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The constant log-MAP decoding algorithm suitable for duo-binary turbo codes. Motivated by an existing algorithm approach, an efficient algorithm is presented ...
Constant log-MAP decoding algorithm for duo-binary turbo codes S. Papaharalabos, P. Sweeney and B.G. Evans The constant log-MAP decoding algorithm suitable for duo-binary turbo codes. Motivated by an existing algorithm approach, an efficient algorithm is presented, which is superior in terms of frame=bit error rate performance and has approximately the same computational complexity. Compared with log-MAP decoding, the proposed algorithm has negligible performance degradation, exactly as for binary turbo codes.

Introduction: Duo-binary turbo codes have attracted much interest because of the advantages compared to binary codes for equivalent implementation complexity [1, 2]. For instance, the path error density is lowered and the decoder latency is divided by two, the influence of puncturing is less crucial, owing to constituent encoders with higher coding rates, and both intra-symbol and intersymbol interleaving are supported. For these reasons, they have been adopted by the Digital Video Broadcasting (DVB) Project to provide full asymmetric twoway communications over the return channel for satellite and terrestrial networks (DVB-RCS and DVB-RCT, respectively) [2]. Relevant work on the standardised DVB-RCS turbo code [3] can be found in [2, 4, 5] related to the encoder design and also in [6–8] related to the decoder design. In the remainder of this Letter we focus attention on [8], in which the constant log-MAP decoding algorithm for duo-binary turbo codes was presented for the first time. An important advantage of constant log-MAP decoding for binary turbo codes [9] is that it makes use of a look-up table (LUT) of two values, instead of the more usually assumed eight values. This reduces implementation complexity against the log-MAP decoding algorithm with negligible performance degradation, e.g. 0.05 dB at high to moderate bit error rate (BER) values. In this Letter, we first take the constant log-MAP decoding algorithm from [8] and then propose an efficient adaptation, which is found to have approximately the same computational complexity, but has superior performance and is very close to log-MAP decoding, exactly as for binary turbo codes. Constant log-MAP decoding for duo-binary turbo codes: Assume a trellis diagram consisting of four distinct paths per node, as two input bits are entering the constituent encoders at each time. A symbol transition occurs from a trellis state s0, at time instant k  1, to a trellis state s, at time instant k. The goal of the symbol-based log-MAP algorithm is to compute the log-likelihood ratio (i.e. soft-output) L{uˆ k(i)} of the transmitted symbol uk(i), i ¼ 0, 1, 2 and 3, given the observation of the received sequence r, as [8]: P expf~ak1 ðs0 Þ þ g~ k ðs0 ; sÞ þ b~ k ðsÞg Pðuk ¼ ijrÞ ðs0 ;sÞuk ¼i ¼ ln P Lf^uk ðiÞg ¼ ln Pðuk ¼ 0jrÞ expf~ak1 ðs0 Þ þ g~ k ðs0 ; sÞ þ bk ðsÞg n

ðs0 ;sÞuk ¼0

o ¼ max* a~ k1 ðs Þ þ g~ k ðs ; sÞ þ b~ k ðsÞ ðs0 ;sÞuk ¼i n o  max* a~ ðs0 Þ þ g~ ðs0 ; sÞ þ b~ ðsÞ ; ðs0 ;sÞuk ¼0

0

k1

k

Table 1: Complexity estimation of one simplified max operation of type-I constant log-MAP algorithm applied to duo-binary turbo codes Type-I constant log-MAP max(x, y, z, w) þ c0

Max operations Additions Comparisons 3

jaj ¼ x  max(x, y, z, w)

1 1

jcj ¼ z  max(x, y, z, w)

Extra operations From 1 to 3 (average ¼ 2)

1

jbj ¼ y  max(x, y, z, w)

1

max(jaj, jbj, jcj) < 2

2

Total

5

1 4

From 1 to 3 (average ¼ 2)

1

Efficient decoding algorithm: We have found that it is better to apply the simplified max* operator over pairs of arguments, rather than four arguments. The resulting algorithm, denoted as type-II constant log-MAP, operates as max*ðx; y; z; wÞ ¼ max*fmax*ðx; yÞ; max*ðz; wÞg ¼ maxfmaxðx; yÞ þ c1 ; maxðz; wÞ þ c2 g þ c3 ð4Þ where the three correcting factors c1, c2 and c3 depend on two input values x1, x2 and are computed from [9] as   3=8; if jx1  x2 j < 2 c¼ ð5Þ 0; otherwise In this way, the proposed algorithm makes use of the binary constant log-MAP algorithm three times. The overall complexity estimation of one type-II constant log-MAP operation is shown in Table 2.

Table 2: Complexity estimation of one simplified max* operation of type-II constant log-MAP algorithm applied to duo-binary turbo codes Type-II constant log-MAP max(x1, x2) þ c

Max operations Additions Comparisons 1

jx1  x2j < 2 Total (applied three times)

0

k

where jaj,  jbj and jcj are three values among x  max(x, y, z, w), y  max(x, y, z, w), z  max(x, y, z, w) or w  max(x, y, z, w). The overall complexity estimation of one type-I constant log-MAP operation is summarised in Table 1. We note that, after the max(x, y, z, w) computation, we need to identify which of the four values among x, y, z or w it corresponds to. For this reason, one to three extra operations may occur in a serial mode. This correspondence is then used to compute the exact values of jaj, jbj and jcj. In the worst case of max(x, y, z, w) ¼ w, three extra operations for this identification are needed with x, y and z, respectively.

