RECENT DEVELOPMENT OF ERROR CONTROL CODES FUTURE COMMUNICATION AND STORAGE SYSTEMS FOR Polar Codes for Soft Decode-and-Forward in HalfDuplex Relay Channels Fangliao Yang, Kai Niu, Chao Dong, Baoyu Tian Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications Beijing 100876, China Abstract: Soft decode-and-forward (DF) can combine the advantages of both amplify-and-forward and hard DF in relay channels. In this paper, we propose a low-complexity soft DF scheme based on polar codes, which features two key techniques: a low-complexity cyclic redundancy check (CRC) aided list successive cancellation (CALSC) decoder and a soft information calculation method. At the relay node, a low-complexity CALSC decoder is designed to reduce the computational complexity by adjusting the list size according to the reliabilities of decoded bits. Based on the path probability metric of the CALSC decoder, we propose a method to compute the soft information of the decoded bits in CALSC. Simulation results show that our proposed scheme outperforms the soft DF based on low-density parity-check codes and the soft DF with belief propagation or soft cancellation decoder, especially in the case when the source-relay channel is at the high signal-to-ratio region. Keywords: Decode and forward; soft information relaying; Polar codes; relay channels I. INTRODUCTION The relay channel, introduced by Van der Meulen [1], has attracted significant attentions for over two decades. Various coding techChina Communications • August 2017 niques have been studied [2-6], [26-27] and applied to relay channels or communication network. Besides, various relaying protocols have been proposed to get a trade-off between power consumption, processing delay and spectral efficiency. Two of the most widely used relaying protocols are amplify-and-forward (AF) and decode-and-forward (DF) [7]. Under AF, the relay retains the soft information of the received signal and does not make any hard decisions. However, it will amplify the channel noise. Under DF, the relay first decodes the received signal and re-encodes the decoded bit to generate an additional part of parity bits. As DF can eliminate the effect of noise amplification, it can provide a substantial improvement in performance. However, when the source-relay channel condition is poor, decoding errors will happen at the relay and error propagation will deteriorate the system performance if the error bits are transmitted to the destination. In conclusion, neither AF nor DF outperforms the other in all scenarios. In order to combine the advantages of both AF and DF, an advanced relaying protocol called the soft DF or soft information relaying (SIR) was proposed [8], in which the relay performs the decoding/re-encoding process in a soft manner and transmits the soft information to the destination (Similarly, we refer to Received: Feb. 28, 2017 Revised: May 21, 2017 Editor: Ming Xiao 22 the DF with hard decision at the relay as the hard DF). In [8], [9], a soft-input soft-output (SISO) encoder with modified BCJR algorithm was proposed. The rate-compatible low-density parity-check (LDPC) codes were introduced in the soft DF scheme in [10]. In [11], a soft decode-compress forward scheme was proposed. It has been demonstrated that the soft DF can outperform AF and hard DF substantially. Polar codes [12], which have been proven to be a family of capacity-achieving codes for the arbitrary binary-input discrete memoryless channel (B-DMC), have shown better performance than turbo codes and LDPC codes under cyclic redundancy check (CRC) aided decoding algorithms with short-to-medium block length [13], [14], [25]. Polar codes have been good choices in many communication scenarios, including relay channels. Polar codes were first studied in the relay channels in [15]. In the remarkable work [16], polar codes have been proven to achieve the capacity of the relay channels with orthogonal receiver components by using a nested structure. In [17], polar codes were used for the Gaussian degraded channel with DF and show substantial performance gain compared with LDPC codes. In [18], a smart DF scheme was proposed, in which the relay employs a threshold of log likelihood ratio (LLR) to determine whether to transmit the signal to the destination or to remain silent. In [19], polar codes based on Plotkin’s construction are employed to achieve coded cooperation between two users. In [20], a class of new DF strategies named generalized partial information relaying protocol is developed for the degraded multiple-relay networks that have orthogonal receiver components based on the nested structure of polar codes. In [21], a