A New List Decoding Algorithm for Short-Length TBCCs With CRC

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1 Received May 15, 2018, accepted June 11, 2018, date of publication June 14, 2018, date of current version July 12, Digital Object Identifier /ACCESS A New List Decoding Algorithm for Short-Length TBCCs With CRC JAE-WON KIM, JUN-WOO TAK, HEE-YOUL KWAK, AND JONG-SEON NO, (Fellow, IEEE) Department of Electrical and Computer Engineering, INMC, Seoul National University, Seoul 08826, South Korea Corresponding author: Jae-Won Kim (kjw702@ccl.snu.ac.kr) This work was supported by the National Research Foundation of Korea through the Korean Government (MSIP) under Grant NRF-2016R1A2B ABSTRACT In this paper, a new list decoding algorithm for tail-biting convolutional codes (TBCCs) with a cyclic redundancy check (CRC) is proposed, where the CRC is considered as a concatenated outer code. The main idea of the proposed algorithm is to modify the list decoding procedure of the TBCC by using the CRC. Two algorithms are proposed for the list decoding of the TBCC with the CRC. The first proposed algorithm is a new initial state estimating algorithm using re-encoded CRC bits and having the low computational complexity. The other proposed algorithm is a modified list Viterbi algorithm, where trellis paths are fixed by re-encoded CRC bits and some CRC bits are used for the error correction. For the TBCC concatenated with the CRC code defined in the long-term evolution standard, the proposed decoding scheme by partially using CRC bits outperforms the conventional list decoding algorithms for the list size L = 4 even though the proposed algorithm has the lower decoding complexity. INDEX TERMS Cyclic redundancy check (CRC), list decoding algorithm, list Viterbi algorithm (LVA), tail-biting convolutional codes (TBCCs), Viterbi algorithm. I. INTRODUCTION Convolutional codes have been widely used since the maximum likelihood (ML) decoding method which estimates the trellis path with the low complexity was proposed by Viterbi, called Viterbi algorithm [1]. When encoding convolutional codes conventionally, the all zero bit sequence is padded and thus the initial state and the final state of the convolutional codes are fixed to the all zero state. However, padding the additional zero bit sequence causes rate loss of the convolutional code. To deal with this problem, a tailbiting convolutional code (TBCC) was proposed [2] and its decoding method was researched [3]. Since the initial state of the TBCC is unknown, several decoding algorithms for the TBCC have been studied. Since the ML decoder performing Viterbi decoding for each initial state requires the very large decoding complexity, there are a lot of researches on reducing the computational complexity [4] [8]. The wrap-around Viterbi algorithm and the bidirectional Viterbi algorithm were proposed in [9] and the reverse trellis algorithm was also proposed in [10]. In order to improve the decoding performance of the TBCC, a list Viterbi algorithm (LVA) was proposed, which generates multiple codeword candidates instead of one codeword [11]. The TBCC with cyclic redundancy check (CRC) is used as a standard for the Long-Term Evolution (LTE) system, where the CRC can be considered as a concatenated outer code. Furthermore, the short-length TBCC with CRC had been considered as a standard candidate for the control channel of the 5G cellular system. The decoding method using the LVA and the CRC was studied for the TBCC with CRC [12], where the initial state should be estimated before the decoding procedure because of the nature of the TBCC whose initial state is not known. Initial state estimating algorithms of TBCCs were studied in [13] [15] and the undetected error probability of concatenated coding systems was studied in [16] and [17]. In this paper, we first propose a method to effectively estimate the initial state of the TBCC with CRC using the structural characteristics of the CRC. Next, we propose a decoding method partially using the CRC bits in the LVA to improve the block error rate (BLER) performance. This paper is organized as follows. The system model and the background knowledges are given in Section II. A new initial state estimating method and a modified LVA for the TBCC with CRC are given in Section III. In Section IV, the simulation results of the proposed list decoding algorithms are compared with those of the existing list VOLUME 6, IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See for more information

