Threshold Analysis of Non-Binary Spatially-Coupled LDPC Codes with Windowed Decoding

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1 1 Threshold Analysis of Non-Binary Spatially-Coupled LDPC Codes with Windowed Decoding Lai Wei, Toshiaki Koike-Akino, David G. M. Mitchell, Thoas E. Fuja, and Daniel J. Costello, Jr. Departent of Electrical Engineering, University of Notre Dae, Notre Dae, IN, U.S {lwei1, david.itchell, tfuja, Mitsubishi Electric Research Laboratories (MERL), Cabridge, MA, U.S. {wei, arxiv: v1 [cs.it] 14 Mar 2014 Abstract In this paper we study the iterative decoding threshold perforance of non-binary spatially-coupled low-density parity-check (NB-SC-LDPC) code ensebles for both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC), with particular ephasis on windowed decoding (WD). We consider both (2, 4)-regular and (3, 6)-regular NB-SC-LDPC code ensebles constructed using protographs and copute their thresholds using protograph versions of NB density evolution and NB extrinsic inforation transfer analysis. For these code ensebles, we show that WD of NB-SC-LDPC codes, which provides a significant decrease in latency and coplexity copared to decoding across the entire parity-check atrix, results in a negligible decrease in the nearcapacity perforance for a sufficiently large window size W on both the BEC and the BIAWGNC. Also, we show that NB- SC-LDPC code ensebles exhibit gains in the WD threshold copared to the corresponding block code ensebles decoded across the entire parity-check atrix, and that the gains increase as the finite field size q increases. Moreover, fro the viewpoint of decoding coplexity, we see that (3, 6)-regular NB-SC-LDPC codes are particularly attractive due to the fact that they achieve near-capacity thresholds even for sall q and W. I. INTRODUCTION Non-binary low-density parity-check (NB-LDPC) block codes constructed over finite fields of size q > 2 outperfor coparable binary LDPC block codes [1], in particular when the blocklength is short to oderate; however, this perforance gain coes at the cost of an increase in decoding coplexity. A direct ipleentation of the belief-propagation (BP) decoder [1] has coplexity O(q 2 ) per sybol. More recently, an ipleentation based on the fast Fourier transfor [2] was shown to reduce the coplexity to O(q log q). Beyond that, a variety of siple but sub-optial decoding algoriths have been proposed in the literature [3] [4]. As for coputing iterative decoding thresholds, a non-binary extrinsic inforation transfer (NB-EXIT) analysis was proposed in [5] and was later developed into a corresponding version P-NB- EXIT [6] suitable for protograph-based codes. A protograph [7] is a sall Tanner graph, which can be used to produce a structured LDPC code enseble by applying a graph lifting procedure, such that every code in the enseble aintains the structure of the protograph, i.e., it has the sae degree distribution and the sae type of edge connections. Figure 1 illustrates a (3, 6)-regular protograph, Variable node B Check node 3 3 Fig. 1. A (3, 6)-regular protograph and its corresponding base-atrix representation. which can be used to produce a (3, 6)-regular LDPC block code enseble. A protograph with (c b) check nodes and c variable nodes can be represented equivalently by a base (parity-check) atrix B consisting of non-negative integers, in which the (i, j)-th entry (1 i c b and 1 j c) is the nuber of edges between check node i and variable node j. To calculate the BP threshold of a protograph-based code enseble, conventional tools ust be adapted to take the edge connections into account. Although soe freedo is lost in the code design when the protograph structure is adopted, one can use these odified protograph-based analysis tools to find good protographs with better BP thresholds than corresponding unstructured ensebles with the sae degree distribution. Spatially-coupled LDPC (SC-LDPC) codes, also known as terinated LDPC convolutional codes [8], have been shown to exhibit a phenoenon called threshold saturation [9], where, as the terination length grows, the BP decoding threshold saturates to the axiu a-posteriori (MAP) threshold of a (d v, d c )-regular underlying LDPC block code enseble, which in turn iproves to the channel capacity as the density (d v and d c ) of the parity-check atrix increases. Iterative decoding threshold results on the binary erasure channel (BEC) for nonbinary SC-LDPC (NB-SC-LDPC) code ensebles have been reported by Uchikawa et al. [10] and Pieontese et al. [11], and the corresponding threshold saturation was proved by Andriyanova et al. [12]. In each of these papers, the authors assued that decoding was carried out across the entire paritycheck atrix of the code; for siplicity, this will be referred to as the flooding schedule (FS) in our paper. A ajor proble with FS decoding of SC-LDPC codes is

2 2 latency. To resolve this issue, a ore efficient technique, called windowed decoding (WD), was proposed in [13]. Copared to FS decoding, WD exploits the convolutional nature of the SC parity-check atrix to localize the decoder and thereby reduce latency; under WD, the decoding window contains only a portion of the parity-check atrix, and within that window BP decoding is perfored. In this paper, assuing that the binary iage of a codeword is transitted, we analyze the WD threshold perforance of (2, 4)-regular and (3, 6)-regular NB-SC-LDPC code ensebles based on protographs. In particular, 1) for the BEC, we develop the NB density evolution (NB- DE) analysis as proposed in [14] into a protograph version, which we call P-NB-DE, and 2) for the binary-input additive white Gaussian noise channel (BIAWGNC) with binary phase-shift keying (BPSK) odulation, we apply the P-NB-EXIT [6] analysis (originally proposed for NB-LDPC block codes) to NB-SC- LDPC codes. The finite field size q is constrained to be 2, where is a positive integer. In both cases, our priary contribution lies in the scenario when WD is ipleented, so that decoder latency can be reduced at the cost of a sall loss in decoder perforance. For three NB-SC-LDPC enseble exaples, we show in Sections III and IV that WD provides two of the ensebles with non-decreasing threshold perforance as W and/or increases. In fact, their WD thresholds are nuerically capacity-achieving for sufficiently large W and. As for the third enseble, although its WD threshold diverges slightly fro capacity when is large (observed on the BEC), it is the strongest candidate for low-latency and/or low-coplexity applications due to its excellent perforance when W and are both sall; this conclusion is further strengthened in our analysis of WD coplexity in Section V. In all, the results of this paper provide theoretical guidance for designing and ipleenting practical NB-SC-LDPC codes for WD. II. NB-SC-LDPC CODE ENSEMBLES An SC-LDPC code enseble can be constructed fro a LDPC block code enseble using an edge spreading technique [15], which can be described conveniently by protographs. As shown in Figure 1, let B denote a block base atrix of size (c b) c, which corresponds to a protograph representation of an LDPC block code enseble with design rate R = b/c. An SC base atrix corresponding to an SC- LDPC code enseble can then be constructed using ( s + 1) coponent base atrices {B i } s i=0, each of size (c b) c, where the edges of B are spread such that s B i = B, i=0 and s is the eory size. The resulting SC base atrix is given in its transpose for in (1) at the botto of this page, where L is called the terination length. The design rate of the code is R L = 1 (c b)(l + s). cl As a result of the terination, there is a rate loss copared to the block code design rate; however, this diinishes as L increases, i.e., R L R = b/c when L. In WD, the window size W is defined as the nuber of colun blocks of size c covered by the decoding window, which slides over a portion of B SC of fixed size W (c b) by W c (in sybols, see [13] for