Construction of Binary LDPC Convolutional Codes

5/22/2018 Construction of Binary LDPC Convolutional Codes

IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 6, JUNE 2012 897

Construction of Binary LDPC Convolutional Codes

Based on Finite Fields

Liwei Mu, Xingcheng Liu, Member, IEEE, and Chulong Liang

AbstractUsing a finite field approach, a novel algebraicconstruction of low-density parity-check (LDPC) convolutionalcodes with fast encoding property is proposed. According to thematrices of quasi-cyclic (QC) codes constructed based on themultiplicative groups of finite fields and the algebraic propertythat a binary circulant matrix is isomorphic to a finite ring,we first generate a polynomial-form parity-check matrix of anLDPC convolutional code under a given rate over a given finitefield. Then some related modifications are made upon the originalpolynomial-form matrix to obtain the new one with fast encodingproperty. Simulation results show that the proposed LDPCconvolutional codes have good performance with the iterativebelief propagation decoding algorithm.

Index TermsFinite fields, low-density parity-check (LDPC)

convolutional codes, fast encoding, belief propagation.

I. INTRODUCTION

LOW-DENSITY parity-check (LDPC) convolutionalcodes, which can be considered as convolutionalcounterparts of the well-known LDPC block codes and

have better performance than block codes [1], are attracting

increasing attention in the field of coding. Although

LDPC convolutional codes can be derived from LDPC block

codes [1], their algebraic construction methods remain limited.A class of algebraically constructed LDPC convolutional

codes [2] based on circulant matrices for designing LDPC

block codes were presented and their performance is superiorto the corresponding block codes. The LDPC convolutional

codes with highly structured graphs [3] were also proposed,whose polynomial-form matrices can be derived from the

generator matrices of the quasi-cyclic (QC) codes. But

many entries in the matrix are multinomial. Based on

graph-cover methods [4], families of time-invariant and

time-varying LDPC convolutional codes were derived, which

are favorable for high-rate very-large-scale-integration (VLSI)implementations.

According to the algorithm for constructing QC LDPC

codes based on the multiplicative groups of finite fields [5],

a novel construction method for LDPC convolutional codes is

presented in this letter. The QC LDPC codes perform wellover the additive white Gaussian noise (AWGN) channel,

and the resulted codes have shown very low error floors [5].Under a given code rate over a given finite field GF(q), we

Manuscript received November 17, 2011. The associate editor coordinatingthe review of this letter and approving it for publication was M. Lentmaier.

The work was supported by the National Natural Science Foundation ofChina (Grant No. 60970041, 61173018).

The authors are with the Department of Electronic and CommunicationsEngineering, Sun Yat-sen University, Guangzhou, 510006, China, and withthe State Key Laboratory of Integrated Services Networks, Xidian University(e-mail: [email protected], [email protected]).

Digital Object Identifier 10.1109/LCOMM.2012.040912.112352

first generate a base matrix of a QC LDPC code. Then the

original polynomial-form parity-check matrix of an LDPC

convolutional code is obtained by replacing each entry o

base matrix with a polynomial. Finally, some modification

are done over the original polynomial-form matrix to obtain

the new one with fast encoding property. Simulation results

show that the proposed codes have good performance.

I I . THE R ELATIONSHIPB ETWEENLDPC

CONVOLUTIONALC ODES AND QC BLOCK C ODES

It can be concluded from [5] that the multiplicative r-fold

dispersion matrix H(r)

QC

FrJrL

2 is equivalent to JL

submatrices, where each submatrix is either arrzero matrixor a circulant permutation matrix Ix. Ix is an identity matrixwith rows shifted cyclically to the right by x positions, wherex is the exponent of each nonzero entry of matrix W basedon the multiplicative groups of finite fields [5]. If the nonzero

entry in the first row of each r r matrix Ix is representedby a unique polynomial, the binary circulant matrix of size

r r forms a ring isomorphism to the ring of polynomial odegree less than r, F2[D]/ < Dr 1>. We have found thaeach nonzero Ix can be denoted by the polynomial Dx. Andif the zero matrix of size r r is represented by 0, we can

associate the scalar parity-check matrix H(r)QC F

rJrL2 with

the polynomial parity-check matrix H(r)QC(D) (F2[D]/ )JL. Interpreting the polynomial-form parity-check

matrix H(r)QC(D) of a QC block code as the same matrix of

an LDPC convolutional code [2], H(D) = H(r)QC(D), we can

obtain a polynomial-form parity-check matrix H(D) of anLDPC convolutional code, which is given by

H(D) =

h1,1(D) h1,2(D) h1,L(D)h2,1(D) h2,2(D) h2,L(D)

......

. . . ...

hJ,1(D) hJ,2(D) hJ,L(D)

, (1

where hi,j(D) (i= 1, 2,...,J,j = 1, 2,...,L) is the i-th row

and j -th column ofH(D).

III . CONSTRUCTION OFLDPC CONVOLUTIONALC ODES

The LDPC block codes require splitting the data to be sen

into frames, such as the ones we proposed in [6], which may bedisadvantaged in some practical applications. However, LDPC

convolutional codes are able to deal with data of arbitrarylengths and perform encoding continuously in real time with

the help of a shift-register based encoder [1]. This section

describes the construction method of LDPC convolutiona

codes with fast encoding characteristic under a given rate.

