International Symposium on Information Theory and its Applications, ISITA2006Seoul, Korea, October 29–November 1, 2006

On the Decoding of LDPC Codes in IEEE 802.16e Standardsfor Improving the Convergence Speed

Min-Ho Jang†, Beomkyu Shin†, Woo-Myoung Park†, Jong-Seon No†, and In San Jeon‡

† School of EECS.Seoul National University, Seoul, Korea

E-mail: {mhjang, thechi, ppakoo}@ccl.snu.ac.kr,jsno@snu.ac.kr

‡ SoC Design Research DepartmentETRI, Daejeon, Korea

E-mail: isjeon@etri.re.kr

Abstract

In this paper, the modified iterative decoding algorithmby partitioning check nodes is applied to low-densityparity-check (LDPC) codes in IEEE 802.16e standards,which gives us the improvement for convergence speedof decoding. Also, the new method of check node par-titioning which is suitable for decoding of the LDPCcodes in IEEE 802.16e system is proposed. The im-provement of convergence speed in decoding reducesthe number of iterations and thus the computationalcomplexity of the decoder. The decoding method bypartitioning check nodes can be applied to the LDPCcodes whose decoder cannot be implemented in thefully parallel processing.

1. INTRODUCTION

Low-density parity-check (LDPC) codes which werefirst introduced by Gallager [1] in the 1960’s show thegood performance approaching closely the Shannon’stheoretical limit for various channels. Also, the proba-bilistic iterative decoding based on low density of a par-ity check matrix is easily implemented in comparisonwith the past. The contemporary technology of hard-ware enables LDPC codes to be used in practice; recenttechnologies have contributed to achieve the reliabilityrequired by today’s high-speed digital systems. Thus,recently, the researchers of error correcting codes havebeen attracted by LDPC codes. In practice, LDPCcodes draw attention as standards of various fields suchas communication, broadcast, and storage.

The various studies related to LDPC codes can bedivided into two parts; one is the method of code designwith efficient encoding structure and the other is the

This work is financially supported by the Ministry of Edu-cation and Human Resources Development (MOE), the Ministryof Commerce, Industry and Energy (MOCIE) and the Ministryof Labor (MOLAB) through the fostering project of the Lab ofExcellency, by BK21, and by the ETRI.

method of decoding with low computational complex-ity. To decrease encoding complexity, the efficient en-coding structure is designed in finite length and block-type codes using protograph codes [2]. LDPC codes inIEEE 802.16e systems [3] have the efficient encodingstructure in the same manner.

Belief propagation (BP) [4], [5] and min-sum ap-proximation (MSA) [6] iterative decoding algorithm arewell-known decoding methods of LDPC codes. But, inthe BP and MSA iterative decoding algorithm, a largenumber of iterations which cause high computationalcomplexity are demanded to recover the reliable infor-mation. Thus, we would pay attention to a modifieddecoding algorithm [7] in order to improve the conver-gence speed, which means reducing the computationalcomplexity of the decoder. It can be used to imple-ment the practical decoder in the wireless communica-tion systems.

The paper is organized as follows: in Section II, af-ter a brief review of conventional decoding algorithm,we introduce a modified decoding algorithm to improvethe convergence speed and propose the method of checknode non-repetition partitioning which is suitable fordecoding of LDPC codes in IEEE 802.16e system. Thesimulation results are shown in Section III and con-cluding remarks are given in Section IV.

2. DECODING METHOD FOR IMPROVINGCONVERGENCE SPEED

A brief review of conventional BP and MSA decod-ing algorithm is provided to understand the modifieddecoding algorithm for improving convergence speed.See [5] and [6] in detail.

The iterative decoding algorithm whose examplesare BP and MSA decoding algorithm is the procedureof error correcting in which each variable and checknode of an LDPC code interchange messages iterativelyusing message updating operations. In BP algorithm,

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we assume that all messages are used in log-likelihoodratio (LLR) values. Then, in case of a check node,each output message is represented as the form of com-plicated tanh function. Similarly, in a variable node,each output message is the summation of all incom-ing messages including channel output message exceptfor the incoming message on the edge where the out-put message will be sent. However, in BP algorithm,the check node updating operation is so complicatedthat it cannot be implemented easily. To solve thiscomputational complexity, MSA decoding algorithm isintroduced. As a result of MSA decoding, the checknode updating operation can be simplified with a littledegradation of performance.

