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Convolutional Codes in Two-Way Relay Networks with Physical-Layer Network Coding

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  • 1. 2724 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 Convolutional Codes in Two-Way Relay Networks with Physical-Layer Network Coding Duc To and Jinho Choi, Senior Member, IEEE Abstract—We study the application of convolutional codes to two-way relay networks (TWRNs) with physical-layer net- work coding (PNC). When a relay node decodes coded signals transmitted by two source nodes simultaneously, we show that the Viterbi algorithm (VA) can be used by approximating the maximum likelihood (ML) decoding for XORed messages as two- user decoding. In this setup, for given memory length constraint, the two source nodes can choose the same convolutional code that has the largest free distance in order to maximize the performance. Motivated from the fact that the relay node only needs to decode XORed messages, a low complexity decoding scheme is proposed using a reduced-state trellis. We show that the reduced-state decoding can achieve the same diversity gain as the full-state decoding for fading channels. Index Terms—Two-way relay network, physical-layer network coding (PNC), multiple access channel, convolutional code, Viterbi algorithm. I. INTRODUCTION IT is known that due to the broadcast nature of wireless channels, it can be useful to perform network coding [1] in the physical layer. Physical-layer network coding (PNC) was proposed by Zhang et al. [2] as a protocol for information exchange between two nodes with the help of a relay in a two- way relay network (TWRN). The information exchange con- sists of two phases, multiple access (MAC) and broadcast (BC) phases. In the MAC phase, two source nodes simultaneously transmit signals to a relay node. Then, in the BC phase, the relay node broadcasts an XORed codeword, which is encoded from the two source nodes’ messages, to two source nodes. It was also reported in [3] that using protocols with two phases in TWRNs can achieve a higher spectral efficiency than using three or four phases. In [4], the notion of PNC was refined and it was demonstrated that how an XORed codeword is obtained by PNC. Note that there could be various ways to represent the XORed codeword, e.g. superimposed XOR was used in [5]. In terms of practical design, there has been extensive research activities on effective modulation and coding schemes for PNC. In [6], when no channel code is considered, an adaptive scheme is proposed to decide detection regions in the MAC phase and to choose a proper constellation in the BC phase. With channel coding, in [7], PNC is performed at message level and the decoding algorithm is redesigned for repeat accumulate (RA) code to decode XORed messages. However, to the best of our knowledge, there is no work studying the Manuscript received February 1, 2010; revised April 6, 2010; accepted July 6, 2010. The associated editor coordinating the review of this letter and approving it for publication was J. M. Shea. The authors are with the School of Engineering, Swansea University, Singleton Park, Swansea, UK (e-mail: {392510, J.Choi}@swansea.ac.uk). Digital Object Identifier 10.1109/TWC.2010.072110.100138 use of convolutional coding for PNC yet, where the decoding can be effectively performed by Viterbi algorithm (VA) [8]. In this letter, the application of convolutional codes for TWRNs using a PNC protocol is investigated. At the re- lay node, VA can be applied for an approximate maximum likelihood (ML) decoding. The resulting decoding becomes two-user decoding over classical MAC [9]. In addition, it is noteworthy that the relay node only needs to decode the XORed messages from