1089-7798/12$31.00 c 2012 IEEE


898 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 6, JUNE 2012

A. Construction of Original LDPC Convolutional Codes

Consider the Galois field GF(q) with q elements, where qis a prime or power of a prime. Let be a primitive element

of GF(q). Then = 0, 0 = 1,,...,q2 give all the

elements of GF(q) and q1 = 1. The nonzero elementsin GF(q) form the multiplicative group of GF(q) under themultiplication operation.

In order to obtain the parity-check matrix of LDPC convo-lutional codes under a given rate R = (LJ)/Lover a givenfinite field GF(q), we first form a J L base matrix M ofa QC LDPC code over the multiplicative group of the finite

field GF(q) [5]. The details are described as follows:1) Find a nonzero integer m, such that m 1 = q2.2) Form a J L base matrix Mover GF(q) given by

M =

mL+1 1 m1 1 m 1mL+2 1 m 1 m+1 1

......

. . . ...

mL+J 1 m+J2 1 m+J1 1

.

According to [5], the multiplicative(q 1)-fold dispersionof matrix M satisfies the column-row (RC) constraint.

3) Let the power exponent of each entry Mi,j (i =1, 2,...,J,j = 1, 2,...,L) of matrix M be Ei,j that takes amaximum value of q2, here. Then according to SectionII, we can obtain a polynomial-form parity-check matrix

H(D) = [hi,j(D)]JL of an LDPC convolutional code,where hi,j(D) = DEi,j .

This construction method is referred to as the original

method. Matrix H(D) is called the original polynomial-formmatrix and the corresponding code is called the original code.

If there is no zero entry in H(D), the LDPC convolutional

codes are called (J, L)-regular codes since the weight sum isexactly the same in each row in the polynomial-form matrixH(D), and this also holds for each column. Otherwise, theyare called irregular LDPC convolutional codes.

B. Construction of LDPC Convolutional Codes with Fast

Encoding Property

The fast encoding property, which is one of the advantages

of LDPC convolutional codes, makes LDPC convolutional

codes have better advantage than LDPC block codes on the

encoding complexity. For a binary matrix, when the last Jlinearly independent columns ofJLmatrix H0 are chosen

as the J J identity matrix, a systematic encoder with fastencoding property can be obtained [1] and the designed rateof an LDPC convolutional code can be kept unchanged.

The modified form of matrix H(D) with fast encodingproperty is presented in this section. If matrix H(D) hasno element D0 in all rows (or columns), the diagonal ele-ments of last J columns ofH(D) will be replaced by D0.Otherwise, through matrix permutation to set D0 to be thediagonal elements of its last Jcolumns. When the remainingdiagonal elements of last J columns ofH(D) have no D0,they are replaced by D0, then a modified matrix HI(D)can be obtained. If the corresponding base matrix M of

matrix HI

(D) satisfies the -multiplied row-constraints [5],

TABLE IAVERAGE( PE RS YMBOL NODE) NUMBER OF CYCLES COMPARISON

BETWEEN THEFE-I AND THEFE-IICONVOLUTIONAL CODES( CC S).

(3,6) FE-I CC over GF(128)



(3,6) FE-II CC over GF(128)

Code

6 0.4297 1.2578 5.9453

6 1.0758 0.6256 1.5640

6 0.9735 0.4834 2.2649

6 0.0132 0.5232 1.0397

6 0.0095 1.1374 2.4834

6 0.1875 0 1.1484

g gN g+2N 4gN +



g is the girth of corresponding code. Ng is the average number of cycles of length g. Ng+2 is the average number of cycles of length (g+ 2). Ng+4 is the average number of cycles of length (g+ 4).

the proposed method can be used to generate a polynomial

form matrix with fast encoding property, which is referred to

as FE-I (Fasting Encoding I).

If the corresponding base matrix M

cant satisfy the-multiplied row-constraints, elementary transformation omatrixH(D)can be used to obtain matrix HII(D)by settingthe diagonal elements to D0 in last J columns of matrixH(D). This method is referred to as FE-2 (Fasting Encoding2).

Example 1: We construct a R = 1/2, (3, 6)-regular LDPCconvolutional matrix over GF(128) with the original method

FE-I and FE-II, respectively.

The original polynomial-form parity-check matrix H(D)isgiven by

H(D) =

D86 D55 D117 D12 D101 D126D55 D117 D12 D101 D126 D1

D117 D12 D101 D126 D1 D103

.

The FE-I polynomial-form parity-check matrix HI(D) isgiven by

H

I(D) =

D86 D55 D117 D0 D101 D126

D55 D117 D12 D101 D0 D1

D117 D12 D101 D126 D1 D0

.

