of LDPC and RS CodesChaudhry Adnan Aslam, Student Member, IEEE, Yong Liang Guan, Member, IEEE,
Kui Cai, Senior Member, IEEE
Abstract— This paper presents two low-complexity edge-basedscheduling schemes, referred to as the e-Flooding and e-Shuffledschedules, for the belief-propagation (BP) decoding of low-densityparity-check and Reed–Solomon codes. The proposed schedulesselectively update the edges of the code graph based on the run-time reliability of variable and check nodes. Specifically, newmessage update is propagated exclusively along the unreliableedges of the code graph. This reduces the decoding complexity ofBP algorithm as only a partial set of message updates is computedper decoding iteration. Besides, restricting the flow of messageupdates may also precludes the occurrence of some short graphcycles, which helps to preserve the BP message independence atcertain variable and check nodes. Using numerical simulations,it is shown that the proposed edge-based schedules reduce theBP decoding complexity by more than 90% compared with theprior-art BP schedules, while simultaneously improving the error-rate performance, at medium-to-high signal-to-noise ratio overadditive white Gaussian noise channel.
THE belief-propagation (BP) decoding is an iterativemessage-passing algorithm which is being used to decode
a variety of linear block codes, such as the low-densityparity-check (LDPC) and the Reed-Solomon (RS) codes.In BP decoding, message (reliability) updates are exchangedbetween the variable and check nodes along the edges of thebi-partite code graph. A BP schedule, which defines the orderof message-passing over the code graph, plays an importantrole as it affects the complexity, convergence speed and error-rate performance of the BP decoder.
The Flooding [1] is the conventional BP schedule in whichall the variable-to-check (V2C) and check-to-variable (C2V)message updates are simultaneously passed in each iteration.
Manuscript received May 18, 2016; revised September 26, 2016 and Novem-ber 24, 2016; accepted December 1, 2016. Date of publication December 9,2016; date of current version February 14, 2017. This work was supported bythe NTU Research Scholarship, and in part by the SUTD-MIT InternationalDesign Center (IDC). The associate editor coordinating the review of thispaper and approving it for publication was K. Abdel-Ghaffar.
C. Adnan Aslam and Y. L. Guan are with the School of Electricaland Electronics Engineering, Nanyang Technological University, Singapore639798 (e-mail: [email protected]; [email protected]).
K. Cai is with the Science and Math Cluster, Singapore University ofTechnology and Design, Singapore 487372 (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TCOMM.2016.2637913
To improve the decoding convergence speed, and equivalentlyto reduce the number of iterations required to achieve theerror-rate convergence, individual nodes can be updated insequential order. Analytical studies reveal that the sequentialschedules converge in half the number of iterations comparedto the Flooding schedule [2]. The Shuffled [3] and Layered [4]are two commonly used sequential BP schedules, whereinthe variable and check nodes are updated according to afixed sequential order. The column-weight (CW) [5] androw-weight (RW) [6] based sequential schedules furtherimprove on the Shuffled and Layered schedules, respectively,in which the high degree variable and check nodes are updatedfirst in each iteration. Although, the decoding convergenceimproves with CW, and RW sequential schedules, the asymp-totic error-rate performance remains unchanged.
Besides, to reduce the BP iteration cost, and subsequentlythe decoding complexity, the forced convergence (FC) [7],variable-node lazy (VLS) [8] and check-node lazy (CLS) [9]schedules update a partial set of variable or check nodesper decoding iteration. Similarly, for reducing the BP iter-ation cost for the non-binary LDPC codes, the check-nodeand the variable-node reliability-based scheduling schemesare reported in [10] and [11], respectively. Apart fromreliability-based scheduling, the probabilistic [12] and graph-based [13] scheduling schemes regulate message passingin-accordance with the length of the cycles in the codegraph. Nevertheless, all these existing BP schedules eitherreduce the decoding complexity or improve on the error-rateperformance.
In addition to fixed order BP schedules, there are alsoresidual-based dynamic schedules reported in [14]–[17],which regulate the message-passing by using the messageresiduals. These dynamic BP schedules achieve lower errorrates because they enable the BP decoder to overcome manytrapping sets encountered during the decoding process [18].On the bleak side, these schedules incur significant addi-tional complexity, striving to compute the message-residuals.In order to reduce the complexity cost associated with dynamicschedules [14]–[17], a reliability-based dynamic BP scheduleis proposed in [19] which offers faster error-rate convergenceat reduced complexity. Inspired by the dynamic schedules, amaximum-mutual-information based fixed decoding schedule,which predicts the next message update based on the mutualinformation increase, is proposed in [20]. Again, this algorithmimproves on the error-rate performance, however, the resultant
526 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 2, FEBRUARY 2017
hardware implementation complexity and the storage require-ment may become prohibitively large as it has to record thecomplete sequence of message updates.
