Error Floor Approximation for LDPC Codes in the AWGN Channel

Transcription

1 submitted to IEEE TRANSACTIONS ON INFORMATION THEORY VERSION: JUNE 4, Error Foor Approximation for LDPC Codes in the AWGN Channe Brian K. Buter, Senior Member, IEEE, Pau H. Siege, Feow, IEEE arxiv: v2 [cs.it] 9 Jun 2013 Abstract This paper addresses the prediction of error foors of ow-density parity-check (LDPC) codes with variabe nodes of constant degree in the additive white Gaussian noise (AWGN) channe. Specificay, we focus on the performance of the sumproduct agorithm (SPA) decoder formuated in the og-ikeihood ratio (LLR) domain. We hypothesize that severa pubished error foor eves are due to the manner in which decoder impementations handed the LLRs at high SNRs. We empoy an LLR-domain SPA decoder that does not saturate near-certain messages and find the error rates of our decoder to be ower by at east severa orders of magnitude. We study the behavior of trapping sets (or near-codewords) that are the dominant cause of the reported error foors. We deveop a refined inear mode, based on the work of Sun and others, that accuratey predicts error foors caused by eementary tapping sets for saturating decoders. Performance resuts of severa codes at severa eves of decoder saturation are presented. Index Terms Low-density parity-check (LDPC) code, beief propagation (BP), error foor, inear anaysis, Marguis code, absorbing set, near-codeword, trapping set. I. INTRODUCTION A very important cass of modern codes, the ow-density parity-check (LDPC) codes, was first pubished by Gaager in 1962 [1], [2]. LDPC codes are inear bock codes described by a sparse parity-check matrix. Decoding agorithms for LDPC codes are generay iterative. The renaissance of interest in these codes began with work by MacKay, Nea, and Wiberg in the ate 1990s [3] [5]. Progress has been rapid, with information-theoretic channe capacity essentiay reached in some appications of LDPC codes and standardization compete for their commercia use in others, e.g., DVB-S2 for sateite broadcast and IEEE 802.3an for 10 Gbit/s Ethernet. The graph of an iterativey decoded code s error rate performance versus the channe quaity is typicay divided into two regions. The first region, termed the waterfa region, occurs at poorer channe quaity, cose to the decoding threshod, and is characterized by a rapid drop in error rate as channe quaity improves. The second region, caed the error foor region, is the focus of the present paper. The error foor appears at higher channe quaity and is characterized by a more gradua decrease in error rate as channe quaity improves. For This work was presented in part at the Forty-Ninth Annua Aerton Conference, Aerton House, Iinois, Sept The authors are with the Department of Eectrica and Computer Engineering, University of Caifornia, San Diego (UCSD), La Joa, CA USA (e-mai: buter@ieee.org, psiege@ucsd.edu). This work was supported in part by the Center for Magnetic Recording Research at UCSD and by the Nationa Science Foundation (NSF) under Grant CCF and Grant CCF message-passing iterative decoders the error foor is apparenty determined by sma structures within the code that are specific to the seected graphica description of the code. The understanding of the LDPC code error foor has progressed significanty, but issues remain. For the binary erasure channe (BEC), the structures that imit the performance of message-passing iterative decoders as channe conditions improve are known as stopping sets [6]. These sets have a combinatoria description and can be enumerated to accuratey predict the error foor for the BEC. Other memoryess channes are more difficut to characterize and the presence of error foors has imited the adoption of LDPC codes in some appications. For exampe, in the DVB-S2 standard, an outer agebraic code is concatenated with an LDPC code to ensure a ow error foor. For Marguistype codes, MacKay and Posto found that the error foors seen in the additive white Gaussian noise (AWGN) channe with sum-product agorithm (SPA) decoding were caused by near-codewords in the Tanner graph [7]. Richardson wrote a semina paper on the error foors of memoryess channes shorty afterward [8]. In it, he caed near-codewords trapping sets and defined them with respect to the decoder in use as the error-causing structures. The parameters (a, b) are used to characterize both near-codewords and trapping sets, where a is the number of variabe nodes in the set and b is the number of unsatisfied check nodes when just those variabe nodes are in error. The (a, b) parameters of error-foor-causing structures are typicay sma. In [8], Richardson emphasized error-foor anaysis techniques for the AWGN channe. He detaied a methodoogy to estimate a trapping set s impact on the error rate by simuating the AWGN channe in the neighborhood of the set. His semianaytica technique was shown to be accurate at predicting the error foors given that the trapping sets were known. The method invoved significant simuation, but much ess than standard Monte Caro simuation. Roughy speaking, error foors can be measured down to frame error rates of about 10 8 in standard computer simuation and about in hardware simuation, depending on code compexity and the computationa resources avaiabe. Richardson s method aows us to reach orders of magnitude further in characterizing the error foor. Further work on trapping sets incudes agorithms to enumerate the occurrence of sma trapping sets in specific codes [9] [13] and the characterization of trapping set parameters in random ensembes of LDPC codes [14], [15]. Severa other significant works on error foors in the AWGN channe exist. For exampe, Doecek et a. noted empiricay