3

1 1

1

6

3

i ¼ 1; 2; and 3 ð1Þ

where a˜, b˜ and g˜ represent the forward recursion, backward recursion and branch transition probability in the logarithmic domain, respectively. The goal of the constant log-MAP decoding algorithm is to simplify the max* operator by using a correcting factor with two possible values. Existing decoding algorithm: In [8], the simplified max* operator is applied over four arguments, i.e. equal to the number of branch transitions per trellis node. This algorithm, denoted as type-I constant log-MAP, operates as max*ðx; y; z; wÞ ¼ maxðx; y; z; wÞ þ c0 The correcting factor c0 is computed from   5=8; if maxðjaj; jbj; jcjÞ < 2 c0 ¼ 0; otherwise

ð2Þ

ð3Þ

Discussion: The computational complexity of the simplified max* operator depends on the real implementation=architecture, e.g. see [9]. One can propose alternative solutions to the above constant logMAP algorithms but the overall complexity comparison remains approximately the same. For instance, the correcting factor of (3) can be implemented by taking the maximum and minimum among the four values x, y, z and w and then taking the absolute difference between them and comparing with the threshold value. In this case, nine total operations are needed for the type-I constant log-MAP. Furthermore, the subtraction x1  x2 in (5) can be eliminated since it is already involved in the corresponding max(x1, x2) operation. This reduces the total operations of type-II constant log-MAP to nine. Therefore, the computational complexity of the two algorithms is still the same. Performance evaluation results: Computer simulations have been carried out assuming the DVB-RCS turbo code, three coding rates (R ¼ 1=3, 2=3, 4=5), QPSK modulation and the AWGN channel. Eight decoding iterations are considered with the max-log-MAP, type-I=

ELECTRONICS LETTERS 8th June 2006 Vol. 42 No. 12

type-II constant log-MAP and log-MAP algorithms. Both frame error rate (FER) and BER results with MPEG frames (i.e. 188 bytes) are shown in Fig. 1, respectively. 100 max-log-MAP type-I constant log-MAP type-II constant log-MAP log-MAP

10−1

# The Institution of Engineering and Technology 2006 3 April 2006 Electronics Letters online no: 20061036 doi: 10.1049/el:20061036

10−2 FER/BER

improvement of up to 0.2 dB was observed at high to moderate FER=BER values compared to a previously known algorithm [8], with approximately the same computational complexity. Finally, it was verified that the proposed decoding algorithm can approach very close to log-MAP decoding but with decoding complexity savings.

S. Papaharalabos, P. Sweeney and B.G. Evans (Centre for Communication Systems Research (CCSR), University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom)

10−3 10−4

E-mail: [email protected]

10−5 R = 1/3

10−6

R = 2/3

R = 4/5

References 10−7

0

0.5

1.0

1.5

2.0 2.5 Eb/No, dB

3.0

3.5

4.0

4.5

1 2

Fig. 1 FER (dashed lines) and BER (solid lines) performance of DVB-RCS turbo code with different coding rates (R) and decoding algorithms (MPEG frame size (i.e. 188 bytes) and eight decoding iterations)

3 4

From Fig. 1 it is noticed that the type-II constant log-MAP provides a maximum performance improvement of 0.2 dB at high to moderate FER=BER values compared to the type-I constant log-MAP. However, performance improvement becomes smaller as the coding rate is increased. This is explained by the puncturing technique that makes the code less powerful. On the other hand, it is only the type-II constant log-MAP that performs very close to log-MAP decoding. The performance degradation is approximately 0.02 dB, similar to the binary case [9]. As a comparison, the performance results with max-log-MAP iterative decoding from the literature are in agreement with those presented in Fig. 1.

5

6

7 8 9

Conclusions: An implementation guide to constant log-MAP decoding for duo-binary turbo codes has been given. A performance

Berrou, C., and Jezequel, M.: ‘Non-binary convolutional codes for turbo coding’, Electron. Lett., 1999, 35, (1), pp. 39–40 Douillard, C., and Berrou, C.: ‘Turbo codes with rate-m=(m þ 1) constituent convolutional codes’, IEEE Trans. Commun., 2005, 53, (10), pp. 1630–1638 Digital Video Broadcasting (DVB): Interaction channel for satellite distribution systems, ETSI EN 301 790, V 1.3.1, 2003. Yu, J., et al.: ‘Interleaver parameter-selecting strategy for DVB-RCS turbo codes’, Electron. Lett., 2002, 38, (15), pp. 805–807 Ould-Cheikh-Mouhamedou, Y., Crozier, S., and Kabal, P.: ‘Distance measurement method for double binary turbo codes and a new interleaver design for DVB-RCS’. Proc. IEEE GLOBECOM 2004, Dallas, TX, USA, November=December 2004, pp. 172–178 Ould-Cheikh-Mouhamedou, Y., et al.: ‘Enhanced max-log-APP and enhanced log-APP decoding for DVB-RCS’. Proc. 3rd Int. Symp. on Turbo Codes and Related Topics, Brest, France, September 2003, pp. 259–262 Papaharalabos, S., et al.: ‘Max=max* operation replacement to improve the DVB-RCS turbo decoder’. Proc. AIAA Int. Communications Satellite Systems Conf. (ICSSC), Rome, Italy, September 2005 Soleymani, M.R., et al.: ‘Turbo coding for satellite and wireless communications’ (Kluwer Academic, Dordrecht, The Netherlands, 2002) Gross, W.J., and Gulak, P.G.: ‘Simplified MAP algorithm suitable for implementation of turbo decoders’, Electron. Lett., 1998, 34, (16), pp. 1577–1578

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