129966863002202240[1]

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129966863002202240[1]

  1. 1. This article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author’s benefit and for the benefit of the author’s institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues that you know, and providing a copy to your institution’s administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution’s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier’s permissions site at: http://www.elsevier.com/locate/permissionusematerial
  2. 2. Author's personal copy Modeling the bit-level stochastic correlation for turbo decoding Yi-Nan Lin b , Wei-Wen Hung a,*, Tsan-Jieh Chen a , Erl-Huei Lu b a Department of Electrical Engineering, Mingchi University of Technology, Taipei, Taiwan b Department of Electrical Engineering, Chang Gung University, Tao-Yuan, Taiwan Received 19 February 2006; received in revised form 26 June 2006; accepted 26 June 2006 Available online 24 July 2006 Abstract After passing a systematic bit through a turbo encoder, the encoding process will introduce some extent of correlation between a sys- tematic bit and its associated parity bits. However, this correlation is neglected in the subsequent turbo decoding process so as to reduce its computational complexity. In this paper, we try to explore the feasibility of modeling the bit-level stochastic correlation for the iter- ative turbo decoding. By properly adjusting the parameter of the correlation model, we can approximate various degrees of the under- lying correlation within the received codewords. Reduction in bit error rate (BER) then may benefit from a more accurate capture of the correlation information at the cost of requiring only a small additional computation complexity. Experimental results indicate that incor- porating the correlation model into the turbo decoding process can achieve better BER performance than that of conventional turbo decoders over AWGN channels. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Turbo encoder/decoder; Bit-level stochastic correlation model; Bit error rate (BER); Computation complexity; AWGN channel 1. Introduction In recent years, considerable interest has been devoted to soft-output decoding schemes that achieve near-Shan- non limit performance. A parallel concatenated convolu- tional code (PCCC), named Turbo code, was first proposed in 1993 by Berrou et al. [1,2]. In the seminal paper, they used a parallel concatenation of two Recursive Systematic Convolutional (RSC) encoders interconnected through an interleaver. Each RSC encoder produces a sys- tematic output that is equivalent to the original informa- tion sequence, as well as a stream of parity information. The two parity sequences can then be punctured before being transmitted along with the original information sequence to the decoder module. The underlying interleav- er is designed to make the two encoded data sequences sta- tistically independent of each other [3,4]. At the decoder module, two RSC decoders are employed. Each decoder receives an a-priory soft input and generates an a-posteriori soft output, which serves as feedback information to the other decoder. These soft inputs and outputs indicate not only whether a received bit was a ‘‘0’’ or a ‘‘1’’, but also the likelihood that the bit has been correctly decoded. The RSC decoders operate iteratively and the quality of the a priori information will improve gradually until some