2 2 submitted to IEEE TRANSACTIONS ON INFORMATION THEORY VERSION: JUNE 4, 2013 that the trapping sets dominating the error foor, termed absorbing sets, had a combinatoria description [16]. Their study made use of hardware-oriented SPA decoders. A variety of works describe techniques to overcome error foors. With respect to these, we note that [17] proposes severa different techniques, [18] sows decoder convergence using averaged decoding, [19] seectivey biases the messages from check nodes, [20] empoys informed scheduing of nodes for updating, [21] adds check equations to the parity-check matrix, and [22] repaces traditiona BP with a difference-map message passing agorithm. In contrast to these, we present in this paper an error foor reduction technique that makes no changes to the standard SPA decoding aside from fixing common numerica probems. Foowing the pioneering work of Richardson, severa authors have proposed anaytica techniques to predict error foors in AWGN. In his Ph.D. thesis [23], Sun deveoped a mode of eementary trapping set behavior based on a inear state-space mode of decoder dynamics on the trapping set, using density evoution (DE) to approximate decoder behavior outside of the trapping set [23], [24]. This work has gained itte attention, even though it reaches a concusion contrary to prior resuts. Specificay, Sun s mode shows no error foor for eementary trapping sets (excuding codewords) in reguar, infinite-ength LDPC codes if the variabe-degree of the code is at east three and the decoder s metrics are aowed to grow very arge. Sun is abe to make simiar caims for irreguar LDPC codes under certain conditions. In these cases, Sun shows that the graph outside of the trapping set wi eventuay correct the trapping set errors; in his exampe, using a (3, 1) trapping set, this occurred at a mean og-ikeihood ratio (LLR) of about after about 40 iterations. Sun s anaytica deveopment was asymptotic in the number of iterations and the bock ength of the code. When appied to finite-ength codes with cyces, we show that the mode breaks down due to strongy correated LLR messages when the decoder is non-saturating. Schege and Zhang [10] extended Sun s work by adding time-variant gains and channe errors from outside of the trapping set to the mode. For the 802.an code, using a mode specific to the dominant (8, 8) trapping set, they then compared their anayticay predicted error foors to error foors found by FPGA-based simuation and importance samping simuation. Their resuts show that the externa errors have very itte impact and that the error foor can be predicted quite accuratey. We generaize their mode and compare predicted error foors to simuations for four different codes with severa LLR imits. We find the mode estimates the error foor to within 0.15 db for three of the four codes that we examine. We aso show how a mode that breaks down for finite-ength codes in a non-saturating scenario may work fairy we when saturated LLRs are introduced. Brevik and O Suivan deveoped a new mathematica description of decoder convergence on eementary trapping sets using a mode of the probabiity-domain SPA decoder [25]. Whie considering just the channe inputs, they are abe to find regions of convergence to the correct codeword, regions Frame Error Rate (FER) 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 Marguis LDPC Code n=2640, Frame Error Rate vs. Eb/No MacKay & Posto Non-sat. SPA, 200 max iterations 1.E Eb/No (db) Fig. 1. FER vs. E b /N 0 for (2640,1320) Marguis LDPC code in AWGN. of convergence to the trapping set, and regions of nonconvergence typified by an osciatory behavior. Xiao et a. [26] have deveoped a technique to estimate the error foor in the quantized decoder case using a binary input AWGN channe. In contrast, the present paper focuses on unquantized decoders and different foor-estimation techniques. Severa authors have found empiricay that increasing LLR imits in the SPA decoder owers the error foor [27] [30]. Our findings concur with theirs. Furthermore, our resuts show that when care is taken to impement the SPA decoder without saturation, error foors can be owered quite dramaticay, often by as much as severa orders of magnitude. In the cases we examine, the error foors are reduced by many orders of magnitude. Recent independent work by Zhang and Schege confirms this [31], [32]. Fig. 1 iustrates the improvement in performance that nonsaturating decoders can provide. The curves represent the frame error rate (FER) reported in [7] for the (2640, 1320) Marguis code and the corresponding resuts obtained from our simuation of a non-saturating SPA decoder. The error foor found in [7] starts at an FER of about 10 6 with an E b /N 0 of 2.4 db. The dominant errors corresponded to near-codewords. (It shoud be noted that others have reported even higher error foors for this code, with an FER of 10 6 appearing at an E b /N 0 of 2.8 db [8], [17].) Our simuation, on the other hand, shows no evidence of an error foor. The owest simuated point at an FER of represents 154 error events observed in 8600 hours of foating-point simuation, with a maximum of 200 iterations per decoded frame. Just one of the error events was a (14, 4) near-codeword, which is one of the two dominant errors foor causing structures in the previousy reported resuts. Thus, we see very itte of the usua eary signs that an error foor is deveoping, suggesting that any error foor woud manifest itsef ony at a substantiay ower FER. The framework we deveop in this paper wi hep to expain the empiricay observed error foor properties described above. Our focus wi be on the prediction of error foors for binary LDPC codes with constant variabe node degree when used on the AWGN channe. The resuts presented here are an extension of those discussed in [8], [10], [23],

3 BUTLER AND SIEGEL: ERROR FLOOR APPROXIMATION FOR LDPC CODES IN THE AWGN CHANNEL 3 [31]. Section II introduces the genera terminoogy reated to nonnegative matrices, graphs, trapping sets, and SPA decoders. We introduce message saturation as a technique to avoid numerica overfow in the SPA decoder and then describe the non-saturating decoding agorithm that wi provide our performance benchmarks. Section III deveops the state-space mode of decoder dynamics on the trapping set we wi be using for the anaytica resuts of this paper. It aso deveops approximations to the dominant eigenvaue required to characterize the system, using connections to graph theory. Section IV deveops the probabiity that the state-space mode fais to converge to the correct decoding soution and the reated error-rate estimates for the error foor. Section V discusses the use of the Gaussian approximation for DE to mode the unsatisfied-check LLR vaues to determine the utimate error foor bounds of non-saturating decoders. Section VI appies a modified version of Richardson s semi-anaytica technique to estimate the error foor of a non-saturating SPA decoder. Section VII demonstrates the accuracy of the statespace mode s predictions of saturated decoder performance by comparing them to simuation resuts for four LDPC codes. Finay, we draw our concusions. Severa additiona reated topics are covered in the appendices. A. Nonnegative Matrices II. PRELIMINARIES In this subsection we introduce the terminoogy of Perron- Frobenius theory of nonnegative matrices [33] [36]. A vector v R n is said to be positive if every eement is stricty greater than zero. Likewise, the vector v is said to be nonnegative if every eement is greater than or equa to zero. In the present paper, we use coumn vectors by defaut. A