Stopping-Set Enumerator Approximations for Finite-Length Protograph LDPC Codes

Transcription

1 Stopping-Set Enumerator Approximations for Finite-Length Protograph LDPC Coes Kaiann Fu an Achilleas Anastasopoulos Electrical Engineering an Computer Science Dept. University of Michigan, Ann Arbor, MI , USA Abstract Asymptotic analysis of low-ensity parity-check LDPC coe weight an stopping-set enumerators, for coewors an stopping sets which grow linearly with coelength, has aie in esigning coes with linear minimum istance an low error floors. However, the analysis cannot capture the behavior of coewors an stopping sets that grow sublinearly with coelength. Thus, it is unclear how well the analysis escribes behavior for finite coelengths, particularly for short coes. In this paper, we provie another perspective on protograph-base an stanar LDPC ensemble enumerators, base on analysis of stopping sets with sublinear growth, which brings new insight into sublinear stopping-set behavior, protograph structure, an precoing. Using approximations to the stopping-set enumerators, we show that for stopping sets that grow at most logarithmically with coelength, the enumerators follow a polynomial relationship with coelength, unlike the exponential relationship for linearlygrowing stopping sets. Further, we analyze for what stopping-set sizes an coelengths the approximations apply. I. INTRODUCTION Low-ensity parity-check LDPC coes in conjunction with iterative ecoing base on message-passing algorithms can achieve excellent performance, e.g., in [1] where ensity evolution is use to analyze the asymptotic, infinite-coelength performance. Despite their excellent threshol performance, LDPC coes with finite coelengths exhibit error floors, which limit the smallest error rates achievable by a coe. Investigating error floors is more tractable for the binary erasure channel BEC where stopping sets [2] are use to analyze the iterative-ecoing LDPC performance. Ensemble stopping-set enumerators capture the average performance of an LDPC ensemble. These stopping-set enumerators behave similarly to weight enumerators, an both enumerators are useful in analyzing error-floor performance, e.g., in [3], [4]. Protograph-base LDPC ensembles possess several avantages, incluing smaller memory requirements ue to simplifie graph representation, high-spee ecoing by utilizing the coe s parallel structure, an lower error floors while maintaining goo threshols [3] [7]. These ensembles use protographs [5] to provie aitional structure to LDPC coes, similar to multi-ege type coes [7], an are generate by a copy-an-permute operation on a base graph [5]. First, copies of the protograph are generate. Then, for each egetype k in the base protograph, the enpoints of the type-k eges are permute between the variable an check noes to which the eges are connecte. Asymptotic analysis of weight an stopping-set enumerators for protograph-base LDPC ensembles has been complete for the case when coewor weight an stopping-set size grow linearly with coelength n [3], [4]. This analysis has aie in the esign of protograph-base LDPC coes with linear minimum istance an with goo threshol an error-floor performance [4], [6]. However, the analysis oes not capture the behavior of stopping sets that grow sublinearly with n. Also, the question arises of the valiity of the enumerator approximations in realistic scenarios when the coelength is finite. For instance, for a coe of length n, it is unclear if size-v stopping sets belong to the class of stopping sets which grow linearly or sublinearly with n. In this paper, we provie another perspective on enumerators for protograph-base an stanar LDPC ensembles, base on analysis of stopping sets with sublinear growth with coelength, which brings new insight into 1 how sublinear stopping-set enumerators behave, 2 how this behavior impacts coe esign, 3 how protograph structure can improve performance over stanar ensembles, an 4 how precoing can improve error-floor performance. We first obtain tractable approximations to the stopping-set enumerators. Then, we aress the question of when the approximations are vali, i.e., for what stopping-set sizes an coelengths o the approximations apply. The resulting analysis shows that for stopping sets that grow at most logarithmically with coelength, the enumerators follow a polynomial relationship with coelength, unlike the exponential relationship for linearly-growing stopping sets. II. ENSEMBLE STOPPING-SET ENUMERATORS First, consier the stanar LDPC ensemble, i.e., where all possible ege permutations between variable an check noes are equiprobable, with maximum variable check noe egree v c, variable-noe an check-noe egree istributions Lx an Rx from the noe perspective, an corresponing normalize variable-noe an check-noe egree istributions lx an rx. The ensemble stopping-set enumerator, i.e., the expecte number of stopping sets of size v in the stanar ensemble of LDPC coes with coelength n, is given by [8] sn, v = L 1 e=0 v k S v nli k i coef { c } [1+x i ix] Ri, x e L 1 e where coef{px, x e } is the coefficient of the x e term in the polynomial px, e is the number of eges 1