details). In this paper, we use the following three protographs as exaples, where C denotes the SC-LDPC code ensebles and B denotes the underlying LDPC block code ensebles: 1) B [2,4] and C [2,4] : The block base atrix B representing a (2, 4)-regular LDPC block code enseble B [2,4] and the coponent atrices used to construct an SC base atrix B SC representing an SC-LDPC code enseble C [2,4], are given by B = [ 2 2 ] B 0 = B 1 = [ 1 1 ]. As noted above, the value of an entry in B (resp. B SC ) is equal to the nuber of edges connecting the corresponding check node and variable node in the protograph for B (resp. C). 2) B and : The block base atrix B corresponding to a (3, 6)-regular LDPC block code enseble B and the coponent atrices B 0 and B 1 corresponding to an SC-LDPC code enseble are given by B = [ 3 3 ] B 0 = [ 2 1 ], B 1 = [ 1 2 ]. 3) : B = [ 3 3 ] B 0 = B 1 = B 2 = [ 1 1 ]. For each exaple, the terination length is chosen as L = 100, so that R L is close to R. We will refer to the (2, 4) group as the collection of ensebles B [2,4] and C [2,4] and the (3, 6) group as B,, and. In practice, an NB-SC-LDPC code is generated fro B SC in two steps, siilar to the procedure for generating an NB- LDPC block code fro B [7]: 1) Lifting : Replace the nonzero entries in B SC by an M M perutation atrix (or a su of non-overlapping B SC = B 0 B 1... B s B 0 B 1... B s (1) B 0 B 1... B s cl (c b)(l+ s)

3 3 M M perutation atrices), and replace the zero entries by the M M all-zero atrix; M is called the lifting factor. In this way, the structure of B SC is aintained in the lifted SC-LDPC atrix, so the threshold analysis of the SC-LDPC code enseble C can be carried out directly based on B SC. 2) Labeling : Randoly assign to each non-zero entry in the lifted parity-check atrix a non-zero eleent in GF(q), where q = 2 is the finite field size. After the lifting step, the parity-check atrix is still binary, i.e., the non-binary feature does not arise until labeling. Both the perutation atrices and the selection of labels can be optiized in order to obtain a good code [6], but this is not our ephasis here, since we are interested in a threshold analysis of the general non-binary enseble, where the diension of the essage odel used in the threshold analysis depends on the size of the finite field [5] [14]. III. THRESHOLD ANALYSIS OF NB-SC-LDPC CODE ENSEMBLES ON THE BEC A. P-NB-DE Analysis on the BEC We extended the NB-DE algorith for the BEC [14] to a protograph version, which we denote P-NB-DE, siilar to the procedure used to extend NB-EXIT to P-NB-EXIT in [6]. Since edge connections are taken into account, P-NB-DE is essentially the BP algorith perfored on a protograph. The resulting BP threshold is denoted ɛ BP (δ) with δ [0, 1], which is the largest channel erasure rate such that all transitted sybols can be recovered successfully with probability at least (1 δ) when the nuber of iterations goes to infinity. Our results are obtained for δ = 10 6 ; however, we note that setting δ = provides siilar results. B. Nuerical Results In this section, we present the nuerical results for the BEC, with ephasis on the threshold perforance of NB-SC- LDPC code ensebles when WD is used. As a benchark, Figure 2 first copares the FS threshold perforance of the (2, 4) group and the (3, 6) group, where FS decoding is carried out on the entire parity-check atrix and not restricted to a window. We observe that the NB-SC-LDPC codes perfor extreely well copared to their block code counterparts, in particular for large field size. Unlike the block code case, always outperfors C [2,4], and the thresholds of these two ensebles increase onotonically with. However, this onotonic increase is not observed for, where the obtained threshold actually decreases very slightly for > 5. (See the related discussion of Figure 3(b) below.) Figure 3(a) shows that, for a sufficiently large window size W, WD provides threshold perforance nearly the sae as the FS, i.e., the perforance loss is negligible while the decoder benefits fro greatly reduced delay. We define W to be the sallest window size such that WD provides a threshold within 3% of the FS threshold universally for all field sizes. 