances, N = 0 4. Fg. 9. Error probablty performances of Code B and the code n [9] for dfferent decoder nose varances, N = 0 4. degree dstrbutons of ths code are λ(x) = x+0.59x 3, ρ(x) = 0.553x x 5. Fg. 9 shows the smulaton results of Code B and the desgned code n [9] for dfferent decoder nose varances. When σ d = (sold curves), Code B shows better performance compared to the code n [9]. Smlar observatons are made when the decoder nose varance (dashed curves) ncreases to. or decreases to 0.8 (dotted curves). In the above desgns, we consdered two consecutve check degrees for a far comparson wth the code n [9]. However, the presented approach may be used to desgn rregular LDPC codes wth arbtrary check degrees as D c ρ = ρ x. = The constrants n (5) reman lnear and hence the same procedure stll apples. We emphasze that the range of decoder nternal nose power that we consdered here (between 0 and.) s rather conservatve. For a scalar quantzaton of a Gaussan random varable, a sngle bt change n the quantzaton btrate scales the quantzaton nose varance by a factor of about 3.4 [3]. Accordng to our smulaton results, the proposed robust LDPC code desgn ndcates a hgher gan when the decoder nternal nose power s stronger. VI. CONCLUSIONS We consdered a nosy message-passng scheme for the decodng of LDPC codes over AWGN communcaton channels. We modeled the nternal decoder nose as addtve whte Gaussan and observed the nconsstency of the exchanged message denstes n the teratve decoder. For the nconsstent LDPC decoder, a densty evoluton scheme was formulated to track both the mean and the varance of the messages exchanged n the decoder. We quantfed the ncrease of the decodng threshold SNR as a consequence of the nternal decoder nose based on a densty evoluton analyss. We also analyzed the performance of the nosy decoder usng EXIT charts. We ntroduced an algorthm based on the computed EXIT charts to desgn robust rregular LDPC codes. The desgned codes partally compensate the performance loss due to the decoder nternal nose. In ths work, we modeled the decoder nose as AWGN on the exchanged messages, however, an nterestng future step s to ncorporate other nose models possbly drectly obtaned from practcal decoder mplementatons. One may also use ths research to desgn codes that are robust to decoder nternal nose whose power may vary n a gven range dependng on possble types of mplementaton. REFERENCES [] R. Gallager, Low-densty party-check codes, IRE Trans. Inf. Theory, vol. 8, no., pp. 8, Jan 96. [] M. Spser and D. Spelman, Expander codes, IEEE Trans. Inf. Theory, vol. 4, no. 6, pp. 70 7, Nov 996. [3] D. J. MacKay and R. M. Neal, Near shannon lmt performance of low densty party check codes, Electroncs letters, vol. 3, no. 8, pp , 996. [4] D. J. MacKay, Good error-correctng codes based on very sparse matrces, IEEE Trans. Inf. Theory, vol. 45, no., pp , 999. [5] M. Eroz, F. Sun, and L. Lee, DVB-S low-densty party-check codes wth near shannon lmt performance, Int. Journal of Satellte Communcatons and Networkng, vol., no. 3, 004. [6] Part 6: Ar nterface for fxed and moble broadband wreless access systems amendment for physcal and medum access control layers for combned fxed and moble operaton n lcensed bands, IEEE P80.6e, Oct 004. [7] S.-Y. Chung, T. J. Rchardson, and R. L. Urbanke, Analyss of sumproduct decodng of low-densty party-check codes usng a Gaussan approxmaton, IEEE Trans. Inf. Theory, vol. 47, no., pp , 00. [8] S. ten Brnk, G. Kramer, and A. Ashkhmn, Desgn of low-densty party-check codes for modulaton and detecton, IEEE Trans. Commun., vol. 5, no. 4, pp , Aprl 004. [9] A. Ashkhmn, G. Kramer, and S. ten Brnk, Extrnsc nformaton transfer functons: model and erasure channel propertes, IEEE Trans. Inf. Theory, vol. 50, no., pp , Nov 004.

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