terminating criterion is met. The effectiveness of the turbo decoding scheme is based on iterating the maximum a posteriori (MAP) algorithm [5], applied to each constituent code. In general, the MAP algorithm is implemented by means of a soft-in soft-out (SISO) decoder. This SISO decoder computes the a poste- riori probability (APP), i.e., a reliability value, for each received information symbol. However, this computation is extremely complex owing to the multiplications and exponential operations required for the forward and backward recursions in the trellis diagram. In order to reduce the decoding complexity of the MAP algorithm, 0140-3664/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2006.06.023 * Corresponding author. Tel./fax: +886 02 29061780. E-mail addresses: ynlin@ccsun.mit.edu.tw (Y.-N. Lin), wwhung@ ccsun.mit.edu.tw (W.-W. Hung), spider@sig.com.tw (T.-J. Chen), lueh@ mail.cgu.edu.tw (E.-H. Lu). www.elsevier.com/locate/comcom Computer Communications 29 (2006) 3856–3862
  3. 3. Author's personal copy researchers have developed other SISO decoders, which are less complex and can be used instead of the MAP algo- rithm. Two of such algorithms are the Max-Log-MAP and the Log-MAP algorithms. The Max-Log-MAP algorithm was proposed by Koch and Baier [6] and Erfanian et al. [7]. This technique simpli- fied the original MAP version by transferring the recur- sions into the logarithmic domain and by invoking an approximation for the sake of dramatically reducing the associated computation complexity. Due to the approxima- tion, the performance of the Max-Log-MAP algorithm is sub-optimal compared to that of the MAP algorithm. However, Robertson et al. [8] presented the Log-MAP algorithm, which corrected the approximation used in the Max-Log-MAP algorithm and hence attained a perfor- mance virtually identically to that of the MAP algorithm at a fraction of its complexity. In this paper, we try to modify the branch metrics involved in the calculations of the forward and backward recursions employed in the Max-Log-MAP algorithm. This modification is intended to reflect various extent of bit-level stochastic correlation within the received codewords. The paper is organized as follows. We first give a brief review of the basic iterative decoding scheme of the Max-Log- MAP algorithm in the next section. The subsequent section then explains the formulation of modeling the embedded correlation for a turbo decoder based on the Max-Log- MAP algorithm. The modified branch metrics for forward and backward recursions are also described. Finally, in section 4, simulation results over AWGN channels are reported to demonstrate the effectiveness of the Max-Log-MAP-based turbo decoder when the bit-level stochastic correlation is properly modeled. 2. Review of the Max-Log-MAP algorithms In the brief review of the Max-Log-MAP algorithm, we follow the notations used in [9,10]. Given the received codeword sequence y, the MAP algorithm computes the a posteriori Log Likelihood Ratio (LLR) L(ukjy) for each decoded bit uk, where LðukjyÞ ¼ ln Pðuk ¼ þ1jyÞ Pðuk ¼ À1jyÞ ! : ð1Þ Incorporating the code’s trellis, this may be written as LðukjyÞ ¼ ln P ðs0;sÞ;uk¼þ1PðSkÀ1 ¼ s0 ^ Sk ¼ s ^ yÞ P ðs0;sÞ;uk¼À1PðSkÀ1 ¼ s0 ^ Sk ¼ s ^ yÞ ! ; ð2Þ where Sk is the state at time k, (s0 ,s) is the set of ordered pairs corresponding to all state transition from the previ- ous state SkÀ1 = s0 to the present state Sk = s caused by in- put bit uk at some specific values. Using