A study on turbo decoding iterative algorithms

Transcription

1 Buletinul Ştiinţific al Univerităţii "Politehnica" din Timişoara Seria ELECTRONICĂ şi TELECOMUNICAŢII TRANSACTIONS on ELECTRONICS and COMMUNICATIONS Tom 49(63, Facicola 2, 2004 A tudy on turbo decoding iterative algorithm Horia Baltă, Maria Kovaci 2 Abtract - In the paper, a tudy of ome turbo decoding iterative algorithm: MAP, MaxLogMAP, LogMAP, i preented. For the correction of the approximation ued in the MAxLogMAP algorithm two method are propoed obtaining two LogMAP algorithm variant. All algorithm variant have been imulated to make poible a comparion from the bit error rate (BER point of view, in order to provide an optimization for each algorithm. The imulation were made for AWGN channel. Two component code with generator matrix: G =[, 5/7] and two interleaver type: peudo-random [] and S- interleaver (S=29 are ued. The interleaving length i N=784. The number of emitted block in one imulation depend of ignal to noie ration,, to obtain a good preciion of reulted curve. Keyword: Turbo code, MAP algorithm, trelli. I. INTRODUCTION The Maximum A-Poteriori (MAP algorithm, propoed by Bahl, Cocke, Jelinek and Raviv (974, i frequently ued after the turbo code dicovery realized by Berrou and al. []. Eentially, MAP algorithm, [2], calculate the Log Likelihood Ratio, LLR, under the form: ( ( ( α ( ( u ŝ, = k- ŝ γ k ŝ, βk + L u = k k y ln ( ŝ ( ŝ, ( ( α ( u ŝ, = k- γ k βk k where:α k- (ŝ = P(S k- = ŝ y j<k i the probability that the encoder trelli wa in ŝ tate at intant k- and the received channel equence, before thi moment, i y j<k, ( = (, ˆ ( ˆ αk γk αk (2 all ˆ β k ( = P(y j>k S k = i the probability that, having been given the trelli tate at intant k, the received channel equence, after thi moment, to be y j>k, ( ˆ (, ˆ ( β γ β k = k k (3 all ˆ γ k (ŝ, = P({y k S k = } S k- = ŝ i the probability that the encoder trelli took the tranition from tate ŝ to tate and the received channel equence for thi tranition i y k. γ k ( ŝ, = C e( u k L( u k / 2 n exp Eb 2 a yki xki 2σ 2 i= (4 In relation (4, u k i the value of information bit for the trelli branch, L(u k i the extrinic information for the k-th bit, E b i the energy of the information bit, σ 2 i the noie power, y ki and x ki repreent correponding value of all the bit attached to the branch which make the liaion of tate ŝ and, from reception (y ki, repectively, emiion (x ki. Fig. preent the computation way of forward-α and backward-β coefficient, for a part of trelli of convolutional code with contraint length K=3. S k- S k-2 S k- S k S k+ y j<k y k y j>k α k- (ŝ γ k- (ŝ, β k ( Fig. The code trelli with G = [, 5/7]. The continuou line correpond of the input bit value. Due to the exponential and logarithm operator in relation ( and (4, the MAP algorithm i difficult to be implemented. Like an alternative, the MaxLogMAP algorithm i eaier to be implemented, due to the approximation:,2 Facultatea de Electronică şi Telecomunicaţii, Departamentul Comunicaţii Bd. V. Pârvan Nr. 2, Timişoara, balta@etc.utt.ro; kmaria@etc.utt.ro

