A Comparison of Soft In/Soft Out Algorithms for Turbo Detection

Transcription

1 A Comparion of Soft In/Soft Out Algorithm for Turbo Detection Gerhard Bauch 1, Volker Franz 2 1 Munich Univerity of Technology (TUM), Intitute for Communication Engineering (LNT) D Munich, Germany gerhard@lnt.e-technik.tu-muenchen.de, WWW: 2 Siemen AG, OEN MN ER 51 Hofmanntr. 51, D Munich, Germany Volker.Franz@oen.iemen.de Abtract In turbo detection the turbo principle i applied to joint equalization and decoding. The performance of a turbo cheme trongly depend on the quality of the oft value paed between the oft in/oft out decoder. In thi paper we decribe the difference between optimum and uboptimum oft in/oft out algorithm for equalization and decoding and compare them in a turbo detection cheme concerning complexity and performance for perfect and mimatched channel etimation. Furthermore, a poibility to improve the oft value of uboptimum algorithm which tend to be too optimitic i mentioned. I. INTRODUCTION AND PRINCIPLE OF TURBO DETECTION The turbo principle which wa firt applied to parallel concatenated convolutional code ( turbo code ) in [1] can be applied to many detection and decoding problem. The idea of turbo code i to build a trong code by concatenation of imple component code o that decoding can be performed in tep uing algorithm of manageable complexity. Decoding of each component code i done uing a oft in/oft out algorithm (Figure 1) which beide the channel value accept an additional a priori information a input and deliver oft a poteriori value of the information bit and, if required, oft a poteriori value of the coded bit (COD decoder). For binary data it i ueful to compute the oft value a log likelihood ratio (L value): input log - likelihood a priori value for information bit channel value for code bit L(u) = log L(^x) Le(^x) a poteriori value extrinic value for code bit 6 6for code bit L(u) Le(^u) - Soft-In/ - Soft-Out L(~x) or - L(^u) Decoder - Lcy or y P (u = +1) P (u =?1) : (1) output log - likelihood extrinic value for information bit a poteriori value for information bit Fig. 1: Soft-In/Soft-Out - Decoder From the a poteriori value an extrinic information for each bit can be obtained by ubtracting the intrinic information about the current bit from the correponding a poteriori L-value. The intrinic information conit of the input L value, the extrinic information i the incremental information about the current bit obtained through the decoding proce from all information available for the other bit in a block. In the next component decoder thi extrinic information i ued a a priori information. Having decoded all component code once, extrinic information i fed back to the firt component decoder and decoding i done again uing the extrinic information a a priori information. A the proceed output of the outer or parallel decoder i ued a a priori input for the next iteration - imilar to a turbo engine - thi feedback i called the turbo component. In turbo detection ([2], [3]) the mobile radio channel with interymbol interference (ISI) including tranmit and receive filter i regarded a a (time varying) code, erially concatenated to the channel encoder which in mobile communication ytem uually i a convolutional code. 1 : Interleaver _ 1: Deinterleaver Fig. 2: Turbo detection _ MAPequalizer COD-MAPdecoder The ISI channel i to be decoded by a oft in/oft out equalizer. We can ue the ame oft in/oft out algorithm for equalization and decoding of the convolutional code and apply the turbo principle to joint equalization and decoding (ee Figure 2). The performance of a turbo cheme not only depend on hard deciion but trongly depend on the quality of the oft value paed between the oft in/oft out decoder. The optimum algorithm concerning minimization of bit error rate and computation of a poteriori information i the BCJR 1 Symbol by Symbol MAP algorithm (BCJR 1 Bahl, Cocke, Jelinek, Raviv L(u)