the Bayes’ rule and splitting the received codeword sequence y into three sec- tions, i.e., the codeword associated with the present transi- tion yk, the codeword sequence prior to the present transition yj<k and the codeword sequence after the present transition yj>k, we can thus rewrite the individual term: PðSkÀ1 ¼ s0 ^ Sk ¼ s ^ yÞ ¼ PðSkÀ1 ¼ s0 ^ Sk ¼ s ^ yj<k ^ yk ^ yj>kÞ; ¼ Pðyj>kjSk ¼ sÞ Á PðSk ¼ s ^ ykjSkÀ1 ¼ s0 Þ Á PðSkÀ1 ¼ s0 ^ yj<kÞ; ¼ bkðsÞ Á ckðs0 ; sÞ Á akÀ1ðs0 Þ; ð3Þ where ckðs0 ; sÞ PðSk ¼ s ^ ykjSkÀ1 ¼ s0 Þ ¼PðukÞ Á PðykjxkÞ ð4Þ is the branch metric that given the trellis was in state SkÀ1 = s0 at time k À 1, it moves to state Sk = s and the re- ceived codeword for this transition is yk. Equivalently, the transition probability values ck(s0 ,s) can be calculated from the product of the a priori probability P(uk) for the input bit uk, and the conditional joint probability P(ykjxk) that given the codeword xk = (xk1, . . . ,xkl, . . . ,xkN) associated with the transition was transmitted we received the code- word yk = (yk1, . . . ,ykl, . . . ,ykN), where N denotes the number of bits in each codeword. The forward path metric is defined as akÀ1ðs0 Þ PðSkÀ1 ¼ s0 ^ yjkÞ: ð5Þ It means the probability that the trellis is in state SkÀ1 = s0 at time k À 1 and the received codeword sequence up to this point is yjk, and can be computed recursively as akðsÞ ¼ X all s0 akÀ1ðs0 Þ Á ckðs0 ; sÞ: ð6Þ On the other hand, the definition of the backward path metric is expressed as bkðsÞ PðyjkjSk ¼ sÞ: ð7Þ It gives the probability that given the trellis is in state Sk = s at time k the future received codeword sequence will be yjk, and can be computed recursively as bkÀ1ðs0 Þ ¼ X all s bkðsÞ Á ckðs0 ; sÞ: ð8Þ From (2) and (3) we can write for the conditional LLR L(ukjy) of the decoded uk, given the received codeword yk LðukjyÞ ¼ ln P ðs0;sÞ;uk¼þ1akÀ1ðs0 Þ Á ckðs0 ; sÞbkðsÞ P ðs0;sÞ;uk¼À1akÀ1ðs0Þ Á ckðs0; sÞbkðsÞ ! : ð9Þ The MAP algorithm is extremely complex due to the exponential and natural logarithm operations required to calculate L(ukjy) using (9). To simplify the calculation, the Max-Log-MAP algorithm transfers the forward path metric ak(s), the backward path metric bk(s) and the branch metric ck(s0 ,s) into the log arithmetic domain and then uses the approximation ln X i eXi % max i fXig; ð10Þ where max i ðXiÞ means the maximum value of Xi. Then, with Ak(s), Bk(s) and Ck(s0 ,s) defined as follows: Y.-N. Lin et al. / Computer Communications 29 (2006) 3856–3862 3857
  4. 4. Author's personal copy AkðsÞ lnðakðsÞÞ; ð11Þ BkðsÞ lnðbkðsÞÞ; ð12Þ and Ckðs0 ; sÞ lnðckðs0 ; sÞÞ; ð13Þ we can change (6) and (8), respectively into AkðsÞ % max s0 ðAkÀ1ðs0 Þ þ Ckðs0 ; sÞÞ ð14Þ and BkÀ1ðs0 Þ % max s ðBkðsÞ þ Ckðs0 ; sÞÞ: ð15Þ Finally, from (9) we can write for the a posteriori LLR L(ukjy) which the Max-Log-MAP algorithm calculates LðukjyÞ % max ðs0;sÞ;uk¼þ1 ðAkÀ1ðs0 Þ þ Ckðs0 ; sÞ þ BkðsÞÞ À max ðs0;sÞ;uk¼À1 ðAkÀ1ðs0 Þ þ Ckðs0 ; sÞ þ BkðsÞÞ : ð16Þ 3. The stochastic correlation model For simplicity, the correlation among the individual bits yk1, . . . ,ykl, . . . ,ykN within the received codeword yk is neglected in the turbo decoding. However, various extent of correlation is incorporated into each transmitted codeword xk during the encoding process. Thus, yk1, . . . ,ykl, . . . ,ykN are correlated. Based on the known fact, the conditional joint probability P(ykjxk) required to calculate the branch metric ck(s0 ,s) is bounded by [11] YN l¼1 