2 e x i max( (5 ln i So, the computing relation, in the MaxLogMAP algorithm cae, are the following: Ak Γ k ( ˆ ln( α ( max( ( ŝ + Γ ( ŝ, Bk i = k Ak k (6 ŝ x i ( ˆ ln( β ( ( ( + Γ ( ŝ, = k Bk k (7 ( ŝ, = ˆ ln( γ ( k L(u k y max ( u ŝ, = = Ĉ + u k L u k 2 Lc n y ki 2 i= ( + xki (A k- (ŝ + Γ k (ŝ, + B k ( k = + max (A k- (ŝ + Γ k (ŝ, + B k ( ( u ŝ, k = (8 (9 The implementation implification price of the MaxLogMAP algorithm i the reduction of the performance (of the BER with 0,2 db veru the MAP algorithm. LogMAP Algorithm, propoed by Roberton and al. [3], correct the approximation ued by MaxLogMAP algorithm and i a little bit more complicated than it. ln( e x + e x 2 x =max(x,x 2 +ln(+ e x 2 = max(x,x 2 + ƒ c ( x -x 2 II. THE TRELLIS CLOSING (0 In function of the trelli cloing the alpha and beta coefficient are initialized. The initial trelli cloing mean the coder initialization with a predefined tate. Thi tate i alo known by the decoder. So, the initialization of the alpha and beta coefficient can be done. With the exception of circular coding, thi initial tate i zero. The final trelli cloing i more difficult to be realized. It i done (excepting the circular code with the price of inertion of the M (the code memory redundant bit in the information equence. Thi fact realize the reduction of the tranmiion rate from /2 to the following value: R cc = (N M / 2 N. ( The trelli cloing give the advantage of the initial tate knowledge (and/or of the final tate, fact which lead to the firm knowledge of the alpha coefficient (at the beginning of trelli and beta coefficient (at the end of the trelli. In the cae of the uncloed trelli thee coefficient can only be predicted probabilitically. A turbo code implie at leat two coder, C and C2. At each coder correpond a trelli. Different trelli cloing technique can be ued for thee coder. In thi paper we invetigate few trelli cloing method for a turbo code (parallel. The table preent thee method. Table Variant Start Final Coding rate C, C2 C, C2 0 0, 0 0,? (N-M/3N 0, 0?,? /3 C Sx, Sy Sx, Sy /3 0. In thi cae, the firt coder cloe the trelli on both extremitie, it inert M redundant bit after the N-M information bit. The econd coder can not do the ame final cloing due to the interleaving of the input equence. So, the econd trelli i not cloed. The firt decoder initialize the alpha coefficient, which correpond at the front end of the trelli to the zero tate, with the probability, and the other alpha coefficient with the probability zero. The firt decoder treat the beta coefficient in the ame way at the end of the trelli. The econd decoder act in the ame way, like the firt, for the alpha coefficient. The beta coefficient of the econd decoder, can be initialized by the one of the following method: 0. The beta coefficient are met equal with the value of the alpha coefficient obtained at the lat iteration. Thi i called the oft initialization; 0h. Thi method initialize the beta coefficient, that correpond to the tate with the highet alpha coefficient, at probability and the other beta coefficient at the probability zero. Thi i called the hard initialization; 0e. Thi method make the beta coefficient to have the ame probability. Thi i called the equal probability initialization;. None of the trellie i final cloed. The advantage, in thi cae, i a higher coding rate. But thi coding rate increae can not be oberved if N>>M. Both decoder mut initialize the beta coefficient in one of the three way enounced above. In thi paper we implement only the oft initialization. The cae of both trelli final cloing i poible only with ome modification of the interleaving between the two coder. C. There i the poibility, uing a pre coding technique, to find, for any data equence x, an initial tate S o of the coder identically with it final tate. So the coding become circular. The decoder doe not know the tate S 0, but know that it can ue the final tate like initial tate. So, it mut to do at leat a forward recurrence. We are implemented and imulated the following variant of circular turbo code: C the decoder realize a forward recurrence and compute a final tate, S 0. The alpha and beta coefficient are initialized with S 0, the backward recurrence i made and the forward recurrence i

