The Application Study of Non-binary LDPC Code

Yuping Wu1, a, Danfeng Zhao1,b ,Ningning Tong1 2,c 1 College of Information & Communication Engineering, Harbin Engineering University, China

2 Department of Electronic Engineering, Heilongjiang Institute of Technology, China

awyp@hrbeu.edu.cn,

bzhaodanfeng@hrbeu.edu.cn,

ctongningning_82@yahoo.com.cn

Keywords:Non-binary LDPC code; MSK modulation; bandwidth efficiency

Abstract:Most practical communication systems need higher bandwidth efficiency; this article

increased the bandwidth efficiency through code rate and modulation mode, by combining

Non-binary LDPC code with MSK modulation and making practical use of them in communication

systems. The simulation result showed that the combination of Non-binary LDPC code and MSK

modulation possessed not only a stronger error correcting capability but also a higher bandwidth

efficiency, which will probably be more applicable in practical communication systems.

1.Introduction

Gallager’s binary low-density parity-check (LDPC) codes[1]are excellent error-correcting codes

which achieve performance close to the benchmark predicted by the Shannon theory [2]. Davey and

Mackay first investigated the extension of LDPC to a non-binary Galois field GF(q) over the binary

input additive-white-Gaussian-noise (AWGN) channel [3]. It was shown empirically that non-binary

LDPC can potentially have better performance than binary irregular LDPC codes [3]. This had

motivated active studies on non-binary LDPC codes ever since.

The simplest LDPC codes are cycle codes [4], as their parity check matrices have column weight

j = 2. An interesting finding [5],[6] is that the mean column weight of non-binary LDPC codes must

approach 2 when the field order q increases; that is, the best non-binary LDPC codes for very large q

tend to be cycle codes over GF(q). It has also been proved [7] that cycle GF(q) codes can achieve

near-Shannon-limit performance as q increases. Further, numerical results [7] demonstrate that cycle

GF(q) codes can outperform other LDPC codes, including degree-distribution optimized binary

irregular LDPC codes.

Because most practical communication systems need possess higher bandwidth efficiency which

can be increased from two aspects, i.e. code rate and modulation mode. Compared with binary LDPC

codes, non-binary LDPC codes are more capable of rectifying burst and random noises, are suitable

for high-order modulation system and can meet the high requirements of modern telecommunication.

MSK modulation is a continuous-phase modulation mode, and due to higher bandwidth efficiency

and small radiation outside the band, etc., it is robustly applicable in the practical communication

systems. This article conducts the simulation verification on the combination, applied in the practical

communication system, of non-binary LDPC code and MSK modulation.

2. Non-binary LDPC codes

A. The definition of non-binary ldpc codes

LDPC codes are linear block codes, whose check matrixH is a sparse matrix so that there are very

few nonzero elements inH . The elements of H are from Galois field ( )GF q , where q is the order of

Galois field. For binary LDPC codes, the nonzero element is only ‘1’; but for non-binary LDPC codes,

the number of nonzero elements is 1q − . For every usable codeword c , we have

T=Hc 0 (1)

The column and row weights of H are defined as the number of nonzero entries in the column

and row, respectively.

Advanced Materials Research Vols. 403-408 (2012) pp 2543-2546Online available since 2011/Nov/29 at www.scientific.net© (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.403-408.2543

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B.Non-binary ldpc codes encoding algorithm

The Tanner graph of non-binary LDPC codes is similar to the binary LDPC code’s, but there are q

types of values of variable nodes, and the structure constraint of parity-check nodes are more

complicated. In this paper, we first generate a binary LDPC code parity check matrix, and then replace

the element “1” in binary check matrix with randomly selected element in {1, 2, …q-1}. Let q=2p, so

that we can use p-bit binary bits to transmit a q-ray symbols. As for non-binary LDPC codes, the

encoding algorithm is similar to the binary LDPC’s, but now all the operators to follow the operation

rules over GF(q).

Example: Let q= 5. The set {0, 1, 2, 3, 4} of integers under modulo-5 addition and multiplication

given by Table 1and Table 2 form a field GF(5) of five elements.

Table 1. Modulo-5 addition Table 2. Modulo-5 multilication

⊕ 0 1 2 3 4

0 0 1 2 3 4

1 1 2 3 4 0

2 2 3 4 0 1

3 3 4 0 1 2

4 4 0 1 2 3

⊗ 0 1 2 3 4

0 0 0 0 0 0

1 0 1 2 3 4

2 0 2 4 1 3

3 0 3 1 4 2

4 0 4 3 2 1

The addition table is also used for subtraction. For example, suppose that 4 is subtracted from 2.

First we use the addition table to find the additive inverse of 4, which is 1. Then we add 1 to 2 with

modulo-5 addition. This gives the element 3.

For division, we use the multiplication table. Supposed that we divide 3 by 2. From the

multiplication table, we find the multiplicative inverse of 2, which is 3. We then multipy 3 by 3 with

modulo-5 multiplication. The result is 4.

AS H is a larger sparse check matrix, we can get the generator matrix G from check matrixH by

the algorithm of gaussian elimination, and further create LDPC code words.

C. Codes design

In this paper, We design the binary and non-binary LDPC codes using the progressive

edge-growth (PEG) algorithm [7]. The flexibility of the PEG code construction method enables us to

freely select any code length, field size, and column weight. To compare the performance of

non-binary LDPC codes with BPSK modulation and non-binary LDPC codes with MSK

modulation,we choose the field sizes and column weights of the PEG-LDPC codes to be the same in

the two modulation modes. Therefore, we design a (758, 1536) PEG-LDPC code over ( )GF q with

rate 1/2 with the values ofq being 16, 64 respectively. Note: the unit of codeword length over

different fields is bit.