With regards to soft decoding of RS codes, the conventionalBP decoder offers poor performance. This is because theRS codes have inherently dense parity-check matrices witha large number of short cycles. Consequently, the BP decoderis often stuck therein due to only a few unreliable bits.The adaptive belief-propagation (ABP) decoder [21] resolvesthis problem, wherein the sub-matrix corresponding to theunreliable bit positions is reduced to a sparse matrix, followedby the BP decoding. However, since the reliable sub-matrixpart remains non-sparse, the ABP decoder needs to computea large number of message updates per iteration, leading tovery high decoding complexity. Furthermore, the modifiedsub-matrix, yet, contains a large number of short cycleswhich produce sub-optimal error-rate performance. A low-complexity variant of the ABP decoder is proposed in [22]which constructs a reduced density parity-check matrix beforethe use of ABP decoding. To further improve on the error-rateperformance, a residual-based layered BP (LRBP) scheduleis recently reported in [23] which strives to propagate moremessage updates towards the unreliable bit positions.
In this paper, we present a novel edge-based BP schedulingstrategy for decoding LDPC and RS codes. The most promi-nent feature of the proposed scheme is that it reduces theBP decoding complexity and simultaneously improves on theerror-rate performance. In the proposed scheme, the messagepropagation is constrained over a limited number of graphedges. The basic principle behind the proposed schedulingscheme is that when the reliability of a check node and itsimmediate neighboring variable node is sufficiently high, theupdate operation of the graph edge connecting the two nodescan be avoided. To this end, we define the check node reliabil-ity by using the parity-check equation and the log-likelihood-ratio (LLR) information of the neighboring variable nodes.Based on that, we divide the set of check nodes into differentgroups. Among them, we only update the edges originatingfrom the less reliable check nodes. Since, there are partialset of graph edges updated in each iteration, the total numberof message updates, and consequently the overall decodingcomplexity, is significantly reduced. As a further consequence,the adverse effect of graph cycles is also diminished leadingto a better error-rate performance.
The remainder of this paper is organized as follows. TheBP decoding algorithm is briefly reviewed in the beginning ofSection II. This is followed by the introduction of the proposededge-based scheduling. The BP decoding complexity, error-rate performance, decoding convergence and code graph cycleeffects under the proposed scheduling scheme are analyzed inSection III. The conclusions are drawn in Section IV.
II. EDGE-BASED BELIEF-PROPAGATION (BP)SCHEDULING
A. Notations and Overview of BP Decoding
A binary (N ,K ) LDPC code defined over GF(2) canbe described by an M × N parity-check matrix H,
where M = N − K . Similarly, an algebraic (N , K ) RS codedefined over GF(2m) can be expended into an equivalent binaryimage [24], and represented with an M × N parity-checkmatrix H, where m is the number of bits per RS symbol,N = Nm and M = (N − K )m. The LDPC and RS codes canbe equivalently represented by using a bi-partite code graphwith N variable nodes, vn for 1 ≤ n ≤ N , and M check nodes,cm for 1 ≤ m ≤ M , where the matrix element hm,n = 1defines a bi-directional edge �em,n between cm and vn . TheBP algorithm iteratively computes and passes new messageupdates from variable node vn to check node cm , denoted bymvn→cm , and from check node cm to variable node vn , denotedby mcm→vn , along the edges �em,n in the code graph. Thesemessage updates can be computed as [1]
m(i)vn→cm
= Lvn +∑
c j ∈M (vn)\cm
m(i)c j →vn
(1)
m(i)cm→vn
= 2 tanh−1
⎛
⎝∏
v j∈N (cm)\vn
tanh
(m(i−1)v j→cm
2
)⎞
⎠ (2)
where i is the iteration count, Lvn is the intrinsic channelinformation, also referred to as the log-likelihood-ratio (LLR),of variable node vn , M (vn) \ cm = {c j : h jn = 1, j �= m} isthe set of neighbors of variable node vn excluding check nodecm and N (cm) \ vn = {v j : hmj = 1, j �= n} is the set ofneighbors of check node cm excluding variable node vn . Theoutput LLR �
(i)vn corresponding to variable node vn can be
computed as
�(i)vn= Lvn +
∑
c j ∈M (n)
m(i)c j →vn
(3)
A hard-decision on code-word bit an , denoted by a(i)n , ismade based on the sign of LLR �
(i)vn . For a valid decoded
code-word a(i) = (a(i)1 , a(i)2 , ..., a(i)N ), the M parity-check(syndrome) equations must be satisfied, given by
s(i)cm=
∑
n:vn∈N (m)
hm,na(i)n = 0 (4)
The BP decoding stops once these parity-check equationsare satisfied or when the maximum iteration count, denotedby Imax, is reached. For ease of notation, we will remove theiteration count superscript i in all the subsequent equations.
B. Motivation
The main objective of the proposed edge-based schedulingis to reduce the computational complexity and improve thedecoding convergence rate of BP decoder by partially updatingthe edges of the code graph. In the conventional BP decod-ing, such as the Flooding [1], Shuffled [3] and Layered [4]schedules, all the edges in the code graph are updated onceper decoding iteration. This involves bi-directional messagepassing from C2V and V2C nodes. However, it is oftenobserved that some of the graph edges converge (achievelarge LLR values) within a few decoding iterations and donot require message updates thereafter. Thus, in the proposedscheduling, we monitor the run-time convergence status of thegraph edges and update only the non-converged (less reliable)
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ASLAM et al.: EDGE-BASED DYNAMIC SCHEDULING FOR BP DECODING OF LDPC AND RS CODES 527
ones in the next decoding iteration, while skipping those thathave already converged.