cooperative polar coding scheme based on partially perfect bit channels retransmission is proposed. However, the current works for polar codes in relay channels are all based on hard DF, which are easily suffered from error propagation. That is to say, if the condition of the source-relay channel is poor, the codes will be erroneous- 23 ly decoded and affect the distributed code construction at the relay, no matter Plotkin’s construction [19], [21] or nested construction [15], [16], [17], [18], [20]. The soft DF may be a good solution. The main purpose of this paper is to investigate the use of soft DF with polar codes in the half-duplex relay channels. Here, we address the main challenge that arises when adopting the polar codes under the soft DF scheme. As the soft DF requires that the processing at the relay should be in a soft manner, the intuitive idea is to utilize the SISO decoding algorithms, like the belief propagation (BP) and soft cancellation (SCAN) algorithms [22] to generate the soft information directly. However, they have some disadvantages. On the one hand, the complexity of BP is considerable due to the iterative decoding process. On the other hand, as the message transmission procedure of BP and SCAN are all based on the factor graph representation, their performance are limited by the short-length loops and inferior to other improved SC decoders, like the list successive cancellation (LSC) decoders and CRC-aided LSC decoder (CALSC) [14]. The main challenge is that we cannot employ LSC or CALSC at the relay directly since it cannot generate soft information. The contributions of this paper are listed in the following: First, we propose a soft DF scheme by utilizing a CALSC decoder at the relay instead of the BP and SCAN. The relay first decodes the received signal with a CALSC decoder, then computes the soft information of decoded bits of CALSC and finally encodes the soft information with an SISO polar encoder. Secondly, considering that the relay is usually energy-constrained in practical applications, we propose a low-complexity CALSC decoder to reduce the computational complexity. In the traditional CALSC decoder, the number of decoding paths (list size) in each decoding level (for each bit) is equal. Motivated by the conjecture that if the reliabilities of the decoded bits are high enough, the number of decoding paths can be reduced with minor China Communications • August 2017 performance loss, we propose a low-complexity CALSC decoder with dynamic list size. Simulation results show that the complexity of the proposed scheme reduces significantly as the channel condition becomes better. Finally, considering that the CALSC decoder employs a hard decision for each decoding path and cannot generate soft information of the decoded bits, we propose a method to compute the soft information based on the path probability metric of the CALSC decoder. Aided by the check results of CRC, we modify the soft information by flipping the sign of the LLRs according to the decoded bits which pass the CRC. Simulation results show that the soft DF scheme based on the proposed method can outperform BP and SCAN. Especially in the case when the condition of source to relay channel is good, the performance gain is remarkable. The rest of the paper is organized as follows. In section II, preliminaries and the system model are introduced. The proposed soft DF scheme is developed in section III. In section IV simulation results are provided to demonstrate the improved performance of our proposed scheme. Finally, conclusions are drawn in section V. II. PRELIMINARIES AND SYSTEM MODEL 2.1 Notation conventions In this paper, we use bold letters, such as , to denote matrix. We use calligraphic characters, such as , to denote sets, and to denote the cardinality of sets. Let denote the complementary set of . We use notation to denote an N-dimen- sion vector and subvector of We write to denote a , . to denote a subvector . length of polar codes be , . , where Encoding: the uncoded bits, denotes denotes the coded bits, , where is an N×N permutation matrix, is the kernel matrix, and denotes the n-th Kronecker power of . By recursively applying the techniques of channel combining and splitting, N independent uses of the B-DMC W can be transformed into uses of synthesized channels, denoted . The transition probabili- as ty of is expressed as (1) where It is proven that the synthesized channels are polarized to reliable ones and unreliable ones with the increasing code length [12]. The sets of reliable and unreliable channels are referred to as the information set and the frozen set , respectively. We can utilize