Advanced Decoding Algorithms for Satellite Broadcasting

Transcription

1 Advanced Decoding Algorithms for Satellite Broadcasting Meritxell Lamarca (1), Josep Sala (1), Eduardo Rodríguez (2), Alfonso Martínez (3) (1) Dept. of Signal Theory and Communications, Universitat Politècnica de Catalunya UPC Campus Nord- Mòdul D5, c/jordi Girona, 1-3, Barcelona (Spain) Phone : (+34) , Fax : (+34) {xell,alvarez}@gps.tsc.upc.es (2) Alcatel Espacio Parque Tecnológico de Madrid (PTM), c/einstein 7, Tres Cantos, Madrid (Spain) eduardo.rodriguez_perez@alcatel.es (3) European Research and Technology Centre European Space Agency P.O.Box 299, 2200 AZ Noordwi ZH, The Netherlands alfonso.martinez@ieee.org Abstract: This paper presents an iterative decoding algorithm for concatenated codes consisting of a Reed-Solomon and a convolutional code, focusing on the code of the DVB-S standard. Existing solutions for the different decoding stages and their interfaces are discussed and their performance is compared. Besides, a new procedure is proposed to define the feedbac signal from the output of the Reed-Solomon decoder to the input of the convolutional decoder that provides a significant improvement over existing solutions. 1. INTRODUCTION The channel coding stage in the DVB-S standard is based on a concatenated code composed of the r=1/2 =7 convolutional code(cc), a convolutional interleaver of depth 12 and a (188,204) Reed-Solomon (RS) code This scheme has been used in a wide range of applications thans to its good complexity vs performance trade-off. The typical decoder for this concatenated code consists of a Viterbi algorithm that delivers hard decisions at its output followed by an errors-only algebraic RS decoder. Although, this decoding scheme is simple, its performance is very far from the potential of such a concatenated code. The losses introduced by this scheme stem from the use of a CC decoder delivering hard-decisions and from the treatment of the inner and outer coding stages as separate stages rather than as a concatenated code. Therefore, it is expected that the performance of the concatenated code could be greatly improved if the principles of soft processing were applied in the decoding process. While some wors were done on the design of advanced decoding schemes for this common type of concatenated codes (e.g. [10,Chap.11], [1] ), existing algorithms are far from achieving optimum or nearly optimum performance.. In this paper we propose an iterative decoder for the DVB-S concatenated code that results in large reductions of the BER. It is based on the application of a soft-input Reed-Solomon decoder and the definition of a RS soft-output that is used as the feedbac signal in the iterative procedure. Although all the results are described for the specific code in the DVB-S standard, the main results would apply to any other system with the same concatenated code structure.. This wor was partially financed by the European Space Agency (ARTES-1 program) and by the following research proects of the Spanish/Catalan Science and Technology Commissions (CICYT/ CIRIT): TIC C02-01 and 2001SGR-00268