In addition to complexity reduction, we also demonstratebetter error-rate performance under the proposed edge-basedscheduling. In BP decoding, all message update operationsare performed on the basic assumption that the messagespassed to a given node are statistically independent, whichonly holds when the underlying code graph is cycle-free. How-ever, for short-to-moderate length LDPC and RS codes, theunderlying code graph does contain short cycles which violatethis message independence assumption, leading to sub-optimaldecoder performance. This behavior is even more prominentin high-rate LDPC and RS codes, which are the focus inthis manuscript. Thus, curtailing the flow of message updatesunder the proposed edge-based schedules also precludes theoccurrence of graph cycles, resulting into an improved error-rate performance.
C. Graph Edge Convergence in BP Decoding
To perform the edge-based BP scheduling, we need todescribe the convergence (reliability) criterion for a graphedge. In this direction, we first define the reliability measureof a check node cm , denoted by ηcm . This is given by thesmallest LLR magnitude observed among the neighboringvariable nodes of cm , computed as
ηcm = minvn∈N (cm)
{∣∣�vn
∣∣} (5)
where |.| is the absolute function. Note that the LLR magnitude∣∣�vn
∣∣ indicates the amount of confidence level on the hard-decision decoding of bit an . Thus, larger values of ηcm reflecta high level of confidence on the set of tentatively decodedcode-word bits associated with the neighboring variable nodesof cm , {an|vn ∈ N (cm)}. Using the check node reliabilitymeasure ηcm and the parity-check (syndrome) value scm , wesplit the M number of check nodes into 3 distinct types,namely the Type-1, Type-2 and Type-3 check nodes, as follows
• Type-1 (most unreliable): All the check nodes with unsat-isfied parity-check, given by scm �= 0, will be classifiedinto Type-1 category. These nodes are among the mostunreliable set of check nodes because the non-zero parity-check condition certainly indicates an error among the setof decoded code-word bits {an|vn ∈ N (cm)}. Therefore,we will pass new message update towards all the variablenodes vn ∈ N (cm).
• Type-2 (unreliable): All the check nodes correspond tosatisfied parity-check, scm = 0, and having reliabilitymeasure smaller than a pre-defined threshold, that isηcm ≤ T , will be categorized as Type-2 check nodes.Although, the parity-check equation is satisfied, the lowreliability measure of Type-2 check node leads to aweaker level of confidence and hence sheds doubt onthe tentatively decoded code-word bits. It suggests thatthe Type-2 check node may still have some incorrectlydecoded neighboring variable nodes. To this end, we willidentify the set of t minimum-reliability neighbors ofcheck node cm , denoted by Nt -min (cm), treating them asamong the likely erroneous variable nodes. Subsequently,
TABLE I
EDGE-STATE(εm,n
)UPDATE CRITERION
we will pass new message updates only towards theseprobable erroneous variable nodes vn ∈ Nt -min (cm).
• Type-3 (reliable): All the remaining check nodes, wherescm = 0 and ηcm > T , will be classified as Type-3 checknodes. These are among the most reliable set of checknodes because all the neighboring variable nodes haveachieved high reliability. Therefore, we will not updateany of the neighboring variable nodes of Type-3 checknode in the next iteration.
In this work, we select the value of t by using numericalsimulations such that the decoded error-rate is minimized,whereas we choose the reliability threshold T based onthe check-node-type probability, as explained in the nextsection. The minimum-reliability neighbor set Nt -min (cm) isgiven by
Nt -min (cm) ={
arg1
minvn∈N (cm)
{∣∣�vn
∣∣}, arg2
minvn∈N (cm)
{∣∣�vn
∣∣}, ...,
argt−1min
vn∈N (cm){∣∣�vn
∣∣}, argt
minvn∈N (cm)
{∣∣�vn
∣∣}}
(6)
where functiont
min{.} finds the t-th minimum element. Basedon the check node categorization, we represent the conver-gence status of an edge by using a binary edge-state vector, S,given by
S = {εm,n|1 ≤ m ≤ M, vn ∈ N (cm)} (7)
where εm,n ∈ {0, 1}. This edge-state vector will be usedto determine whether an edge is to be updated at the nextiteration. The value εm,n = 0 indicates that the edge �em,n
has converged and will not be updated, whereas, εm,n = 1indicates that the corresponding edge has not converged andwill be updated at the next iteration. The edge-state updatecriterion is formally described in Table I.
In summary, at each decoding iteration, we will update theedge-state vector according to Table I, and propagate newmessage updates only where εm,n = 1, as shown in Fig. 1.Using this edge-state vector, we regulate the message-passingunder the proposed e-Flooding and e-Shuffled schedules asdescribed in Algorithm 1 and 2, respectively. The countervariables k and z in Algorithm 1 and 2 count the averagenumber of V2C/C2V message updates and average num-ber of edge-state vector (S) updates performed per decodedcode-word, respectively.