2.1. Decoding Algorithm with Improved Con-vergence Speed

Now, we consider the modified message-passing it-erative decoding algorithm [7] to improve the conver-gence speed of performance. Each iteration in the con-ventional iterative decoding algorithm consists of twosteps. At the first step, all variable nodes carry outthe updating calculation of messages and send them toall neighboring check nodes. Similarly, at the secondstep, all check nodes calculate the messages by updat-ing operation and send them to all neighboring variablenodes. At each step, all the operations are performedsimultaneously.

For convenience, in the modified decoding algo-rithm, we assume that the check nodes are partitionedinto p subsets. The messages from variable nodes tothe check nodes in the first subset are updated andthen the messages from the check nodes in the firstsubset to their neighboring variable nodes are updated.This procedure corresponds to one iteration for the firstsubset of check nodes. The decoding process is sequen-tially applied to the remaining p − 1 subsets of checknodes, which corresponds to one iteration of decoding.In other words, one iteration in the modified decodingalgorithm means the above sequential message updat-ing and passing for all variable nodes and all subsets ofthe check nodes. Therefore, it is clear that the amountof computation for one iteration in the modified decod-ing algorithm is the same as that of one iteration in theconventional decoding algorithm.

2.2. New Method of Check Node Partitioning

How to partition the subsets of check nodes hasinfluence on the convergence speed of the modified de-coding algorithm. Here, it is described that the efficientmethod of check node partitioning can improve the con-vergence speed in the decoding. The simplest method

is to divide sequentially all check nodes into subsetswhich contain the same number of check nodes. It iscalled sequential partitioning.

We would like to propose the non-repetition parti-tioning as the efficient method of check node partition-ing. Row permutations in the parity check matrix ofLDPC codes do not change the characteristic of codes.Thus, it is possible to change the ordering of rows inparity check matrix. Using this fact, we can constructthe equivalent parity check matrix according to thefollowing partitioning criterion. The partitioning cri-terion for subsets of check nodes is as follows: Withinthe same subset of rows, each column contains the com-ponent ‘1’ less than or equal to one. In other words,variable nodes connected by check nodes within a sub-set have the only connection with this subset. Notethat the number of check nodes within each subset ofcheck nodes can be different and for convenience of im-plementation, it is determined as the minimum numberof subsets satisfying the partitioning criterion.

The reason why the above criterion is adopted isthat for the present iteration, the updating messagesfrom check nodes in a subset to variable nodes aresent and sequentially the variable nodes send updatedmessages to check nodes in another subset. But othervariable nodes that are not connected with the checknodes in a subset and thus do not update the messagesfrom the subset send the messages used at previousiteration. For example, let Si, 1 ≤ i ≤ p, be the i-th subset of check nodes. During the l-th iteration inthe modified decoding algorithm, the l-th updated mes-sages from S1, S2, · · ·, Si−1 and the (l − 1)-st updatedmessages from the remaining subsets are used for thel-th message updating from the variable nodes to thecheck nodes in Si. Therefore, in terms of propagat-ing many internal updating messages through edges ofgraph corresponding to LDPC codes, the above crite-rion – variable nodes connected by check nodes withina subset have the single connection with this subset –is optimum. After all, the messages interchanged morefrequently among nodes cause faster convergence speedof performance.

3. SIMULATION RESULTS

In this section, the modified iterative decoding algo-rithm by partitioning check nodes is applied to LDPCcodes in IEEE 802.16e standards and is examined interms of the convergence speed of decoding perfor-mance. We consider an additive white Gaussian noise(AWGN) channel and the maximum number of itera-tions is limited to 50 times.

In Fig. 1, it is shown the block-type 12× 24 parity

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Figure 1: Block-type parity check matrix of LDPCcodes with rate 1/2 in IEEE 802.16e standards.

check matrix of a rate 1/2 LDPC code in the IEEE802.16e standards [3] using protograph codes. Eachblank in the block-type parity check matrix denotes az× z all-zero matrix, and each block with a shift valuerepresents a z×z circular right shifted matrix of a z×zidentity matrix. In particular, each block with a 0 valuedenotes a z × z identity matrix. Therefore the LDPCcodes with various code lengths can be defined by theadjustment of the shift values according to z.