The FE-II polynomial-form parity-check matrix HII(D) isgiven by

H

II(D) =

D74 D43 D105 D0 D89 D114

D116 D11 D100 D125 D0 D102

D54 D116 D11 D100 D125 D0

The delay decomposition of the polynomial parity-check

matrix

H(D) = H0+ H1D+ +HmsDms FJL2 [D]

leads to a scalar description of the time-invariant LDPC

convolutional code by a semi-infinite parity-check matrixH


MU et al.: CONSTRUCTION OF BINARY LDPC CONVOLUTIONAL CODES BASED ON FINITE FIELDS 899

which can be represented as

H=

H0

H1 H0

... H1

Hms

... . . .

Hms

. . .

. . .

, (2)

where memoryms is the largest exponent in H(D) such thatHms = 0 and each submatrix Hi(i = 0,...,ms) is a J Lmatrix.

It is known that the constraint graphs of LDPC convolu-

tional codes have girths lower-bounded by the girth of the

corresponding QC LDPC constraint graphs [2]. Hence, the

proposed LDPC convolutional codes have girth at least 6.

Table I shows the average number (per symbol node) of

a given cycle length per constraint length, i.e., (ms + 1)Lfor the Tanner graphs of some LDPC convolutional codes

constructed based on the method FE-I and FE-II. It shows thatthe Tanner graphs of the proposed FE-I LDPC convolutional

codes have fewer small cycles than those of FE-II ones. So it

can be concluded from Table I that the FE-I codes have better

performance. Although the considered examples show that FE-

I codes behaves nicely, the proof for their cycle improvement

is still missing, since FE-I changes the fundamental codestructure. But a lot of statistics (e.g., Table I) have shown

that the Tanner graphs of FE-I codes have fewer small cycles.

For the original method, the input information sequences

can be encoded using the generator matrix obtained from

the original polynomial-form parity-check matrix H(D) by

Gaussian elimination method, so the original one requiresmore storage space and higher computational complexity. For

FE-I and FE-II, although they all have fast encoding property,

it is shown that FE-I codes have better performance and only

FE-I can obtain maximum encoding memory ms= q2overGF(q). So when satisfying the -multiplied row-constraints,FE-I is the first choice.

As described in Section II, when the matrix H(D) overGF(q) is given, each entry hi,j(D)ofH(D)can be replacedby a(q 1) (q 1) right-shift circulant permutation matrixIx (wherex is the exponent ofhi,j(D)) or a (q 1) (q1) zero matrix (if hi,j(D) = 0), In this way, we can get a

corresponding binary parity-check matrix H

(q1)

QC of a QCblock code. Based on [4], we can also yield the matrix of a

time-varying LDPC convolutional code from the binary matrix

H(q1)QC .

IV. SIMULATION R ESULTS

In this section, the proposed time-invariant LDPC convo-

lutional codes are obtained. Computer simulations are con-

ducted for a sliding window message-passing decoder [1] on

AWGN channels with BPSK modulation. All decoders are

allowed a maximum of 50 iterations. In order to compare thebit error rate (BER) performance with the same processing

complexity, the length of LDPC block codes is N=v, where

0.5 1 1.5 2 2.5 3 3.5 4 4.5

106

105

104

103

102

101

Eb/No (dB)

BER

Original BC

FEII BCFEI BC

Original CCFEII CC

FEI CC

Fig. 1. Performance ofR = 2/5, (3,5)-regular LDPC convolutional code(CCs) and their corresponding block codes (BCs) of length N= 1050 oveGF(211) constructed with original method, FE-I and FE-II, respectively.

v= (ms+1)Lis constraint length of the LDPC convolutionacodes.

Under a given rate R = 2/5 over finite field GF(211), Fig1 shows the performance of (3,5)-regular LDPC convolutiona

codes constructed with the original method, FE-I and FE-II

respectively. Simulation results show that the performance of

the LDPC block code generated with FE-I is at least 0.7dB better than that generated with the original method (the

method in [5]) or FE-II when BER < 1 106. For theLDPC convolutional code, the gain is about 0.4 dB. Whether

they are LDPC convolutional codes or LDPC block codes, the

codes constructed with the original method and FE-II have

the same performance. So FE-I codes behave best among the

codes of interests. The good performance of the FE-I codesis mainly attributed to the few short-cycles of the equivalenTanner graphs.

As shown in Fig.1, these proposed FE-I codes have almosthe same performance as those codes constructed in [2] (Fig

12) or [4]based on the Tanner unwrapping technique, although

the girths of corresponding block matrices for the former and

the latter are 6 and at least 8, respectively.

V. CONCLUSIONS

A novel algorithm for constructing LDPC convolutionacodes with fast encoding property is proposed. The proposed

method, which is based on finite fields, can generate LDPCconvolutional codes under any given code rate. Comparedwith the existing construction methods of LDPC convolutiona

codes derived from QC LDPC block codes, the proposed codesalso have good performance. These characteristics demonstrate

the feasibilities of our proposed schemes in practical applica-

tions.

VI . ACKNOWLEDGMENT

The authors would like to thank the editor and anonymou

reviewers for their constructive comments in improving thisletter. We would also like to thank Professor Xiao Ma for his

valuable suggestions.


900 IEEE COMMUNICATIONS LETTERS, VOL. 16, NO. 6, JUNE 2012

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Construction of Binary LDPC Convolutional Codes