1 For C [2,4], we find W = This 3% value is actually loose for oderate to large. In the cases we exained, the WD threshold with W = W typically lies within % of or is even nuerically identical to the FS threshold for > B [2,4] B C [2,4] Fig. 2. Coparison of the thresholds based on the flooding schedule (FS) for the (2, 4) and (3, 6) groups on the BEC. Siilar observations can be ade for WD of and in Figure 3(b) and 3(c): For, W = 7. The WD threshold grows to a value within 0.1% of the channel capacity when = 5, and then decreases very slightly as increases further. Nevertheless, the WD threshold reains very close to capacity even for large. For, W = 10. The WD threshold does not degrade as increases, but instead saturates to a value nuerically indistinguishable fro capacity. However, when W is sall (e.g., W = 5), does not perfor as well as due to its larger eory size, which increases the delay required to ake reliable decisions (see [13]). To suarize, for the three considered NB-SC-LDPC code ensebles, the gain introduced by spatial coupling copared to the corresponding uncoupled NB-LDPC block code ensebles grows with increasing field size for sufficiently large window size W. Furtherore, the thresholds of the C [2,4] and NB-SC-LDPC code ensebles saturate to a value nuerically indistinguishable fro capacity for large with either FS decoding or WD with a sufficiently large W. This is analogous to the case where the thresholds of binary (d v, d c )-regular SC-LDPC code ensebles saturate to capacity as d v and d c get large. It is interesting to note that for binary ensebles the graph density (d v and d c ) ust get large to approach capacity, whereas for non-binary ensebles capacity can be approached for fixed density by increasing the field size. In other words, the increase in coplexity needed to approach capacity is different in the two cases. IV. THRESHOLD ANALYSIS OF NB-SC-LDPC CODE ENSEMBLES ON THE BIAWGNC A. P-NB-EXIT Analysis on the BIAWGNC We use the P-NB-EXIT algorith presented in [6] to analyze the threshold perforance of NB-SC-LDPC code ensebles on the BIAWGNC, assuing that the binary iage

4 B [2,4] C [2,4] C [2,4], W = 6 C [2,4], W = 10 C [2,4], W = 20 (a) B [2,4] and C [2,4] Threshold: Eb/N0 (db) B [2,4] B C [2,4] (a) Coparison of the (2, 4) and (3, 6) groups: FS B, W = 3, W = B, W = 3, W = 5 Threshold: Eb/N0 (db) (b) B and (b) B and 5 Fig. 4. FS thresholds of the (2, 4) and (3, 6) groups and WD thresholds of on the BIAWGNC. Fig B, W = 3, W = 5, W = 10 (c) B and FS and WD thresholds of the (2, 4) and (3, 6) groups on the BEC. protograph, where the essages represent utual inforation (MI) values, a odel obtained by approxiating the distribution of the log-likelihood ratios as (jointly) Gaussian. The threshold is obtained by deterining the sallest signal-tonoise ratio E b /N 0 such that decoding is successful, i.e., the sallest value of E b /N 0 such that the a-posteriori MI between each variable node and a corresponding codeword sybol goes to 1 as the nuber of iterations increases. B. Nuerical Results Figure 4(a) copares the FS thresholds of the (2, 4) and (3, 6) groups on the BIAWGNC and Figure 4(b) shows the WD thresholds of for different W. 2 Both figures illustrate siilar behavior as the BEC results presented in Section III-B, and the sae is true for the WD thresholds of C [2,4] of a codeword is transitted and that BPSK odulation is used. Siilar to the P-NB-DE analysis on the BEC, the P- NB-EXIT analysis is also a BP algorith perfored on the 2 Due to coputational coplexity, the BIAWGNC thresholds were calculated only up to = 8. However, siilar to the approach taken by Uchikawa et al. in [10], the BIAWGNC threshold perforance for = 9 and 10 is conjectured to be consistent with the corresponding BEC results.