2 The maor constraint in the application of advanced decoding schemes to the DVB-S standard and other similar codes lies in its complexity. The large constraint length of the CC leads to a large number of trellis states. Besides, the large alphabet size of the RS code precludes the use of maximum lielihood (ML) or near-to-ml decoders and rules out the application of most existing algorithms for soft decoding of Reed-Solomon codes. In this paper we review the application of the existing algorithms for convolutional and Reed-Solomon decoding to the DVB-S code. In the following sections the decoding stages are analysed separately and the performance of the combined stages is evaluated by Monte Carlo simulations. In particular, the following algorithms are considered for the convolutional code: SOVA, log-map at the byte level, log-map at the bit level and Max-log-MAP at the bit level. For the Reed-Solomon code three approaches have been considered: the GMD algorithm described in [5], an extended distance algorithm based on [4] and the Kötter-Vardy (KV) algorithm in [6]. The proposed decoder is compared with the existing iterative decoder based on state pinning and with a non-iterative solution based on the KV algorithm. 2. DECODER ARCHITECTURE Figures 1,2 depict the bloc diagram of the encoder and proposed decoder for the concatenated code in the DVB-S standard. Note that the stage labeled as Bit LLR to Byte LLR is only required when the convolutional decoder delivers soft decisions at the bit level and its purpose is to compute the byte-level log-lielihood ratios (LLR s) required by the RS decoder based on these values. Information Sequence R-S Encoder π Conv. Encoder x 1 x 2 Coded Output Fig.1. Bloc diagram of the encoder Channel Input A Priori y 1 y 2 Conv. Decod π 1 π Bit LLR to Byte LLR Feedbac R-S Decod Decod Output 3. CONVOLUTIONAL DECODER Fig.2: Bloc diagram of the iterative decoder Several soft-input soft-output convolutional decoders were considered in this analysis. The best performance is obtained using a log-map decoder delivering at its output log-lielihood ratios (LLR s) at the byte level, i.e. a log-map decoder operating over an equivalent trellis where each trellis step corresponds to an information byte [3]. However, as the complexity of that decoder is very high, other decoder algorithms were considered that deliver one LLR per information bit. The performance of the log-map, Max-log-MAP and SOVA algorithms [11] for the binary trellis was compared with the byte-level log-map in terms of the error probability at the RS decoder output. The log-map decoders were based on the BCJR algorithm and a training window of sufficient length (240 bits) was used to compute the initial values of the bacward probabilities (β s) in the BCJR algorithm, since according to the DVB-S standard the trellis is never reset to the zero state. Simulations showed that the iterative decoder was more sensitive to the window length than the conventional one due to the use of a soft-input RS decoder that is sensitive not only to CC decoder decisions but also to the statistics of the CC output, so the window length had to be larger than the traditional value of five times the constraint length of the code. In all decoding algorithms, the a priori probability for the information bits was set to zero for the first iteration and was taen as the information provided by the RS decoder in subsequent iterations. This a priori information was subtracted from the output LLR s to deliver to the RS decoder the extrinsic soft-values, as is usually done in iterative decoding for serial concatenated turbo codes [2].

3 4. CONVOLUTIONAL TO RS DECODERS INTERFACE When the CC decoder provides one LLR per bit, a mapping must be used to compute the LLR s per byte required by the RS decoder. Two different approaches were considered to map the bit LLR s probabilities to byte LLR s, as suggested in [10,Chap.11]. Let B and {b i } i=1,,8 denote a byte and its corresponding bits. Then the two approaches are formulated as follows: Product ( prod ): if the bit soft values at the convolutional decoder output were independent, the byte probability could be computed as ( B = [ a1,, a8 ]) = b i = a i ) 8 p K (1) Minimum ( min ): The dependence between the bit soft values due to the convolutional code memory can be taen into account assuming that one byte is reliable only if the eight bits are also reliable, that is: p ( B = [ a1, K, a8 ] ) = min b = a ) (2) i= 1 i= 1, K, 8 Neither (1) nor (2) allow an exact computation of the byte soft values at the convolutional decoder output; this can only be accomplished through the use of the byte-level MAP decoder. However, they can be used to obtain approximate values that, according to the simulation results, yield small losses compared with the optimum scheme including a byte log-map decoder. The product approach has the advantage that it reduces significantly the computational burden of the mapping from bit to byte LLR s at the input of the RS decoder, since explicit computation of the byte LLR s can be avoided. 5. REED-SOLOMON DECODERS The development of Reed-Solomon decoders that accept soft-input values and deliver soft-output values while having ML or nearly-ml performance is still an open problem. This is specially evident for long RS codes such as the one in DVB-S, defined in GF(2 8 ). The design proposed here is divided in two stages (see figure 3), as described next Soft-input Reed-Solomon decoders Among the RS decoders accepting soft-input data, only the GMD in [5][9] and the algorithm in [4] have moderate complexity for the RS (204, 188) code. Unfortunately, the performance of these decoders is far from that of an ML decoder. The design of near-ml decoders had a maor brea-through with [6], but its application to real DVB-S decoders is not feasible yet due to the lac of efficient algorithms that reduce its computational complexity to acceptable values. In this paper the KV algorithm in [6] has been implemented in order to use it as a benchmar. Besides, two RS decoders have been implemented in the iterative decoder: the improved GMD decoder proposed in [9] (hereafter GMD) and an extended distance decoder based on [4] (hereafter BD+1). Both algorithms correspond to algebraic bounded distance decoders that carry out several decoding attempts with different numbers of erasures and compare the Euclidean distance between the candidate codeword and the received data with a threshold that defines the acceptance criterion (AC). The difference between the two decoders relies on their symbol error correction capability: 2 e+ s r=16 for the GMD and 2 e+ s r+1=17 for BD+1, with e, s and r the number of errors, erasures and redundancy symbols respectively. In both algorithms, the erasures location was decided according to the reliabilities of the input bytes, following the same procedure proposed in [9]. Both algorithms start with a decoding attempt with s=r=16 erasures and if the candidate codeword that is obtained does not satisfy the AC they progressively reduce the number of erasures down to 0 (considering only even values of s for GMD and all of them for BD+1). For each value of s they chec whether a new candidate is found and whether this candidate satisfies the AC. If it does, the decoding process stops and the candidate is delivered at the decoder output. If none of the candidates verifies the AC, the candidate with smallest distance and its corresponding value of s are stored. In order to define the feedbac signal for the iterative decoding process, the output of the GMD and the BD+1 decoders is described with three parameters: i i

