IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 12, DECEMBER 2014 2089

EXIT Chart Analysis of Puncturing for Non-Binary LDPC CodesKuntal Deka, A. Rajesh, and Prabin Kumar Bora

Abstract—This letter presents an EXtrinsic Information Trans-fer (EXIT) chart tool for the puncturing of non-binary (NB)low-density parity-check (LDPC) codes. With the help of this tool,we study the dependence of the performance of various bitwise andsymbolwise puncturing patterns on the degree of a variable node(VN). The grouping algorithm (GA) is a useful technique for theshort-length codes to find the recoverable VNs. By incorporatingthe GA under the EXIT chart model, we propose a method to findthe optimum recoverable puncturing pattern.

Index Terms—Non-binary low-density parity-check (NB LDPC)codes, puncturing, grouping algorithm (GA), EXIT chart.

I. INTRODUCTION

FOR communication systems with time-varying channelconditions, rate-compatibility is a desirable feature of a

code. Ha et al. [1] introduced a rate-compatible (RC) puncturingscheme for short-length LDPC codes. They proposed group-ing and sorting algorithms to select the puncturing bits. Thegrouping algorithm (GA) partitions the variable nodes (VNs)into K + 1 groups G0,G1, · · · ,GK , where G0 is the set ofunpunctured VNs and Gi (i �= 0) is the set of VNs guaranteedto be recovered exactly after i iterations. VNs in Gi with loweri are preferred for puncturing. Within each group, the order ofpuncturing is determined using the sorting algorithm.

Klinc et al. [2] proposed a puncturing scheme for the non-binary (NB) LDPC codes. The GA from [1] was found to beuseful for the short-length NB LDPC codes also. For codesdefined over GF(2s) (for some integer s), one can puncturea certain number of bits (bitwise puncturing) or all the bits(symbolwise puncturing) corresponding to a VN. They recom-mended to use the GA for selecting the recoverable VNs andthen perform bitwise puncturing on the recoverable VNs.

In [3] and [4], the authors analyzed the puncturing schemesby calculating the decoding thresholds via Monte-carlo baseddensity evolution (DE). They showed that the degrees of theVNs play a crucial role in the performances of different punc-turing patterns. They considered an irregular code over GF(24)and found optimum puncturing distributions for different rates.For this optimization, the GA was not taken into consideration.Through different simulations for short-length codes, we haveobserved that the GA prefers not to puncture the VNs of higherdegrees. In that case, the optimized patterns in [3] and [4]cannot be directly used along with the GA as these patternstake into account all the VNs including the higher-degree ones.

Manuscript received June 18, 2014; accepted October 21, 2014. Date ofpublication October 31, 2014; date of current version December 8, 2014. Theassociate editor coordinating the review of this paper and approving it forpublication was H. Saeedi.

The authors are with the Department of Electronics and Electrical Engineer-ing, Indian Institute of Technology Guwahati, Guwahati 781039, India (e-mail:kuntal@iitg.ac.in; rajesh@iitg.ac.in; prabin@iitg.ac.in).

Digital Object Identifier 10.1109/LCOMM.2014.2366119

The EXtrinsic Information Transfer (EXIT) chart is an alter-native to DE for analyzing and designing codes for optimumasymptotic performance. In the EXIT chart, the mutual infor-mation (MI) transfer characteristics of the VN decoder (VND)and the check node decoder (CND) are plotted to calculatethe threshold graphically [5]. The minimum Eb/N0 value forwhich the VND curve just touches the CND curve is consideredas the threshold (Eb

N0

∗). The EXIT chart model for the NB

LDPC codes was developed by Bennatan and Burshtein [6]for an arbitrary discrete-memoryless channel. Using this model,Dolecek et al. have generalized the protograph EXIT (PEXIT)chart [7] to the non-binary case and formulated the NB-PEXITchart [8], [9]. With the help of this tool, short-length NBprotograph LDPC codes have been designed [8], [10].

The motivation of the letter is three-fold: (1) to devise anEXIT chart model to study different puncturing patterns for NBLDPC codes, (2) to consider the GA in the EXIT chart analysisand (3) to devise a procedure of finding the optimum puncturingpattern with the recoverable VNs using the EXIT chart model.