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528 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 2, FEBRUARY 2017
TABLE II
COMPLEXITY ANALYSIS OF FIXED AND DYNAMIC BP SCHEDULES (PER k NUMBER OF C2V/V2C UPDATES)
Fig. 1. Message passing from check-to-variable and variable-to-check nodesunder the proposed edge-based scheduling scheme.
D. Selection of Reliability Threshold (T )
To select the check node reliability threshold T , we definethe check-node-type probability. This is given by the prob-ability of a check node being selected as either Type-2, orType-3, denoted by p (Type-2) and p (Type-3), respectively,as a function of the neighboring variable node LLR (�) andthe check node degree
(dc
). These probability expressions are
derived in the Appendix A, given by
p (Type-2) =⎡
⎣1−(
e�
1+e�
)dc
⎤
⎦
⎡
⎣1
2+ 1
2
dc∏
i=1
(1 − 2e�
1 + e�
)⎤
⎦
(8)
p (Type-3) =(
e�
1 + e�
)dc
⎡
⎣1
2+ 1
2
dc∏
i=1
(1 − 2e�
1 + e�
)⎤
⎦ (9)
In Fig. 2, we plot these two expressions for dc = 10,dc = 20 and dc = 30. It can be observed that, withincreasing LLR, the probability of Type-2 check node tendsto decrease, whereas the probability of Type-3 check nodetends to increase. To balance between the two, we will setthe value of T such that both type of check nodes will havean equal probability. This leads to the intersection point oftwo probability curves, as shown in Fig. 2. Therefore, we will
Algorithm 1 Proposed e-Flooding Schedule
1 Initialize all mvn→cm = Lvn ;2 Reset k = 0, z = 0;3 Set kmax = Mdc Imax;4 while k < kmax do5 Update edge-state vector S according to Table I;6 for m = 1 : M do7 for every vn ∈ N (cm) do8 if εm,n = 1 then9 Generate and propagate mcm→vn (2);
10 for n = 1 : N do11 for every cm ∈ M (vn) do12 if εm,n = 1 then13 Generate and propagate mvn→cm (1);14 k = k + 1;
15 z = z + 1;16 if {scm = 0|1 ≤ m ≤ M} then17 Terminate decoding;
18 End
equate (8) and (9) and solve for T , as follows
p (Type-2)|�=T = p (Type-3)|�=T
1 −(
eT
1 + eT
)dc
=(
eT
1 + eT
)dc
T = ln
⎛
⎝( 1
2
)1/dc
1 − ( 12
)1/dc
⎞
⎠ (10)
Using the above expression, we plot the reliabilitythreshold T as a function of average check node degree dc
in Fig. 3. We can notice that the value of T is an increasingfunction of dc. Thus, given a fixed value of dc for a particularLDPC or RS code, we choose the reliability threshold Taccording to (10) such that the two type of check nodes will
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ASLAM et al.: EDGE-BASED DYNAMIC SCHEDULING FOR BP DECODING OF LDPC AND RS CODES 529
Algorithm 2 Proposed e-Shuffled Schedule
1 Initialize all mvn→cm = Lvn ;2 Reset k = 0, z = 0;3 Set kmax = Mdc Imax;4 while k < kmax do5 Update edge-state vector S according to Table I;6 for n = 1 : N do7 for every cm ∈ M (vn) do8 if εm,n = 1 then9 Generate and propagate mcm→vn (2);
10 for every cm ∈ M (vn) do11 if εm,n = 1 then12 Generate and propagate mvn→cm (1);13 k = k + 1;
14 z = z + 1;15 if {scm = 0|1 ≤ m ≤ M} then16 Terminate decoding;
17 End
Fig. 2. Check-node-type probability p (Type-2) and p (Type-3), plottedagainst the variable node log-likelihood-ratio (�) for check node degree
dc = 10, dc = 20 and dc = 30.
have an equal opportunity to participate in the next decodingiteration.
III. PERFORMANCE EVALUATION OF E-FLOODING
AND E-SHUFFLED BP SCHEDULES
In this section, we analyze the decoding complexity, error-rate performance and decoding convergence speed of BP algo-rithm under the proposed e-Flooding and e-Shuffled schedules.We mainly target medium-to-high rate LDPC and RS codes,since their parity-check matrices are relatively non-sparse andthe underlying code graphs contain a number of short cycles.
Fig. 3. Illustration of reliability threshold (T ) versus the average check nodedegree (dc): plotted using (10).
To this end, we construct a regular column-weight 3 LDPCcode with code-rate of 0.50. Besides, we also construct threeregular column-weight 4 LDPC codes with code-rate of 0.70,0.80 and 0.90, respectively. All LDPC codes are constructedwith fixed block-length of 1000 bits by using the progressive-edge-growth (PEG) algorithm [25]. This algorithm is well-known to construct finite-length LDPC codes with very gooderror correcting performance by ensuring a large girth of thecode graph. Furthermore, we select two RS codes, (25,31)code over GF(25) with code-rate of 0.80 and (55,63) codeover GF(26) with code-rate of 0.87. For BP decoding ofRS codes, we apply the binary image expansion to convertthe non-binary parity-check matrix into a binary form [24].Then, we decode the binary-expended RS codes under theadaptive belief-propagation (ABP) by employing the proposede-Flooding schedule.