2 MAP) ([4],[5],[3]). However, the complexity of the BCJR MAP i very high. Therefore, uboptimal olution have to be conidered, e.g. the BCJR Max Log MAP ([6], [7]), the Soft Output Viterbi Algorithm (SOVA) ([8]), and the Soft Output Viterbi Equalizer (SOVE) ([6],[9]). In thi paper we will decribe the difference between thee algorithm and compare them in a turbo detection cheme concerning performance in bit error rate for perfect and mimatched channel etimation and complexity. Furthermore, we will how that the uboptimum algorithm tend to deliver too optimitic L value and we will give a poibility to improve the quality of the L value and therefore the performance of turbo detection. II. SOFT IN/SOFT OUT ALGORITHMS FOR EQUALIZATION AND DECODING The goal of the conidered oft in/oft out algorithm i to etimate the a poteriori L value L(^u k ) = ln P (u k = +1jy) P (u k =?1jy) ; (2) where y i the equence oberved at the receive filter output. In the following three quantitie are important (ee Figure 3): tate with forward probabilitie α k-1 ( ) k tate with backward y = (y... y ) probabilitie β k () u =+1 (x...x ) u =-1 (x...x ) Fig. 3: Trelli butterfly k () meaure the forward probability p(; y jk ) that the trelli i in tate at time k. k () meaure the backward probability p(y j>k j) that the trelli i in tate at time k. The only difference in the metric of the equalizer and of the channel decoder appear in the calculation of the branch tranition probability k. For the equalizer we obtain ln k =? jy k? M i=0 g i u k?i j L a(u k )u k : (3) M denote the memory of the dicrete time channel model with complex valued time varying coefficient g i (t). The complex valued ymbol y k wa received at time k and the equalizer ue an a priori information L a (u k ). The variance of the additive white Gauian noie in inphae and quadrature component i 2. In a erially concatenated cheme the output log-likelihood of the inner decoder are input to the outer decoder. If L(~x k;i ) i the input for the i-th code bit tranmitted for the information bit u k we obtain for the decoder ln k = N i=1 for a code of rate 1 N. A. BCJR Log MAP Algorithm 1 2 L(~x k;i)x k;i L a(u k )u k (4) The BCJR MAP compute the a poteriori information for each bit taking into account information from all bit in a block. To avoid number repreentation problem the BCJR MAP ha to be implemented in the log domain a the equivalent BCJR Log MAP. The output of the BCJR Log MAP i ([7]) L(^u k ) = ln? ln e ln k?1( 0 )+ln k +ln k ()? e ln k?1( 0 )+ln k +ln k () ; (5) where 0 and denote the tate of the trelli at level k? 1 and k, repectively (Figure 3). The um are to be taken over all exiting tranition in the trelli from tate 0 to tate labelled with the information bit u k = +1 or u k = 1, repectively. and can be calculated by recurive formulae: ln k () = ln exp(ln k ( 0 ; ) + ln k?1 ( 0 )); (6) 0 ln k ( 0 ) = ln exp(ln k+1 ( 0 ; ) + ln k+1 ()): (7) The equation (5), (6), (7) can be evaluated exactly uing the Jacobian logarithm: ln[exp( 1 ) + exp( 2 )] = max( 1 ; 2 ) + f c (j 1? 2 j); (8) where f c (j 1? 2 j) = ln(1 + exp(?j 1? 2 j)) (9) i a correction term which can be implemented uing a lookup table. The evaluation of (5) can be implified for trelli butterfly tructure where the two path with ame u k merge in one tate. Thi i true for feedforward convolutional code and for the equalizer. The oft output of the BCJR Log MAP then i: L(^u k ) =lne ln k()+ln k ()? ln e ln k()+ln k () ; (10) However, concerning the number of ADD and COMP operation to be performed for a COD Max Log MAP decoder which alo compute a poteriori information about the code bit it i better to compute the oft output for all information and code bit according to (5).