matrix M R m n is said to be a positive matrix if every entry is stricty greater than zero. Likewise, the matrix M is said to be a nonnegative matrix if every entry is greater than or equa to zero. Moreover, the nonnegative matrix M is said to be a (0, 1)-matrix if every entry beongs to the set {0, 1}. A permutation matrix P is a square (0, 1)-matrix, in which each row and each coumn contains a singe 1 entry. The symmetric permutation of the square matrix M by P is PMP T. The nonnegative square matrix M is said to be a reducibe matrix if there exists a permutation matrix P such that PMP T = [ ] X Y 0 Z where X and Z are square submatrices. Otherwise, the matrix M is said to be an irreducibe matrix. Given M R m m, the set of distinct eigenvaues is caed the spectrum of M, denoted σ(m) = {µ 1,..., µ n }, where µ k C. The spectra radius of M is the rea number ρ(m) = max { µ : µ σ(m)}. The spectra circe is a circe in the compex pane about the origin with radius ρ(m). The foowing are a few important resuts from the theory of nonnegative matrices. Let M be an irreducibe nonnegative matrix. The matrix M has a simpe, rea eigenvaue r on the spectra circe. Aso, there exists a positive vector w 1 such that w1 T M = rw1 T, which is the unique positive eft eigenvector of M up to a positive scae factor. If M has ony one eigenvaue on the spectra circe, then it is said to be primitive. If M has h > 1 eigenvaues with moduus equa to ρ(m), then it is said to be imprimitive or a cycic matrix. The vaue h is known as the index of imprimitivity or period. B. Genera Graph Theory An undirected graph G = (V, E) consists of a finite set of vertices V and a finite coection of edges E. Each edge is an unordered pair of vertices {v i, v j } such that the edge joins vertices v i and v j. Given the edge {v i, v j }, we say that vertex v i is adjacent to vertex v j, and vice versa. A vertex and edge are incident with one another if the vertex is contained in the edge. We do not give further consideration to graphs with sefoops. A sef-oop is an edge joining a vertex to itsef. Loopess graphs are commony known as mutigraphs, which may contain parae edges. Parae edges are mutipe incusions of an edge in the edge coection. A simpe graph has neither sef-oops nor parae edges. The order of a graph is the number of vertices and the size is the number of edges. The degree d(v i ) of vertex v i is the number of edges incident with v i. Euer s handshaking emma states that the sum of a degrees of a graph must be twice its number of edges. It foows from the fact that each edge in the graph must contribute 2 to the sum of a degrees. Thus, in equation form for the undirected graph G = (V, E), it may be stated that order (G) = V and size (G) = E = 1 d(v i ). 2 v i V A reguar graph is a graph whose vertices are a of equa degree. In an undirected graph G, a wak between two vertices is an aternating sequence of incident vertices and edges. The vertices and edges in a wak need not be distinct. The number of edges in a wak is its ength. The vertices v i and v j are said to be connected if the graph contains a wak of any ength from v i to v j, noting that every vertex is considered connected to itsef. A graph is said to be connected if every pair of vertices is connected, otherwise the graph is said to be disconnected. The vertices of a disconnected graph may be partitioned into connected components. Unique edges are those edges which appear ony once in the edge coection of the graph. A wak that backtracks is a wak in which a unique edge appears twice or more in-a-row in the wak. A cosed wak is a wak that begins and ends on the same vertex. A cyce is a cosed wak with no repeated edges or vertices (except the initia and fina vertex) of ength at east two. The girth of a graph is the ength of its shortest cyce, if it has cyces. If every vertex in a mutigraph has at east degree two, then the graph contains one or more cyces. A eaf is a vertex of degree one. A tree is a connected graph without cyces. Trees, with at east two vertices, have eaves. We ca an undirected graph eafess if it does not contain eaves. We wi say that two graphs are identica if they have equa vertex sets and equa edge coections. Two graphs are said

4 4 submitted to IEEE TRANSACTIONS ON INFORMATION THEORY VERSION: JUNE 4, 2013 to be isomorphic if there exists between their vertex sets a one-to-one correspondence having the property that whenever two vertices are adjacent in one graph, the corresponding two vertices are adjacent in the other graph. This correspondence reationship is caed an isomorphism. The isomorphism is a reabeing of the vertices that preserves the graph s structure. The adjacency matrix A(G) of any simpe graph G of order n is the n n symmetric (0, 1)-matrix whose (i, j) entry is 1 if and ony if {v i, v j } is an edge of G. More generay, the adjacency matrix A(G ) of any mutigraph G is the symmetric nonnegative matrix whose (i, j) entry indicates the number of edges joining v i and v j in G. The number of waks of exacty ength p between vertices v i and v j, in graph G, is the (i, j) entry of A(G) p. Two graphs are isomorphic if and ony if there exists a symmetric permutation that reates their adjacency matrices. The incidence matrix N(G) of any mutigraph G of order n and size m is the n m (0, 1)-matrix whose (i, j) entry is 1 if and ony if the ith vertex is incident with jth edge of G. It then foows that the mutigraph s adjacency matrix may be stated as A(G) = N(G)N(G) T diag (d(v 1 ), d(v 2 ),..., d(v n )). A cyce graph, denoted C n, is a graph of order n 2, in which the distinct vertices of the set {v 1, v 2,..., v n } are joined by the edges from the coection {{v 1, v 2 },..., {v n 1, v n }, {v n, v 1 }}. A cyce graph is a singe cyce. A bipartite graph B = (V, C, E) is a specia case of an undirected graph in which the graph s vertices may be partitioned into two disjoint sets V and C. Each edge e E of a bipartite graph joins a vertex from V to a vertex from C. Hence, bipartite graphs cannot have sef-oops by definition. The parity-check matrix H over F 2 describes a bipartite graph B, caed the Tanner graph of H, in which the vertices are known as variabe nodes V and check nodes C. Tanner graphs of binary codes do not have parae edges. The parity-check matrix H is reated to the adjacency matrix A(B) of the Tanner graph B as [ ] 0 H T A(B) =. (1) H 0 A d v -variabe-reguar Tanner graph is a Tanner graph whose variabe nodes a have equa degree d v, and a (d v, d c ) reguar Tanner graph is both d v -variabe-reguar and has check nodes a of degree d c. Given a subset of the variabe nodes S V, we use N (S) to denote the