We numerically optimize the value of t (number ofminimum-reliability neighbors) to be t = 2 for all LDPCcodes, and t = 8 and t = 30 for (25,31) and (55,63) RScodes, respectively. We set the value of T for different LDPCand RS codes based on the check node degree dc by using (10).Throughout this section, we assume that the binary LDPC andRS code-word bits, an ∈ {0, 1} for 1 ≤ n ≤ N , are BPSKmodulated by mapping into {+1,−1} voltage levels. TheseBPSK modulated bits are transmitted over an AWGN channelwith two-sided power spectral density of N0/2. The receivedoutput samples yn are converted into LLR information,Lvn = 4yn
N0, for the initialization of BP decoder.
A. Average Decoding Complexity
The BP decoding complexity is measured as the totalnumber of computations that are performed up to the error-rate convergence. It includes the variable-to-check (V2C) andcheck-to-variable (C2V) complexity, denoted by V and C ,respectively, and the scheduling complexity. In Table II, wereport a detailed complexity analysis of the proposed and prior-art scheduling schemes. Note that the scheduling complexityfor the proposed schedules, denoted by ψ , comprises of realvalue comparisons that are required to identify the set of tminimum-reliability neighbors, Nt -min (cm), corresponding toall the satisfied check node equations scm = 0. The value of ψcan be approximately computed by
ψ ≈ M
⎡
⎣1
2+ 1
2
dc∏
i=1
(1− 2eT
1 + eT
)⎤
⎦[t(dc − 1
) + 1]
z (11)
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530 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 2, FEBRUARY 2017
In order to define a complete complexity cost, it is impor-tant to incorporate the scheduling complexity in the overalldecoding complexity. To this end, we notice that a real valuecomparison can be realized in hardware by using a full-adder circuit [26], thus it has an equivalent complexity asthat of an addition operation. In this perspective, we includethe scheduling complexity ψ as part of the V2C updateoperations,1 and write the overall V2C and C2V complexityfor the proposed edge-based scheduling as
Vproposed = dvk + ψ (12)
Cproposed = k (13)
where k is the average number of V2C/C2V update operationscomputed per decoded code-word to attain the error-rate con-vergence, as shown in Algorithm 1 and 2. Note that the C2Vcomplexity dominates the overall BP decoding complexity asit involves multiplication and tanh(.) operations.
We compare the complexity of the proposed edge-basedschedules with that of the prior-art schedules. Firstly, we makecomparisons between the proposed e-Flooding and conven-tional Flooding [1] schedules. Furthermore, since the proposede-Shuffled schedule is a sequential based algorithm, we alsocompare it with well-known sequential schedules such as Shuf-fled [3], Layered [4], variable-node lazy (VLS) [8], check-nodelazy (CLS) [9], node-wise RBP (NW-RBP) [14], informedvariable-to-check RBP (IVC-RBP) [15], silent-variable-node-free (SVNF) [16], oscillating variable-node RBP (OV-RBP)[17] and reliability RBP (Rel-RBP) [19] schedules. Note thatthe conventional Flooding [1], Shuffled [3] and Layered [4]schedules do not involve run-time scheduling computations asthey follow a fixed order of message propagation. In contrast,the VLS and CLS schedules incur additional run-time realcomparisons, as shown in Table II. Thus, the overall V2C andC2V complexity of the lazy schedules can be computed as
VVLS = dvk + k/dv (14)
CVLS = k (15)
VCLS = dvk + k/dc (16)
CCLS = k (17)
To define the complete complexity cost for the dynamicschedules [14], [16], [17], [19], we include the residualcomputation cost into their respective C2V complexity as allthese schedules are C2V residual based dynamic schedules.2
Furthermore, similar to (12) and (14), we define an equivalentV2C complexity by combining the cost of V2C messageupdates and the number of addition/subtraction operations,which include the residual computation2 and real comparison.Thus, using Table II, we write the V2C and C2V complex-ity cost of NW-RBP [14], SVNF [16], OV-RBP [17] and
1According to (1), each V2C update operation requires dv additions.2Each residual computation ||m(i)cm→vn −m
(pre)cm→vn || requires one C2V mes-
sage pre-computation m(pre)cm→vn by using (2), and one subtraction operation.
The min-sum approximation [27] for the computation of m(pre)cm→vn , instead
of sum-product equation (2), affects the error-rate performance of dynamicschedules and also incurs more real comparison operations.
Fig. 4. Average variable-to-check (V2C) and check-to-variable (C2V) decod-ing complexity of the proposed e-Flooding schedule, compared (normalizedby) w.r.t. Flooding [1] schedule: employing rate 0.50, 0.70, 0.80 and 0.90LDPC codes.