PðykljxklÞ 6 PðykjxkÞ 6 min l fPðykljxklÞg; ð17Þ where min l ðPðykljxklÞÞ means the minimum value of P(ykljxkl). Ideally, P(ykjxk) would be calculated directly. One of the possible adequate joint probabilistic models, which can simulate both the lower and upper bounds of inequality (17), is defined as PðykjxkÞ ¼ exp À XN l ½À lnðPðykljxklÞފF #1 F 0 @ 1 A; ð18Þ where 0 P(ykljxkl) 6 1. It can be shown that (18) reduces to the lower bound of inequality (17), when the parameter F = 1, and to the upper bound when F fi 1. However, the exponential and natural logarithm operations involved in (18) always make above joint probabilistic model impractical. From the inequality (17), we can observe that there are at least two ways of obtaining an approximate of P(ykjxk). In the first way using the lower bound, the indi- vidual bits yk1, . . . ,ykl, . . . ,ykN within the received code- word yk are assumed to be independent or weakly correlated. The conditional joint probability P(ykjxk) can be approximated as PðykjxkÞLB ¼ YN l¼1 PðykljxklÞ ¼ YN l¼1 1 ffiffiffiffiffiffiffiffi 2Áp p Ár Áexp À Eb 2Ár2 Áðykl ÀxklÞ 2 ; ð19Þ where Eb is the transmitted energy per bit and r2 is the noise variance. Upon substituting (19) in (4), we have the branch metric using lower bound: cLB k ðs0 ; sÞ ¼PðukÞ Á PðykjxkÞLB ¼PðukÞ Á YN l¼1 1 ffiffiffiffiffiffiffiffiffi 2 Á p p Á r Á exp À Eb 2 Á r2 Á ðykl À xklÞ2 ¼PðukÞ Á 1 ð ffiffiffiffiffiffiffiffiffi 2 Á p p Á rÞN Á exp À Eb 2 Á r2 Á XN l¼1 ðy2 kl þ x2 kl À 2 Á ykl Á xklÞ ¼CLB Á PðukÞ Á exp Eb r2 Á XN l¼1 ðykl Á xklÞ ; ð20Þ where CLB ¼ 1 ð ffiffiffiffiffiffiffiffiffi 2 Á p p Á rÞ N Á exp À Eb 2 Á r2 Á XN l¼1 ðy2 kl þ x2 klÞ ¼ 1 ð ffiffiffiffiffiffiffiffiffi 2 Á p p Á rÞ N Á exp À Eb 2 Á r2 Á XN l¼1 ðy2 klÞ þ N : ð21Þ By means of (21), the log branch metrics CLB k ðs0 ; sÞ using the lower bound in the recursive formulas of (14) and (15) de- rived for Ak(s) and BkÀ1(s0 ), respectively, can be written as CLB k ðs0 ; sÞ lnðcLB k ðs0 ; sÞÞ ¼ ln CLB Á PðukÞ Á exp Eb r2 Á XN l¼1 ðykl Á xklÞ ¼^CLB þ ln½Pðukފ þ Lc 2 Á XN l¼1 ðykl Á xklÞ; ð22Þ where ^CLB ¼ lnðCLBÞ and the channel reliability value Lc = 2ÆEb/r2 . The term ^CLB does not depend on the sign of the bit uk or the transmitted codeword xk and so is a constant in (16) and cancels out. The log branch metrics CLB k ðs0 ; sÞ derived from the lower correlation bound is wide- ly used in the turbo decoding. This has the advantage of low model complexity, but the disadvantage losing the bit-level stochastic correlation among systematic bits and the associated parity bits. The second way is to use the upper bound, which is valid if there is a strong correlation within a codeword. The min- imum value of fPðykljxklÞ; 1 6 l 6 Ng will dominate the calculation of conditional joint probability P(ykjxk) and PðykjxkÞUB ¼ min l ðPðykljxklÞÞ: ð23Þ Assume the ith term P(ykijxki) has the smallest value among fPðykljxklÞ; 1 6 l 6 Ng, then PðykjxkÞUB ¼ 1 ffiffiffiffiffiffiffiffiffi 2 Á p p Á r Á exp À Eb 2 Á r2 Á ðyki À xkiÞ 2 ð24Þ and the corresponding branch metric using upper bound: 3858 Y.-N. Lin et al. / Computer Communications 29 (2006) 3856–3862
