Simplified Criteria for Early Iterative Decoding Termination

Transcription

1 Simplified Criteria for Early Iterative Decoding Termination Spyros Gidaros and Vassilis Paliouras Electrical and Computer Engineering Department University of Patras, PATRAS, Greece phone: + (30) , fa: + (30) , paliuras@ee.upatras.gr web: Abstract- Two novel stopping criteria for iterative decoding of turbo codes are introduced in this paper. The proposed criteria are shown to substantially reduce the required computational compleity, while the achieved bit-error rate is not significantly affected, compared to results based on previously published stopping criteria. Computational compleity reduction is achieved by reducing the number of iterations. Further compleity reduction is achieved by reducing the volume of the data on which the criteria operate. The proposed criteria are shown to reduce the required computational compleity by a factor of ten for cases of practical interest. Two solutions are presented that allow the eploitation of a performance-compleity trade-off. I. INTRODUCTION Turbo coding [][] is a channel coding technique that copes with the errors introduced to communication signals. Since its introduction, it has attracted a lot of attention as the error correction performance achieved is close to the Shannon limit. However, due to its iterative nature, the computational compleity and the delay of turbo decoding are directly proportional to the number of the iterations. Recently, many researchers have proposed criteria in order to minimize the number of decoding iterations by stopping the procedure in an earlier stage. Shao et al. [3] introduced two stopping criteria, namely the sign-echangeratio (SCR) and hard-decision-aided () criteria. Both criteria modify the cross-entropy (CE) criterion proposed by Hagenauer [4] and are effectively reduced to sign-change counts of the etrinsic information and log-likelihood-ratio (LLR), respectively. Wu et al. [5] proposed the signdifference-ratio (SDR) criterion, an etension of the SCR criterion. Shibutani et al. [6] described a criterion which utilizes a cyclic-redundancy-check (CRC) code of the frame to terminate the procedure. The criterion proposed by Leung et al. is also CRC-based [7]. Bokolamulla and Aulin count the number of soft decisions that are less than a threshold and if this count eceeds another threshold, the procedure terminates [8]. Lee and Park use the number of absolute LLR values less than a threshold and the number of hard decision l's in order to complete the iterations [9]. AlMohandes and Elmasry describe a criterion based on the criterion and the CRC code of the frame used []. We thank the European Social Fund (ESF), Operational Program for Educational and Vocational Training II (EPEAEK II), and particularly the Program PYTHAGORAS II, for funding the above work /06/$ IEEE Zhai and Fair describe the mean-estimate (ME), the meansign-change (MSC), and the MSC-CRC criteria []. The ME criterion is based on the mean of the absolute values of the LLR. The MSC criterion uses both the information used in ME and the sign changes of the LLR values, while MSCCRC uses the MSC criterion in addition with the CRC code. Wang et al. proposed four criteria, the first two of which consider the absolute values of the etrinsic information and LLR, respectively, and if the values eceed a threshold, the iteration procedure terminates []. The third criterion (Wang3) counts the number of hard-decision 's and stops the iterations if the count remains unchanged between two consecutive iterations. Finally the fourth criterion (Wang4) compares the signs of etrinsic information and LLR values in the same iteration and terminates the procedure if they identify. In this paper we introduce two new stopping criteria that modify the criterion as follows: the required computations are applied on a restricted number of data per frame rather than on the complete frame. The proposed method is calculated faster and with fewer computations per iteration without, however, degrading the performance in terms of the number of iterations and the bit error rate in comparison to. The proposed criteria reduce the compleity by a factor of ten, when compared to the criterion, without compromising BER performance. The main concept on which the proposed termination criterion is based, is to monitor a part of the data frame. It is noted that the particular concept can be applied to the SCR criterion as well, resulting in substantial compleity reduction. The remainder of this paper is organized as follows: Section II reviews the basics of turbo decoding. Section III introduces the proposed stopping criteria. Section IV discusses the compleity of the proposed techniques in comparison to proposals published in the literature. The eperimental procedure is detailed in section V. Finally, conclusions are discussed in Section VI. II. REVIEW OF TURBO DECODING A generic parallel turbo encoder consists of three main branches, as shown in Fig.. The data pass through the first branch, unchanged. They are coded by a recursive systematic convolutional (RSC) encoder in the second branch 09