Rel-RBP [19], as follows
VNW-RBP = dv(dv − 1
)k + (
dv − 1) (
dc − 1)
k
+ (E − 1) k/dc (18)
CNW-RBP = (dv − 1
) (dc − 1
)k (19)
VSVNF = dv(dv − 1
)k + (
dv − 1) (
dc − 1)
k
+ [dv
(dc − 1
) − 1]
k (20)
CSVNF = (dv − 1
) (dc − 1
)k (21)
VOV-RBP = dvk + (dc − 1
)k + φ (22)
COV-RBP = (dc − 1
)k (23)
VRel-RBP = dv(dv − 1
)k + (
dv − 1)
2k
+ 2(dv−1
) [2
(dc−1
)+1]
k+(M−1) k (24)
CRel-RBP = (dv − 1
)2k (25)
where the average number of comparison operations forOV-RBP schedule, denoted by φ, is computed through simu-lations. In contrast, the IVC-RBP [15] is a V2C residual baseddynamic schedule. Thus, we include the residual computationand real comparison cost into the V2C complexity, given by
VIVC-RBP = dv(dv − 1
) (dc − 1
)k + (E − 1) k (26)
CIVC-RBP = [(dv − 1
) + (dc − 1
)]k (27)
In order to demonstrate the complexity reduction thatthe proposed edge-based BP schedules offer over the exist-ing schedules, we normalize the V2C and C2V complex-ity of the proposed schedules. To be precise, we simulatethe LDPC codes under different schedules until the errorrates converge, or up to a maximum count of 50 iterations(Imax = 50) and plot the ratio
VproposedVprior-art
andCproposedCprior-art
, as shownin Fig. 4, 5 and 6. Note that the baseline comparison (normal-ization factor Vprior-art and Cprior-art) in Fig. 5 and 6 is variedw.r.t the prior-art schedules. It can be observed that the averageV2C and C2V complexity is significantly reduced comparedto all the prior-art schedules. For instance, for rate-0.90 LDPCcode at Eb/No = 5.50 dB, the proposed e-Flooding scheduleoffers V2C and C2V complexity reduction of up to 45% and75%, respectively, compared to the Flooding [1] schedule.Similarly, the proposed e-Shuffled schedule outperforms all
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ASLAM et al.: EDGE-BASED DYNAMIC SCHEDULING FOR BP DECODING OF LDPC AND RS CODES 531
the well-known sequential fixed and dynamic BP schedules indecoding complexity. In particular, for rate-0.90 LDPC codeat Eb/No = 5.50 dB, the V2C complexity cost is reducedby up to 50%, 40%, 30%, 30%, 98%, 98%, 85%, 95% and95%, whereas the C2V complexity cost is reduced by up
Fig. 7. Average variable-to-check (V2C) and check-to-variable (C2V)decoding complexity of the ABP decoding algorithm employing the proposede-Flooding schedule, compared (normalized by) w.r.t. ABP Flooding [21] andABP LRBP [23] schedules: employing (25,31) rate 0.80 and (55, 63) rate0.87 RS codes.
to 80%, 75%, 68%, 68%, 98%, 92%, 92%, 95% and 60%,compared to the Shuffled [3], Layered [4], VLS [8], CLS [9],NW-RBP [14], IVC-RBP [15], SVNF [16], OV-RBP [17] andRel-RBP [19] schedules, respectively.
For soft BP decoding of RS codes, we perform the ABPdecoding employing the proposed e-Flooding schedule andcompare the resultant decoding complexity with that of theABP Flooding [21] and ABP LRBP [23] schedules. TheLRBP scheme concatenates the dynamic BP decoding withthe algebraic hard-decision decoding [28]. To make a faircomparison, we only decode the RS codes under the dynamicBP algorithm. Using Table II, we write the total V2C and C2Vcomplexity cost of the LRBP schedule as
VLRBP = dv(dv − 1
)k + (
dv − 1) (
dc − 1)
k
+ (dc − 1
)k (28)
CLRBP = (dv − 1
) (dc − 1
)k (29)
The normalized V2C and C2V complexity curves are shownin Fig. 7. It shows that, for both (25, 31) and (55,63) RScodes at Eb/No = 7.0 dB, the C2V complexity is reduced bymore than 90% compared to the Flooding [21] and LRBP [23]schedules. The V2C complexity required by the proposede-Flooding schedule is also lower at moderate-to-high SNR.
B. Error-Rate and Decoding Convergence
Since the BP algorithm is a sub-optimal decoding algorithmfor finite-length block codes, particularly for short-lengthcodes, the scheduling strategy not only influences the decod-ing complexity but also affects the error-rate performance.To demonstrate the improvement in the error-rate performancethat the proposed e-Flooding and e-Shuffled schedules canprovide over the prior-art schedules, we plot the frame-error-rate (FER) curves against the channel SNR for the LDPC and
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532 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 2, FEBRUARY 2017
Fig. 8. Frame-error-rate (FER) performance of the proposed e-Floodingschedule, compared w.r.t. Flooding [1] schedule at Imax = 50: employingrate 0.50, 0.70, 0.80 and 0.90 LDPC codes.