set of adjacent check nodes. Given a Tanner graph B = (V, C, E) and any set of variabe nodes S V, et B S represents the induced subgraph of S. That is, B S = (S, N (S), E S ) is a bipartite graph containing the variabe nodes S, the check nodes N (S), and the edges between them E S. We wi frequenty refer to these induced subgraphs as Tanner subgraphs with parity-check submatrix H S. The submatrix H S is formed by seecting the coumns of H as indexed by the members of the set S and optionay removing any resuting a-zero rows. The ine graph L(G) of a mutigraph G is the graph whose vertices are the edges of G. Two vertices of L(G) are adjacent if and ony if their corresponding edges in G have a vertex (or two) in common. In fact, two parae edges connect vertices in L(G) if and ony if their corresponding edges in G are parae. We find that the ine graph s adjacency matrix is reated to the incidence matrix of G by A(L(G)) = N(G) T N(G) 2 I. For the ine graph L(G) of mutigraph G = (V, E), order (L(G)) = E = 1 d(v i ) and (2) 2 v i V size (L(G)) = 1 d(v i ) (d(v i ) 1) (3) 2 = 1 2 v i V d(v i ) 2 E. v i V The spectra radius ρ(g) of a mutigraph G = (V, E) is defined to be the spectra radius of the matrix A(G), and it is bounded by 1 V v i V d(v i ) ρ(g) max v i V d(v i). (4) Note that d(v i ) is just the ith row (or coumn) sum of A(G). Since the matrix A(G) is symmetric, the ower bound foows by appying the Rayeigh quotient to A(G) with an a-one vector [33, pp ]. The upper bound appies since the matrix A(G) is nonnegative, and is due to Frobenius [33, p. 492]. The ower bound hods with equaity if and ony if the a-one vector is an eigenvector of A(G) corresponding to the eigenvaue ρ(g). If G is connected, then the upper bound hods with equaity if and ony if G is reguar. When G is connected these conditions are equivaent. A directed graph or digraph D = (Z, A) consists of a set of vertices Z and a coection of directed edges or arcs A. Each arc is an ordered pair of vertices (z i, z j ) such that the arc is directed from vertex z i to vertex z j. For arc (z i, z j ), we ca the first vertex z i its initia vertex and the second vertex z j its termina vertex. We wi focus on simpe digraphs which excude sef-oops and parae arcs. In a digraph D, a directed wak is an aternating sequence of vertices and arcs from z i to z j in D such that every arc a i in the sequence is preceded by its initia vertex and is foowed by its termina vertex. A digraph D is said to be strongy connected if for any ordered pair of distinct vertices (z i, z j ) there is a directed wak in D from z i to z j. For exampe, the digraph in Fig. 4d is strongy connected. The adjacency matrix A(D) of any simpe digraph D is the (0, 1)-matrix whose (i, j) entry is 1 if and ony if (z i, z j ) is an arc of D. Lemma 1. A (0, 1)-matrix is irreducibe if and ony if the associated digraph is strongy connected. Proof: See [36, p. 78]. The outdegree d + i of vertex z i is the number of arcs in digraph D with initia vertex z i. Likewise, the indegree d i of vertex z i is the number of arcs with termina vertex z i. For

5 BUTLER AND SIEGEL: ERROR FLOOR APPROXIMATION FOR LDPC CODES IN THE AWGN CHANNEL 5 the digraph D = (Z, A), we note that order (D) = Z and size (D) = A = z i Z d + i = d i. z i Z A reguar digraph is a digraph whose vertices are a of equa indegree and outdegree. The spectra radius ρ(d) of a digraph D is defined to be the spectra radius of its matrix A(D), and it is bounded in the foowing emma. Lemma 2. Let the ith vertex z i of digraph D = (Z, A) have outdegree d + i. Then the spectra radius ρ(d) of D is bounded above and beow by min z i Z d+ i ρ(d) max z i Z d+ i. (5) If D is strongy connected, then the inequaities are strict uness the digraph has reguar outdegree. Anaogous statements hod for the indegrees. Proof: See [33, p. 492] and [35, pp. 8 and 22]. The interested reader is referred to [36] [39], etc., for a more compete treatment of the subject. Our use of graph theory has paraes to [25]. C. LDPC Codes, the AWGN Channe, and SPA Decoding LDPC codes are defined by the nu space of a parity-check matrix H. The codewords are the set of coumn vectors C, such that c C satisfies Hc = 0 over a particuar fied. A given code can be described by many different parity-check matrices. The H matrix over F 2 may be associated with a bipartite graph B = (V, C, E) termed a Tanner graph described in the previous subsection. The set of variabe nodes V represent the symbos of the codeword that are sent over the channe and correspond to the coumns of the parity-check matrix. The set of check nodes C enforce the parity-check equations represented by the rows of H. Assumption 1. We are ony concerned with LDPC codes over the binary fied F 2 and with binary antipoda signaing over the AWGN channe. The SPA decoder woud be optima with respect to minimizing the symbo error rate, if the Tanner graph had no cyces. In any good finite-ength code, the Tanner graph wi indeed have cyces [40], but we generay find that the SPA decoder does quite we. Like many, we prefer to impement our decoder simuation in the LLR-domain, as it is cose to the approximations typicay used in buiding hardware. Since we wi assume that the reader is famiiar with SPA decoding of LDPC codes over the AWGN channe (see [41, 5.4] and [42]), the remainder of this subsection is presented merey to carify notation. We take the channe SNR to be 1/σ 2 = 2RE b /N 0. We denote the intrinsic channe LLR for the ith received symbo by λ i, and perform check node updates according to λ [i j] = 2 tanh 1 k N (j)\i tanh λ[k j] 1 2, (6) which computes the message to be sent from the jth check node to the adjacent ith variabe node, during the first-haf of the th iteration. We update variabe nodes in the usua way during the second-haf of every iteration. In (6), we use N (j)\i to indicate a variabe nodes adjacent to check node j, excuding variabe node i. We denote the sum of the decoder s intrinsic and extrinsic LLRs at the ith variabe node at the [i] competion of the th iteration by λ. D. SPA Decoder without Saturation Direct impementation of (6) yieds numerica probems at high LLRs. As specified in IEEE 754 [43], doube-precision foating-point (i.e., 64-bit) computer computations maintain 53-bits of precision, whie the remaining bits are for the sign and exponent. Thus, such a computer impementation resuts in tanh(λ/2) being rounded to ±1 for any LLR magnitude λ > = 55 n 2. As an argument of ±1 wi cause the tanh 1 function to overfow, protection from high magnitude LLRs must be added to (6) or the variabe node update expression to ensure numerica integrity or an aternative soution