Comparison of MAP decoding methods for turbo codes

POSTER 2016, PRAGUE MAY 24 1 Comparison of MAP decoding methods for turbo codes Vitor ĎURČEK 1, Tibor PETROV 2 1,2 Dept. of Telecommunications and Multimedia, Faculty of Electrical Engineering, University of Žilina, Univerzitná 8215/1, 010 26 Žilina, Slovaia vitor.durce@fel.uniza.s, tibor.petrov@fel.uniza.s Abstract. BCJR algorithm is the first algorithm for MAP decoding of turbo codes. It contains complex mathematical operations, so simpler derivates were created. This paper explores the original BCJR algorithm along with its simplified version: Max-Log-APP algorithm. Their strong and wea points are mentioned and their performance is also compared in a simulation, where a turbo encoder based on design from 3GPP/UMTS and 3GPP/LTE standards is used. registers lined bac to the input of the encoder in addition to the output lins. Partial codes of a turbo encoder need to be recursive, because it maximizes its interleaver gain. Interleaver gain does not exist for non-recursive constituent convolutional encoders [7]. By choosing the right recursive constituent encoders, the final turbo code gets higher additional error-correcting performance (originating from its interleaver length) than it would get while using nonrecursive constituent encoders. Keywords Turbo codes, BCJR, Turbo decoding, Trellis termination. 1. Introduction Turbo codes, a class of error-correcting codes capable of performing near the Shannon limit in terms of BER (bit error rate), were first introduced in 1993 [1]. Because of their excellent performance, they have found applications in the Consultative Committee for Space Data Systems (CCSDS), 3GPP/UMTS and 3GPP/LTE standard, Digital Video Broadcasting Return Channel Satellite and Terrestrial (DVB-RCS and DVB-RCT), 3GPP2/cdma2000 wireless communication systems, and IEEE.802.16 WiMAX standards [2]. Turbo codes, as originy defined by C. Berrou, A. Glavieux and P. Thitimajshima, can be described as parel concatenated convolutional codes (PCCC). A turbo encoder consists of two recursive systematic convolutional encoders which are usuy identical, but not necessarily. Both convolutional encoders are fed the same input bits, but the order of these bits is changed by an interleaver before they enter the second convolutional encoder. A part of an output sequence of a systematic channel code is always identical to the input sequence which it originates from [3]. In convolutional codes this is possible when copies of input bits exit the encoder directly as output bits in addition to them entering the shift registers too (see Fig. 4). Convolutional codes, which are part of a turbo code, have to be systematic because it maes decoding of the turbo code easier [4] and it is one of the fundamental parts of the turbo code design. Recursive convolutional codes have some of the memory blocs of their encoder s shift 2. MAP decoding of turbo codes In 1974, an algorithm for APP (a posteriori probability) decoding of convolutional codes, now ced the BCJR algorithm, was introduced by Bahl, Coce, Jeline, and Raviv. It is an iterative soft-decision decoding algorithm. It has not replaced the Viterbi algorithm in decoding of convolutional codes, because it is more complicated and its performance is not much better when used to decode these codes. But it proved to be an integral part of turbo code decoding [8]. BCJR algorithm embodies complex mathematical operations and was mostly avoided in turbo decoder implementations. Its derivates such as Log-APP and Max-Log-APP algorithms (also referred to as Log-MAP and Max-Log-MAP; named after the maximum a posteriori criterion) are usuy used. There exist multiple versions of the Log-APP algorithm, but in general it is the BCJR algorithm implemented in logarithmic domain [9]. BCJR algorithm is an algorithm for symbol-wise MAP (Maximum A Posteriori) decoding. From this point the use of binary symbols will be assumed. Main goal of the algorithm is to compute LLR (Log-Lielihood Ratio) values L(c ) of bits c in a data word from received systematic bits x and encoded bits z and z' in a received code word. Logarithmic a posteriori probability ratio L(c ) (often referred to as LLR) is defined as: N P c 1 y 1 Lc ( ) log N P c 1 y 1 In the numerator the probability that a certain decoded bit c equals +1 given bits y 1 y N in the received code word can be seen. Conditional probability that the decoded bit c equals -1 is in the denominator. Bipolar mapping is used (1)