RS codes. All the FER curves are plotted at Imax = 50.In Fig. 8, we make comparison between the proposede-Flooding schedule and the conventional Flooding [1] sched-ule and show that the proposed schedule outperforms thelatter. In Fig. 9, we compare the FER performance of theproposed e-Shuffled schedule with Shuffled [3], Layered [4],VLS [8], CLS [9], NW-RBP [14], IVC-RBP [15], SVNF [16],OV-RBP [17] and Rel-RBP [19] schedules. Note that theShuffled, Layered, VLS and CLS schedules have almost thesame asymptotic FER performance. Thus, we only plot theFER performance of Shuffled schedule in Fig. 9. It can beobserved that the proposed e-Shuffled schedule outperformsall of them, except for the SVNF schedule, which the proposedschedule yields a slightly inferior performance. However, notethat decoding complexity is the main motivation of this paper,and the proposed e-Shuffled schedule significantly outperformsthe SVNF schedule in this regard as shown in Fig. 5 and 6.For RS codes, the proposed e-Flooding schedule achievesperformance gain by up to 0.25 dB and 0.5 dB compared tothe LRBP [23] and Flooding [21] schedules, respectively, asshown in Fig. 10. In general, the proposed schedules providelarger improvement for higher rate codes. This is because, fora fixed block length, the underlying code graphs contain alarge number of short cycles. These graph cycles degrade theerror correcting capability of BP decoder, while the proposededge-based schedules diminish the effect of graph cycles byprecluding the occurrence of such graph cycles.
Arising from the lower decoding complexity, the proposededge-based schedules also provide faster decoding conver-gence, which determines the speed at which the decoderattains error-rate convergence. To show that, we plot the FERperformance of different schedules by varying the maximumnumber of C2V updates (equivalently the C2V complexity C )at Eb/No = 2.80 dB for rate 0.50 LDPC code and Eb/No =5.25 dB for rate 0.90 LDPC code, as shown in Fig. 11 and 12,respectively. It can be observed that the proposed e-Shuffled
Fig. 9. Frame-error-rate (FER) performance of the proposed e-Shuffledschedule, compared w.r.t. Shuffled [3], Layered [4], VLS [8], CLS [9],NW-RBP [14], IVC-RBP [15], SVNF [16], OV-RBP [17] and Rel-RBP [19]schedules, respectively: employing rate 0.50, 0.70, 0.80 and 0.90 LDPC codes.
schedule outperforms all the existing BP schedules. Amongthe residual-based schedules, the Rel-RBP scheme offers fastererror-rate convergence as compared to the NW-RBP, IVC-RBP,SVNF and OV-RBP schemes.
C. Code Graph Cycles in BP Decoding
As discussed earlier, the BP algorithm operates on the basicassumption that the incoming message updates at every checkand variable nodes are statistically independent. If the codegraph is cycle-free, throughout the iteration process, all theincoming message updates remain independent of each other.However, in code graph with cycles, such as the case ofLDPC and RS codes, dependencies are created particularlywhen the number of iterations is large. To be precise, for a
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ASLAM et al.: EDGE-BASED DYNAMIC SCHEDULING FOR BP DECODING OF LDPC AND RS CODES 533
graph cycle of length s, the incoming message update becomescorrelated after s/2 iterations [12], [13]. In this situation, theBP algorithm becomes sub-optimal and consequently yieldsinferior error-rate performance. For instance, the code graphshown in Fig. 13 induces a cycle of length s = 6 at variablenode v1, given by v1 ↔ c1 ↔ v2 ↔ c2 ↔ v4 ↔ c4 ↔ v1.Here, the intrinsic information of v1 is propagated towardscheck node c1 and c4. After 3 decoding iterations, the samecorrelated information is passed back to v1 through this graphcycle. Subsequently, all the incoming message updates at v1become statistically dependent. In contrast, as the messageflow is constrained under the proposed edge-based scheduling,the BP decoder may skip the update operation along certainedges of the code graph. Consequently, some of the graphnodes may not receive the correlated information after passings/2 iterations, as illustrated in Fig. 13. Here, it is assumedthat the check node c2 belongs to Type-2 and the neighboringedge �e2,2 has achieved high reliability (ε2,2 = 0). In thissituation, the graph edge �e2,2 will not be updated in thesubsequent iteration. As a result, the intrinsic informationof v1 received at v2, via v1 → c1 → v2, will not bepropagated beyond v2. Meanwhile, the intrinsic informationof v1 received at c2, via v1 → c4 → v4 → c2, will not be
Fig. 13. Code graph cycle experienced in BP decoding: the conventionalBP schedule receives correlated message update at variable node v1 after 3iterations via cycle v1 ↔ c1 ↔ v2 ↔ c2 ↔ v4 ↔ c4 ↔ v1; the proposededge-based BP schedule avoids the correlated message passed back to variablenode v1 by skipping update operation along edge �e2,2.
Fig. 14. Message independence violation under the proposed e-Flooding andconventional Flooding [1] schedules: higher message dependence is observedunder the conventional Flooding schedule.
propagated beyond c2. Consequently, node v1 will not receivethe correlated information either from check node c1 or fromcheck node c4 after 3 decoding iterations, thus protecting themessage independence assumption at v1.