not using tanh 1 must be found. Thus, preventing tanh 1 overfow by imiting LLRs wi resut in an upper imit on LLR magnitude. This is what we refer to as saturating in an LLR-domain SPA decoder. Our examination of pubished error foor resuts suggests that LLR saturation is commony empoyed. Decoder hardware impementations aso typicay saturate the LLRs, athough for different reasons. Hardware saturation is usuay at LLR eves ess than the 38.1 mentioned above. This eads to the situation where both foating-point simuation and hardware designs produce error foors at performance eves with minor variation. The efforts to expore beyond these barriers have primariy appeared ony in the very recent iterature, incuding [32], [44], [45]. The non-saturating SPA decoder simuation we have used is based on a pairwise check node reduction [46], [47] which incudes a sma approximation [48]. Its impementation in doube-precision foating-point contains no numerica issues unti LLR magnitudes reach approximatey For a detaied examination of the numerica issues of the SPA decoder in severa domains of foating-point computation, see [45]. E. Trapping Sets and Absorbing Sets Definition 1 (Richardson [8]). A trapping set is an errorprone set of variabe nodes T that depends upon the decoder s [i] input space and decoding agorithm. Let λ (r) denote the ith soft output symbo at the th iteration given the decoder input vector r. We say that symbo i is eventuay correct if there [i] exists L such that, for a L, λ (r) is correct. If the set of symbos that is not eventuay correct is not the empty set, we ca it a trapping set, T (r). It is caed an (a, b) trapping set if the set has a = T variabe nodes and the induced subgraph B T contains exacty b check nodes of odd degree. From here on this paper wi use the term trapping set, unquaified, to mean the trapping sets defined above with

6 6 submitted to IEEE TRANSACTIONS ON INFORMATION THEORY VERSION: JUNE 4, 2013 respect to the AWGN channe decoded by an LLR-domain SPA decoder with saturating LLRs. Note, that whie nearcodewords are distinct from codewords, the trapping set definition incudes codewords. Definition 2. A Tanner subgraph or trapping set is caed eementary if a the check nodes are of degree one or two [49]. Eementary trapping sets are simpe enough to be modeed with inear systems and aso account for most of the error foors seen in practice. In [8], [14], [17], [49] [51], the authors observe that the majority of trapping sets contributing to the error foors of beief propagation decoders are eementary trapping sets. Exampe 1. Four sampe eementary Tanner subgraphs are shown in Fig. 2 for a code of variabe-degree three. These are a trapping sets if they satisfy the not eventuay correct condition of Definition 1. We wi define absorbing sets as they dominate the error foor of SPA decoders with saturating LLRs. Absorbing sets are a particuar type of near-codeword or trapping set. The fuy absorbing definition adds conditions beyond the subgraph itsef to the remaining Tanner graph. Definition 3 (Doecek et a. [16]). Let S be a subset of the variabe nodes V of the Tanner graph B = (V, C, E) and et S induce a subgraph having a set of odd-degree check nodes N o (S) and a set of even-degree check nodes N e (S). If S has cardinaity a and induces a subgraph with N o (S) = b, in which each node in S has stricty fewer neighbors in N o (S) than neighbors in N e (S), then we say that S is an (a, b) absorbing set. If, in addition, each node in V has stricty fewer neighbors in N o (S) than neighbors in the set C \N o (S), then we say that S is an (a, b) fuy absorbing set. Fuy absorbing sets are stabe structures during decoding using the bit-fipping agorithm [16]. This resuts from every variabe node of the set neighboring stricty more checks which reinforce the incorrect vaue than checks working to correct the variabe node. III. STATE-SPACE MODEL In this section we refine and justify the existing inear system mode which cosey approximates the non-inear message passing of the LLR-domain SPA decoder described earier, as appied to an eementary trapping set. Since this is a inear bock code, with no oss of generaity, we wi frequenty assume for convenience that the a-zero codeword has been sent. The state-space mode was introduced in [23], [24] to anayze eementary trapping sets. Identifying and modeing trapping sets are of interest so as to expain the observed phenomenon of the trapping sets variabe nodes being decoded incorrecty, whie the variabe nodes outside of the set are eventuay decoded correcty by an LLR-domain SPA decoder. Assumption 2. A trapping sets of interest are eementary and contain more than one variabe node. We do not consider subgraphs with a singe variabe node as they woud not have states in the state-space mode. For trapping-ike behavior of singe degree-1 variabe nodes, see [52, p. 24]. First, we review the faiure state of an eementary trapping set. Let the variabe nodes of the trapping set S V induce the Tanner subgraph B S. In ater iterations the condition is reached where the variabe nodes S within the trapping set are in error and those variabe nodes outside the trapping set (i.e., V \S) are correct. In this condition, the subgraph s check nodes of degree one are unsatisfied and those of degree two are satisfied. The vector of messages from the degree-one (i.e., unsatisfied) check nodes λ (ex) at iteration are taken to be stochastic system inputs to the state-space mode and are therefore modeed separatey. We wi use density evoution and simuation techniques, described in ater sections, to mode this input vector. The other system input vector is the intrinsic information λ provided by the channe to be used in variabe node updates at every iteration. For the AWGN channe mode, each eement of λ is an independent and identicay distributed (i.i.d.) Gaussian random variabe. Both of these system inputs wi be treated as coumn vectors of LLRs; the vector λ has a entries and the vector λ (ex) has b entries for an (a, b) trapping set. A. Check Node Gain Mode Sun s inear mode incuded the asymptotic approximation that every degree-two (i.e., satisfied) check node output message is equa to the input message at the other edge [23]. This is based on the conditions of correct variabe nodes and very high LLR vaues outside of the trapping set. Schege and Zhang introduced more accuracy to this mode by appying a mutipicative gain g at the degree-two check nodes, where 0 < g 1 [10]. This gain modes the effect of the d c 2 externa variabe nodes owering the magnitude of LLR messages as they pass through the degree-two check nodes. This is iustrated in Fig. 3. This approach adds accuracy to modeing the eary iterations, and after severa iterations these gains approach 1. Assumption 3. We wi assume that the externa inputs to the trapping set s degree-two check nodes are i.i.d. This is generay a reasonabe assumption to make as the gain computations are most significant at eary iterations. By the iteration count in which significant message correation is present on the check nodes inputs, the gains are approximatey unity. Schege and Zhang compute the mean gain ḡ corresponding to each iteration count 1 [10]. Thus, the gain mode turns the state updates into a time-varying process. We have tried introducing a gain variance to the mode, but found the effect to be very sma, so we present their mean gain mode 1, for which the gain of the jth check node during the th iteration 1 The foowing incudes a correction to the gain expressions of [10] reated to the iteration count dependencies