the reliable channels to transmit the information bits and the unreliable ones to transmit fixed bits which are known to both encoder and decoder in advance. Decoding: the decoding can be performed with successive cancellation (SC) algorithm, which is described as follows: let the estimation of ly from to , denote is decoded successive- and is given by (2) where 2.2 Polar codes (3) Let denote a B-DMC with input alphabet , ouput alphabet , and transition probability , , . Let the 2.3 System model China Communications • August 2017 In this paper, we consider a typical three-node relay channel. It consists of a source node, a 24 relay node, and a destination node, which are denoted as s, r and d, respectively. The transmission on the channel is divided into twotime slots. In the first time slot, the source encodes the information and broadcasts it to the relay and destination simultaneously. In the second time slot, the relay processes the signal received from the first time slot and transmits the signal to the destination while the source node remains silent. For simplicity, we assume that each node has just one antenna and the modulation scheme is binary phase shift keying modulation (BPSK). Besides, the s-r channel, the r-d channel, and the s-d channel are all assumed to be the additive white Gaussian noise (AWGN) channels. At the source node, the modulated signal is denoted as , where is the length of the signal. At the relay node, the received signal and the re-transmitted signal are denoted as and respectively. is the transmission power of the source and is the noise of the s-r channel. Relay BP SCAN BP/SCAN SCA Decoder Decod w1N AWG GN N Channel AWGN Channel Destination Source Polar encoder c1N BPSK x1N AWGN AWG N Channel Pola olar Decooder LLR Calculation Fig. 1a The intuitive soft DF scheme with BP/SCAN decoder Relay CALS SC Decoder Decod Sof Soft Info f rmation fo Calculation Polar encod ncoder c1N Destination BPSK BPS K x1N AWG AWGN Channel Cha LL LR Calculation Calculatio Fig. 1b The proposed soft DF scheme with CALSC decoder 25 w1N AWGN AWG Channeel AWGN AWG Chann annel Source SISO O Encoder Encode Polar Decoder and respectively, which are given by (5) (6) where is the transmission power of the re- lay, and are the noise of s-d and r-d channel respectively. and are all ze- ro-mean real Gaussian random variables with variance and , respectively. Throughout this paper we will assume . III. THE PROPOSED SOFT DF SCHEME BASED ON POLAR CODES In this section, we will first briefly introduce the framework of the proposed soft DF scheme based on polar codes. Then, we will describe the key techniques in detail. 3.1 System framework is given by (4) where At the destination node, the received signals from the source and relay are denoted as Unlike the hard DF in which hard decision is made at the relay node, the soft DF scheme requires that the signals are decoded and re-encoded in a soft manner. Therefore, the polar decoder and encoder should be a SISO version or be able to generate the soft information. An intuitive idea is to utilize a BP or SCAN decoder at the relay, in which the soft information is directly obtained after the iterative processing back and forth on the factor graph and can be transmitted to the destination, as illustrated in Fig. 1a. However, the message transmission procedure of both BP and SCAN are based on the factor graph, the performance are limited by the short-length loops of the factor graph. In other words, there is performance gap between the BP/SCAN and other improved SC decoders, like LSC and CALSC. Furthermore, the complexity of BP is considerable due to the iterative decoding process. Considering that the CALSC decoder is of superior performance, we propose the soft DF scheme based on CALSC decoder, as illustrated in Fig. 1b. At the relay node, the signal is first fed into China Communications • August 2017 SC Decoding g LSC decod ecoder with L1 lists Threshold old Decision n LLRs with SC LLRs with GA 10 2 10 1 10 0 10 -1 0 20 40 60 80 100 information bit 120 140 Fig. 3a The LLRs of information bits with erroneous decoding (SNRsr=2.0 dB, N=256, K=128) 10 3 LLRs with SC LLRs with GA 10 2 LLR In practical applications, the relay is usually energy-constrained. Therefore, we propose a low-complexity CALSC decoder at the relay to reduce the complexity of the relay. The CALSC decoding is the combination of LSC and CRC decoding. As LSC decoding takes up most of the processing delay, we focus on how to reduce the complexity of LSC. The LSC decoding can be regarded as a path searching process on a code tree. For polar codes with length , there are levels on the code tree. In