2 decoding algorithms. In Section V, some future work is listed. Finally, conclusions are given in Section VI. II. SYSTEM MODEL AND BACKGROUND A. TBCC WITH CRC The TBCC with CRC in the LTE standard [18] can be considered as a concatenated code of the TBCC and the CRC code. The inner code is the TBCC of the code rate 1/3, the number of memories m = 6, and generator polynomials (133, 171, 165) in the octal format. The outer code is the CRC code of length r = 16 (CRC-16-CCITT, the generator polynomial x 16 +x 12 +x 5 +1). The rate matching is conducted based on the circular buffer and interleaving is used as in the LTE standard. B. LIST DECODING ALGORITHMS FOR TBCC WITH CRC List decoding algorithms [19], [20] for the TBCC concatenated with the CRC code consist of three steps, where the list size is L. First, the L s initial state candidates should be selected. Second, for each initial state candidate, perform the LVA, which produces L c codeword candidates [11]. Last, select the best L out of L s L c codeword candidates and the codeword having the lowest path metric value and satisfying the CRC is selected for the final decoding output. If all of the L codeword candidates do not satisfy the CRC, the decoding failure or erasure is declared. In the LVA, there are three kinds of list Viterbi decoding methods, that is, sequential decoding, parallel decoding, and per-stage decoding. Among these, the parallel LVA is concerned in this paper, where the best L trellis paths are selected among 2L paths at each state. The overall system model of the TBCC with CRC in this paper is summarized in Fig. 1. III. PROPOSED ALGORITHMS Two algorithms are proposed for the list decoding of the TBCC concatenated with the CRC code. The first one is a new initial state estimating algorithm using the CRC code. The other one is a modified LVA which fixes trellis paths in the stages of CRC bits. A. NEW INITIAL STATE ESTIMATING ALGORITHM In this section, a new initial state estimating algorithm with the low computational complexity is proposed. The purpose of the proposed initial state estimating algorithm is to reduce the computational complexity with the minor performance degradation. Specifically, the proposed algorithm has the similar performance to that in [20] but has the lower computational complexity. Since the purpose of this step (Step 1 in Section II-B) is to find initial state candidates, it is not needed to find the correct codeword. In fact, it is enough to find the last m bits of input messages of the TBCC (i.e. the last m bits of the CRC bits) because the initial state of the TBCC is determined by them, where it is assumed that m is less than or equal to the CRC size as in the LTE standard. The main idea of the proposed algorithm is to estimate the initial state of the TBCC by using the CRC. If we have reliable information bits and do CRC re-encoding with these information bits, the last m bits of the re-encoded CRC bits will also be reliable as the signal to noise ratio (SNR) goes high. Thus, we can estimate the last m bits of input messages of the TBCC encoder by reliably decoding the information bits and CRC re-encoding. Algorithm 1 A New Initial State Estimating Algorithm Input: The received sequence, information size, code rate, CRC code generator polynomial, TBCC generator polynomials, and L s. Output: L s initial state candidates. Step 1) Select the starting point of the Viterbi decoding in the trellis diagram of the TBCC as the first parity bit of CRC and repeat the Viterbi decoding over the trellis diagram twice. Step 2) The information bits are obtained by tracebacking the path of the state having the lowest path metric value at stage a in Fig. 2 (a red straight line in Fig.2). Step 3) CRC re-encoding is performed with the decoded information bits and the last m bits of the reencoded CRC bits are checked if they match the state reached by traceback in the decoding process (a red dotted circle in Fig. 2). Step 4) If they are matched, select the state reached by traceback as an initial state candidate. If they are not matched, the state is not selected as an initial state candidate. Do the same process for the states at stage a in order of the states having the lower path metric values until there are L s initial state candidates. Step 5) If L s initial state candidates are not selected after the above process, a circular Viterbi decoding algorithm is applied to the trellis of the additional stages of CRC bits and the states are added to the initial state candidate group in order of the path metric values at stage b (a green straight circle at stage b in Fig. 2). The TBCC can be decoded from any stage of the trellis diagram and the bits located in the initial