  5. 5. Author's personal copy cUB k ðs0 ; sÞ ¼PðukÞ Á PðykjxkÞUB ¼CUB Á PðukÞ Á exp Eb r2 Á yki Á xki ; ð25Þ where CUB ¼ 1 ð ffiffiffiffiffiffiffiffiffi 2 Á p p Á rÞ Á exp À Eb 2 Á r2 Á ðy2 ki þ x2 kiÞ ¼ 1 ð ffiffiffiffiffiffiffiffiffi 2 Á p p Á rÞ Á exp À Eb 2 Á r2 Á ðy2 ki þ 1Þ : ð26Þ By means of (26), the log branch metrics CUB k ðs0 ; sÞ using the upper bound can be written as CUB k ðs0 ; sÞ lnðcUB k ðs0 ; sÞÞ ¼ ln CUB Á PðukÞ Á exp Eb r2 Á ðyki Á xkiÞ ¼^CUB þ ln½Pðukފ þ Lc 2 Á ðyki Á xkiÞ; ð27Þ where ^CUB ¼ lnðCUBÞ. The term ^CUB is also regardless of the sign of the bit uk or the transmitted codeword xk and so can be considered a constant in (16) and omitted. How- ever, there are N correlation components contributed to the calculation of the log branch metrics Ck(s0 ,s) in (22) whereas only one component in (27). It’s suggested to incorporate a gain factor N into the correlation term in (27). Thus, we have CUB k ðs0 ; sÞ ¼ ^CUB þ ln½Pðukފ þ Lc 2 Á ðN Á yki Á xkiÞ: ð28Þ To normalize the correlation terms for the log branch metrics using lower bound and upper bound in (22) and (28), we have CLB k ðs0 ; sÞ ¼ ^CLB þ ln½Pðukފ þ Lc 2 Á N Á 1 N Á yk1 Á xk1 þ Á Á Á þ 1 N Á ykN Á xkN ð29Þ and CUB k ðs0 ; sÞ ¼ ^CUB þ ln½Pðukފ þ Lc 2 Á N Á ½yki Á xkiŠ: ð30Þ Observing the difference between (29) and (30), we can find that the contribution made by each correlation component yklÆxkl in the correlation term PN l¼1ðykl Á xklÞ can be used to indicate the degrees of assumed bit-level correlation. There- fore, we can define the correlative log branch metrics CCORR k ðs0 ; sÞ, which takes account of the correlation be- tween each systematic bit and the associated parity bits, as CCORR k ðs0 ; sÞ ¼ ^CCORR þ ln½Pðukފ þ Lc 2 Á N Á XN l¼1 ðll Á ykl Á xklÞ; ð31Þ where ^CCORR is a constant, and the weights {ll, 1 6 l 6 N} are non-negative and satisfy the condition XN l¼1 ll ¼ l1 þ Á Á Á þ ll þ Á Á Á þ lN ¼ 1: ð32Þ Apparently, by properly adjusting the weights, we can achieve various extent of bit-level correlation. When the weights are equally distributed with fll ¼ 1 N ; 1 6 l 6 Ng, then the individual bits within a codeword are assumed to be weakly correlated or even independent. In this case, the term CCORR k ðs0 ; sÞ can be formulated as CLB k ðs0 ; sÞ in (22). On the other hand, neglecting the correlation components with higher conditional probability values and assigning unity to the weight corresponding to the correlation component with the lowest conditional probability value can approxi- mate the strongest bit-level correlation. 4. The effect of correlative turbo decoder In this section, we conduct a series of experiments to illustrate the effectiveness of the stochastic correlation model we proposed for turbo decoding. The system architecture for adaptation of the correlative turbo decoder is shown in Fig. 1. Information bits uk are grouped into blocks of bits and encoded with a turbo encoder consisting of a parallel concatenation of two recursive systematic convolutional (RSC) codes. The RSC component codes are K = 3 codes with generator polynomials G0 = 7 and G1 = 5 in octal representation. These generator polynomials are optimum in terms of maximizing the minimum free distance of the component codes [12]. The standard interleaver used between the two component RSC codes is a 1000-bit random inter- leaver with odd–even separation [13]. Puncturing of par- ity bits up1 k and up2 k can be performed to achieve the required one-third code rate. The coded bits are modu- lated using binary phase shift keying (BPSK) and white Gaussian noise n(t) with a double-sided power spectral density of N0/2 is added to the modulated signal s(t). At the decoder, only eight iterations are carried out, as no significant improvement in performance is obtained with a higher number of