4 ĉ: Candidate codeword S {0,1, 16,*}: Minimum number of erasures that could be used to deliver the candidate codeword for the received data. The value * is assigned to S when the hard-decision of the received data is a valid RS codeword, so there is no need to decode it. T={true, false}: Compliance with the acceptance criterion: T=true if the candidate codeword satisfies the extended distance criterion and T=false otherwise. LLR GM D/BD+1 RS Decoder ĉ S T Feedbac evaluation APri (ĉ ) Decoder output Feedbac 5.2. Reed-Solomon soft-output evaluation Fig.3. Reed-Solomon decoder The design of an iterative decoder requires the computation of the reliability of the RS coded bytes that will be used as a priori information for the CC decoder in the iterative procedure. Ideally, we would lie to compute the LLR s for each coded byte, but no algorithm has been found to compute them. The solutions published in the literature are scarce, particularly for long RS codes. In [8] the soft values of the coded bits were replaced by a binary signal that led to state pinning of the convolutional code. In [1], the same problem was solved giving to the a priori information in the convolutional decoder a weight that was adusted empirically. In this paper we propose a new procedure for the computation of the feedbac signal that, even though it is not the true LLR of the coded bytes, it captures the reliability information that can be obtained from the GMD and the BD+1 decoder and obtains significant BER reductions compared with state pinning procedure, especially for moderate SNR values. In this section we evidence the difficulties computing the LLR s and show how the values of the parameters S, T defined in the previous section can be used to infer some information about the reliability of the coded bytes. The proposed procedure can be used in any scheme based on errors and erasures decoders where the reliability of RS codewords needs to be computed. Let c=(c 1,,c 204 ), LLR and LLR rx denote the RS word, the corresponding log-lielihood ratios and the set of actual soft values at the RS decoder input. If a soft-output MAP symbol decoder were applied to the RS code it would deliver the following soft values: c = m LLR= LLR rx ) m=0,...,255 =1,...,204 (3) or equivalently c = m LLR= LLR rx ) (4) for all possible RS codewords m. However, this is not feasible in the GMD and the BD+1 decoders, since the received data is only compared with a small subset of the possible codewords. In spite of that, if the value (4) were nown for the candidate codeword, it could be used to infer some information about the lielihood of the symbols in the codeword: c = i LLR = LLR rx ) = c c c : c = i c = c LLR = LLR rx ) = i LLR = LLR rx ) c = LLR = LLRrx ) = i LLR = LLR rx ) 1 c = LLR = LLR rx ) i = 0,...,255 if if = 1,...,188 Thus, this value can be used to get a bound on the symbol probability that is tight if the codeword is very liely and, therefore, it can be used to derive a very conservative bound on the symbol LLR: LLR ( ) c = LLR = LLRrx ) max 0, log 1 c = LLR = LLRrx) 0 if if = i i = m m (5) (6)