2 in the stopping set, k i is the number of egreei variable noes in the stopping set, an S v = { k : 0 k i nl i, 1 i v ; v k i = v; } v ik i = e. A similar expression can be erive for protograph-base ensembles where for each ege-type k, all possible permutations of the type-k eges are equiprobable an inepenent of all other ege types. Theorem 1: For an LDPC ensemble generate by copies of a protograph with M variable-noe types an J check-noe types, the expecte number of size-v stopping sets is sn, v = 1 v,i n i J c,j c,j coef 1 + x k x k c,j, x nν j,k k 2 where the coelength n = M, n i is the number of type-i variable noes in the stopping set, v,i c,j is the egree of the ith jth variable-noe check-noe type, ν j,k is the kth variable-noe type { to which the jth check-noe type is connecte, an S p = n {0,..., } M : } M n i = v. The proof is omitte see [9], but some intuition behin 2 can be seen as follows. The set S p contains all possible istributions of variable-noe types that v variables can take, an the coefficient term in 2 represents the number of ways to connect the v variable noes to type-j check noes such that a stopping set is forme. Equation 2 is an exact expression an can be shown to be equivalent to the expression in [4]. By expressing the enumerator with the coefficient term in 2, we are able to escribe a close-form combinatorial expression which is then approximate in Section III. In [4], the equivalent term is calculate recursively using multinomial z-transforms. III. ENSEMBLE ENUMERATOR APPROXIMATIONS In orer to obtain simpler, more tractable enumerator expressions, approximations are erive by taking upper an lower bouns on the combinatorial expressions in the exact enumerator an then combining the results to fin an approximation. The proofs are omitte see [9]. A. Stanar Ensembles Theorem 2: For a stanar LDPC ensemble with c > 2, coelength n, an stopping-set size v such that v xn min 1 i v:l i 0 1 j c:r j 0 {l i, 21 Rr j / v }, 3 for any constant x [0, 1/ v, the expecte number of stopping sets of size v is approximate by sn, v = min{ vv,l 1} e=v Ov log v v e/2 ± n log n. 4 Since we are examining sublinear stopping sets, the conition on v is not restrictive. The approximation in Theorem 2 can be simplifie when the stopping-set growth is constraine. For example, for stopping sets that grow at most logarithmically with coelength, i.e., v β log n for an Ov log v ± appropriately-restricte constant β, the error term n log n in 4 simplifies to n ±Olog[log n]. Further, for stopping sets whose size v is a fixe, finite constant, the error term simplifies to O1. See [9] for more etails. B. Protograph-Base Ensembles First, we introuce a few terms. Let V be the set of all variable noes in the graph an fix a subset V v V such that the variable-noe types in V v are istribute accoring to n S p. Now, several quantities will { be efine for a particular check-noe type j. Let B = b 1, b 2,..., b c,j {0, 1} c,j : } c,j b k 1. This set represents the possible connections a single type-j check noe can have to V v such that it is connecte at least twice or not at all. For each k, b k is 1 if the kth ege emanating from the check noe is connecte to V v an is 0 otherwise. Enumerate the elements in B = {β 0, β 1,...,β B 1 } where β 0 is the all-zero vector an all other vectors are enumerate in any fashion. Let β h,k be the kth element of β h. Let m h be the number of type-j check noes whose connections are escribe by β h an let m = m 0,..., m B 1. Finally, for a given m, let w j be the number of type-j check noes connecte to V v. Theorem 3: For an LDPC ensemble generate by copies of a protograph with M variable-noe types, J check-noe types, coelength n = M, an stopping-set size v < n, the expecte number of size-v stopping sets is approximate by sn, v = Ov log v ± log n 5 v e+w where e is the number of eges emanating from stopping sets istribute accoring to n. The quantity w = J { w j where wj = max m Sm {w j } an S m = m {0,...,} B : B 1 h=0 m h = ; B 1 h=0 β h,km h = } n νj,k k = 1,..., c,j for each j {1,...,J}. The quantity w represents the largest possible number of check noes connecte to stopping sets istribute accoring to n. The ifficulty in applying Theorem 3 is in etermining w for a given v an n. By restricting the stopping-set growth, the approximation in Theorem 3 can be simplifie, similar to the stanar-ensemble simplifications in Section III-A. C. Insights into Sublinear Stopping-Set Enumerator Behavior Theorems 2-3 show that the enumerator approximations have a polynomial relationship to the coelength n. In contrast, the asymptotic analysis in [3], [4] show that the enumerator follows an exponential behavior with coelength, for linearlygrowing stopping sets. This ifference inicates that our analysis captures sublinear behavior of the stopping sets which coul not be capture in [3], [4]. Thus, the results here help to more fully capture stopping-set behavior an may be more useful in gaining insight for finite coelengths, particularly for short coes where the ominating stopping-set size v = δ min n in the linear analysis may be too small e.g., small enough to