7 BUTLER AND SIEGEL: ERROR FLOOR APPROXIMATION FOR LDPC CODES IN THE AWGN CHANNEL 7 Fig. 2. (a) a = 3, b = 1 (b) a = 4, b = 2 (c) a = 5, b = 3 (d) a = 4, b = 4 Four sampe eementary Tanner subgraphs of a code with d v = 3. Odd-degree check nodes are shown shaded. VN i... Check node j d c -2.. VN i 1 2 Fig. 3. Iustration of a check node of degree two in the trapping set and degree d c in the Tanner graph. is g [j] k N (j)\{i 1,i 2} tanh ( ) [k j] λ 1. (7) 2 Then the expected vaue of the check node s gain over a reaizations of the channe noise vector n and a[ check nodes [j]] c j, such that d(c j ) = d c, is ḡ (d c ) = E n,j g. Using Assumption 3, one forms the foowing simpification with respect to ḡ (d c ) ( )] [k j] dc 2 λ 1 ḡ (d c ) = E n,j,k [tanh, (8) 2 over a j such that d(c j ) = d c and a k N (j). The appication of gain at the check node may be justified using a Tayor series expansion of (6) that begins with etting λ out f(λ) = 2 tanh 1 [tanh ( λ 2 ) g ], (9) where g is the check node s gain (7), λ is the check node s input LLR from the trapping set, and λ out is the check node s output LLR on the other edge within the trapping set. Noting that the first two partia derivatives of (9) evauated at λ = 0 are f(λ) 2 f(λ) λ = g and λ=0 λ 2 = 0, λ=0 one produces the truncated series approximation λ out gλ. The approximation is good for λ 2 tanh 1 (g). The exact expression for λ out (9) saturates at an output of ±2 tanh 1 (g), whie the approximation is inear in λ. Since we expect the LLRs outside of the trapping set to grow quite fast, we beieve this to be an adequate approximation. We now extend the check node gain mode to incude inversions through the degree-two check nodes caused by erroneous messages from outside of the trapping set. This occurs primariy during eary iterations and is iustrated in Fig. 3. When an odd number of the d c 2 messages from outside of the trapping set into a degree-two check node within the trapping set are erroneous, then that check node wi invert the poarity of the LLR message sent from variabe node i 1 to i 2, as we as the message sent in the opposite direction. We wi present a simpification of the method used by Schege and Zhang to account for these inversions [10]. Moreover, they present a sma rise in their predicted error foor instead of owering it as we find with our technique. Schege and Zhang injected a stochastic canceation signa into the state update equations, whie we choose to modify the mean check node gains. During the first iteration, messages from variabe nodes may contain inversions due to channe errors. The probabiity that a specific input to the check node contains an inversion during iteration = 1 is just the uncoded symbo error rate ( ) 2REb P e,1 = Q. (10) Thus, utiizing Assumption 3, the probabiity of a poarity inversion in a specific check node is given at iteration by P inv, = ( ) dc 2 P k e,(1 k P e, ) dc 2 k k odd (11) = 1 (1 2P e,) dc 2, 2 which is the probabiity of an odd number of errors in the d c 2 input messages at iteration. The fina simpification in (11) is from [2, p. 38]. For additiona iterations we can use density evoution [53], [54] to predict an effective E b /N 0 at the output of the variabe nodes and reuse equations (10) and (11). When there is a channe inversion, the output message magnitude wi ikey be very ow, as suggested in [10]. Hence, to mode this we wi equate random channe inversions to randomy setting the check node gain to zero with probabiity P inv,. We introduce the check node s modified mean gain as simpy N 0 ḡ = ḡ (1 P inv, ). (12) For an LDPC code with reguar check node degree d c, the scaar gain ḡ of (12) may be appied at each iteration to mode the externa infuence on the subgraph s degree-two

8 8 submitted to IEEE TRANSACTIONS ON INFORMATION THEORY VERSION: JUNE 4, 2013 check nodes. For an LDPC code with irreguar check degree, we have a few options. If the check node degree spread is sma, we may generaize (8) and (12) by aso taking the expectation over the degree distribution of a the degree-two check nodes in the subgraph. Aternativey, we may maintain severa versions of gain ḡ, one for each check-degree that appears in the subgraph. Assumption 4. We assume that a the degree-two check nodes in the subgraph may be modeed by the modified mean gain ḡ. One may criticize this gain mode with inversions as doubecounting. First, inversions from outside the trapping set wi reduce the check node gain through (8). Then, the same inversions reduce the check node gain again in (12). Heuristicay, one may consider this as just adding weight to the significance of these inversions. We find it justified empiricay in Section VII and note that our earier arguments paraeed those in [10]. B. Linear State-Space Mode Assumption 5. We consider ony codes described by d v - variabe-reguar Tanner graphs with d v 3. For an examination of irreguar codes, see [23]. Assumption 6. In justifying the mode it is assumed that the messages within the trapping set under study have itte impact on the rest of the Tanner graph. This seems reasonabe given that the number of unsatisfied check nodes which form the main interface between the trapping set and the rest of the Tanner graph is sma. The state-space modeing equations are shown beow for input coumn vectors λ (ex) and λ and for output coumn vector λ, whose entries are the LLR-domain soft output decisions at iteration [10], [23]. x 0 = Bλ x = ḡ Ax 1 + Bλ + B ex λ (ex) for 1 (13) λ = ḡ Cx 1 + λ + D ex λ (ex) for 1 (14) The centra part of the state-space mode, of course, is the updating of the state x. The state vector x represents the vector of LLR messages sent aong the edges from the subgraph s variabe nodes toward the