each level, there are nodes, . Each node has two succeeding branches each of which stands for a decoded bit 0 or 1. A decoding path consists of the continuous branches from the root node to a leaf node. The path probability metric (PPM) 10 1 can be measured using a posteriori probability [14] given by 10 0 (7) where ) 10 3 3.2 The low-complexity CALSC decoder of a decoding path LSC decoder with L2 lists wi Fig. 2 The proposed low-complexity SCL decoder ( LLR a CALSC decoder. As the output of CALSC is all hard information, we then propose a soft information calculation method based on Bayes rule to calculate the probabilities of decoded bits. Finally, the calculated soft information is fed into a SISO polar encoder. In order to reduce the complexity, a low-complexity CALSC decoder is further proposed. It is noted that the SISO encoder is the soft version of the encoder at the source node, which yields an overall repetition coding structure. At the destination node, the LLRs of the signals received from the source and relay are first calculated. After combination, the LLRs are then passed into the polar decoder for signal reconstruction. The detailed descriptions are given in the following subsections. denotes the received symbols and can be calculated using the re- cursive formula in [12]. In the traditional LSC decoder, the number of surviving decoding paths, namely the list size, is set to a fixed valChina Communications • August 2017 0 20 40 60 80 100 120 140 information bit Fig. 3b The LLRs of information bits with correct decoding (SNRsr=2.0 dB, N=256, K=128) ue, say . In each level, decoding paths with the largest PPMs are kept while the others are 26 deleted. LSC decoding can mitigate the effect of error propagation in SC decoding which has just one decoding path and cannot correct the past decoded bits once they were erroneously decoded. We argue that if the reliabilities of decoding bits are high enough, it is unnecessary to employ a large number of decoding paths to reserve the decoding information. Thus, the complexity can be reduced. Motivated by this conjecture, we propose an LSC decoding algorithm with dynamic list size, as illustrated in Fig. 2. Two LSC decoders with different list sizes are employed in our algorithm. If the reliabilities of the decoded bits are higher than the predesigned threshold, which indicates a high probability of correct decoding, an LSC will be applied. decoder with a small list Conversely, if the reliabilities of decoded bits are smaller than the threshold, an LSC decoder with a large list will be applied. In the following, we will elaborate how to measure the reliabilities and how to design the threshold. First, we utilize an SC decoding to obtain the real-time LLRs of decoded bits. As depicted in Fig. 3, the red square points stand for the absolute LLRs of the decoded bits calculated by Gaussian approximation (GA) [23] in increasing order, while the blue dots stand for the absolute LLRs calculated by the SC decoding. The LLRs of GA are calculated point by point offline in the given SNR range. We can infer that if the LLR curve of SC is far away from GA, as illustrated in Fig. 3a, erroneous decoding tends to occur. Conversely, if the LLR curve of SC approaches GA, the probability of correct decoding will be high and we can employ an LSC decoder with a small list to do the decoding. It is noted that the SC decoding is an error propagation decoding process, which means that if the previous bits are correctly decoded the LLR of current decoded bit is reliable, otherwise it is not reliable and cannot be used for soft information relaying. Here, we use the deviation degree of LLR curve of SC and GA to measure the average degree of reliability and to help to select the 27 LSC decoder with list size or . We define a deviation metric (DM) to quantize the deviation degree between the LLRs of SC and GA, which is given by (8) and represent the LLR of SC where and GA, respectively. Here, is used to take the high reliabilities and low reliabilities into the same measure scale. A small DM means a high probability of correct decoding and high average reliabilities of decoded bits. We use the DM as the measurement of the reliabilities of decoded bits when performing the real-time decoding process and use the threshold of DM to determine whether to apply a small-list LSC decoder or a large-list one. Here, we give a straightforward method to design the threshold. We perform the SC decoding off-line and record the DM each time when correct decoding occurs. Then we calculate the mean and variance of DM, denoted as and respec- tively. Assuming the distribution of DM is