3 be avoie by an intelligent permutation algorithm to provie meaningful results. We efine two categories of behavior: Category P contains stopping sets which follow the polynomial behavior in Theorems 2 an 3, which inclues finite stopping sets an stopping sets which grow at most logarithmically with coelength. Category E contains stopping sets which follow the exponential behavior in [3], [4], which inclues stopping sets which grow linearly with coelength. By analyzing 4 an 5 for category-p stopping sets, we fin the ominate terms of the ensemble enumerators such that lim n sn, v = αn ES,P where α is some constant an E S an E P are the ominating exponents for stanar an protograph-base ensembles, respectively: E S = e min /2 v = v,min v/2 v 6 E P = e v where e = min {e w }. 7 Equation 6 shows that the minimum variable-noe egree v,min is the key factor in error-floor performance of stanar ensembles. If v,min = 2, as is the case for many goo LDPC coes, then E S = 0 an there always exists a positive probability that small stopping sets exist in the ensemble. For protograph-base coes, w e/2. Thus, e e min min 2 an hence, M n i v,min 2 v,min v = 2 E P = e v v,min v/2 v = E S. 8 Since large exponents are esirable to make the exponent of n more negative, this result shows that protograph-base ensembles perform at least as well as stanar ensembles an, in fact, have the possibility for achieving exponents which are strictly more negative. Maximizing E P requires minimizing w which is epenent solely on the protograph structure. Intuitively, the protograph enforces structure such that w an hence the exponent E P can be strictly less than for stanar ensembles, resulting in better error-floor performance. Aitional insight provie by Theorems 2-3 inclues the following. For finite stopping sets, 1 if v e + w < 0, then the expecte number of such stopping sets can be mae arbitrarily small by choosing a large enough n, or 2 if v e+w 0, then the expecte number of such stopping sets is lower boune by a positive finite number for all n. If v,min 3, as in [10], then v e+w < 0 for all stopping sets in the ensemble. Thus, the expecte number of all finitesize stopping sets can be mae arbitrarily small by choosing a large enough n, an such coes are guarantee to have goo error-floor performance for category-p stopping sets. IV. REGION OF APPROXIMATION VALIDITY To help etermine whether stopping sets behave polynomially or exponentially, we examine when the approximations of Section III are vali. Specifically, for what values of stoppingset size v an coelength n o the approximations apply? We A B C Fig. 1. Protograph P analyze in Section V. The protograph contains 6 variable-noe types circles an 3 check-noe types squares. will only present the analysis for protograph-base ensembles for which at most one ege type connects any variable-noe type to any check-noe type in the protograph. For all other protographs, one can simply expan the protograph to meet this conition an then apply the analysis given here. Given n S p an m S m, the corresponing term in the stopping-set enumerator in Theorem 1 can be expresse as where c = lns n,m n, v = lnc + v e + w lnn + A 9 [ ] 1 v,i 1 J 1 n i! β 1 h=1 m h! + M v e w is inepenent of n an [ M ni 1 A = ln 1 k ] 1 v,i + J w j 1 ln 1 k. Comparing the error term A to v e+w ln n will show when the approximation in Section III is vali. Theorem 4: Given a small fraction γ > 0 an any constant a 0, 1, let N 0 be the solution of n in the following equation n lnn = Mv γ 1 + a 21 a v,avg 2 where v,avg is the smallest average variable-noe egree in stopping sets of size v. Then, for all v an n satisfying the conitions v/n a/m an n > N 0, the error term is upper boune by A γ v e + w ln n an hence, for small γ, the stopping-set enumerator can be approximate by lns n,m n, v lnc + v e + w lnn. 10 The proof is omitte see [9]. Since v is proportional to N 0 lnn 0, the ratio of v/n 0 monotonically increases as v increases. Also, there may exist values of n < N 0 such that the upper boun on the error term still hols. Aitional work to tighten the bouns use to generate the approximations are necessary to capture a larger portion of the region of valiity. V. AN ILLUSTRATIVE EXAMPLE Comparing the regular 3,6 LDPC stanar ensemble to the ensemble generate by protograph P in Fig. 1 provies an illustrative example of insights provie by the sublinear analysis in Sections II-IV. The protograph-p ensemble is a subset of the stanar ensemble. Among other restrictions, protograph P s structure prevents size stopping sets from forming. Examining the behavior of category-e stopping sets as in [3], both ensembles have the same δ min, the largest δ = v/n