5 5 and (not included in the figure due to space liitations). To suarize, sall gains are observed for C [2,4] copared to B [2,4] until the field size gets large, whereas nuerically capacity-achieving WD thresholds that are significantly better than the corresponding block code thresholds are observed for both and. Moreover, we find that W = 10 for C [2,4] and, while is a better choice for WD, since W = 7. V. DECODING COMPLEXITY In practice, we would like to copare the perforance of NB-SC-LDPC codes and the corresponding NB-LDPC block codes when their decoding latency is the sae. Since it is assued that the binary iage of a codeword is transitted, it is convenient to easure the latency in ters of bits, denoted as W b, which is the nuber of coluns in the window for WD of SC-LDPC codes and the blocklength of LDPC block codes, both easured in bits (instead of GF (q) sybols). To be ore specific, if the lifting factor (see Section II) is M for SC-LDPC codes and M for LDPC block codes, then the equal latency condition is given by W b = W c M = c M, i.e., M = W M, which eans that SC-LDPC codes ust use perutation atrices W ties saller than LDPC block codes to aintain the sae latency, where c is the nuber of coluns in the B and B i atrices. For the codes we considered fro the (2, 4) and (3, 6) groups, c = 2. For fixed M, W b then depends on W. (The threshold analysis corresponds to the case when M.) As stated in [3] and the references therein, for NB-LDPC codes, if the BP algorith eploys the fast Fourier transfor, then the coputational coplexity at a check node is O(q) = O(q log 2 q) per sybol while that at a variable node is O(q). In our case, however, due to the constraint of equal latency, the decoding coplexity should be estiated per window for an SC-LDPC code, or equivalently, per blocklength for an LDPC block code. Like the protograph exaples in this paper, an SC-LDPC code is typically derived fro a (d v, d c )-regular LDPC block code. Consequently, if the window size is oderate to large, the part of the SC parity-check atrix covered by the window can be considered as (d v, d c )-regular as well and thus has (approxiately) the sae nuber of non-zero entries as the parity-check atrix of an LDPC block code. This indicates that the decoding coplexity of an SC-LDPC code and an LDPC block code is the sae when the decoding latency is the sae. 3 The total nuber of non-zero entries in the window is d v W b /, so the decoding coplexity per window is ( ) Wb O d v (q + q) = MO (W c d v (q + q)). (2) 3 In fact, due to the check-node irregularity at the beginning of the window and the variable-node irregularity at the end of the window, the actual decoding coplexity of the SC-LDPC code is slightly lower than the LDPC block code. Nevertheless, we keep this regularity assuption for siplicity. Other factors that influence the decoding coplexity, such as the nuber of iterations, are not considered. Decoding Coplexity , W=5, =5 C [2,4], =10 and, =5 C [2,4], W=10, =10, W=10, =5 d v = 2 d v = 2, W = 5 d v = 2, W = 10 d v = 3 d v = 3, W = 5 d v = 3, W = 10 Fig. 5. The order of decoding coplexity O (W c d v (q + q)) when an LDPC block code and an SC-LDPC code have the sae decoding latency and thus have the sae decoding coplexity. Ignoring the M factor on the right-hand side of (2), Figure 5 shows the order of the decoding coplexity when d v = 2 and 3 with FS decoding and WD (W = 5 and 10); note that FS decoding is equivalent to WD with W = L + s L = 100, i.e. decoding corresponds to an increase in the window size by approxiately an order of agnitude and to a corresponding order-of-agnitude increase in decoding coplexity. The five specific points highlighted in the figure are all cases when FS decoding or WD threshold of an SC-LDPC code enseble nuerically achieves (or is very close to) capacity (recall that the FS threshold of an LDPC block code enseble cannot be capacity-achieving, as shown in Figures 2 and 4(a)). We observe that Coparing FS decoding to WD for the sae enseble, both the coplexity and the latency are significantly reduced by adopting the latter. Coparing C [2,4] (W = 100), (W = 50), and (W = 25) for WD, both the coplexity and the latency of are less than the other two ensebles for the sae perforance (near-capacity). As a result, the (3, 6)-regular construction (especially ) is better than the (2, 4)-regular construction when designing an NB-SC-LDPC code with decoding latency and coplexity constraints and stringent perforance requireents. This result is supported by decoding perforance siulations of finite-length codes (see [17]). VI. CONCLUSIONS This paper analyzed the windowed decoding threshold perforance of several ensebles of non-binary spatially coupled LDPC codes; this was done for both the binary erasure channel and the BPSK-odulated additive white Gaussian

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