2 V. ĎURČEK, T. PETROV, NUMERICAL MODEL OF OPTICAL SWITCH BASED ON 2D MEMS where bit value 0 changes to 1 and bit value 1 changes to - 1. Turbo decoder maes its final decision about decoded bits based on the last LLR. When the LLR is positive the decoded bit value is +1 and when it s negative the bit value is -1 [5-7]. In BCJR-based decoding the log-lielihood ratios are computed from three types of metrics: forward state metric, bacward state metric and transition metric ɣ. So the equation for LLR in (1) can be written as: Lc s '. s. s ', s s '. s. s ', s 1 c log 1 c LLR of a data bit c is therefore logarithm of a ratio, where the nominator contains sum of transition metrics of trellis state transitions that imply the bit value is +1, multiplied by forward metric of their starting states s and also by bacward metric of their ending states s. In the denominator the same sum can be found, but it is of transition metric values that imply the decoded bit value to be -1. Notions and suggest that the metrics are normalized as opposed to and notions. Normalization prevents them from reaching values that are too high. Transition metric can be computed as s', s e exp 0.5L t,1 t,1 0.5 Lc x,1 t,1. q.exp 0.5L z i2 t c, i1, i where t,1 t,q are bits (both systematic and redundant) related to a certain state transition in trellis of used turbo code. Systematic output bits of turbo encoder are referred to as x,1 and redundant bits as z,i (depicted in Fig. 3). L e (t,1) is log-lielihood ratio of the -th bit (in a data word) passed from another decoder so ced extrinsic value. Lc is a channel-dependent value calculated from SNR (Signalto-Noise Ratio) and received signal. Forward state metric can be computed as: s 1 s' s', s s' (2) (3) (4) It is computed from state transitions that lead from previous trellis states (states on the left side in trellis) to the current state s and is a sum of these transitions metrics multiplied by normalized forward metrics of a state which each of them originates from. Bacward state metric is computed as: 1 s ( s') s s', s (5) It is computed from products of transition metrics of transitions that originate in the current s and normalized bacward metrics of states that these transitions continue to (memory states to the right side of the state s in trellis) [5-7]. Despite its complex mathematical operations, it is now possible to implement a turbo decoder using the original BCJR algorithm on modern hardware. FPGA implementation of such decoder can be seen in [10]. Max-Log-APP decoding algorithm simplifies the original BCJR algorithm so it is more practical to implement, but at the cost of lower bit error rate. It is also less sensitive to SNR mismatch compared to the original BCJR algorithm which requires accurate estimation of the noise variance. The following simplifications are introduced in Max-Log-APP. The algorithm wors with the metrics in the logarithmic domain after this there is no need to use non-linear exponent operation. The sum operation is approximated using max operation, which can be seen here: A B C A B C ln... ln max,,,... (6) Equation (3) in a form used in Max-Log-MAP (after applying natural logarithm) can be seen here: ', e 0.5 q s s L t t,1,1 0.5L x t 0.5L z t c,1,1 i2 c, i1, i Natural logarithm cancels the exponent operation. Simplified form of equation (4) for calculation of forward state metric is: s max s' s', s s' 1 (7) (8) Sum of metric values was replaced by the maximum element of the sum. Similar adjustment needs to be made to the equation (5) for the bacward state metric: s s s s 1 ' max ', (9) s Forward and bacward state metrics are also initialized by logarithmic values of initializations for BCJR metrics. The equation (2) for LLR is also changed: max ', 1 ' L c s s s s s, s ' 1 s, s ' 1 s s 1 s s max ', ' (10) There is no need for logarithmic operation, because the metric values are already in the logarithmic domain [5-7]. 3. Trellis termination Trellis termination is an important method for ensuring that error-correcting performance of a code is not degraded at the end of a code word. For optimal performance of trellis-based decoding, the starting and the ending state of trellis need to be nown. The nown starting state is usuy enforced by filling shift-registers with zeroes before beginning of encoding of every data

POSTER 2016, PRAGUE MAY 24 3 word (also referred to as data bloc). For non-recursive convolutional codes the nown final state is often secured by adding m zeroes to the tail of every data word, where m is the number of encoder s memory blocs. For recursive convolutional codes a special set of bits dependent on the data word needs to be added to the tail of said word in order to fill memory blocs with zeroes and mae the final trellis state -zero. These methods solve the problem of unreliable set of bits at the end of every word for convolutional codes. They are ced trellis termination and are relatively straightforward [7]. However the problem of trellis termination is more complex in turbo codes because of the presence of interleaver [9]. Reordered input bits of the second constituent encoder present new chenges in termination of trellises and there exist multiple methods. The following methods can be used with a pseudorandom interleaver. 