Gaussian, we utilize the -law, which holds , to set the threshold to . The performance and the complexity of the proposed LSC decoder will be given in section IV. 3.3 Soft information calculation of the LSC decoder The output of the LSC decoder is a list of decoding paths with PPMs. However, there is no soft information about the probabilities of the decoded bits. In this subsection, we propose a method to calculate the soft information based on to the PPMs of the decoding paths. Let denote the set of decoded bit sequences. Let denote the j-th bit sequence, where and . We use to denote the i-th bit in the j-th sequence, and use to denote the PPM of the j-th bit sequence, abbreviated . As the PPMs of the decoding paths are normalized to 1 when performing the LSC decod- China Communications • August 2017 ing, we have (9) By using the Bayes rule, we deduce the LLRs of the decoded bits given by It is noted that , . The signal transmitted to the destination can be expressed as (15) where is the power normalization factor which can be calculated as (16) (10) where is the indicative function. The calculated soft information is then fed to the SISO encoder which will be introduced in the next subsection. With CRC aiding, we modify the sign of LLRs of decoded bits by the decoded sequence which passes the CRC decoding, given by (11) where denotes the i-th bit in the decoded bit sequence which pass the CRC. 3.4 The SISO encoder based on polar codes The processing unit of the SISO polar encoder consists of four nodes, two input nodes and , two output nodes and , as depicted in Fig. 4. The soft information is given by (12) and (13) By applying (12) and (13) recursively, we obtain the LLRs of the encoded bits. It is noted that the unit has lots of similarities with the BP, however, the SISO processing unit merely has the forward information propagation without the backward one. As the value of LLR is with high-dynamic, it is difficult for the relay to forward these signals directly. We use another representation of the LLR, which is given by 3.5 Calculation of LLRs at the destination In our proposed DF scheme, the destination receives two kinds of signals, i.e., the noised BPSK modulation symbols from the source and the soft information from the relay. As the encoder at the relay is a soft version of the one at the source and no interleavers are employed, the overall code structure at the destination is a simple repetition coding. In the traditional hard DF scheme with repetition coding, the signals at the destination are usually combined with maximum ratio combing (MRC) method. However, in the soft DF scheme, the two kinds of signals transmitted by the source and the relay are not the same type. We have to adopt different ways to calculate the LLRs for the two kinds of signals. For the signal from the source, the LLR of each bit is computed as (17) For the signals from the relay, the calculation of LLRs is not so straightforward. We have to model the channel between the correct first. bit and Employing the method in [8], we first model the error between the output of the SISO encoder and the correct bits to be an equivalent Gaussian channel as (18) where is the equivalent noise introvI (2i 1) vI (2i ) r2i 1, I r2i , I + = r2i 1,O r2i ,O vO (2i 1) vO (2i ) (14) Fig. 4 Processing unit of SISO polar encoder China Communications • August 2017 28 duced by soft decoding and encoding. The mean and variance of can be calculate d o ff l i n e a s and , respectively. is the output of the The received signal cascaded channel of the equivalent Gaussian channel and the r-d channel. For simplicity, we give the expression of the LLR directly [9] as (19) Finally, is added to passed to the polar decoder. and IV. SIMULATION RESULTS AND ANALYSIS In this section, we study the performance and the complexity of the proposed soft DF scheme with CALSC decoder. To justify the good performance, we take four systems for comparison, which are listed as follows: a. AF scheme where MRC is employed to combine the received signal from the relay and the source node. b. Hard DF scheme with nested structure [16], abbreviated the nested hard DF. In the nested structure, the information set of polar codes designed for s-r channel is a subset of the one designed for r-d channel. Therefore, the relay extracts and forwards the information which cannot be correctly decoded by the destination with . c. The competitive soft DF scheme utilizing a BP/SCAN decoder at the relay, abbreviated the soft DF with BP/SCAN. As illustrated in Fig. 1a, at the relay node, the received signal is decoded with a BP or SCAN decoder. After iterative decoding, the LLRs of the