When the sequential partitioning with the numberof subsets of check nodes p = 6 is applied to the LDPCcode, each subset of check nodes corresponding to the(1, 2), (3, 4), (5, 6), (7, 8), (9, 10), or (11, 12)-th row in12 × 24 matrix is fixed. Also, for the non-repetitionpartitioning with p = 6, each subset corresponding tothe (1, 10), (2, 11), (3, 5), (4, 6), (7, 9), or (8, 12)-th rowin matrix is determined according to the criterion ofnon-repetition partitioning. First of all, we comparethe decoding performance by sequential partitioningwith that by non-repetition partitioning applying eachto modified decoding algorithm by partitioning checknodes. Fig. 2 shows the frame error rate (FER) perfor-mance for LDPC codes in the IEEE 802.16e standardswith rate 1/2 and code length 2304 using sequentialpartitioning and non-repetition partitioning. Specially,for sequential partitioning p is set to 1 (conventionalBP), 2, and 6.

For the case of sequential partitioning, the conver-gence speed of performance is faster as p increases. Butthe modified decoding with the large p value cannot beimplemented due to latency. Because the performanceby non-repetition partitioning with p = 6 approachesto that by sequential partitioning with p = 1152 whichcorresponds to the maximum number of check nodepartitioning, non-repetition partitioning is optimum forboth performance and implementation. In particular,the performance is rapidly improved by non-repetitionpartitioning for the small number of iterations.

Now, we apply the modified iterative decoding algo-rithm using non-repetition partitioning to LDPC codes

1.00 1.25 1.50 1.75

10-4

10-3

10-2

10-1

100

FE

R

Eb/No(dB)

Conventional BP,50

Seq. Partitioning (p=2),15

Seq. Partitioning (p=2),25

Seq. Partitioning (p=2),50

Seq. Partitioning (p=6),15

Seq. Partitioning (p=6),25

Seq. Partitioning (p=6),50

Non-rep. Partitioning,15

Non-rep. Partitioning,25

Non-rep. Partitioning,50

Figure 2: The FER performance for LDPC codes withrate 1/2 and code length 2304 according to iterationsapplying sequential and non-repetition partitioning.

with various lengths and rates in IEEE 802.16e stan-dards and confirm the improvement of convergencespeed of it. In the case of rates 2/3A, 3/4A, and3/4B LDPC codes, each subset corresponds to each rowblock in the block-type parity check matrix and thusthe numbers of subsets of check nodes are p = 8, p = 6,and p = 6, respectively. Also, in the case of rate 2/3BLDPC codes, for the non-repetition partitioning withp = 4, each subset corresponding to the (1, 4), (2, 7),(3, 6), or (5, 8)-th row block in matrix can be selected.

1.0 1.5 2.0 2.5 3.0 3.5

10-4

10-3

10-2

10-1

100

FE

R

Eb/No(dB)

N=576(Conv. BP,50)

N=576(Partioning,25)

N=576(Partioning,50)

N=1152(Conv. BP,50)

N=1152(Partioning,25)

N=1152(Partioning,50)

N=1728(Conv. BP,50)

N=1728(Partioning,25)

N=1728(Partioning,50)

N=2304(Conv. BP,50)

N=2304(Partioning,25)

N=2304(Partioning,50)

Figure 3: The FER performance according to itera-tions applying modified BP decoding method by non-repetition partitioning to rate 1/2 LDPC codes.

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First, we start with modified decoding methodbased on check node updating operation of BP decod-ing. Fig. 3, Fig. 4, and Fig. 5 show the FER per-formance of LDPC codes with rate 1/2, 2/3, and 3/4,respectively. Next, we consider the modified decod-ing method based on check node updating operationfor MSA decoding algorithm by partitioning the checknodes [6]. Optimum normalization constant γ = 1/αless than 1 and offset constant β for each rate are de-termined through the brutal search to obtain the per-formance of MSA decoding as a reference (Table 1).

2.0 2.5 3.0 3.5 4.0

10-4

10-3

10-2

10-1

100

FE

R

Eb/No(dB)

N=576(Conv. BP,50)

N=576(Partioning,25)

N=576(Partioning,50)

N=1152(Conv. BP,50)

N=1152(Partioning,25)

N=1152(Partioning,50)

N=1728(Conv. BP,50)

N=1728(Partioning,25)

N=1728(Partioning,50)

N=2304(Conv. BP,50)

N=2304(Partioning,25)

N=2304(Partioning,50)

Figure 4: The FER performance according to itera-tions applying modified BP decoding method by non-repetition partitioning to rate 2/3A LDPC codes.

2.5 3.0 3.5 4.0 4.5

10-4

10-3

10-2

10-1

100

FE

R

Eb/No(dB)

N=576(Conv. BP,50)

N=576(Partioning,25)

N=576(Partioning,50)

N=1152(Conv. BP,50)

N=1152(Partioning,25)

N=1152(Partioning,50)

N=1728(Conv. BP,50)

N=1728(Partioning,25)

N=1728(Partioning,50)

N=2304(Conv. BP,50)

N=2304(Partioning,25)

N=2304(Partioning,50)

Figure 5: The FER performance according to itera-tions applying modified BP decoding method by non-repetition partitioning to rate 3/4A LDPC codes.