4 Protograph-P Ensemble Stanar Ensemble TABLE I BOUNDS FOR APPROXIMATION VALIDITY FOR STOPPING SETS OF SIE v IN THE PROTOGRAPH-P LDPC ENSEMBLE sn,v v=5 v= Coelength n -3 v=2 v=2 v=4 v=3 v=1 v=4 v=5 Fig. 2. Ensemble stopping-set enumerator vs. coelength for the regular 3,6 stanar ensemble an protograph-p ensemble. such that the enumerator still ecays exponentially with n an hence, an important factor in error-floor performance. However, the category-p stopping-set behavior is ifferent for the two ensembles, as will be shown next. From 6 an 7, the ominating exponents E S an E P, respectively, can be calculate for category-p stopping sets. For the regular 3,6 stanar ensemble, 3v v E S v = v + = For the protograph-p ensemble, it can be shown see [9] that w = 3 v/2 an hence, v v E P v = v + 3v 3 = 2v A comparison of the two results shows that when v is even, the exponents are the same for both ensembles. However, when v is o, E P v E S v = 1 an thus, the protograph-p ensemble has better exponents than the stanar ensemble. The ensemble stopping-set enumerators for the two ensembles are shown in Fig. 2 as a function of coelength for stopping-set sizes v = 1 to 5. The values plotte are exact, numerically-evaluate values calculate from 1 an 2. This figure provies several key observations. First, the stopping-set enumerators follow the category-p behavior: the log-log plot is linear an the slope agrees with the exponent values E P an E S calculate above. Secon, protograph P greatly reuces the number of stopping sets of o size. For o v, the log-log slope was both analytically calculate an numerically evaluate to be one less for the protograph-p ensemble than for the stanar ensemble. For even v, the protograph-p ensemble has the same log-log slope but higher offset than the stanar ensemble. This leas to several questions: 1 is there some conservation of stopping-set enumerators an 2 are there regions where the effect of the offset will ominate over the effect of the slope an vice versa? Thir, the linear behavior in the log-log omain preicte v N 0 v/n for category-p stopping sets seems to hol even for very small coelengths, e.g., n 10v. Since the v/n ratio of 0.1 is significantly larger than the δ min values of in [3], [4], this suggests that category-p behavior may exten to stopping-set sizes which grow faster than logarithmically with coelength. Further, the smallest stopping set in an ensemble may fall into the range where category-p behavior is applicable, which woul imply that investigations into category-e stopping sets may not be the critical analysis. However, these are simply conjectures base on a couple ensemble examples with small v an n values. It is unclear whether this pattern will hol for larger v an n values an for all ensembles. Lastly, if we fix δ = v/n an trace the behavior as n grows, we begin to see ifferent trens in the plot [9]. There appears to be an increase in the enumerators when v = 0.1n an a ecrease when v = 0.01n. This generally agrees with the trens preicte for category-e stopping sets in [3], [4]. Applying the analysis of Section IV to protograph P provies an illustrative example of when the approximations of Section III hol. Specifically, using a = 0.25 an γ = 0.02, we fin the values of stopping-set size v an coelength n for which the approximation is accurate within 2%. Applying Theorem 4, we obtain values for N 0 in Table I. This table shows, for example, that for stopping sets of size v = 5, the stopping sets follow category-p behavior for all coelengths n N 0 = 222, i.e., for all ratios v/n v/n 0 = Since v/n 0 increases monotonically with v, all stopping sets of size v 0.023n follow category-p behavior for all n 222. In the analysis of category-e stopping sets in [4], the largest v/n ratio for a negative stopping-set enumerator exponent is δ min = Thus, accoring to this analysis, the stoppingset sizes of interest are aroun 0.023n. However, Table I shows that for all n 222, these stopping sets actually fall uner category-p behavior. This example has shown that category-p analysis is necessary to more fully capture the ensemble enumerator behavior. VI. PRECODING Definition 1: Let P be a protograph with M variable noes an J check noes. To precoe [4] a variable noe v i for some i {1,...,M}, first a a check noe c J+1 which is connecte to v i via 2 eges. Then, a a variable noe v M+1 which is connecte to c J+1 via 1 ege. Finally, puncture v i. The resulting protograph P has M + 1 variable noes one of which is puncture, J + 1 check noes, an E + 3 eges. The technique of precoing has been shown experimentally an numerically to improve error-floor performance [4].

