Available online at www.sciencedirect.com Available online at www.sciencedirect.com
Procedia Engineering
Procedia Engineering 00 (2011) 000–000 Procedia Engineering 15 (2011) 1617 – 1621 www.elsevier.com/locate/procedia
Advanced in Control Engineering and Information Science
Hybrid decoding scheme based on WED and BF algorithms Dong Zijian, Xu Hongyan a* School of Electronic Engineering, Huaihai Institute of Technology, Lianyungang Jiangsu, China, 222005
Abstract This contribution studies the application of Weighting Erased Decoding (WED) algorithm in the decoding of LDPC codes. WED is a kind of maximum likelihood algorithm used in the decoding of linear block codes with quantized channel outputs. Based on WED algorithm and Bit Flipping (BF) algorithm, a hybrid decoding scheme is proposed, which takes advantage of the simplicity of BF and the utilizing of reliability of received symbols. Simulation results show that the proposed scheme is feasible and effective, but the performance improvement is limited and depends on a parameter similar to the minimum Hamming distance in block codes.Click here and insert your abstract text.
© 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. Selection and/or peer-review under responsibility of [CEIS 2011] Weighting Erased Decoding (WED); Bit Flipping (BF); Low-Density Parity-Check (LDPC)；Hybrid Decoding
1. Introduction Iterative decoding of Low-Density Parity-Check (LDPC) codes proposed by Gallager has been studied extensively in recent years [1]. Bit-flipping (BF) decoding is simple, but brings about considerable performance loss compared to the BP decoding [2]. In order to improve decoding performance, weighted BF (WBF) decoding and its variants were proposed [3-5], which effectively utilize the reliability of received symbols. Weighting Erased Decoding (WED) algorithm is a kind of maximum likelihood algorithm used in the decoding of linear block codes with quantized channel outputs [6,7]. This algorithm is suitable for the
* Dong Zijian. Tel.: 086-518-8518366; fax: 086-518-8518364. E-mail address:
[email protected].
1877-7058 © 2011 Published by Elsevier Ltd. Open access under CC BY-NC-ND license. doi:10.1016/j.proeng.2011.08.301
1618 2
Dong Zijian and Hongyan / Procedia Engineering 15000–000 (2011) 1617 – 1621 Dong Zijian ，Xu et al/ Procedia Engineering 00 (2011)
binary input and Q-ary output channel. WED algorithm can improve the decoding performance compared to the algebraic decoding, because it can take advantage of the reliability of received symbols. Since LDPC codes are a class of linear block codes, we try to adapt the WED algorithm to the decoding of LDPC codes. The decoding scheme is to combine the WED and the BF/WBF decoding. The idea of the hybrid algorithm is obvious, but in order to improve decoding performance, there are still many issues to be resolved, which is discussed in this paper. The remainder of this paper is organized as follows. We begin by discussing the BF algorithm in section 2 and WED algorithm in section 3. In Section 4, we present the hybrid decoding scheme. Simulation results are presented in section 5, followed by conclusions. 2. BF and WBF Assume an LDPC code defined by a sparse check matrix H = h ij
C
c0 , c1 ,..., cN
channel. Let r
1
is transmitted with BPSK modulation, additive white Gaussian noise (AWGN)
r0 , r1 ,..., rN
1
be the soft-decision received sequence and v
the corresponding hard-decision sequence. Let the check node z , m
. The coded sequence M N
n
m
n, hmn
v0 , v1 ,..., vN
1
be
1 be the set of bits connected with
1 be the set of checks connected with the bit node vn .
m, hmn
The decoding process of BF is as follow: 1. Compute the syndrome T
S = Hv
s0 , s1 ,..., sM
T 1
. If S = 0 , then the iteration is
terminated and the decoding is declare successful. If not, go to the second step; 2、For each message node, compute the error metric:
Tn
En
S T hn
(1)
sum(Tn )
（2）
3 、Turn the bit with highest E value, and go to step 1. k
As there are many types of weighted BF algorithms, we give here only the reliability-ratio based weighted BF (RRWBF) algorithm, to illustrate the principle of utilizing symbol reliability of various WBF. The decoding process of RRWBF is as follow: T . If S = 0 , then the iteration is terminated 1. Compute the syndrome T
S = Hv
s0 , s1 ,..., sM
1
and the decoding is declare successful. If not, go to the second step; 2、Find the most unreliable message node associated with each check node :
rmmax
max
n ,n
rn
m
(3)
3、For each message node, compute the error metric:
En
m
n
2 sm
1
rmmax rn
(4)
1619 3
Dong ZijianAuthor and Xuname Hongyan / Procedia Engineering 15 (2011) 1617 – 1621 / Procedia Engineering 00 (2011) 000–000
Where the variable
is a normalization factor introduced for ensuring
rn n
rmmax
m
1
.