II. BACKGROUND

In the EXIT chart model of [6], the MIs between the trans-mitted code symbol C at an average VN and different messagesW related to that VN are computed. For brevity, we refer theMI between C and W as the MI for W. Let IA,VND representthe MI for the CN → VN messages. For a VN of degree i, theEXIT function is approximated as

IE,VND(IA,VND, i) ≈ JR

(√(i− 1) [J−1(IA,V ND)]2

)(1)

where J(·) and JR(·) are the MI functions for a symmetricGaussian-distributed log-likelihood ratio (LLR) vector and aright-bound (VN → CN) message respectively. The detailedprocedures of obtaining J(·) and JR(·) are described in [6].The effective VND EXIT function is given by

IE,VND(IA,VND) =∑dmax

v

i=2λiIE,VND(IA,VND, i) (2)

where (λ2, · · · , λdmaxv

) is the edge-perspective degree distribu-tion for the VNs.

Let IA,CND represent the MI for the VN → CN messages.For a degree-j CN, j − 1 number of incoming messages aregenerated as described in [6] to calculate the left-bound (CN →VN) message. Suppose the EXIT function for a degree-j CNis IE,CND(IA,CND, j). Then the effective CND EXIT functioncan be written as

IE,CND(IA,CND) =∑dmax

c

j=dminc

ρjIE,CND(IA,CND, j) (3)

where (ρdminc

, · · · , ρdmaxc

) is the edge-perspective degree distri-bution for the CNs.

1089-7798 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

2090 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 12, DECEMBER 2014

Fig. 1. Thresholds for punctured quasi-regular codes with r0 = 0.5, N = 142 at rate rl = 0.6. (a) GF(26); (b) GF(25); (c) GF(24).

III. PROPOSED EXIT CHART FOR

PUNCTURED NB LDPC CODES

We call the puncturing to be regular if the same numberof bits are punctured for all the VNs in a Tanner graph andirregular otherwise. We first develop the EXIT chart model forregular puncturing with random selection of the VNs.

A. Random Selection of the VNs and Regular Puncturing

Let b ∈ {1, · · · , s} denote the number of punctured bits perVN. Suppose the mother code is of rate r0. Then for a targetrate rl, the number of punctured VNs is given by

N lp =

[N × s× (rl − r0)

rl × b

](4)

where N is the total number of VNs in the Tanner graph and [·]is the nearest integer function.

Two extreme cases of (4) arise:1) When b = s (symbolwise puncturing), N l

p becomes min-imum. We puncture the minimum possible number ofVNs. But these VNs are punctured completely.

2) When b = 1 (if allowable), N lp is the largest. We puncture

the maximum number of VNs. However, each of theseVNs undergoes the minimal puncturing.

Such conflicting cases can be investigated with the help ofthe EXIT chart. The methods to obtain the VND and the CNDcurves are discussed below.

VND Curve:Let pv denote the probability that a VN is punctured. pv can

be expressed as

pv =N l

p

N(5)

Let JR1and JR2

denote the MI for a right-bound messageflowing from an unpunctured and a punctured VN respectively.Then the effective JR can be found as

JR = (1− pv)JR1+ pvJR2

(6)

The VND curve can now be obtained using (1) and (2).CND Curve:

Consider a CN c of degree j connected to the neighboringVNs {v1, v2, · · · , vj}. Without loss of generality, we considerthe message flowing along the edge c → v1 to find the CNDcurve. We have to generate the messages from the j − 1 VNsin the set N = {v2, v3, · · · , vj}. Let d ∈ {0, · · · , j − 1} denotethe number of punctured VNs in N . The probability that the setN contains d number of punctured VNs is given by

pN (d) =

(j − 1

d

)(pv)

d(1− pv)j−d−1 (7)

Suppose IE,CND(IA,CND, j, d) is the EXIT function for aCN of degree j when the number of punctured VNs in N is d.The effective EXIT function for a degree-j CN can be obtained as

IE,CND(IA,CND, j) =

j−1∑d=0

pN (d)IE,CND(IA,CND, j, d) (8)

Finally IE,CND(IA,CND) can be computed using (3).The proposed EXIT chart model is used to investigate the

effects of puncturing patterns for quasi-regular NB LDPC codeswith different values of mean column weight (t). We calculatethe thresholds for punctured codes over GF(26) with r0 = 0.5and rl = 0.6. We consider N = 142 to obtain the value of pv .Fig. 1(a) shows these thresholds against b for different valuesof t. Several interesting observations can be made from thefigure. When t = 2, puncturing a less number of bits per VNgives better thresholds. At t = 2.6, the scenario is completelyopposite. In between these two contrasting cases, there existsa point around t = 2.2, where the thresholds appear to beunaffected by b. We call this t as the crossover point (tc). FromFig. 1(b) and 1(c), it can be observed that the values of tc forthe quasi-regular codes over GF(25) and GF(24) are 2.38 and2.5 respectively.