3 B. BCJR Max Log MAP Algorithm From the BCJR Log MAP we obtain a imple approximation by diregarding the correction term in (8) when evaluating (5), (6), and (7). The algorithm now conit of two Viterbi Algorithm, one running forward through the trelli to compute the and one running backward through the trelli to compute the. The oft output of the BCJR Max Log MAP i L(^u k ) = max fln k?1 ( 0 ) + ln k ( 0 ; ) + + ln k ()g?? max fln k?1 ( 0 ) + ln k ( 0 ; ) + + ln k ()g; (11) ln k () = maxfln k ( 0 ; ) + ln k?1 ( 0 )g; (12) 0 ln k ( 0 ) = max fln k+1( 0 ; ) + ln k+1 ()g: (13) For trelli butterfly tructure where the two path with ame u k merge in one tate we obtain from (11): L(^u k ) = max fln k () + ln k ()g?? max fln k () + ln k ()g: (14) C. Soft Output Viterbi Equalizer (SOVE) The BCJR Max Log MAP can be further implified by omitting the backward recurion thu diregarding the. To make ure that before the deciion i done the bit u k ha paed all tap of the dicrete time channel model, at time k the oft output for bit k-m i computed. Thu, the SOVE i a Viterbi algorithm with deciion delay = M. The output of the SOVE i computed according to L(^u k?m ) = max fln k?1 ( 0 ) + ln k ( 0 ; )g? u k?m =+1? max fln k?1 ( 0 ) + ln k ( 0 ; )g; (15) u k?m =?1 ln k () = maxfln k ( 0 ; ) + ln k?1 ( 0 )g: (16) 0 The SOVE i a very imple olution but the hort deciion delay limit performance. Therefore, in [9] an improvement of the SOVE wa propoed employing expanded memory length. The memory of the dicrete time channel model i expanded to M = M ch + M exp (Figure 4) where M ch + 1 i the contraint length of the channel and M exp i the expanion of the memory leading to a higher number of tate and a longer deciion delay. By expanding the memory the hard deciion and the quality of the oft value are improved. Computational requirement do not increae dramatically becaue the number of different tranition probabilitie doe not increae. However, the required memory increae Fig. 4: Dicrete time channel model for SOVE with expanded memory exponentially with the number of tate. Therefore, a good choice of the expanion M exp would be 2. The SOVE i only conidered a equalizer. D. Soft Output Viterbi Algorithm (SOVA) The Soft Output Viterbi Algorithm i a Viterbi algorithm with deciion delay uually choen to = 5(M + 1) which in the computation of oft value only conider path which merge with the ML path within the deciion delay. The difference in the metric of the urviving and competing path are calculated for all tate in each decoding tep. The output oft value i the minimum along the ML path to thoe competing path, which would lead to a different hard deciion: L(^u k ) ^u k min l=0::: l k ; (17) u ( 1 ) 6=u ( 2 ) k k k = (1) k? (2) k ; (18) (i) k () = (i) k?1 (0 ) + ln k ( 0 ; ): (19) Compared to the SOVE the SOVA ha a longer deciion delay. Furthermore, update equence are required which indicate whether the competing path would have lead to a different deciion. E. Improvement of the Soft Output Only the BCJR MAP and the BCJR Log MAP deliver a poteriori log likelihood which really correpond to the oberved bit error probabilitie. The uboptimum algorithm tend to deliver too optimitic L value and therefore the performance in a turbo detection cheme degrade. The BCJR Max Log MAP perform better than the SOVA. Uing the SOVA for both equalization and decoding i better than replacing the equalizer by a SOVE. However, when uing a SOVE with expanded memory due to the better quality of the L value the performance of turbo detection can be better than when uing a SOVA equalizer depite the after the equalizer i till wore. In [11] for the AWGN channel it wa propoed to improve the performance of turbo decoding with the SOVA by multiplying the extrinic information by the factor c = m e 2, 2 e