Specifically, to observe the dependencies among messageupdates under the proposed e-Flooding and conventionalFlooding [1] schedules, we simulate the LDPC codes andrecord the average number of variable nodes where the incom-ing message updates become statistically dependent. To thisend, we compute the message independence violation (IV ),given by following ratio
IV = Variable nodes forming a graph cycle in BP decoding
Total # of variable nodes in the code graph(30)
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534 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 2, FEBRUARY 2017
This quantity is plotted against the channel Eb/N0 inFig. 14. We can notice that the proposed e-Flooding scheduleviolates the message independence at fewer variable nodes,compared to the Flooding schedule, which helps to achievea better error-rate performance. It should be noted that, ingeneral, the number of variable nodes violating the messageindependence asymptotically approaches to zero with increas-ing channel SNR. This is because the number of channelinduced errors asymptotically approaches to zero with increas-ing channel SNR, which consequently leads to the convergenceof BP algorithm in a single decoding iteration. Hence, themessage independence is preserved at all the variable nodes.
IV. CONCLUSION AND FUTURE WORK
Dynamic belief propagation (BP) schedules achieve lowererror rates than fixed BP schedules but suffer high decod-ing complexity. To overcome this shortcoming, two noveledge-based scheduling strategies for the BP decoding oflow-density parity-check (LDPC) and Reed-Solomon (RS)codes are presented in this paper. The proposed schedulingschemes are aimed to lower the BP decoding complexityand simultaneously improve on the error-rate performance.In these schedules, the reliability of individual graph edgesis monitored and new message updates are propagated onlyalong the less reliable edges in the code graph. In the firstproposed schedule, the unreliable graph edges are updatedsimultaneously, whereas in the second proposed schedulethey are updated sequentially. Numerical simulations are per-formed employing the proposed edge-based schedules for awide collection of medium- and high-rate regular LDPC andRS codes. Simulation results show that the proposed edge-based scheduling schemes outperform all the well-known fixedand dynamic BP schedules, including [1], [3], [4], [8], [9],[14]–[17], [19], [23], in terms of both decoding complexity andrate of decoding convergence in the range of medium-to-highSNR. Meanwhile, the error-rate performance also improveswith increasing channel SNR.
Possible future work includes an analytical determination ofthe set of minimum-reliability edges in the code graph to fullyexploit the potential of the proposed edge-based schedulingschemes. Furthermore, combining scheduling strategy withlow hardware complexity decoding algorithms, such as min-sum decoder [27] and reliability-based iterative majority-logicdecoder [29], is also an interesting topic for future research.
APPENDIX
CHECK-NODE-TYPE PROBABILITY p(TYPE)
Consider a set of dc independent binary random variables{a1, a2, ..., adc
}, with pi1 = Pr(ai = 1) and pi0 = Pr(ai = 0),that are connected with check node cm . Given that, we cancompute the probability that the set contains an even numberof 1s (check node is satisfied), given by [30]
⎡
⎣1
2+ 1
2
dc∏
i=1
(1 − 2 pi1
)⎤
⎦ (31)
Then, all dc variables have probability pi1 is given by(pi1)
dc , and atleast one variable that has probability not equalto pi1 is given by 1 − (pi1)
dc . Multiplying these pro-babilities
with (31) and replacing pi1 with LLR representation pi1 =e�
1+e�, where � = log(
pi0pi1) and pi0 = 1 − pi1 , we get the
expression for p(Type-2) and p(Type-3), as
p (Type-2) =⎡
⎣1−(
e�
1 + e�
)dc
⎤
⎦
⎡
⎣1
2+ 1
2
dc∏
i=1
(1− 2e�
1 + e�
)⎤
⎦
p (Type-3) =(
e�
1 + e�
)dc
⎡
⎣1
2+ 1
2
dc∏
i=1
(1 − 2e�
1 + e�
)⎤
⎦
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Chaudhry Adnan Aslam (S’13) received the B.E.degree (Hons.) from the Mehran University of Engi-neering and Technology, Pakistan, and the M.S.degree from the University of Southern California,Los Angeles, CA, in 2005 and 2009, respectively.He is currently pursuing the Ph.D. degree from theSchool of Electrical and Electronics Engineering,Nanyang Technological University, Singapore. Hisresearch interests include coding theory and signalprocessing for data storage and wireless communi-cations systems.
Yong Liang Guan (M’99) received the B.E. degree(Hons.) from the National University of Singapore,Singapore, and the Ph.D. degree from the ImperialCollege of London, London, U.K. He is currentlyan Associate Professor with the School of Electricaland Electronic Engineering, Nanyang Technolog-ical University, Singapore. His research interestsinclude modulation, coding and signal processing forcommunication systems, and information securitysystems.
Kui Cai (SM’11) received the B.E. degree ininformation and control engineering from ShanghaiJiao Tong University, Shanghai, China, the M.Eng.degree in electrical engineering from the NationalUniversity of Singapore, and the joint Ph.D. degreein electrical engineering from the Technical Univer-sity of Eindhoven, The Netherlands, and the NationalUniversity of Singapore. She has been with DataStorage Institute, Singapore, since 1999, where shewas the Program Leader of Non-Volatile MemoryCoding and Signal Processing. She is currently an
Associate Professor with the Singapore University of Technology and Design.She is the Vice-Chair (Academia) of the IEEE Communications Society,Data Storage Technical Committee. She is a recipient of the 2008 IEEECommunications Society Best Paper Award in coding and signal processingfor data storage. Her research interests include coding theory, communicationtheory, and signal processing for various data storage systems and digitalcommunications.