bits are obtained directly without passing into an SISO encoder. The soft bits are computed by equation (14) and (15). d. A soft decode-compress-forward relaying scheme based on LDPC codes [11]. This scheme employs distributed LDPC coding and 29 generates soft parity symbols at the relay node. The received signal from the source and relay are defined by a large parity check matrix. The polar decoder applied at the relay in nested hard DF is traditional CALSC decoder with list size . In soft DF with BP, the maximum iteration number of BP decoder is set to 60. As the performance of SCAN decoder has very a limited gain when the iteration number is larger than 16, we set the maximum iteration number to 16. In our proposed soft DF scheme with CALSC, the list size of the low-complexity LSC is set to and respectively, and the threshold of DM is calculated off-line. To provide a fair comparison, the polar decoder employed at the destination in all schemes is traditional CALSC decoder with list size . In [11], the overall LDPC code has a block length of N=1224, the overall code rate at the destination is 1/3. To provide a fair comparison, the length of polar codes in source node is set to N1=512, the length of information bit is set to K=341, a CRC-16 code with generator polynomial g(D)=D16+D12+D5+1 is attached. As the overall code structure at the destination is a simple repetition coding, the overall polar code has a block length of N=N1+N2=1024, the overall code rate is R=K/(N1+N2)=1/3, where N1=512 and N2=512 are the code length at the source and relay, respectively. In our simulations, the s-r, r-d and s-d channels are all assumed to be AWGN channels. We fix the signal-to-ratio (SNR) of the s-r link at different values to represent both poor and good channel conditions, and fix SNRrd=SNRsd, just as the configurations in [11]. Before the real-time transmission takes place, the mean and variance of the equivalent noise are computed numerically and stored at the destination, so the relay doesn’t need to transmit extra information. Fig. 5 shows the bit-error-rate (BER) performance of the proposed scheme when the SNR of s-r channel is fixed to 2.8 dB (poor channel condition). The x-axis represents the SNR of s-d channel. It can be seen that our China Communications • August 2017 China Communications • August 2017 10 0 AF Nested hard DF Soft DF with BP Soft DF with SCAN Proposed scheme (L1 =2,L 2 =16) 10 -1 10 -2 Proposed scheme (L1 =16,L2 =16) Bit Error Rate LDPC based scheme [11] 10 -3 10 -4 10 -5 10 -6 10 -7 1 1.5 2 2.5 3 3.5 SNRsd /dB 4 4.5 Fig. 5 Bit-error-rate performance of the proposed soft DF scheme under SNRsr=2.8 dB. The simulations used BPSK over AWGN links, with SNRsd=SNRrd, N=1024, R=1/3 10 0 10 -1 10 -2 Bit Error Rate proposed scheme outperforms AF, the nested hard DF and the soft DF with BP/SCAN. The performance of soft DF with SCAN is slightly better than BP because SCAN decoding performs better than BP decoding. Besides, the proposed scheme has a performance gain of about 0.35 dB compared with the LDPC-based soft DF scheme when the BER is at 10-5. Fig. 6 shows the BER performance of the proposed scheme when the SNR of s-r channel is fixed to 5.0 dB (good channel condition). The proposed schemes with low-complexity CALSC, BP, and SCAN all outperform the LDPC-based soft DF because CRC-aided decoding is employed and can greatly improve the performance of polar codes. Unfortunately, for soft DF with BP and SCAN, there are error floors at the high SNR region, because the two kinds of decoding algorithms are all iterative decoding based on the factor graph of polar codes and there are lots of short-loops on the factor graph. In other word, when the s-r channel is at good condition, error floors will occur and the decoding error at the relay node will propagate to the destination. Conversely, when the SNR of s-r channel is at the low region, as illustrated in Fig. 5, BP/SCAN decoding does not reach the point where error floor occurs. At BER=10-6, our proposed scheme with CALSC has no error floor and has a performance gain of more than 1.0 dB compared with the ones with BP and SCAN. In addition, Fig. 5 and Fig. 6 both demonstrate that the proposed soft DF with low-complexity CALSC ( and ) has neglected performance loss compared to the soft DF with traditional CALSC decoder ( and ) applied at the relay. Because when the s-r channel condition is poor, the LSC decoder with a large list size will be more often used than the LSC decoder with a small list , the performance of the proposed low-complexity CALSC decoder approaches the traditional one with list size and . When the s-r channel condition is good, the performance of LSC decoder with different list size ( ) are almost the same, as illustrated in Fig. 1 in [24]. 