4 where m e and e 2 are the expectation value and the variance of the magnitude of extrinic information L e in the block. Depite in the derivation of thi factor a gauian ditribution of the output log likelihood i preumed, which i not true for turbo detection, we have hown that the performance of turbo detection can alo be improved in thi way when uing a BCJR Max Log MAP or SOVA decoder. F. Dependence on channel tate information Uing the BCJR Log MAP for both equalization and decoding deliver the bet reult in turbo detection for perfect and mimatched channel etimation. However, the gain compared to the BCJR Max Log MAP decreae when channel etimation i conidered. Thu, the BCJR Log MAP i more enitive to mimatched channel etimation than the uboptimum algorithm. Particularly, the BCJR Log MAP i the only algorithm which require etimation of the variance 2 of the Gauian noie. The output of the uboptimum algorithm can be written a the product of a caling factor 1 and a econd factor independent of 2. A the ame caling 2 factor appear in the firt term of (3), in the L value paed to the decoder and in the extrinic information of the decoder we can replace 2 in (3) by a contant for all Viterbi like algorithm (BCJR Max Log MAP, SOVA, SOVE) without degradation in the performance of turbo detection. G. Complexity Analyi In thi ection a comparion of the number of operation to be performed for the algorithm decribed above i given. We only conider ADD and COMP operation. We ue parameter imilar to GSM TCH/FS but aume that all L=260 bit of a block are encoded by a 16 tate convolutional code of rate 1. The channel memory taken into account by the 2 equalizer i aumed to be M e = 4, the deciion delay of the SOVA i = 25. Furthermore, we aumed that channel etimation i not improved in the iteration and therefore the computation of the firt term in (3) ha to be done only once. Normalization factor neceary to avoid number repreentation problem are neglected. Table I give the ratio of the complexity of turbo detection when uing the BCJR Log MAP for both equalization and decoding and the complexity of turbo detection when uing the other algorithm. Table II give the ratio of the complexity of turbo detection and the complexity of equalization and decoding without iteration. When no turbo detection i applied the decoder i not required to deliver oft output. Therefore, the SOVA decoder i replaced by a Viterbi decoder (VA). III. SIMULATION RESULTS Simulation where done for the GSM peech channel at full rate (TCH/FS) diregarding delay retriction in order to obtain a priori information for all bit of cla 1 in each Equalizer/ ADD COMP Decoder It. 0 It. 5 It. 10 It. 0 It. 5 It. 10 Max-Log-MAP 1,7 2,4 2,5 1,3 1,6 1,6 SOVA/SOVA 2,1 4,0 4,4 1,2 1,1 1,1 SOVE/SOVA 2,2 4 4,5 1,3 1,3 1,2 TABLE I Complexity of turbo detection compared to turbo detection with Log MAP equalizer and decoder: C(Log MAP Equalizer/Log MAP Decoder) C(Equalizer/Decoder) Equalizer/ ADD COMP Decoder It. 1 It. 5 It. 10 It. 1 It. 5 It. 10 Log-MAP 1,8 4,9 8,8 2,3 7,6 14,2 Max-Log-MAP 1,5 3,6 6,2 2,1 6,4 11,8 SOVA/SOVA, SOVA/VA 1,3 2,6 4,2 2,5 8,6 16,3 SOVE/SOVA, SOVE/VA 1,3 2,7 4,4 2,6 9,2 17,3 TABLE II Complexity of turbo detection compared to equalization and decoding without iteration: C(Turbo Detection) C(no Turbo Detection) 1.0e db 2.7 db 2.5 db etimated: 0 iteration perfect: 0 iteration E/No in db Fig. 5: in cla 1 for turbo detection with perfect and mimatched channel etimation, BCJR Max Log MAP equalizer and decoder time lot (ee [12]). We tranmitted data over a time invariant channel with tap g 0 = 0:227; g 1 = 0:46; g 2 = 0:688; g 3 = 0:46; g 4 = 0:227. Reult for turbo detection with perfect channel knowledge are compared to turbo detection with correlative channel etimation. Channel etimation i done only once, no improvement of channel etimation in the iteration i conidered. Figure 5 how the bit error rate of the cla 1 bit for turbo detection uing the BCJR Max Log MAP for both equalization and decoding. Turbo detection work for perfect channel knowledge and mimatched channel etimation. However, the gain achieved by turbo detection i reduced when channel etimation i conidered. However, we can achieve an additional gain when channel etimation i improved during the iteration a hown in [13]. Figure 6 how imilar reult for turbo detection with a SOVE equalizer and a SOVA