3.1 No trellis termination This is the most basic approach where both trellises are left to end in unnown final states. No additional bits are added to the tail of data words, so there are no changes to the over coding rate. Also bits in a data word are interleaved. However this method results in the weaest decoding performance at the end of trellis [11]. In Fig. 1 a graphical representation of this method can be seen. Each white square represents one bit of a data word, where the upper data word is input for the first constituent encoder and the lower word is input for the second constituent encoder. Bits of the second data word are reordered by interleaver (indicated by arrows). Fig. 1. Information sequences of turbo code without trellis termination 3.2 Termination of the first constituent encoder With this strategy tail bits are added to the end of every data word. These cause trellis of the first constituent encoder to end in a nown state (most often the -zero state). The same data word that forces the aforementioned first trellis to end in a nown state is then fully interleaved (including the added tail bits) and maes the second trellis end in an unnown state. In spite of the second trellis not being terminated, the error-correcting performance is still notably improved. However the coding rate is lowered, because of the added trellis-terminating bits [12]. Graphical representation of this method can be seen in Fig. 2, where data bits are depicted as the white squares and tail bits as the blac squares. It can be seen that bits in a data word are interleaved in this method. Fig. 2. Information sequences of turbo code with first trellis terminated in a now state More methods of trellis termination exist, but they genery cannot be combined with a pseudorandom interleaver. 4. Simulations The turbo encoder used in simulations is based on turbo encoder from the 3GPP/UMTS [14] and 3GPP/LTE [15] standard. Structure of this encoder is depicted in Fig. 4. It consists of two identical convolutional encoders. Each one of these constituent encoders stores former input bits in 3 memory blocs. Properties of such encoder can be described by numbers (2, 1, 3), where n = 2 is number of output bits, = 1 is number of input bits and m = 3 is the length of shift-register. Its coding rate is therefore 1/2. Lins within the encoder can be described by this transfer function: 3 1DD GD ( ) 1, 1 2 3, (11) D D where 1 represents the systematic bit (the output bit that is identical to the current input bit) and the ratio of polynomials represents the second output bit. In Fig. 3 c 1 is the input of the turbo encoder. These bits also enter the first convolutional encoder in an unchanged order. Interleaver bloc is mared with π while c' 1 are the reordered c 1 input bits at the output of the interleaver. White circles on the left side represent input bits of constituent convolutional encoders (this input can be changed via the recursive lins) while white circles on the right side are output bits of the turbo encoder. Systematic output bits mared as x 1 are identical to the input bits c 1. Encoded output bits of the first and second constituent encoder are mared as z 1 and z' 1, respectively. Memory blocs of shift registers are mared with D (encoding delay). Code rate of the whole turbo code without trellis termination is 1/3. Interleaver for input bit stream of the second constituent encoder is pseudorandom bits in a data bloc are reordered based on a random permutation.

4 V. ĎURČEK, T. PETROV, NUMERICAL MODEL OF OPTICAL SWITCH BASED ON 2D MEMS Fig. 3. Structure of turbo encoder used in simulations In Fig. 4 error-correcting performance of the described turbo code can be seen while using BCJR algorithm for decoding. Soft values calculated during decoding are not quantized. Number of decoding iterations is set to 10. State metric normalization is employed to avoid numerical overflow, where forward and bacward state metric values related to the same time instant are divided by their sum. These normalized values are the used to calculate state metrics related to the next time instant. Values of transition metrics are not changed in this way. Data bloc length is 640 bits and performance of this turbo code is tested with no trellis termination and also with its first trellis terminated. Interleaver length for the case of no trellis termination is 640 bits and for the case of one terminated trellis, its length is 637 bits. Initialization of state metrics values without trellis termination were the following. Forward metric corresponding to the -zero memory state had initial value of zero while the other values were set to zero. All initial values of the bacward metrics were set to 1/8. For the case with one terminated trellis the initial bacward metrics were set to zero except the one corresponding to the -zero memory state. Simulation software is programmed in MATLAB environment without use of built-in tools for turbo encoding and decoding. Performance of the same turbo code with an implementation of the Max-Log-APP algorithm for decoding can be seen in Fig. 5. Soft values were not quantized, no trellis termination was used and number of decoding iterations was 10. Initialization of the state metrics in the case of Max-Log-APP decoding needs to be done in the logarithmic domain. Initial values of forward metrics for the turbo code without trellis termination were set to - (the lowest value) except for the value corresponding to the -zero state this one was set to zero. All initial bacward metric states were set to the natural logarithm of 1/8. BER performance of the turbo code with Max-Log- APP decoding and no trellis termination is clearly inferior to the performance of the same turbo code with BCJR decoding, which is the cost of lowered decoding complexity. Fig. 5 Performance of a 1/3 turbo code with bloc length of 640 bits, pseudorandom interleaver and Max-Log-APP decoding Performance of turbo codes was tested over AWGN (Additive white Gaussian noise) channel. SNR (signal-tonoise-ratio) ranges and lengths of information bit sequences were chosen with consideration of time needed to finish the simulations. Longer information bit sequences need to be processed in order to get accurate BER at higher SNR values. Length of information sequence is 10 7 bits for lower SNR values and 10 8 bits for higher SNR values. 