2 and the signs of these values (i.e., hard decisions) are compared with the ones computed at the previous iteration resembling the criterion. Therefore the proposed procedure terminates the decoding when it holds that sign L(i -)(fak) Fig.. The organization of a generic parallel turbo encoder. and they are interleaved and coded by a second RSC encoder in the third branch. The outputs of the encoders are the corresponding parity bits. The turbo encoder maps N data bits to 3 N code bits resulting in a code rate of /3. The two RSC encoders can be identical without any degradation of the performance of the coding system. The interleaver is used to uncorrelate the two streams of data encoded by the RSC encoders. The generic parallel turbo decoder of Fig., receives the transmitted data and the parity bits produced by the encoders. It consists of two decoders, related to the corresponding encoders, an interleaver identical to the one of the turbo encoder, a deinterleaver performing the inverse function of the interleaver, and a hard-decision module which derives the decoded bits. The basic idea of turbo decoding is that each decoder produces information called etrinsic information (Le), to be used by the other decoder as a priori information. The echanged information helps the decoders to improve their decisions. In fact this loop gave the name "turbo" to this decoding process [3]. III. PROPOSED CRITERIA The criterion [3] terminates iterations when the following relation holds sign (L(-)(Ak)) sign (L() (u)k), where Uik denotes the kth frame data, for all k,,..., N. The criterion compares the hard decisions, i.e., the signs of the LLR values, denoted by L, of the second decoder (decoder in Fig. ) computed at the current (ith) iteration to the ones computed at the (i - l)st iteration. If all hard decisions made at the ith iteration identify with the ones computed at (i - )st iteration, the iterative decoding procedure terminates. Otherwise, another iteration is eecuted. A. The proposed Sampled criterion (S) The proposed stopping criterion works as follows. For each frame which enters the turbo decoder, M of its N data are randomly selected and monitored through out the decoding of the frame. At the end of each iteration the corresponding LLR values of the selected data are calculated sign (L() ()k), () for all k C S, S c {,,...,N}, with IISII M < N. Notation S denotes the cardinality of the set S, i.e., the number the elements of S. The substantial difference between the proposed S and is that the proposed criterion is applied only on a selected portion of M data of the frame and not on the complete frame. A procedure to define M is described in section III-B. The particular simplification substantially reduces the number of computations, the required memory size and number of memory accesses, as well as the time required for the application of the criterion. The eperiments presented in section 5 reveal moderate performance degradation due to this simplification in terms of bit error rate. In addition the average number of iterations is reduced. The proposed criterion has similar performance with the criterion as the selected portion of the frame's data qualifies as a representative sample of the complete frame. B. Derivation of M The cross entropy between the probability distribution at the output of the second decoder obtained at ith and (i- )st iteration, based on assumptions made by Shao et al. [3] is given by TI (i) Z fukall kca (<) ) (ilk) +log+ P ( L9 L() + ep( ~~~~~ (iuk))j (3) where A {,,..., N}. criterion stops when the sum in (3) approaches zero, or in other words, when the L values grow large and AL become negligible. The proposed S criterion monitors M samples taken from the entire population of A, M < N. The proposed S criterion can ehibit inferior performance compared to, when the monitored M samples produce high L values and negligible AL, while this condition is not met by some of the N -M samples, which