3 remade. It memorize the new tate S 0 for the tarting of the next iteration. C2 the decoder realize the both recurrence in the oft variant and retain, for the next iteration, with the role of S 0, the beta coefficient value from the end of the backward recurrence. C3 the decoder realize the both recurrence in the oft variant plu one for the alpha coefficient, only. The initial tate for that econd recurrence i done by the final value of beta coefficient of the lat iteration. The final coefficient of the econd forward recurrence give the tate to be tored for the next iteration. C4 the decoder realize the forward recurrence and build a S 0 tate in a hard deciion (it earche the alpha coefficient maximum. It retain thi tate for the next iteration and alo make the backward recurrence and remake the forward recurrence. III. THE LOGMAP ALGORITHM. IMPLEMENTATION. The variant of LogMAP algorithm differ by the correction term approximation way, decribed in equation (0: f c (x = ln(+e -x, x 0 (2 The function that approximate f c (x mut be eay to implement and they mut reproduce the mot exactly poible the form of thi function. Two approximation were propoed in thi paper, indicated in Fig. 2 and Fig.3. f c (x g c (x Fig.2. The rectangular approximation way. f c (x h c (x x f c (x. The value of the function g c (x are in the et {0.6, 0.3, 0.4, 0.065, 0.03, 0.04, 0.005, 0.002, 0}. The linear variant correpond to an approximation of f c (x of the form: x, x x h ( x = o x (3 c o 0, x > x o and by numerical approximation wa obtained the value x o = 2,347 for which h c (x realize the better approximation of f c (x. IV. EXPERIMENTAL RESULTS In the figure 4 are preented the curve BER( obtained with the three MAP algorithm variant 0 plu the MAP algorithm. Depite the fact that for ignal to noie ratio inferior to db the performance are identical, up to thi value the reult how that the variant 0e i better. It i followed, in order, by the variant: 0, and 0h. Thee reult how that at low ignal to noie ratio the error are produced excluively by the bad election of the path in the trelli and up db the error due to the trelli non cloing have a higher weight. In figure 5 are repreented the BER( curve obtained with the four variant of the circular MAP algorithm already defined in comparion with the bet MAP algorithm: 0e. The firt three circular MAP variant have imilar performance, inferior to the performance of the variant 0e. Tacking into account all the reult already preented it reult that the hard variant i not a good olution. The imulation reult realized with /3 rate RSC turbo code (parallel with G=[,5/7], which utilize in the variant 00 the LogMAP algorithm are compared in Fig.6 with the reult obtained with the bet MAP variant: 0e. From figure reult that all the two LogMAP variant are better than the MAP at leat for value of the ignal to noie ratio inferior to db. Up thi value the curve are not very accurate but obviouly the performance are imilar. The curve reduced preciion i due to the reduced volume of imulation. - Depite the fact that practical implementation of the LogMAP algorithm are fater than thoe of the MAP algorithm the imulation program work lower in the cae of the LogMAP algorithm. Between the two LogMAP algorithm variant the reult how that the linear one i better. Thee concluion mut be verified alo for other component code. x Fig.3. The linear approximation way. The rectangular variant propoed by Roberton and all. [3] i a zero order extrapolation of the function

4 BER MAP0 MAP0h MAP0e o MAP Fig. 4 The BER curve obtained with: MAP0, MAP0h, MAP0e, MAP algorithm. BER MAPC MAPC2 MAPC3 o MAPC4 + MAP0e Fig. 5 BER performance of C, C2, C3, C4 algorithm veru MAP0e algorithm. BER MAPC MAPC2 MAPC3 Fig. 6 BER performance of rectangular and linear LogMAP algorithm veru MAP0e algorithm.

5 V. CONCLUSIONS In the paper, a tudy of ome turbo decoding iterative algorithm: MAP, MaxLogMAP, LogMAP, wa preented. For the correction of the approximation ued in the MAxLogMAP algorithm, two method were propoed, obtaining two LogMAP algorithm variant. All algorithm variant have been imulated to make poible a comparion from the bit error rate point of view, in order to provide an optimization for each algorithm. VI. REFERENCES [] C. Berrou, A. Glavieux, P. Thitimajhima Near Shannon limit error-correcting coding and decoding: Turbo-code, Proc.ICC 93, Geneva, Switzerland, May 993, pp [2] L.Hanzo, T.H.Liew, B.L.Yeap, Turbo Coding, Turbo Equaliation and Space-Time Coding for Tranmiion over Fading Channel, John Wiley & Son Ltd, England, 2002 [3] P. Roberton, E.Villebrun, P.Hoeher, A Comparion of Optimal and Sub-Optimal MAP Decoding Algorithm Operating in the Log Domain, Proceeding of the International Conference on Communication, Seattle, USA, pag , iunie 995