5. Conclusion Fig. 4. Performance comparison of a 1/3 rate turbo code with and without using trellis termination. Bloc length is set to 640 bits with pseudorandom interleaver and BCJR decoding. In this paper BCJR and Max-Log-APP algorithms for decoding of turbo codes were described. The differences between a turbo code with no terminated trellis and one terminated trellis were mentioned and depicted. Results of a simulation comparing error-correcting performance of a BCJR-decoded turbo code with and without using trellis termination were presented. Turbo code with one

POSTER 2016, PRAGUE MAY 24 5 terminated encoder had better BER performance with only sm decrease of coding rate (relative to the data bloc length). Results of simulation of the same turbo code without terminated trellises were presented with the Max- Log-APP algorithm used for decoding. When compared to the BCJR algorithm, the BER performance was noticeably lowered due to used approximations and therefore lowered computational complexity. Acnowledgements This wor is supported by the Slova Research and Development Agency under the project APVV-0025-12 ("Mitigation of stochastic effects in high-bitrate -optical networs"). References [1] Berrou, C., Glavieux, A., Thitimajshima, P. Near Shannon Limit Error-Correcting Coding and Decoding: Turbo-Codes, ICC 1993, Geneva, Switzerland, pp. 1064-1070, May 1993. [2] Tasaldiran, M., Morling, R.C.S., and Kale, I. Parel decoding of turbo codes using multi-point trellis termination and collision-free interleavers WTS 2009, Prague, Czech Republic, pp. 1-5, April 2009. [3] Berrou, C., Glavieux, A. Near Optimum Error Correcting Coding and Decoding: Turbo-Codes, IEEE Transactions on Communications, Vol. 44, No. 10, pp. 1261-1271, Oct. 1996. [4] Benedetto, S., Garello, R., and Montorsi, G. A search for good convolutional codes to be used in the construction of turbo codes, IEEE Trans. Commun., vol. 46, no. 9, Sep. 1998, pp. 1101-1105. [5] Sadjadpour, H. R. Maximum A Posteriori Decoding Algorithms For Turbo Codes, Digital Wireless Communication II, Proceedings of SPIE, vol. 4045, 2000, [6] Robertson, P., Villebrun, E., Hoeher, P. A comparison of optimal and sub-optimal MAP decoding algorithms operating in the log domain, IEEE International Conference on Communications, Vol. 2, p. 1009-1013, Jun 1995. [7] W.E. Ryan, S. Lin, "Channel Codes: Classical and Modern", Cambridge University Press, New Yor, 2009, ISBN 978-0-511-64182-4. [8] D.J. Costello, Jr., G.D. Forney, Jr., "Channel coding: The road to channel capacity", Proceedings of the IEEE, Vol. 95, no. 6, pp. 1150-1177, June 2007. [9] K. Gracie, M. Hamon, "Turbo and Turbo-Lie Codes: Principles and Applications in Telecommunications", Proceedings of the IEEE, Vol. 95, no. 6, pp. 1228-1254, June 2007. [10] O. Atar, M.H. Sazli, and H.G. İl, "FPGA Implementation of Turbo Decoders Using the BCJR Algorithm", Przegląd Eletrotechniczny, 2013, vol. 89, no. 2b, pp. 257-260. [11] J. Hofelt, O. Edfors, and T. Maseng, A survey on trellis termination alternatives for turbo codes, in IEEE Vehicular Technology Conf., Houston, TX, vol. 3, May 1999, pp. 2225 2229. [12] M. Kovaci, H. Balta, Performance of trellis termination methods for RSC component encoders of turbo codes, IEEE International Symposium on Electronics and Telecommunications (ISETC), Timisoara, Romania, pp. 111-114, 2012. [13] S. Crozier, P. Guinand, J. Lodge, and A. Hunt, "Construction and performance of new tail-biting turbo codes, in Proc. 6th Int. Worshop on Digital Signal Processing Techniques for Space Applications (DSP 98), Noordwij, The Netherlands, September 1998, ESTEC. [14] 3GPP Technical Specification: Group Radio Access Networ, Universal Mobile Telecommunications System, Multiplexing and channel coding (FDD), Release 12, TS 25.212 v12.1.0, January 2015. [15] 3GPP Technical Specification: Group Radio Access Networ, LTE, Evolved Universal Terrestrial Radio Access, Multiplexing and channel coding, Release 13, TS 36.212 v13.0.0, January 2016. About Authors Author-Vitor DURCEK was born in Bansá Bystrica, Slovaia in the 1990. In 2014 he finished MSc at University of Žilina, Faculty of Electrical Engineering, Department of Telecommunications and Multimedia. Currently he studies doctoral degree at the same department and his research is focused on forward error correction methods for high-speed optical networs. Co-author-Tibor PETROV was born in Plovdiv, Bulgaria in 1990. In 2015 he finished MSc at the University of Žilina, Faculty of Electrical Engineering, Department of Telecommunications and Multimedia. Currently he studies doctoral degree at the same department. His research activities include wireless networs and cooperative technologies in the Intelligent Transportation Systems (ITS) environment.