5 Divsalar has partially proven the benefits of precoing by comparing stopping-set enumerator exponents [4]. However, the partial proof is unable to show when an if precoing improves error-floor performance of the ensemble as a whole. We use sublinear-enumerator analysis here to gain aitional insight into precoing. For a BEC with erasure probability ǫ, the average block error probability of an LDPC ensemble can be upper boune by E G [P IT B n, ǫ] P ub e = n v=1 ǫv sn, v where the boun is tight in the error-floor region see [9]. It can easily be shown that for a coe ensemble with puncture noes, the upper boun for a single term in the summation over v is Pe,puncv ub = ǫ veff n i J c,j c,j coef 1 + x k x k 1 v,i c,j, x nν j,k k 13 where v is the stopping-set size incluing puncture noes an v eff is the number of variable noes in the stopping set excluing puncture noes. When noes are puncture, they are automatically erase an hence, the channel only nees to erase v eff variable noes. If stopping sets of size v follow category-p behavior, then Pe,puncv ub can be approximate by Pe,punc ub v using Theorem 4: Pe,punc ub v = ǫveff v e+w M n J β 1 i! 1 v,i h=1 m h!. 14 Theorem 5: Consier a protograph P whose variable noe v i is precoe to generate the protograph P. Let V P an V P be the sets of stopping-set sizes which ominate the performance of the ensemble base on protograph P an P, respectively, an let V U = {V P V P }. Assume that for all v V U, stopping sets of size v follow category-p behavior for both protograph-base ensembles. Let be large enough such that Theorem 4 hols an 1/ǫ. Then, Pe,P ub e,p v P ub v V U P ub v V U e,pv P ub e,p. 15 Since the upper boun on error probability is tight in the error-floor region, protograph P performs at least as well as protograph P in the error-floor region. Proof: A sketch of the proof is provie here. See [9] for a full proof. Begin with a finite stopping set from the ensemble generate from protograph P, with v variable noes connecte to w check noes via e eges. Expan this stopping set with variable noes of type M + 1 an check noes of type J +1 to create a stopping set in the ensemble generate by protograph P. The expansion results in new stopping-set characteristics enote by the prime variables: v e + w = v e + w n punc n M+1 /2 16 where n punc is the number of puncture variable noes. By upper bouning v V U P ub e,p v an observing that n M+1 2n punc in orer for a stopping set to be forme, the result in 15 can be obtaine. The requirement that the ominating stopping sets must follow category-p behavior may not be restrictive. This is still uner investigation, but the example in Section V showe that at least for the regular 3,6 LDPC ensembles analyze, this requirement is usually satisfie. Theorem 5 shows that if the ominating stopping sets follow category-p behavior, then precoing can only improve the error-floor performance. Intuitively, precoing expans the LDPC graph, while maintaining transmitte coelength, such that more channel erasures are neee to generate ecoer failures; thus, in the error-floor region where ǫ is small, the resulting increase in the exponent of ǫ is a ominating factor in ecreasing error probability. Section III showe that ensembles with all variable-noe egrees v,i 3 have goo error-floor performance since v e + w < 0 for all stopping sets. However, current literature, e.g., [1], [6], inicates that variable noes of egree strictly less than 3 are useful in achieving goo threshol performance. The technique of precoing allows one to inclue egree variable noes to improve threshol performance without sacrificing error-floor performance; in fact, error-floor performance may be improve simultaneously! ACKNOWLEDGMENT This work was supporte in part by a NASA Grauate Stuent Researchers Program Fellowship. The authors woul like to thank the Information Processing Group at the Jet Propulsion Laboratory, California Institute of Technology, Pasaena, CA, in particular, Jon Hamkins, Dariush Divsalar, Sam Dolinar, Kenneth Anrews, an Christopher Jones, for their guiance an avice. REFERENCES [1] T. J. Richarson, M. A. Shokrollahi, an R. L. Urbanke, Design of capacity-approaching irregular low-ensity parity-check coes, IEEE Trans. Information Theory, vol. 47, no. 2, pp , Feb [2] C. Di, D. Proietti, I.E. Telatar, T.J. Richarson, an R.L. Urbanke, Finite-length analysis of low-ensity parity-check coes on the binary erasure channel, IEEE Trans. Information Theory, vol. 48, no. 6, pp , June [3] S. L. Fogal, R. McEliece, an J. Thorpe, Enumerators for protograph ensembles of LDPC coes, in Proc. International Symposium on Information Theory, Aelaie, Australia, Sept [4] D. 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