part of the trellis diagram are likely to be erroneous. Thus, we select the starting point of the Viterbi decoding in the trellis diagram of the TBCC as the first parity bit of CRC as in Fig. 2 because we do not need the first r m bits of CRC for the initial state estimation. To obtain reliable information bits, we repeat the Viterbi decoding over the trellis diagram twice and have the additional stages of CRC bits. In the trellis boundaries, each path metric value is not initialized as 0 and thus retain its value. The detailed description of the proposed algorithm is given in Algorithm 1. In Algorithm 1, it can be an initial state with a high probability if the last m bits of the re-encoded CRC bits from the decoded information bits match the state reached VOLUME 6, 2018

3 J.-W. Kim et al.: New List Decoding Algorithm for Short-Length TBCCs With CRC FIGURE 1. The overall system model of the TBCC with CRC. FIGURE 2. The repeated trellis diagram of the proposed initial state estimating algorithm for the TBCC with CRC. by tracebacking the information bits in the decoding process. If there are less than Ls states satisfying this condition, we just select the states in order of the lower path metric values at the last stage of the trellis. Table 1 shows the probabilities that the actual initial state exists in the initial state candidates obtained from Algorithm 1, where binary phase shift keying (BPSK) is assumed and Ls = 4. In addition, the probability that the actual initial state exists in the initial state candidates obtained from Step 4 of Algorithm 1 is also listed. It can be noted that most of actual initial states are selected by CRC re-encoding in Algorithm 1. Fig. 3 shows the error probabilities that the initial state estimating algorithms do not find the correct initial state, where the information size is 48, Ls = 4, and the Chen-Sundberg (CS) algorithm in [19] is the optimal exhaustive search algorithm which needs the very high computational complexity. From Fig. 3, Algorithm 1 has the similar performance to that in [20] even though Algorithm 1 has the lower computational complexity, which will be explained in Section IV. TABLE 1. The probabilities that the actual initial state exists in the initial state candidates for Ls = 4. FIGURE 3. The error probabilities that the initial state estimating algorithms do not find the correct initial state, where the information size is 48 and Ls = 4. B. MODIFIED LVA After estimating Ls initial state candidates, the LVA is applied to each initial state candidate (Step 2 in Section II-B). As described earlier, the conventional parallel LVA selects L paths for each state and selects Lc codeword candidates in VOLUME 6,

4 order of path metric values at the last stage. After performing the LVA for each initial state candidate, the conventional list decoding algorithm selects the best L out of L s L c and the final decoding output is the decoded information bits of the codeword candidate having the lowest path metric value and satisfying the CRC. CRC is usually used to detect the errors in the decoded output but the CRC can also be used in the error correction as in [21], where the syndrome decoding is considered for the CRC. Since the required error detecting capability of the decoder can be changed based on system constraints and parameters such as the channel SNR, it may not be effective to use the CRC only for error detection of the decoded codewords. Algorithm 2 A Modified LVA Input: The received sequence, information size k, code rate, CRC code generator polynomial, TBCC generator polynomials, initial state, L c, t, and list size L. Output: L c codeword candidates. Step 1) Perform the conventional LVA up to stage k + t. Step 2) Traceback the surviving paths at stage k + t to obtain information bits and do CRC re-encoding for each information bit sequence. Step 3) Based on the re-encoded CRC bits, extend the fixed trellis paths after stage k +t and calculate the path metric values for the fixed trellis paths ended in the corresponding initial state. For extended paths which do not end in the corresponding initial state, those paths are discarded. Step 4) The output list consists of L c codeword candidates, which are selected in order of trellis paths having the lower path metric values. Instead of using the CRC only for the error detection of the decoded codewords, we propose a method to fix trellis paths in stages of the CRC bits using the re-encoded CRC bits as in Fig. 4(b) to find a more reliable codeword candidate. Let k be the number of information bits and t (0 t r = 16) be the number of first CRC bits processed