In [11], the authors have obtained the optimum values oft (topt) for quasi-regular NB LDPC codes over binary-inputadditive white Gaussian (BI-AWGN) channels for a rate of 0.5.For the codes over GF(24), GF(25) and GF(26), the values oftopt are 2.3, 2.2 and 2.1 respectively. The values of tc for therate-0.6 punctured codes over GF(24), GF(25) and GF(26) areapproximately 2.5, 2.38 and 2.2 respectively. It can be seen that

DEKA et al.: EXIT CHART ANALYSIS OF PUNCTURING FOR NON-BINARY LDPC CODES 2091

tc is higher than topt. The gap between tc and topt decreaseswith increasing field order.

B. Selection of the VNs According to theGA and Irregular Puncturing

We formulate the EXIT chart model in the context of usingdifferent b for the VNs of different degrees in the case of irreg-ular codes. We also take into account the GA which imposesupper limits on the number of puncturable VNs of a particulardegree.

Consider the set of recoverable VNs obtained from the GAas given by GR =

⋃Ki=1 Gi. Suppose the number of VNs of

degree di in GR is given by Ngdi, i = 1, 2, . . . , z, where z is

the total number of different degrees of the VNs in GR. LetNdi

and Ndi,b respectively denote the total numbers of degree-di VNs and the degree-di VNs having b punctured bits. For atarget rate rl, the optimum recoverable pattern (Ndi,b) can beobtained by solving the following optimization problem:

minimizeNdi,b

i=1,···,z,b=1,···,s

E∗b

N0

subject toz∑

i=1

s∑b=1

bNdi,b =[N×s×(rl−r0)

rl

]and Ndi,b � Ng

di, ∀i, ∀b

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(9)

The optimization is carried out as follows. First all possiblecombinations of Ndi,b satisfying the constraints in (9) are foundout exhaustively. The thresholds for these combinations arecomputed using the EXIT chart model presented below. Thecombination yielding the minimum threshold is considered asthe optimum recoverable puncturing pattern. Then we select thepunctured VNs from GR according to this optimum pattern.

The formulation of the EXIT curves are shown below.VND Curve:Let J0

R and JbR denote the MI for a right-bound message

of an unpunctured VN and a VN containing b punctured bitsrespectively. Then the MI for a right-bound message emanatingfrom a degree-di VN is given by

JR,di=

∑s

b=1pbv,di

JbR +

(1−

∑s

b=1pbv,di

)J0R (10)

where pbv,di=

Ndi,b

Ndidenotes the probability that a VN of degree

di contains b punctured bits. JR,di= J0

R for di �∈ {d1, d2,. . . , dz}.

The effective MI now is given by

JR =∑dmax

v

di=2λdi

JR,di(11)

CND Curve:Let Ab, b ∈ {1, . . . , s} be the random variable representing

the number of VNs in N with b punctured bits per symbol.Clearly

∑sb=1 Ab = d, where d is the total number of punctured

VNs in N . For a given d, A1, A2, . . . , As are characterized bythe joint probability mass function

pA1,...,As|d(a1, . . . , as) =d!

s∏b=1

ab!

s∏b=1

(pbv)ab (12)

TABLE INUMBER OF VNS IN DIFFERENT GROUPS

TABLE IIOPTIMIZED RECOVERABLE PUNCTURING PATTERNS IN THE CONTEXT OF

THE GROUPING ALGORITHM AND THE CORRESPONDING THRESHOLDS

where pbv is the probability that b bits of a VN are puncturedgiven that the VN is selected for puncturing. pbv is given by

pbv =

∑zi=1 Ndi,b∑z

i=1

∑sb′=1 Ndi,b′

Suppose IE,CND(IA,CND, j, d, a1, . . . , as) is the EXIT func-tion for a degree-j CN when the set N contains d puncturedVNs arranged as (A1 = a1, A2 = a2, . . . , As = as). Then theeffective EXIT function for a degree-j CN is given by

IE,CND(IA,CND, j)=

j−1∑d=0

∑a1,...,as

pN (d)pA1,...,As|d(a1, . . . , as)

× IE,CND(IA,CND, j, d, a1, . . . , as) (13)

with pN (d) as given by (7) and pv =∑z

i=1 λ̂di

(∑s

b=1Ndi,b)

Ndi,

λ̂dibeing the fraction of degree-di VNs.