5 If this LLR estimate was fed bac to the CC decoder, it would add positively to the symbols appearing in very liely codewords, while it would not penalize any other symbol. Unfortunately, the value of LLR=LLR rx c=ĉ) cannot be obtained from the GMD or the BD+1 decoders, so the equation in (6) must be modified. The method proposed in this paper relies on the observation that the number of erasures required for decoding is related to the probability of getting a wrong candidate and the probability of decoding failure and, therefore, it is also related to the reliability of the decoded word. In the case of a classical RS decoder the probability of finding a wrong codeword is nown [7]; in the case of the GMD or the BD+1 decoder the computation of this probability becomes very intricate, but it can be measured by Monte Carlo simulations. As an example, figure 4 depicts the statistics of the RS output for a simulation at EbNo=1.9dB. It shows the fraction of decoded words that were right/wrong according to the values of output parameters S, T (the case of wrong words with T=true is not shown because it is so rare that it never appeared in the simulations). 1,E+00 1,E-01 Fraction of words 1,E-02 1,E-03 1,E-04 1,E Number of erasures (S) Right words, T=true Right words, T=false Wrong words, T=false Fig.4. RS decoder output histogram at 1.9dB for the first iteration of a decoder consisting of: log-map at the bit level, min mapping from bit LLR s to Byte LLR s and GMD. Fraction of the transmitted words that are correct and wrong at the RS decoder output as a function of the number of erasures and the compliance with the acceptance criterion. The proposed reliability measure is the probability: c = ĉ S=i, T= ) which can be derived from the oint probabilities in fig. 4: p ( c = c S = i, i=0,,16,* =true, false c =, S = i, ( c =, S = i, + c, S = i, ˆ = (7) p Note that this probability can be written in terms of the probability of the received codeword as the averaging of that value over all sequences of byte LLR values that would produce output parameters S=i, T= if they were applied to the RS decoder input: c = LLR) LLR) LLR( i, ) c = S = i, = (8) p S = i, T = ( ) Equation (8) evidences that the reliability information provided by the GMD and the BD+1 decoders is not directly lined to the received data (LLR rx ) but to the averaging over all the mentioned sequences. The proposed metric for reliability feedbac is finally defined replacing the unnown term in (6) by its averaged estimate (8) c = S = i, max = ( ) 0, log if m APri 1 c = S = i, (9) 0 if m

6 When this metric is used, the feedbac signal will have an alphabet size equal to at most 17 values, compared with the state pinning approach where only two values were considered. Figure 5 depicts an example of the a priori weights for the first and second decoder iterations that are obtained from the application of equation (9). These are the weights for the words that do not satisfy the GMD criterion, those RS decoded words that satisfy it have the maximum weight, since c ĉ,s=i,t=false) is very small. The probability distributions required for the application of (9) were measured by means of Monte Carlo simulation (see figure 4 for the first iteration values). As the values of c=ĉ,s=i,t=false) and c ĉ,s=i,t=false) for low values of S were very difficult to measure, their values were approximated by second order polynomial interpolation in the log domain. Figure 5 shows that the weights for the second iteration are smaller than for the first one, although this does not traduce into a reduction of the feedbac from the RS to the CC decoder in the second iteration because the number of words that satisfy the GMD criterion and, therefore, have maximum weight is increased. 20,0 18,0 16,0 14,0 Weight 12,0 10,0 8,0 Iter 1 Iter 2 6,0 4,0 2,0 0, Nr. erasures Fig.5. A priori weights for the first and second iteration of a decoder woring at 1.9dB and consisting of decoder consisting of: log-map at the bit level, min mapping from bit LLR s to Byte LLR s and GMD. The weights for the first iteration were computed following equation (9) for the statistics in figure SIMULATION RESULTS Figure 6 depicts the word error rate (WER) of the concatenated code with CC of rate r=1/2 for the first and third decoder iteration for WER down to Although it was not possible to simulate lower values due to limitations in the computational load, it is expected that the behavior shown here would also hold for higher Eb/No values. The results in the first iteration show that the gain obtained by using a soft-input RS decoder is the same for all CC decoders and that the BD+1 provides a certain improvement over GMD. This gain of BD+1 increases with the iterations, as can be seen comparing the performance for the third iteration. Besides, focusing on the CC decoders, the third iteration shows that SOVA exhibits a small performance loss compared with the MAP algorithm. In order to compare the proposed feedbac with previous existing solutions, figure 6 also depicts the WER of the iterative decoder when a state pinning approach (named SP in the figure) is applied that feeds bac either all words satisfying the distance criterion in (9), with maximum weight, or no signal otherwise. The proposed approach clearly improves the performance of that one of state pinning. It can also be seen in the figure that the byte LLR approximations in (1) and (2) have equal performance when the proposed feedbac signal is applied. Although it is not shown in the figure, the Monte Carlo simulations have shown that in the case of SP the performance of the min approach is much better than that of prod because the number of words that satisfy the GMD distance criterion is larger when the min approach is used and, therefore, the amount of words that are fed bac increases.