by the LVA. The conventional LVA selects L paths among 2L paths for each state in order of path metric values to the end of the trellis. However, in the proposed method, when the LVA is proceeded up to stage k + t, CRC re-encoding is performed with the decoded information bits by traceback for each surviving path at stage k + t. Then, each surviving path at stage k + t is extended based on the last r t reencoded CRC bits after stage k + t and the path metric values are further calculated for the fixed trellis paths ended in the corresponding initial state. For the extended paths which do not end in the corresponding initial state, those paths are discarded. It is noted that although we enforce those paths to end in the corresponding initial state, those paths do not satisfy the CRC because the extended path is based on the CRC. FIGURE 4. The example of the modified LVA for L = 2: (a) The conventional LVA. (b) The modified LVA. If the path does not satisfy the CRC, it is never selected as the final decoding output of the list decoding algorithm and thus that path can be discarded. In the proposed LVA, the process of selecting L out of 2L for each state after stage k + t is omitted. The detailed description of the proposed LVA is given in Algorithm 2. From Algorithm 2, we can prevent an error event that a true codeword with a slightly poor path metric value satisfying the CRC is not selected due to an erroneous codeword unsatisfying the CRC with a better path metric value. Finally, CRC check can be done by using the decoded t CRC bits at the last part of the list decoding algorithm. For the list decoding algorithm using the proposed LVA with t and the list size L, there are L TBCC codeword candidates and only the first t bits of the CRC part of each TBCC codeword candidate can detect the errors because remaining r t bits always satisfy the CRC. Then, the error detection probability (i.e. the probability that t bits of the CRC part of each TBCC codeword candidate do not satisfy the CRC given that each TBCC codeword candidate is incorrect) can be approximated as (1 1 2 t ) L and the miss detection probability (i.e. there is at least one TBCC codeword candidate that t bits of the CRC part satisfy the CRC given that each TBCC codeword candidate is incorrect) can be approximated as 1 (1 1 2 t ) L. Thus, the proper value of t can be selected from the required error detecting capability. If the channel SNR is sufficiently large and thus the probability that the incorrect TBCC codeword is selected is low, the small error detection capability is needed and we can choose the small value of t to improve the BLER performance VOLUME 6, 2018

5 IV. ANALYSIS OF PROPOSED ALGORITHMS A. COMPARISON OF COMPLEXITY There are many algorithms dealing with initial state estimating methods for the list decoding algorithm [12], [19], [20], [22], [23]. Among them, the CS algorithm in [19] is the optimal exhaustive search algorithm. Since the purpose of the proposed initial state estimating algorithm is reducing the computational complexity and the algorithm in [20] has the lowest computational complexity among them, we select an initial state estimating algorithm by QUALCOMM [20] for complexity comparison of the proposed one. We also compare the computational complexity between the conventional LVA and the proposed LVA. The initial state estimating algorithm in [20] repeats the trellis diagram three times and performs a circular Viterbi algorithm to find the reliable trellis path for the initial state estimation. Let s be the number of the soft decision bits and the path metric bits used in the Viterbi operation and assume that comparison between two path metric values are s times more complex than comparison between two values of one bit. Based on the bit operation, we compute the complexity of addition and subtraction between two values of c bits as c. We also compute the complexity of one operation of addition modulo 2 as 1. Then, the computed complexity is higher than the actual complexity for CRC re-encoding. Also, we compute the sorting complexity of d variables as ( d 2). First, we consider the computational complexities of the proposed initial state estimating algorithm and the initial state estimating algorithm by QUALCOMM. The proposed initial state estimating method performs addition and comparison of the path metric values for 2k + 3r stages, sorting path metric values in stage a and the last stage in Fig. 2, CRC re-encoding for information bit sequences, and checking the last m = 6 reencoded CRC bits for each information bit sequence. Based on the bit operation, the