10 -3 10 -4 10 -5 10 -6 AF Nested hard DF Soft DF with SCAN Soft DF with BP Proposed scheme (L1 =2,L 2 =16) 10 -7 Proposed scheme (L1 =16,L 2 =16) 10 -8 -1 -0.5 LDPC based scheme [11] 0 0.5 1 SNRsd /dB 1.5 2 2.5 3 Fig. 6 Bit-error-rate performance of the proposed soft DF scheme under SNRsr=5.0 dB. The simulations used BPSK over AWGN links, with SNRsd=SNRrd, N=1024, R=1/3 Then we evaluate the complexity of our proposed scheme. To simplify the complexity evaluation of polar decoding and encoding, we measure the average time complexity in terms of the number of calculation unit which is employed in CALSC, BP, and SCAN. For example, for code length N=512, the time complexity of CALSC decoder with L=16 is LNlog2N=72728. Fig. 7 shows that the proposed soft DF can achieve a significant reduc- 30 3.5 # 10 5 With traditional CALSC (L1 =32,L 2 =32) 3 With traditional CALSC(L1 =64,L 2 =64) With proposed CALSC (L 1 =2,L 2 =16) 2.5 Computation units [2] With traditional CALSC (L1 =16,L 2 =16) [3] With proposed CALSC (L 1 =2,L 2 =32) With proposed CALSC (L 1 =2,L 2 =64) 2 With BP(60 iterations) With SCAN(16 iterations) 1.5 [4] 1 0.5 0 -4 [5] -3 -2 -1 0 SNR sr /dB 1 2 3 4 Fig. 7 The complexity of our proposed soft DF under different SNRs of the source-relay channel [6] [7] tion in complexity with the increasing SNR of s-r channel, compared to the soft DF with traditional CALSC, BP, and SCAN. [8] V. CONCLUSIONS In this paper, we develop a low-complexity soft DF scheme using polar codes in the half-duplex relay channels. This scheme utilizes a low-complexity CALSC decoder at the relay. As the output of CALSC is all hard information, we then propose a soft information calculation method to calculate the probabilities of decoded bits in CALSC. Simulation results show that the proposed soft DF with CALSC decoder outperforms the soft DF with BP/SCAN and the LDPC-based soft DF in terms of BER and achieves a significant reduction in complexity when the source-relay channel condition becomes better. [9] [10] [11] [12] ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (No. 61171099, No.61671080), Nokia Beijing Bell lab. References [1] 31 E. C. van der Meulen, “Three-terminal commu- [13] [14] nication channels,” Advanced Applied Probability, no. 3, pp. 120–154, 1971. I. Hussain, M. Xiao and L. Rasmussen, “Erasure Floor Analysis of Distributed LT Codes”, IEEE Transactions on Communications, vol. 63, No. 8, pp. 2788-2796, August 2015. I. Hussain, M. Xiao and L. 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Guo, “Low-complexity secure network coding against wiretapping using intra/inter-generation coding”, in China Communications, vol. 12, no. 6, pp. 116125, June 2015. China Communications • August 2017 Biographies Fangliao Yang, received the B.S. degree in Electronic Engineering from Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 2010. He is currently working towards the Ph.D. degree at the School of Information and Communication Engineering of BUPT. His research interests include information theory, coding theory and digital signal processing. Email: yangfangliao@ bupt.edu.cn Kai Niu, received the B.S. degree in information engineering and Ph.D. degree in signal and information processing from Beijing University of Posts and Telecommunications (BUPT), Beijing, China, in 1998 and in 2003, respectively. Currently, he is a professor at the School of Information and Communication Engineering of BUPT. His research interests include coding theory and its applications, space-time codes and broadband wireless communication. Email: niukai@bupt.edu.cn Chao Dong, received the B.S and Ph.D. degree in Signal and Information Processing from Beijing University of Posts and Telecommunications (BUPT), Beijing, China in 2007 and 2012, respectively. Since August 2014, he was a lecturer of School of Information and Communication Engineering of BUPT. His research interest is mainly on the MIMO signal processing, multiuser precoding, decision feedback equalizer and the relay signal processing. Email: dongchao@bupt.edu.cn Baoyu Tian, received the B.S. degree in Academy of Military Engineering, Harbin, in 1969, and the M.S. degree in information science from Beijing Institute of Posts and Telecommunications, in 1982. Currently, he is a professor at the School of Information and Communication Engineering of BUPT. His research interests include signal and information processing, digital mobile communication and source coding theory. Email: tianbaoyu@bupt. edu.cn 32

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