4 、Turn the bit with highest E value, and go to step 1. k
3. WED Algorithm and the hybrid decoding scheme Suppose a binary linear (n, k) code with minimum distance dmin. The cannel output is uniformly
2m , these Q regions are assigned weights
quantized into Q levels regions. For Q
wj
j / Q 1 ,0
j
Q 1 . Define 2m
pi
For each received signal, We have the binary representation of the integer For received sequence
r
i 1
/ Q 1, m 1
wl j,
i 0
a lj ,i pi
m 1
j
i 0
r0 , r1 ,..., rn
0
i
m 1
, where the binary tuple
a j , i 2i
a j ,0 , a j ,1 ,..., a j ,m
1
is
.
, we can now construct an m
1
(5)
n binary matrix A for
which the lth column is given by the binary tuple obtained according to the above process.
For 0 i m 1 , the ith row of A is decoded into a codeword by an algebraic decoder, and the m decoded codewords form another m n binary matrix A’. Here, the decoder can use BF or WBF algorithm for LDPC codes. If using RRWBF algorithm, the symbol reliability for each row code still use the original channel output log-likelihood ratio.
m 1 , let fi be the number of places where the ith row of A and the ith row of A’ differ. Define the reliability indicator of the ith row of A’ as [7] For 0
i
Ri
max 0, d min
2 fi
(6)
For the hybrid decoding of WED and BF, the equation above we used is modified as
Ri
max 0, D
fi
(7)
D affect the performance of decoding, and this is confirmed by the simulations we sl sl do. For the lth column of A’, with 0 l n 1 , let 1 and 0 be the index sets of the rows that contain 1 and 0 at bit position l respectively. Then the lth bit is decoded into 0 if Where the parameter
i S0l
Ri pi
i S1l
Ri pi
the hard decision.
, into 1 if
i S0l
Ri pi
i S1l
Ri pi
. In case of equality, the bit decoded according to
1620 4
Dong Zijian and Hongyan / Procedia Engineering 15000–000 (2011) 1617 – 1621 Dong Zijian ，Xu et al/ Procedia Engineering 00 (2011)
4. Simulation Results The LDPC code selected in this paper is a （504,252,3,6）regular LDPC code given in [8]. Fig.1 illustrates the bit error rate performance and Fig.2 illustrates the frame error rate performance. In these two figures, we show the performance of BF, RRWBF, and two hybrid decoding algorithm (marked as WED-BF and WED-RRWBF in figures). Simulations use BPSK modulation in AWGN channel. The maximum number of iterations is fixed at 30 for BF/RRWBF in all cases. The simulation results show that the proposed hybrid decoding scheme is feasible, but only obtains limited performance improvement, and decoding performance is also heavily dependent on the parameters A. How to further improve the decoding performance depends on in-depth research. 5. Conclusion We propose a hybrid decoding scheme for LDPC codes based on WED and BF/RRWBF. This hybrid decoding scheme takes advantage of the simplicity of WED and BF/RRWBF. The simulation results show that the proposed hybrid decoding scheme is feasible, but only obtains limited performance improvement, and decoding performance is also heavily dependent on the parameters A. How to further improve the decoding performance depends on in-depth research.
Fig.1 BER comparison of several algorithms for (504,252) LDPC code
Dong ZijianAuthor and Xuname Hongyan / Procedia Engineering 15 (2011) 1617 – 1621 / Procedia Engineering 00 (2011) 000–000
Fig.2 FER comparison of several algorithms for (504,252) LDPC code
References [1] Gallager R G. Low-Density Parity-Check Codes[J]. IRE Transactions on Information Theory, January 1962, IT-8: 21–28. [2] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, pp. 399–432, Mar. 1999. [3] Kou Y., Lin S. and Fossorier M., Low-density parity-check codes based on finite geometries: a rediscovery and new results, IEEE Transactions on Information Theory, vol. 47, pp. 2711–2736, November 2001. [4] Zhang J., Fossorier M. P. C., A Modified Weighted Bit-Flipping Decoding of Low-Density Parity-Check Codes, IEEE Communications Letters, vol. 8, pp. 165–167, March 2004. [5] F. Guo and L. Hanzo, “Reliability ratio based weighted bit-flipping decoding for low-density parity-check codes,” Electron. Lett., vol. 40, pp. 1356–1358, Oct. 2004. [6] Weldon E J. Decoding binary block codes on Q-ary output channels [J]. IEEE Transactions on Information Theory, 1971, IT-17:713-718. [7] Lin Shu and Costello Daniel. Error Control Coding [M]. Englewood Cliffs, N J: Prentice-Hall, 2004:413-417. [8] Mackay, Encyclopedia of Sparse Graph Codes, http://www.inference.phy.cam.ac.uk/mackay/codes/data.html#l67, Cited 15 Sep 2009.
1621 5