Table 1: Optimum values of γ and β in min-sum ap-proximation decoding for LDPC codes in IEEE 802.16estandards.

γ βRate 1/2 0.83 0.43

Rate 2/3A 0.82 0.44Rate 3/4A 0.76 0.46

Although MSA decoding decreases the computa-tional complexity by simple check node updating op-eration, its performance approaches to that of BP de-coding within 0.1dB. For optimum γ and β for LDPCcodes with various rates and lengths, similar character-istic is appeared. Fig. 6 shows the FER performanceof rate 1/2 LDPC codes using modified MSA decodingalgorithm with non-repetition check node partitioningand optimum γ.

From the above simulation results, we know thatthe performance by non-repetition partitioning with 25iterations is similar to that by conventional (BP andMSA) decoding (p = 1) with 50 iterations for vari-ous lengths and rates. Therefore, we can confirm theimprovement for convergence speed in the modified de-coding algorithm with non-repetition check node par-titioning. Note that the amount of computation forone iteration in the modified decoding algorithm is thesame as that for one iteration in the conventional de-coding algorithm. Because the same performance ofdecoding is guaranteed by less iterations, the compu-tational complexity of modified decoding algorithm bynon-repetition partitioning can be reduced by half.

1.0 1.5 2.0 2.5 3.0 3.5

10-4

10-3

10-2

10-1

100

FE

R

Eb/No(dB)

N=576(Conv. MSA,50)

N=576(Partioning,25)

N=576(Partioning,50)

N=1152(Conv. MSA,50)

N=1152(Partioning,25)

N=1152(Partioning,50)

N=1728(Conv. MSA,50)

N=1728(Partioning,25)

N=1728(Partioning,50)

N=2304(Conv. MSA,50)

N=2304(Partioning,25)

N=2304(Partioning,50)

Figure 6: The FER performance according to itera-tions applying modified MSA decoding method by non-repetition partitioning to rate 1/2 LDPC codes.

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4. CONCLUSIONS

In this paper, the modified iterative BP and MSAdecoding algorithm by partitioning check nodes was ap-plied to LDPC codes in IEEE 802.16e standards andwe confirmed the improvement in convergence speed ofperformance. Because the same performance of decod-ing is guaranteed by less iterations, the computationalcomplexity can be reduced by half. Also, the method ofnon-repetition check node partitioning which is suitablefor LDPC codes in IEEE 802.16e system is proposed.The decoding method by partitioning check nodes canbe applied to the systems which cannot be implementedin the fully parallel processing as an efficient sequen-tial processing method. The modified iterative decod-ing method of LDPC codes using the proposed checknode partitioning method can be used to implementthe practical decoder in the wireless communicationsystems.

References

[1] R. G. Gallager, Low-Density Parity-Check Codes,Cambridge, MA: MIT Press, 1963.

[2] J. Thorpe, “Low-density parity-check (LDPC)codes constructed from protograph,” IPNProgress Reprot, 42-154, JPL, Aug. 2003.

[3] IEEE 802.16 Working Group, “Part 16: Air in-terface for fixed and mobile broadband wirelessaccess systems,” IEEE P802.16e/D8, May 2005.

[4] Frank R. Kschischang, Brendan J. Frey, andHans-Andrea Loeliger, “Factor graphs and thesum–product algorithm,” IEEE Trans. Inform.Theory, vol. 47, no. 2, pp. 533-547, Feb. 2001.

[5] T. J. Richardson and R. L. Urbanke, “The ca-pacity of low-density parity-check codes undermessage-passing decoding,” IEEE Trans. Inform.Theory, vol. 47, no. 2, pp. 599-618, Feb. 2001.

[6] J. Chen, A. Dholakia, E. Eleftheriou, M. P. C.Fossorier, and X.-Y. Hu, “Reduced-complexitydecoding of LDPC codes,” IEEE Trans. Com-mun., vol. 53, no. 8, pp. 1288-1299, Aug. 2005.

[7] Sunghwan Kim, Min-Ho Jang, Jong-Seon No,Song-Nam Hong, and Dong-Joon Shin, “Sequen-tial message passing decoding of LDPC codesby partitioning check nodes,” submitted to IEEETrans. Commun., Sep. 2004.

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