computational complexities of each process are (2k + 3r) 2 m+1 s, (2k + 3r) 2 m s, 2 ( 2 m ) 2 s, 2 m 2k, and 2 m m, respectively, where the average case for CRC re-encoding is assumed. The initial state estimating method by QUALCOMM performs addition and comparison of the path metric values for 3k +3r stages and sorting path metric values in the last stage. Based on the bit operation, the computational complexities of ( each process are (3k + 3r) 2 m+1 s, (3k + 3r) 2 m s, and 2 m) 2 s, respectively. The complexity ratio of Algorithm 1 to that in [20] is 88.6%, 84.2%, and 81.6% for k = 32, 48, 64, respectively, where s = 6. Next, we compute the computational complexities of the proposed LVA and the conventional LVA. The proposed LVA performs addition and sorting of the path metric values for k+t stages, CRC re-encoding 2 m L information bit sequences, checking the last m re-encoded CRC bits, the path metric value addition for the fixed paths, and sorting z path metric values at the last stage, where z is the number of the fixed paths ended in the corresponding initial state. Based on the bit operation, the computational complexities of each process are 2 m+1 L(k + t)s, 2 m (k + t)s ( 2L) 2, 2 m L 2k, 2 m L m, (r t)zs, and ( z 2) s, respectively, where the average case for CRC re-encoding is assumed. The conventional LVA performs addition and sorting of the path metric values for k + r stages. Based on the bit operation, the computational complexities of each process are 2 m+1 ksl + ((r m) 2 m m + 2 m )sL and ( 2 m ks + ((r m) 2 m + 2 m m )s ) ( 2L ) 2, respectively. The complexity ratio of Algorithm 2 to the conventional LVA is 77.5%, 84.6%, and 88.7% for k = 32, 48, 64, respectively, where s = 6, L = 4, t = 0, and expectation of 16z + ( z 2) is calculated by simulation. Note that the complexity comparison for the last part of the list decoding algorithm, that is, selecting the final decoding output satisfying CRC is ignored because that complexity of the conventional list decoding algorithm is obviously higher than that of the proposed list decoding algorithm. The computational complexities of all algorithms corresponding to the first and second steps of the list decoding algorithm are summarized in Table 2. TABLE 2. The computational complexities of the algorithms based on the bit operation for m = 6 and r = 16. B. SIMULATION RESULTS We simulate the TBCC concatenated with the CRC-16 code as mentioned in Section II based on the LTE standard [18], where BPSK is assumed. The BLER is calculated including both detected errors and undetected errors. For the message lengths 32, 48, 64 and the code rates 1/3, 1/6, the proposed list decoding algorithm using the initial state estimating method as Algorithm 1 and the modified LVA (Algorithm 2), denoted as Prop. LVA-Init. outperforms or is similar in performance to the conventional list decoding algorithm using the initial state estimating method by QUALCOMM and the conventional LVA, denoted as LVA-Q in Fig. 5, where L s = 4, L c = 4, t = 0, and L = 4. We choose L s = 4 because the main idea of the initial state estimating method in [20] is to choose four initial state candidates from the best reliable trellis path. Fig. 5 also shows that the list decoding algorithm using the initial state estimating method as the CS algorithm and the proposed LVA (Algorithm 2), denoted as Prop. LVA-CS outperforms the conventional list decoding algorithm using the initial state estimating method as the CS algorithm and the conventional VOLUME 6,

6 J.-W. Kim et al.: New List Decoding Algorithm for Short-Length TBCCs With CRC FIGURE 5. BLER performance for the TBCC with CRC-16, where Ls = 4, Lc = 4, t = 0, and L = 4: (a) Information size 32. (b) Information size 48. (c) Information size 64. LVA, denoted as LVA-CS. ML w.o. list decoding in Fig. 5 means that performing Viterbi decoding for all initial states (2m = 64 states in this case) without using the list decoding. Since it is a concatenated coding system, the ML decoder for the TBCC does not have the optimal BLER performance in the perspective of the overall concatenated coding system. Although the proposed LVA sacrifices some of the detection capability, there is a trade-off between the BLER performance gain and the detection capability depending on t. Fig. 6 shows the BLER performance based on t, where the initial state estimating algorithm is the CS algorithm for the proposed LVA and the conventional LVA. From Fig. 6, it can be noted that the BLER performance of the proposed LVA for non-zero t is degraded compared to the BLER performance for t = 0 but still outperforms the BLER