5 1.0e+00 etimated: 0 iteration perfect: 0 iteration We have decribed oft in/oft out algorithm which can be ued for equalization and decoding in turbo detection. We have hown that there exit a trade off between complexity and the quality of the oft value which determine the performance of turbo detection. However, when channel etimation i conidered the optimum BCJR Log MAP loe it uperiority to the BCJR Max Log MAP due to higher enitivity to mimatched channel etimation and dependence on the variance of the additive noie wherea the degradation of Soft Output Viterbi Algorithm compared to the BCJR Max Log MAP increae. Therefore, the BCJR Max Log MAP eem to be the algorithm bet uited for turbo detection. For better channel the difference in performance of the mentioned algorithm decreae. V. REFERENCES 2.9 db 2.7 db 2.3 db E/No in db Fig. 6: in cla 1 for turbo detection with perfect and mimatched channel etimation, SOVE equalizer, SOVA decoder 1.0e db perfect: Log-MAP MaxLog-MAP SOVE/SOVA etimated: Log-MAP MaxLog-MAP SOVE/SOVA 1.2 db E/No in db Fig. 7: in cla 1 after for turbo detection with perfect and mimatched channel etimation decoder. Figure 7 give a comparion of the after for different algorithm. For perfect channel knowledge the BCJR Log MAP achieve the bet reult due to the good quality of it proceed oft value. However, when conidering channel etimation due to the higher enitivity to mimatched channel etimation the reult for the BCJR Log MAP and the uboptimum BCJR Max Log MAP are approximately the ame. The degradation of the SOVE/SOVA cheme compared to the BCJR Max Log MAP even increae due to channel etimation. IV. CONCLUSIONS [1] C. Berrou, A. Glavieux, and P. Thitimajhima, Near hannon limit error correcting and decoding: Turbo code (1), in International Conference on Communication (ICC), pp , IEEE, May [2] C. Douillard, M. Jézéquel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, Iterative correction of interymbol interference: Turbo equalization, European Tranaction on Telecommunication, vol. 6, pp , September October [3] G. Bauch, H. Khorram, and J. Hagenauer, Iterative equalization and decoding in mobile communication ytem, in The Second European Peronal Mobile Communication Conference (2.EPMCC 97) together with 3. ITG Fachtagung Mobile Kommunikation, pp , VDE/ITG, September/October [4] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, Optimal decoding of linear code for minimizing ymbol error rate, IEEE Tranaction on Information Theory, vol. IT-20, pp , March [5] J. Hagenauer, E. Offer, and L. Papke, Iterative decoding of binary block and convolutional code, IEEE Tranaction on Information Theory, vol. IT 42, pp , March [6] W. Koch and A. Baier, Optimum and ub optimum detection of coded data diturbed by time varying interymbol interference, in Globecom, pp , IEEE, December [7] P. Roberton, E. Villebrun, and P. Höher, A comparion of optimal and ub optimal MAP decoding algorithm operating in the log domain, in International Conference on Communication (ICC), pp , IEEE, [8] J. Hagenauer and P. Höher, A Viterbi algorithm with oft deciion output and it application, in Globecom, pp , IEEE, November [9] T. Nagayau, H. Kubo, K. Murakami, and T. Fujino, A oft output Viterbi equalizer employing expanded memory length in a trelli, IEICE Tranaction on Communication, vol. E80 B, pp , February [10] J. Hagenauer, The turbo principle: Tutorial introduction and tate of the art, in International Sympoium on Turbo Code, pp. 1 11, ENST de Bretagne, September [11] L. Papke, P. Roberton, and E. Villebrun, Improved decoding with the SOVA in a parallel concatenated (Turbo code) cheme, in International Conference on Communication (ICC), pp , IEEE, July [12] G. Bauch and V. Franz, Iterative equalization and decoding for the GSM ytem, in Vehicular Technology Conference (VTC), IEEE, May [13] V. Franz and G. Bauch, Iterative channel etimation for turbo detection, in 2nd ITG Conference Source and Channel Coding, pp , VDE/ITG, March 1998.