iterations. When the received signal r(t) is demodulated, the correlative decoder first estimates the SNR value ^cð^c ¼ Eb=2 Á N0Þ, and then deter- mines the channel reliability value Lc and the correlation weight {ll, 1 6 l 6 N}. In our simulation, there are two bits within the received codeword yk, i.e., N = 2. Assume that Pðyk1jxk1Þ ¼ min l ðPðykljxklÞÞ ¼ minfPðyk1jxk1Þ; Pðyk2jxk2Þg ð33Þ and according to (32) we have l1 þ l2 ¼ 1: ð34Þ If we let l1 = l, then l2 ¼ 1 À l1 ¼ 1 À l ð35Þ and the correlative log branch metrics CCORR k ðs0 ; sÞ in (31) can be rewritten as Y.-N. Lin et al. / Computer Communications 29 (2006) 3856–3862 3859
  6. 6. Author's personal copy CCORR k ðs0 ; sÞ ¼ ^CCORR þ ln½Pðukފ þ Lc 2 Á N Á fl Á yk1 Á xk1 þ ð1 À lÞ Á yk2 Á xk2g: ð36Þ It is worth to note that the weight l can be treated as the con- tribution of a correlation component with the lowest condi- tional probability value. In the case of l = 1, it implies that there exists strong bit-level stochastic correlation within each received codeword. When the weight l is set to a smaller val- ue, the degree of correlation can be further reduced. 4.1. The BERs versus the correlation weights Fig. 2 shows the performance of a turbo decoder using the stochastic correlation model versus different values of the weight l under predetermined Eb/N0 values at À1, 0 and +1 dB. From this figure, it can be seen that the perfor- mance of the correlative turbo decoder with +1 dB is poor at small weights, but improves rapidly as the weight is increased. After a specific weight, the BER performance degrades again. The similar phenomena can also be found in the cases of 0 dB and À1 dB. Apparently, the proposed stochastic correlation model has the effect on the BER per- formance of a turbo decoder. Fig. 2 reveals a fact that mul- tiplying a set of weights to the correlation components fðykl Á xklÞ; 1 6 l 6 Ng in the log branch metrics Ck(s0 ,s) is highly useful in modeling the underlying correlation and helpful in achieving better BER performance. 4.2. The relation between Eb/N0 value and the correlation weight In our simulation, the modulated signal is corrupted by white Gaussian noise in which the Eb/N0 value ranges from À2 dB to +3 dB with 0.2 dB increment. The correlation weight is also increased from 0 to 1 with increment of 0.05. From the experimental results, the relation between Eb/N0 value and the optimum correlation weight can be obtained approximately and is illustrated in Fig. 3. As shown in Fig. 3, the optimum correlation weight that achieves the best BER performance is heavily related to the Eb/N0 value of the underlying environment. In the less noisy condition (Eb=N0 P 1dB), the optimum weight is set to 0.5. At this time, the correlative log branch metrics CCORR k ðs0 ; sÞ is reduced to its lower bound CLB k ðs0 ; sÞ that is equivalent to conventional log branch metrics. When the transmission channel becomes noisier, larger weight is used for the stochastic correlation model in order to obtain opti- mum BER performance. ku Puncturing BPSK Demodulator Interleaver RSC 1 RSC 2 Determine Correlation Weighting factor )(tn )(tr )(ts cL SNR Interleaver Deinterleaver Interleaver kc , p ku Parallel-to-serial BPSK Modulator ky Channel SNR Estimation Block kc yL Serial-to-Parallel )( ,,2,1 Nμμ μ s kc yL Decoder1 Decoder2 1p kc yL 2p kc yL int s yL 12eL 21eL ku k p2 u u 1 k p Fig. 1. The system architecture for adaptation of the correlative turbo decoder. Fig. 2. The performance of a turbo decoder using the stochastic correlation model versus different values of the weight l. 3860 Y.-N. Lin et al. / Computer Communications 29 (2006) 3856–3862