degree-two check nodes. The state is updated once per fu iteration. One may consider that ḡ x 1 represents the LLR messages after passing through the degree-two check nodes during the first haf-iteration, and ḡ Ax 1 represents their contribution to the variabe node update during the second haf-iteration of iteration. Since ad v equas the number of edges of the variabe-reguar Tanner subgraph of a variabe nodes, the number of states in the mode of a subgraph is m = ad v b. For an eementary subgraph, these m edges are incident on the subgraph s degree-two check nodes, forcing m to be an even integer. The m a matrix B and the m b matrix B ex are used to map λ and λ (ex), respectivey, to the appropriate entries of the state vector to match the set of variabe node update equations. Since every variabe node has exacty one eement from λ participating, B has a singe 1 in every row, and is zero otherwise. The number of 1 entries per row of B ex corresponds to the number of degree-one check nodes adjacent to the corresponding variabe node, which may be none. Like a the matrices appearing in the state-space equations, they are (0, 1)-matrices. The m m matrix A describes the dependence of the state update upon the prior state vector. Each nonzero entry [A] ij = 1 indicates the jth edge of the prior iteration contributes to the variabe node update computation of the ith edge. Thus, the matrix A T may be taken as the adjacency matrix associated with a simpe digraph of order m which describes the state-variabe update reationships. Given our d v -variabe-reguar codes, the row weight of A and the row weight of B ex wi sum to d v 1 for every row. Finay, the a m matrix C and the a b matrix D ex are used to map x 1 and λ (ex) entries, respectivey, to the corresponding entry of the soft output decision vector λ. The row weight of C and the row weight of D ex wi sum to the variabe node degree d v for every row. If there is a singe degree-one check node neighboring every variabe node in the subgraph, such as in the (8, 8) trapping set of 802.3an, then (13) degenerates to the case where B = B ex and B has reguar coumn weight as derived in [10]. Furthermore, this degenerate case produces ony A matrices which have uniform row and coumn sums with a dominant eigenvaue r = d v 2. Thus, our deveopment wi be significanty more genera than [10]. Again, note that the check node gains form our approximate mode for the behavior of degree-two check nodes within the trapping set. One drawback to any inear system approach is that saturation cannot be introduced to the state variabes. Any saturation effects must be appied to the inear system inputs. Exampe 2. For the (4, 2) trapping set with d v = 3, we abe with integers a the edges into every degree-two check node as depicted in Fig. 4a. Now, we can set up the matrices used to make the inear system. We have numbered the edges in Fig. 4a to correspond to the order that the edges wi appear in the state vector x. The matrix A describes the updating reationship of the state vector in a fu iteration of the SPA decoder. Since edge 1 depends ony on edges 6 and 9 during one iteration, we fi the first row of A to indicate such. A the matrices associated with the trapping set of Fig. 4 are given beow. Additionay, treating the matrix A T as an adjacency matrix, it describes the simpe digraph shown in Fig. 4d. A = C = B = [ ] D ex = B ex = [ ] If we remove the gains ḡ from the inear state-space equations, we have a inear time-invariant (LTI) system for

9 BUTLER AND SIEGEL: ERROR FLOOR APPROXIMATION FOR LDPC CODES IN THE AWGN CHANNEL (a) Tanner subgraph, B S (b) Mutigraph, G (c) Line graph, L(G) (d) Simpe digraph, D Fig. 4. Severa graphica descriptions of the (4, 2) trapping set with mode s states numbered. Arrows have been added to the Tanner subgraph in (a) ony to emphasize the direction of state variabe fow. The spectra radii of G, L(G), and D are , , and , respectivey. which we can write a transfer function. Let the uniatera Z- transform of the state x be denoted by X(z) and the Z- transform of the input vector λ (ex) be denoted by Λ ex (z), where X(z) = k=0 x kz k [55]. Note that the other input vector λ is not a function of iteration number. Finay, etting the soft output vector λ of the system be denoted in the Z- domain by Λ(z), yieds [ ] Λ(z) = C (zi A) 1 λ B + I 1 ] z 1 (15) + [C (zi A) 1 B ex + D ex Λ ex (z). This makes evident the importance of the eigenvaues of A. As the eigenvaues are the roots of the characteristic equation det(a µi) = 0, they correspond directy to the poes of the discrete-time LTI system in (15). We find that for the trapping sets addressed herein, the dominant poes are on or outside the unit circe. Hence, the LTI system is marginay stabe or (more often) unstabe. We wish to take the recursive state-update equation (13) and deveop it into an expression without feedback. The first two iterations are easy to express term-by-term, as x 1 = ḡ 1ABλ + Bλ + B ex λ (ex) 1 and x 2 = ḡ 1ḡ 2A 2 Bλ + ḡ 2ABλ + Bλ j=1 + ḡ 2AB ex λ (ex) 1 + B ex λ (ex) 2. Generaizing to an arbitrary iteration > 0, we have ( ) x = A Bλ ḡ j + A i Bλ + B ex λ (ex) i ḡ j. i=1 C. Graphica Assumptions and Interreationships j=i+1 (16) We have aready proposed two graphica restrictions in Assumptions 2 and 5. This subsection notes some further assumptions we wish to pace on the trapping sets and describes the interreationships among severa usefu graphica descriptions of the trapping set: the Tanner subgraph, its corresponding undirected mutigraph, the ine graph of the mutigraph, and the simpe digraph that corresponds directy to the state update mode. Finay, we form estimates of the dominate eigenvaue for the trapping set. Lemma 3. There exists a bijective map between the set of d v -variabe-reguar eementary Tanner subgraphs (up to a reabeing of the check nodes) and the set of mutigraphs G = (V, E) with vertex degrees upper bounded by d v, i.e., d(v i ) d v v i V. Proof: Given the eementary Tanner subgraph B S = (S, N (S), E S ), each variabe node v S becomes a vertex v V of G, that is V = S, and the degree-one check nodes are discarded. The check nodes of degree two become the edges of G, each joining a pair of vertices. Aternativey, et paritycheck submatrix H S describe B S ; then A(G) = H T S H S d v I. Given the mutigraph G, each vertex v V of G becomes a variabe node v S of B S. So, again V = S. Each edge of G is repaced by a degree-two check node (with an arbitrary abe) and two edges in B S. Finay, degree-one check nodes are attached with singe edges to variabe nodes as needed unti every variabe node has d v neighbors. In creating the mutigraph G, we have a more basic description of the potentia eementary trapping set. This is iustrated in creating Fig. 4b from Fig. 4a. Either one of these graphs is sufficient to characterize the iteration-to-iteration dependencies among the state variabes in the mode. The mutigraph G = (V, E) corresponding to B S has order (G) = V = a and size (G) = E = ad v b (17). 