performance of the conventional LVA. Thus, by choosing t based on system constraints and parameters such as the required miss detection rate of the decoder and the SNR, the BLER performance can be enhanced through the proposed LVA. V. FUTURE WORK First, it has to be studied how to apply the proposed algorithms for the recursive convolutional encoders because FIGURE 6. BLER performance for the proposed LVA and the conventional LVA, where the code rate is 1/3, k = 32, Ls = 4, Lc = 4, and L = 4. this paper only deals with the feedforward convolutional encoders. Second, since there are many initial state estimating algorithms, they will show some trade-offs between the computational complexities and the performance. To compare these algorithms fairly, the more correct numerical parameters for the fair complexity comparison and a new analysis method have to be researched. VOLUME 6, 2018

7 VI. CONCLUSIONS In this paper, a new list decoding algorithm for the TBCC concatenated with the CRC codes was proposed. First, we proposed a new initial state estimating method using a circular Viterbi algorithm and the CRC. Then, a modified LVA was proposed by fixing the trellis paths using the re-encoded CRC bits. By partially using the CRC bits, the proposed decoding algorithm outperforms the conventional list decoding algorithms in [19] and [20] for the TBCC concatenated with the CRC-16 code defined in the LTE standard even though the proposed algorithm has the lower decoding complexity. REFERENCES [1] A. J. Viterbi, Error bounds for convolutional codes and an asymptotically optimum decoding algorithm, IEEE Trans. Inf. Theory, vol. 13, no. 2, pp , Apr [2] H. Ma and J. Wolf, On tail biting convolutional codes, IEEE Trans. Commun., vol. COMM-34, no. 2, pp , Feb [3] R. V. Cox and C.-E. W. 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Zhao, and G. B. Giannakis, CRC-assisted error correction in a convolutionally coded system, IEEE Trans. Commun., vol. 56, no. 11, pp , Nov [22] M. Handlery, R. Johannesson, and V. V. Zyablov, Boosting the error performance of suboptimal tailbiting decoders, IEEE Trans. Commun., vol. 51, no. 9, pp , Sep [23] J. Wang, M. Korb, K. Zhang, H. Kröll, Q. Huang, and J. Wei, Parallel list decoding of convolutional codes: Algorithm and implementation, IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 64, no. 10, pp , Oct JAE-WON KIM received the B.S. degree in electrical and computer engineering from Seoul National University, Seoul, South Korea, in 2014, where he is currently pursuing the Ph.D. degree in electrical engineering and computer science. His area of research interests includes error-correcting codes, coding theory, and index coding. JUN-WOO TAK received the B.S. degree in information and communications from Korea University, Seoul, South Korea, in He is currently pursuing the Ph.D. degree in electrical engineering and computer science with Seoul National University, Seoul. His area of research interests includes information theory and wireless communication. HEE-YOUL KWAK received the B.S. degree in electrical and computer engineering from Seoul National University, Seoul, South Korea, in 2013, where he is currently pursuing the Ph.D. degree in electrical engineering and computer science. His area of research interests includes error-correcting codes, coding theory, and coding for memory. JONG-SEON NO (S 80 M 88 SM 10 F 12) received the B.S. and M.S.E.E. degrees in electronics engineering from Seoul National University, Seoul, South Korea, in 1981 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, CA, USA, in He was a Senior MTS with Hughes Network Systems from 1988 to He was an Associate Professor with the Department of Electronic Engineering, Konkuk University, Seoul, from 1990 to He joined the Faculty of the Department of Electrical and Computer Engineering, Seoul National University, in 1999, where he is currently a Professor. His area of research interests includes error-correcting codes, sequences, cryptography, LDPC codes, interference alignment, and wireless communication systems. He was a recipient of the IEEE Information Theory Society Chapter of the Year Award in From 1996 to 2008, he served as a Founding Chair of the Seoul Chapter of the IEEE Information Theory Society. He was a General Chair of Sequence and Their Applications, Seoul, in He also served as a General Co-Chair of the International Symposium on Information Theory and Its Applications, Seoul, in 2006, and the International Symposium on Information Theory, Seoul, in He has been a Co-Editor-in-Chief of the IEEE JOURNAL OF COMMUNICATIONS AND NETWORKS since VOLUME 6,