  7. 7. Author's personal copy 4.3. The effectiveness of the correlative turbo decoding Fig. 4 compares the BER performances of the three cases, i.e., the uncoded case (denoted as ‘‘Uncoded’’), the turbo decoder based on the Max-Log-MAP algorithm (denoted as ‘‘TBC’’) and the ‘‘TBC’’ with stochastic corre- lation model (denoted as ‘‘CTBC’’). In the implementation of the ‘‘CTBC’’ case, the optimum correlative weight is adaptively selected block by block according to the follow- ing relation obtained from Fig. 3. lopt ¼ 0:50 if Eb=N0 0:8dB; 0:55 if 0:8dB P Eb=N0 À0:6dB; 0:60 if À 0:6dB P Eb=N0 À1:2dB; 0:65 if Eb=N0 À1:2dB: 8 : ð37Þ From the Fig. 4, we can find that the ‘‘Uncoded’’ and ‘‘TBC’’ cases have an intersection at about Eb/N0 = À0.8 dB. This phenomenon implies that the turbo decoding based on the Max-Log-MAP algorithm is no longer effec- tive when Eb/N0 is less than À0.8 dB. In contrast, the inter- section of the ‘‘Uncoded’’ and ‘‘CTBC’’ cases is at about À0.95 dB. This result verifies the effectiveness of the pro- posed stochastic correlation model. Just as with a conventional turbo decoder based on the Max-Log-MAP algorithm, the correlative turbo decoder needs to calculate the logarithmic forward path metrics Ak(s), the logarithmic backward path metrics Bk(s), and the logarithmic branch metrics Ck(s0 ,s) to find the a-poste- riori LLR L(ukjy). In addition, the correlative turbo decod- er requires N additional multiplication operations to calculate the correlation term Lc 2 Á N Á PN l¼1ðll Á ykl Á xklÞ as shown in (31). For a turbo decoder with code rate of 1/3 (i.e., N = 2), the proposed correlation model introduces only a little extra computation complexity. 5. Conclusion During the encoding process, a turbo encoder always makes a systematic bit and its associated parity bits cor- related. In this paper, a stochastic correlation model is proposed to explore the influence of bit-level correlation on the BER performance of a turbo decoder. By means of the correlation model, it provides us a useful frame- work for modeling the underlying stochastic correlation in a simple way. Experimental results reveal that the correlation effect has significant impact on the perfor- mance of a turbo decoder. By properly adjusting the correlative weights, we can accurately approximate the extent of bit-level correlation within each codeword and by which the robustness of a turbo decoder can be enhanced. References [1] C. Berrou, A. Glavieux, P. Thitimajshima, Near Shannon limit error- correcting coding and decoding. Turbo codes, Proc. Int. Conf. Commun., Geneva, Switzerland, (1993) 1064–1070. [2] C. Berrou, A. Glavieux, Near-optimum error-correcting coding and decoding: turbo codes, IEEE Trans. Commun. 44 (1996) 1261–1271. [3] J. Hokfelt, O. Edfors, T. Maseng, A turbo code interleaver design criterion based on the performance of iterative decoding, IEEE Commun. Lett. 5 (2001) 52–54. [4] F. Daneshgaran, M. Laddomada, Optimized prunable single-cycle interleavers for turbo codes, IEEE Trans. Commun. 52 (6) (2004) 899–909. [5] L.R. Bahl, J. Cocke, F. Jelinek, J. Raviv, Optimal decoding of linear codes for minimizing symbol error rate, IEEE Trans. Inform. Theory 5 (1974) 284–287. [6] W. Koch, A. Baier, Optimum and sub-optimum detection of coded data disturbed by time-varying inter-symbol interference, IEEE Globecom (1990) 1679–1684. [7] J.A. Erfanian, S. Pasupathy, G. Gulak, Reduced complexity symbol detectors with parallel structures for ISI channels, IEEE Trans. Commun. 