are not monitored by the S. It is known that the LLR values follow a Gaussian distribution, the mean value of which grows with each iteration. The number of LLR values due to which the criterion performance degrades, is reduced at each iteration. An L value is considered high if its value eceeds a threshold, defined as follows. Assume an information packet composed of ones. The decoder monitors the mean value and the variance of LLR values. After a number of iterations, the LLR mean value is practically stabilized. When this steady state is reached, a threshold is defined. LLR values that eceed the particular threshold are not considered likely to change. In case of low SNR the mean value,u of LLRs is close to zero, and candidate thresholds are of the form p +

3 P X P Hard Decision Fig.. Basic organizzation of a turbo decoder. k u, 0 < k <. For high SNR values, candidate threshold values are less than the mean value of the distribution, thus assuming a form of,u-kk, 0 < k <. The Gaussian pdf is given by f () ( a Gaussian stochastic variable is represented as X - N(i, (), while a Gaussian stochastic variable which has,u 0 and o- is called standard normal Gaussian variable. Furthermore it can be proven that any Gaussian variable X N(u, u) can be represented as standard - normal Z N(0, ) if Z J. Therefore the standard - normal Gaussian pdf is given by (z) z ep - ) (5) while the distribution function is given by (4) <>(z) J ep( )dt. (6) -00 It is noted that the mean value and variance of a stochastic variable are given by E[X] u and E[(X p)] u respectively. Furthermore, under a symmetry assumption, there is a simple formula which relates the mean value and the variance, as follows [4] u,. (7) The probability that an LLR value X with mean value,u and standard deviation o- is less than a threshold T, is given by T-) P (X < T) p (X < < P (Z<T< ( (J I) TJ) + erf (,)(.) Let Pa P(X < T) and Pb -Pa be the probability that the LLR value is greater than or equal to the threshold. Assuming independence, the probability K that M out of the N LLRs eceed the threshold is K ( -Pa ) M. o- Since the LLRs that eceed the threshold cause disagreement between and S, it is required that the particular event occurs rarely, or probability K is negligible, practically in the same order of magnitude as the required BER. Therefore, a sufficient value of M can be computed as follows ( -Pa)M < K, M > log K () 0-Pa) Inequality () can be used to compute M and it is accurate if M is relatively small compared to N. Inequality () corresponds to the selection of M items, allowing repetitions. A more accurate way to compute the number of the samples M that need to be monitored is based on the selection of M items out of N without repetitions. It is noted that out of the N LLR values, NPb LLRs reach high values, while N-NPb remain low. As mentioned earlier, the S criterion ehibits inferior performance if all of the monitored M samples belong to the samples that reach high LLR values, which are NPb thus missing LLR values which remain low. Therefore, it is required that this probability is less than a value K. Let A' denote the number of different permutations of y different things taken at a time without repetitions. Then NPb (NPb N(N AM NPb< K() AM N )(NPb -)...(NPb -M + ) < K(3) )(N -)...(N -M + ) (NPb)! (N- M)! < K(4) (NPb -M)!.N! Inequality (4) can be numerically solved to give M. As an illustrative eample, assume EbINO 3.07dB, 6 transmitted data bits, and packet size N The threshold for each iteration calculated as follows: For each iteration the mean LLR value and the corresponding standard deviation are calculated. For ten iterations the results presented in Table I are obtained. By selecting a threshold value of T 3, Pa is obtained by (), where,u is measured, o- is computed via (7), and Pb -Pa. By using (), for K -4, the values of M are obtained and tabulated in Table II. The two criteria produce the results shown in Table III. It is shown that comparable BER