2 Tanner graphs with cyces of ength 4, the smaest permitted in a Tanner graph, wi map to parae edges in G. Cyces of ength k in B S map to cyces of ength k/2 in G. The 4-cyces in the Tanner graph are often avoided by code designers. Exampe 3. The mutigraph corresponding to the Tanner subgraph of Fig. 2d is the cyce graph of order 4, C 4. Assumption 7. Tanner subgraphs of interest and their associated mutigraphs are connected. Those trapping sets described by disconnected Tanner subgraphs can instead be anayzed component by component. The main reason for this simpification is to reduce the number of cases in which the state update expression contains a reducibe A matrix. Assumption 8. We further assume that the variabe nodes within the Tanner subgraphs contain from zero to d v 2

10 10 submitted to IEEE TRANSACTIONS ON INFORMATION THEORY VERSION: JUNE 4, 2013 adjacent degree-one check nodes. Equivaenty, we assume that the associated mutigraphs are eafess; that is, they do not contain vertices of degree one. For consideration of mutigraphs with eaves, see Appendix B. A eaf in the mutigraph maps to a variabe node in the Tanner subgraph neighboring one degree-two check node and d v 1 degree-one check nodes. Thus, we require here that every variabe node in the Tanner subgraph must neighbor at east two degree-two check nodes. The mutigraphs aowed by Assumptions 7 and 8 wi be connected, eafess graphs, containing one or more cyces. The conditions described here, Sun termed simpe trapping sets in [23]. Our Assumption 8 is equivaent to eementary absorbing sets for codes of variabe-degree three. However, our Assumption 8 is ess restrictive in some ways than absorbing sets for d v 4 as we aow for more degree-one check nodes per variabe node. Finay, we describe how to create the simpe digraph D = (Z, A) from an undirected mutigraph G = (V, E) which meets Assumptions 7 and 8. This is iustrated in creating Fig. 4d from Fig. 4b. Each edge of a connected eafess mutigraph G corresponds to two vertices in D, representing the two directions of LLR message fow in the origina Tanner graph. That is, the number of vertices m in D is simpy m = order (D) = 2 size (G) = ad v b. The digraph D is a representation of the message updating process with respect to the output of the trapping set s variabe nodes in one fu iteration of the SPA decoder. Let edge e i = {v j, v k } E map to vertices z i, z i Z, with z i representing the direction of e i fowing from v j to v k and z i representing the other direction. No arcs in D join z i to z i. Arcs initiating in z i are directed to vertices corresponding to other edges in G fowing out of v k and arcs terminating in z i are directed from vertices corresponding to other edges in G fowing into v j. Hence, d + i = d(v k ) 1 = d i, (18) d i = d(v j ) 1 = d + i, and (19) size (D) = = d(v j ) [d(v j ) 1]. z i Z d i v j V With respect to a Tanner subgraph that meets Assumption 8, induced from a code of variabe-degree three, the size of the associated digraph simpifies to 6a 4b. In our (4, 2) exampe, the digraph of Fig. 4d has order 10 and size 16. We now describe a second version of the construction of the simpe digraph. From the mutigraph G, shown in Fig. 4b, we create the ine graph L(G) shown in Fig. 4c as described in Section II-B. Next, we perform a specia directed ifting of L(G) to form the simpe digraph D, shown in Fig. 4d. This ifting by a factor of two repaces each vertex with a pair of vertices and each edge with a pair of arcs, oriented in opposite directions. Thus, if the (i, j) entry of A(L(G)) is w it is repaced with a 2 2 submatrix containing w ones, where the ine graph imits w to the vaues 0, 1, and 2. When the 2 2 submatrix which repaces the (i, j) entry of A(L(G)) is determined, the seection of the corresponding TABLE I SUBMATRIX CORRESPONDENCE RULES OF THE DIRECTED LIFTING OF THE LINE GRAPH (FOR i = j, ONLY THE ALL-ZERO MATRIX IS ALLOWED) Entry (i, j) Entry (j, i) ( 1 0 ) ( 0 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (j, i) submatrix foows a set of rues of correspondence, isted in Tabe I, which ensure that the new pair of arcs are oriented in opposite directions. Thus, the subdiagona eements can be easiy determined from the superdiagona eements. The exact construction of either one of these is more invoved and requires a direction assignment consistency between, for exampe, Fig. 4a and the ifting of Fig. 4c. A directed waks in D wi correspond to waks in G. A waks in G that do not backtrack wi correspond to directed waks in D. Backtracking is prohibited here due to the structure of SPA decoder as expressed in the message update rues, which excude an edge s own incoming LLR from its outgoing LLR computation. The adjacency matrix of D is A(D), which is an m m (0, 1)-matrix. The matrix A used in state space mode (13) is equa to A T (D) as D describes the fow of messages within the mode. This connection wi be convenient when discussing the properties of A. Exampe 4. Figs. 2a, 2b, 2c, 2d, and 5c meet Assumptions 7 and 8. Of these, ony Fig. 2d has a reducibe A(D) matrix and it may be anayzed in this paper as expained in Appendix A. The graphs of Figs. 5a and 5b fai Assumption 8 and have reducibe A(D) matrices. In the case of Fig. 5a, the graph may first be anayzed without the eaf, and then the eaf considered as expained in Appendix B. The case of Fig. 5b has a zero eigenvaues as no amount of eaf remova creates a satisfactory structure. Thus, it is not anayzed in this paper. We are now ready to bound the spectra radius of the graphs presented in this subsection in order to cosey approximate the dominate eigenvaues of A(D) and hence A. Using (4) and (17), we may bound the spectra radius of the starting mutigraph G = (V, E) as d v b/a ρ(g) max v i V d(v i), (20) where both equaities hod for connected G if and ony if G is reguar. The maximum vaue of d(v i ) is imited by the parent Tanner graph to be no greater than d v. To bound the spectrum of the ine graph L(G), we may use the reation ρ(l(g)) = ρ(a(g) + D) 2, (21) where D = diag (d(v 1 ), d(v 2 ),..., d(v n )). This foows from the incidence matrix reated expressions for each type of graph and the foowing fact which is easy to show: Given any rea matrix M, the two products M T M and MM T have identica nonzero eigenvaues.