From (13) and (3), the CND curve can be obtained.

IV. SIMULATION RESULTS

We take a code over GF(24) having the optimum degreedistributions considered in [3]:

λ(x) = 0.5376x+ 0.1678x2 + 0.1360x4 + 0.1586x9

ρ(x) = 0.5169x4 + 0.4831x5

We construct a 71 × 142 parity-check matrix by first forminga binary matrix with the help of the progressive edge-growth[12] algorithm and then replacing the 1 s by the randomlychosen elements from GF(24) \ {0}. The numbers of the VNsof degree 2, 3, 5 and 10 are 104, 22, 11 and 5 respectively. Thecodeword bitstream is BPSK modulated and transmitted overa BI-AWGN channel. Decoding is done by the Fast FourierTransform based q-ary sum product algorithm (FFT-QSPA)[13] with the maximum number of iterations set at 50.

Table I shows the number of the VNs in different groupsalong with their degrees as obtained from the GA. The GAdid not select any VN of degree 5 and 10. The total number

2092 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 12, DECEMBER 2014

TABLE IIISCHEME S3 AT rl = 0.7

Fig. 2. Performance comparisons.

of recoverable VNs is 54. Thus the maximum rate attainable bythe GA is 71

142−54 = 0.8068.By computing the thresholds by the proposed EXIT chart

model, the optimum recoverable puncturing patterns are foundout. Table II shows these patterns along with their thresholds.The thresholds of the optimum puncturing distributions withoutany constraints as reported in [3] and [4] are also shown. It canbe seen that for lower rates, the thresholds for the optimum re-coverable patterns are close to the thresholds for optimum punc-turing distributions without any constraints. However at higherrates, because of the small number of the puncturable VNs, thethresholds for the optimum recoverable patterns become high.

We consider four puncturing schemes for performance stud-ies: (1) S1 (proposed scheme)—the optimized recoverable pat-tern in Table II with the selection of the punctured VNs fromGR (2) S2—the optimized recoverable pattern in Table II withrandom selection of the punctured VNs (3) S3—the optimizeddistribution without any constraints [3], [4] with the selection ofas many punctured VNs as possible from GR. If the distributionrequires more number of punctured VNs of a particular degreethan GR can provide, then we select the remaining VNs ofthe same degree randomly from G0 (4) S4—the optimizeddistribution without any constraints with random selection ofthe punctured VNs [3], [4].

Table III shows the pattern of the punctured VNs for S3

at rl = 0.7. We try to ensure the recoverability of the heavilypunctured VNs. For this, we assign the recoverable VNs (GR)starting from b = 4 down to b = 1.

Fig. 2 shows the symbol-error-rate (SER) performances of allthe schemes. It is observed that S1 is better than S2 and S3 isbetter than S4 at all rates. Thus if we select the VNs from GR

rather than selecting randomly, the performance is improvedfor both the optimized recoverable patterns in Table II and theoptimum distributions in [3] and [4]. It is also observed that

at rl = 0.6 and rl = 0.7, both S1 and S2 perform better thanS3 and S4. This happens because, unlike S1 and S2, each ofS3 and S4 includes degree-5 and degree-10 VNs and resultsin a higher number of dead CNs [1]. However, at rl = 0.8,S2 gives the worst result. This is because of the higher valueof the threshold (3.29) for the optimized recoverable patternin Table II compared to the threshold (2.37) for the optimumdistribution in [3], [4]. Although S1 has the same threshold(3.29) as that for S2, it yields the best SER performance as allthe punctured VNs are from GR.

V. CONCLUSION

This letter proposed an EXIT chart model for the puncturingof NB LDPC codes. With the help of the EXIT chart model, weinvestigated the role of the degrees of the VNs in the puncturingperformance. The grouping algorithm is an effective tool to findrecoverable puncturing patterns specifically for short-lengthcodes. We presented a method to obtain optimized puncturingpatterns compatible with the grouping algorithm.

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