4 Iteration a TABLE I EVOLUTION OF LLR STATISTICS AS A FUNCTION OF ITERATIONS. K -4 Iteration Pb TABLE II SAMPLE SIZE M FOR K M Iteration (i -) Iteration i Fig. 3. Operation of TS. This procedure reduces to the application of () for the set of inde values <I< N.Le(i) (iij) < t, (5) where t denotes the adopted threshold value. From the eperiments presented in section 5, the proposed criterion demonstrates superior performance in terms of the bit error rate, and a slight increase in average number of iterations. IV. COMPLEXITY COMPARISON 4. BER Average Iterations TABLE III S IN COMPARISON TO, WITH N S, Algorithm S (M 4) S (M 4) Iteration j Eb No (db) performance is achieved by the proposed S, using only M 4 samples, instead of N 4096 required by the, for the case of EbINo 3.07dB. C. The proposed Threshold S criterion (TS) The Threshold S is a modification of S as described in the following: the monitored data of the frame are not chosen randomly, as in the case of S; instead, the criterion is applied on those data for which the corresponding L' values are less than a predefined threshold in every iteration. At each iteration the hard decisions from the data chosen at the ith iteration are compared to the hard decisions for the same position data, computed using L' available either from the (i- )st iteration or earlier iteration. In Fig. 3, let denote the selected data ui for which Le() (ui) < t. The arrows point to the data computed at (il)st or jth, iteration, j < i-, to which the selected data are compared, using the criterion (). The required number of memory words is bound by N, since a value Le(j) (ul) < t, is stored at position, overwriting any Ljc(j) (ut),.' < j, previously stored at the same position. Ryan discusses an efficient implementation of the turbo decoder [3]. In the particular implementation, at each decoder iteration, the quantity L' (Uk), i.e., the etrinsic information from decoder to decoder (see Fig. ), is directly computed instead of initially computing L (Uk) and subsequently subtracting from it the quantity y /j -+Le (Uk) in order to derive L- (Uk). Similarly the etrinsic information L- (Uk) from decoder to decoder is also directly computed. Therefore, in case this procedure is adopted, stopping criteria compleity should take into consideration the compleity corresponding to the computation of the LLR values for all data of the frame. In general, the overall computational compleity of an iteration, is given by C CLog-map + Cinterleaver + Ccriterion. (6) It is assumed that all stopping criteria share a common compleity of CLog map + Cinterleaver, while they demonstrate different Ccriterion compleities, quantified below. In the following the computational compleity of and proposed S and TS are derived. Assuming that the frame comprises N data, the compleity of the two criteria is summarized in Table IV, where the notation is as follows: states is the number of states of the convolutional code, (*) denotes number of multiplications, operation (/) denotes number of divisions, (+) denotes number of additions or subtractions, ma* denote the implementation of the ma* function computed as ma* (,y) ma(,y) +log( + -l-y) (7) (cf. equation (4.7) of [3]), and finally, R denotes the average cardinality of Si of (5), i.e., R E [ Si ]. Since M can be more than ten times smaller than N, significant compleity reduction is achieved, which any substantial performance loss.

5 Task Select data (logic operations) S N data M data TS R data N N (+) M() M(I) M (+) R(*) R(I) R (+) N M R N M R CalculateN(*) N(I) Calculate LLRvalues Obtain sign (binary operations) Check sign changes (logic operations) - m 3 TABLE IV COMPLEXITY OF STOPPING CRITERIA. I 0 Memory requirements Store LLR values Store Hard decision N data N. 8 bits N bits S M data M 8 bits M bits - Iterations Wang --- Wang 3 v Wang 4 - A -SCR * TSH DA S - TS R data N. 8 bits R bits n-, I Eb/N0 3.5 Fig. 4. BER vs. EbINo performance, for the case of RSC (7,5) constituent codes for various criteria. In this eperiment S monitors M 77 values out of N 4096 for, while the threshold of the TS is set at TABLE V REQUIRED MEMORY SIZE. Log-MAP compleity N(/) initial N. states Computation Alpha Computation { states+(+) (*) ma* } N states { states+ (*) Computation Beta states+4 (+) ma* } N states { (*) 4-states (+) I Soft output { states ma* ~states+4 (N) -' r, m -' - l O 0- Iterations (-) } Wang -----'----- W3ang Wang 4 TABLE VI COMPLEXITY OF LOGMAP. SCR l o-4 * F 3 Wang et al. [] suggest that a sufficient representation of LLR values comprises a 6-bit integral part and a -bit fractional part, i.e., a total of 8 bits. Based on this, required memory size is quantified in Table V. Eperimental results reveal that for low SNRs, the proposed S terminates the decoding procedure up to two iterations earlier than. Notice that each iteration includes the common compleity CLog + Cinterleaver. The computational compleity of LogMAP is summarized in Table VI. It is noted that each iteration requires the computation of two instances of LogMAP. map V. EXPERIMENTAL RESULTS Eperiments were conducted for a four-state RSC code with generating