7 Figure 6 also depicts the performance of the non-iterative decoder consisting of a byte-level MAP and the KV algorithm in [6] with a list of at most 12 candidates. In spite of its lower complexity, the second iteration of the iterative decoders (either the state pinning solution or the feedbac based on equation (9)) outperform the MAPByte-KV non iterative decoder. Other simulations not included in the figure have shown that the KV performance severely degrades if the bitlevel MAP decoder is used and the byte reliabilities are obtained by means of the mappings described in section 4. Finally, figure 7 illustrates the WER reduction in the iterative decoder as a function of the number of iterations, showing that large gains can be achieved if the number of iterations is increased. It seems that in the 6th iteration the slope of the plot decreases, but it is not clear whether this is a typical behaviour of this decoder or it is due to the use of wrong weights (the weights depend on the Eb/No and the iteration and the simulation was run using the weights evaluated for the first iteration at Eb/No=1.9dB) 7. CONCLUSIONS In this paper the application of soft-input soft-output decoding stages to the concatenated code in the DVB-S standard has been analyzed and a new procedure for generating the RS soft-output has been proposed. The proposed algorithm provides large reductions in the bit error rate compared with the classic decoder for this ind of codes (e.g. about 0.4dB at WER=10-1 and 0.6dB at WER=10-3 for the third iteration), while its complexity is moderate (bit-level Max-Log- MAP, equation (1), GMD and the proposed feedbac have all relatively low complexity). REFERENCES [1] O.Aitsab, R.Pyndiah, Performance of a concatenated Reed-Solomon/Convolutional codes with iterative decoding, Proc.GLOBECOM 97, pp [2] S.Benedetto et al, A soft-input soft-output Maximum A Posteriori (MAP) module to decode parallel and serial concatenated codes, JPL TDA Progress Report, Vol , Nov.1996 [3] J.Bermann, On turbo decoding of nonbinary codes, IEEE Comm. Letters, Vol.2, pp.94-96, Apr.1998 [4] E.Berleamp, Bounded distance+1 soft-decision Reed-Solomon decoding, IEEE Trans. Inform. Theory, Vol.42, pp , May 1996 [5] G.D.Forney, Generalized minimum distance decoding, IEEE Trans.Inf.Theory,Vol.IT-12,pp , Apr.1966 [6] R.Kötter,A.Vardy, Algebraic soft-decision decoding of Reed-Solomon codes, Proc. ISIT 2000, pp.61 [7] R.J.McEliece, L.Swanson, On the decoder error probability for Reed-Solomon codes, IEEE Trans. on Inf.Theory, Vol.IT-32, pp [8] E.Paase, Improved decoding for a concatenated coding system recommended by CCSDS, IEEE Trans. on Communications, Vol.38, pp ,august 1990 [9] D.J.Taipale, M.B.Pursley, An improvement to generalized minimum distance decoding, IEEE Trans. Inform.Theory, Vol.IT-37, pp , January 1991 [10] S.Wicer, V.K.Bhargava Ed., Reed-Solomon Codes and Their Applications, IEEE Press, 1994 [11] J.P.Woodward, L.Hanzo, Comparative study of turbo decoding techniques: An overview, IEEE Trans. on Vehicular Technology, Vol.49, pp , Nov.2000

8 1,0E+00 1,0E-01 1,0E-02 WER 1,0E-03 1,0E-04 1,0E-05 1,30 1,40 1,50 1,60 1,70 1,80 1,90 2,00 2,10 2,20 EbNo (db) SOVA-GMD min 1st iter SOVA-GMD min 3rd iter MAPbit-GMD min 1st iter MAPbit-GMD min 3rd iter MAPbit-GMD prod 1st iter MAPbit-GMD prod 3rd iter MAPbit-BD+1 min 1st iter MAPbit-BD+1 min 3rd iter MAPbit-GMD min 2nd iter SP MAPBit-GMD min iter 2 MAPByte-Kötter_Vardy SP MAPBit-GMD min iter 3 MAPByte-GMD 1st iter VitEuclidea+Errors_only RS Fig.6. Word Error Rate of the concatenated code 1,0E+00 1,0E-01 WER 1,0E-02 1,0E Iteration Fig.7. Evolution of WER with iterations for Max-Log-MAP and GMD decoder at Eb/No=1.75dB.