3. Simulation results and analysis

0 0.5 1 1.5 2 2.5

10-5

10-4

10-3

10-2

10-1

100

Eb/N0(dB)

Bit error rate

16-ary LDPC codes with BPSK and MSK

BPSK10

BPSK25

BPSK100

MSK10

MSK25

MSK100

0 0.5 1 1.5 2 2.5

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0(dB)

Bit error rate

64-ary LDPC codes with BPSK and MSK

BPSK10

BPSK25

BPSK100

MSK10

MSK25

MSK100

Figure 1 The bit error rate of 16-ary LDPC

codes under MSK and BPSK modulation

Figure 2 The bit error rate of 64-ary LDPC codes under

MSK and BPSK modulation

2544 MEMS, NANO and Smart Systems

Fig. 1 and Fig. 2 are the bit error rate performance simulation curves of the combination of

non-binary LDPC codes and MSK modulation and BPSK modulation under 16-ary and 64-ary codes

respectively with different iteration numbers. The information length is 768 bits, decoding adopts BP

decoding algorithm, with the iteration number being 10, 25, 100 respectively, and the communication

channel being gaussian white noise. From the figures, no matter whatever the iteration number is, the

bit error rate performance curves of the combination of non-binary LDPC codes and MSK modulation

and that of the combination of non-binary LDPC and BPSK modulation are of superposition

basically, i.e. the error correcting capabilities of non-binary LDPC under two modulations are

basically identical, thus indicating the combination of non-binary LDPC and MSK modulation also

has a much strong error correcting capability.

0 0.5 1 1.5 2 2.5

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0(dB)

Bit error rate

16-ary and 64-ary LDPC codes with BPSK and MSK

64-aryBPSK

64-aryMSK

16-aryBPSK

16-aryMSK

0 0.5 1 1.5 2 2.510

-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N0(dB)

Symbol error rate

16-ary and 64-ary LDPC codes with BPSK and MSK

64-aryBPSK

64-aryMSK

16-aryBPSK

16-aryMSK

Figure 3 The bit error rate of 64-ary LDPC codes

under MSK and BPSK modulation

Figure 4 The symbol error rate of 64-ary LDPC codes

under MSK and BPSK modulation

0 0.5 1 1.5 2 2.5

10-5

10-4

10-3

10-2

10-1

100

Eb/N0(dB)

Frame error rate

16-ary and 64-ary LDPC codes with BPSK and MSK

64-aryBPSK

64-aryMSK

16-aryBPSK

16-aryMSK

Figure 5 The frame error rate of 64-ary LDPC codes

under MSK and BPSK modulation

Figure 6 The normalized power spectrums of MSK and

BPSK

Fig. 3~ Fig. 5 shows the bit error rate performance, symbol error rate performance and frame error

rate performance of non-binary LDPC codes and MSK modulation and BPSK modulation under

16-ary and 64-ary codes respectively with iteration number being 100. The information length is 768

bits, decoding adopts BP decoding algorithm, and the communication channel being gaussian white

noise. From the figures, no matter 16-ary LDPC codes or 64-ary LDPC codes, the bit error rate

performance curves, symbol error rate performance and frame error rate performance of the

combination of non-binary LDPC codes and MSK modulation and that of the combination of

non-binary LDPC and BPSK modulation are of superposition basically, i.e. the error correcting

capabilities of non-binary LDPC under two modulations are basically identical, thus indicating the

combination of non-binary LDPC and MSK modulation also has a much strong error correcting

capability.

Advanced Materials Research Vols. 403-408 2545

What fig. 6 indicates is the normalized power spectrums of MSK and BPSK, from the figure, the

power spectrum of MSK signal is more compact compared with BPSK, showing the frequency

bandwidth ratio occupied by the main lobe of MSK signal power spectrum is narrower than BPSK

signal and being 0.75 times the bandwidth of the main lobe of BPSK signal; apart from the bandwidth

of main lobe, the side lobe of the power spectrum also drops more quickly, showing the MSK signal

power is mainly included in the main lobe. Therefore, we can see that, compared with BPSK signal,

MSK signal is more applicable in the practical communication channel with high bandwidth

efficiency.

4. Conclusions

According to the simulation and analysis above, we know that the combination of non-binary

LDPC code and MSK modulation owns not only a stronger error correcting capability but also higher

bandwidth efficiency, which will be more applicable in the practical communication systems with

high bandwidth efficiency.

References

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

[2] D. J. C. Mackay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans.

Inform. Theory, vol. 45, no. 2, pp. 399–431, Mar. 1999.

[3] M. C. Davey and D. Mackay, “Low-density parity-check codes over GF(q),”IEEE Commun. Lett.,

vol. 2, pp.165–167,June 1999.

[4] D.Jungnickel and S.A. Vanstone, “Graphical codes revisited,” IEEE Trans. Inform. Theory, vol.

43,pp.136–146,Jan. 1997.

[5] M. C. Davey and D. Mackay, “Monte Carlo simulations of infinite low density parity check codes

over GF(q),” in Proc.of Int. Workshop on Optimal Codes and related Topics, Bulgaria, June

9-15,1998.

[6] M. C. Davey, Error-Correction using Low-Density Parity-Check Codes.Dissertation, University

of Cambridge, 1999.

[7] X.-Y. Hu and E. Eleftheriou, “Binary representation of cycle tannergraph GF(2b) codes,” Proc.

International Conference on Communications,vol. 27, no. 1, pp. 528 – 532, June 2004.

2546 MEMS, NANO and Smart Systems

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