polynomials (7,5). A total of 6 information bits are transmitted for each eperiment. The procedure is as follows: the information stream is partitioned into 45 frames of 4096 bits. Each frame is coded and BPSKmodulated. Subsequently, coded information is transmitted TS - S 3.05 Fig Eb/N0 Detail of BER plot presented in Fig through a AWGN channel. The received information enters the turbo decoder and after a number of iterations defined by the appropriate stopping criterion, hard decisions are taken and the corresponding bit-error rate (BER) is computed. Figs. 4 and 5 depict BER vs. Eb/No plots for the following termination criteria:, S, TS, SCR, the first, third, and fourth criteria proposed by Wang, denoted as Wang, Wang 3, and Wang 4 respectively. It can be seen that the BER performance of the S closely resembles the best of the performance of the other criteria, while the TS achieves a BER curve which closely resembles the BER curve obtained by using a fied number of ten iterations. Fig. 6 depicts BER performance of the proposed TS criterion for two values of threshold, namely 0 and Fig. 7 presents the corresponding average iterations vs. Eb/NO. 3

6 * TS thre TS thre0 0 F U 8 -. E Eb/No Fig. 6. BER vs. EbINo for TS. 0 TS thre TS thre0 8.5 *<d 8- MO a5 E 7 - z Fig Eb/No Average number of iterations for TS. Fig. 8 depicts the average number of iterations imposed by the various stopping criteria for various SNR values. It is noted that the proposed S and TS criteria operate on a small percentage of the data on which the previously reported criteria operate, thus further and substantially reducing compleity. VI. CONCLUSIONS Two novel stopping criteria for iterative decoding of turbo codes have been detailed. The proposed criteria were shown to substantially reduce the required computational compleity, while maintaining the BER performance and a small number of decoding iterations. The performance of the proposed criteria has been compared to previously published results, and is shown to offer interesting compleityperformance trade-offs. In particular, and in the case M << N, more than ten times compleity reduction is possible without performance loss. REFERENCES [] C. Berrou, A. Glavieu, and P. Thitimajshima, "Near shannon limit error-correction coding and decoding. turbo codes," in Proc. Int. Conf: Communications, pp , 993. Z Wang v Wang 3 Wang 4 -A-SCR -4- TS S Eb/NO Fig. 8. Required number of iterations for several iteration criteria and various EbINo values. In this eperiment the proposed S monitors M 77 values out of N 4096 required by, while the TS threshold is set at [] C. Berrou and A. Glavieu, "Near optimum error correcting coding and decoding: Turbo codes," IEEE Trans. Commun, vol. 44, no., pp. 6-7, 996. [3] R. Y. Shao, S. Lin, and M. P. C. Fossorier, "Two simple stopping criteria for turbo decoding," IEEE Trans. Commun, vol. 47, no. 8, pp. 7-, 999. [4] J. Hagenauer, E. Offer, and L. Papke, "Iterative decoding of binary block and convolutional codes," IEEE Trans. Inform. Theory, vol. 4, pp , 996. [5] Y. Wu, D. Woemer, and J. Ebel, "A simple stopping criterion for turbo decoding," IEEE Communication Letters, vol. 4, pp , 000. [6] A. Shibutani, H. Suda, and F. Adachi, "Reducing average number of turbo decoding iterations," IEE Electronic Letters, vol. 35, pp , 999. [7] 0. Y. H. Leung, C. W. Yue, C. Tsui, and R. S. Cheng, "Reducing power consumption of turbo code decoder using adaptive iteration with variable supply voltage," in Proc. of ISLPED, (San Diego), pp. 36-4, 999. [8] D. Bokolamulla and T. Aulin, "A new stopping criterion for iterative decoding," in Proc. of Int. Conf Communications (ICC), pp , 004. [9] D. S. Lee and I. C. Park, "A low-compleity stopping criterion for iterative turbo decoding," IEICE Trans. Commun., vol. E88-B, Jan [] I. A. Al-Mohandes and M. I. Elmasry, "Iteration reduction of turbo decoders using an efficient stopping/cancellation technique," in Proc. of IEEE International Symposium on Circuits and systems, pp. I-609- I-6, 00. [] F. Zhai and I. Fair, "Techniques for early stopping and error detection in turbo decoding," IEEE Trans. Commun., vol. 5, pp , Oct [] Z. Wang, H. Suzuki, and K. K. Parhi, "Finite wordlength analysis and adaptive decoding for turbo/map decoders," Journal of VLSI Signal Processing, vol. 9, pp. 09-, 00. [3] W. E. Ryan, "Concatenated convolutional codes and iterative decoding," available on-line: [4] S. ten Brink, "Convergence behavior of iteratively decoded parallel concatenated codes," IEEE Transactions on Communications, vol. 49, pp , Oct