42 (1994) 1661–1671. [8] P. Robertson, E. Villebrun, P. Hoeher, A comparison of optimal and sub-optimal MAP decoding algorithms operation in the log domain, Proc. Int. Conf. Commun. (1995) 1009–1013. Fig. 3. The relation between Eb/N0 value and the optimum correlation weight. Fig. 4. Comparison of BER performances for the ‘‘Uncoded’’ case, the ‘‘TBC’’ case and the ‘‘CTBC’’ case. Y.-N. Lin et al. / Computer Communications 29 (2006) 3856–3862 3861
  8. 8. Author's personal copy [9] Jason P. Woodard, Lajos Hanzo, Comparative study of turbo decoding techniques: an overview, IEEE Trans. Veh. Technol. 49 (6) (2000) 2208–2233. [10] L. Hanzo, T.H. Liew, B.L. Yeap, Turbo Coding, Turbo Equalisation and Space-Time Coding for Transmission over Fading Channels, sponsored by IEEE Communication Society, John Wiley Sons, Ltd, New York, 2002. [11] J. Ming, F.J. Smith, Stochastic correlation model for speech recognition, IEE Electron. Lett. 32 (11) (1996) 970–971. [12] J.G. Proakis, Digital Communication, third ed., McGraw-Hill, New York, 1995. [13] A.S. Barbulescu, S.S. Pietrobon, Interleaver design for turbo codes, IEE Electron. Lett. (1994) 2107–2108. Yi-Nan Lin received his B.S. degree from the Electrical Engineering Department of National Taiwan Institute of Technology in 1989, and the M.S. degree in Computer Science Engineering from the Yuan Ze University in 2000. He joined the Department of Electrical Engineering at Mingchi University of Technology, Taishan, Taiwan, in 1990. He is now a lecturer in the Department of Electronic Engineering. He is also a Ph.D. candidate in the Electrical Engineering Department of Chang Gung University, Taoyu- an, Taiwan. His current research interests include error-control coding, and digital transmission systems. Wei-Wen Hung received his B.S. degree from the Electrical Engineering Department of Tatung Institute of Technology in 1986, and the M.S. and Ph.D. degrees in electrical engineering from the National Tsinghua University in 1988 and 2000, respectively. He joined the Department of Electrical Engineering at Mingchi University of Technology, Taishan, Taiwan, in 1990. He was the Vice Dean of Student Affairs from 2000 to 2002. He was also the Chairman of Department of Electronic Engineering in 2003. He is now a professor in the Department of Electronic Engineering. His current research interests include speech signal processing, wireless communica- tion and embedded system design. Tsan-Jieh Chen is now an undergraduate, majored in electrical engineering in Mingchi University of Technology. His current interests include signal processing, wireless communica- tion, VLSI design and coding theory. Erl-Huei Lu received his B.S. and M.S. degrees in electrical engineering from the Chung Cheng Institute of Technology, Taiwan, in 1974 and 1980, respectively, and the Ph.D. degree in elec- trical engineering from the National Cheng Kung University, Tainan, Taiwan, in 1988. He is now a professor in the Department of Electrical Engi- neering, Chang Gung University, Taoyuan, Tai- wan. His current research interests include error- control coding, network security and systolic architectures. 3862 Y.-N. Lin et al. / Computer Communications 29 (2006) 3856–3862

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