source nodes. Therefore, we propose a reduced-state trellis, an approach has been widely used for sequence detection in frequency-selective channels [10]. Exploiting the reduced-state trellis, we can obtain a low complexity decoding. Furthermore, we show that the reduced- state decoding can achieve the same diversity gain as the full- state decoding for fading channels. II. SYSTEM MODEL The system model is shown in Fig. 1, where two source nodes S1 and S2 exchange their messages with the assist of relay node R using the PNC protocol consisting of MAC and BC phases. In the MAC phase, S1 and S2 encode their messages and transmit coded signals simultaneously. Then, relay node R decodes the XORed version of the two signals at message level and forwards its coded signals to both the source nodes during the BC phase. Since each source node knows its own message, it can decode the message sent by the other node from the XORed message. Suppose that two binary message sequences u = {𝑢1, 𝑢2, . . . , 𝑢 𝐿} and v = {𝑣1, 𝑣2, . . . , 𝑣 𝐿} of the same length 𝐿 are generated by S1 and S2, respectively. They are encoded by two (terminated) convolutional encoders, 𝒞1 and 𝒞2, at the two source nodes. Here, we assume that 𝒞1 and 𝒞2 have the same code rate 𝑟. In addition, assume that 𝑟 = 1/𝑁, where 𝑁 is a positive integer for the sake of simplicity. Denote by b and c the coded sequences of length 𝐿𝑁 at S1 and S2 from message sequences, u and v, respectively. Then, with a slight abuse of notation, we have b = {𝑏1,1 . . . , 𝑏1,𝑁 , . . . , 𝑏 𝐿,1, . . . , 𝑏 𝐿,𝑁 } = 𝒞1(u) c = {𝑐1,1 . . . , 𝑐1,𝑁 , . . . , 𝑐 𝐿,1, . . . , 𝑐 𝐿,𝑁 } = 𝒞2(v). Throughout the letter, binary phase shift keying (BPSK) is used for modulation, where coded bits 0 and 1 become binary data symbols “+1” and “−1”, respectively. The signal sequences transmitted by S1 and S2 can be represented by {1 − 2𝑏𝑙,𝑖} and {1 − 2𝑐𝑙,𝑖}, respectively. In the MAC phase consisting of 𝐿𝑁-symbol duration, both the source nodes transmit their coded signals simultaneously. In particular, if the time index is denoted by 𝑡 = (𝑙 − 1)𝑁 + 1536-1276/10$25.00 c⃝ 2010 IEEE
  • 2. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 2725 encoder modulator modulator encoder BC phase MAC phase decoderdecoder decoder encoder modulator ×+ × ×+ + {h (1) t } × b c S1 S2 R {h (2) t } {n (1) t } {n (2) t }{g (1) t } {g (2) t } {1 − 2bl,i} {1 − 2cl,i} vu ˇv ˇu{1 − 2dl,j} {rt} ˆm d {y (1) t } {y (2) t } {wt} Fig. 1. System model. 𝑊mac(b, c) = 𝐿𝑁∑ 𝑡=1 𝑟𝑡 𝜎R − ℎ (1) 𝑡 √ 𝛾mac,1 (1 − 2𝑏𝑙,𝑖) − ℎ (2) 𝑡 √ 𝛾mac,2 (1 − 2𝑐𝑙,𝑖) 2 . (3) 𝑖, 1 ≤ 𝑙 ≤ 𝐿, 1 ≤ 𝑖 ≤ 𝑁, during the MAC phase, the received signal at R is given by 𝑟𝑡 = ℎ (1) 𝑡 √ 𝐸1 (1 − 2𝑏𝑙,𝑖) + ℎ (2) 𝑡 √ 𝐸2 (1 − 2𝑐𝑙,𝑖) + 𝑤𝑡, 𝑡 = 1, 2, . . . , 𝐿𝑁, (1) where 𝐸 𝑘 and {ℎ (𝑘) 𝑡 }, 𝑘 = 1, 2, are the transmission power and the sequence of channel gains from source node S 𝑘 to relay node R, respectively, and {𝑤𝑡} is the background noise sequence at R, which is assumed to be an independent and identically distributed (iid) sequence of circularly symmetric complex Gaussian (CSCG) random variables with zero mean and variance 𝜎2 R, i.e., 𝑤𝑡 ∼ 𝒞𝒩(0, 𝜎2 R). For the additive white Gaussian noise (AWGN) channel, it is assumed that ℎ (𝑘) 𝑡 = 1∀𝑡, while {ℎ (𝑘) 𝑡 } is an iid random sequence1 for fading channels. In this letter, PNC is performed at message level as same as that in [7], which is referred to as link-by-link coded PNC scheme. Let m = {𝑚1, 𝑚2, . . . , 𝑚 𝐿} be the element-wise XORed message sequence from u and v. That is, the 𝑙th element of m is given by 𝑚𝑙 = 𝑢𝑙 ⊕ 𝑣𝑙, where ⊕ denotes the binary XOR operator. With a slight abuse of notation, we let m = u ⊕ v. Denote by ˆm the decoded version of m at R. Under the assumption that R knows the channel state information (CSI), i.e., {ℎ (𝑘) 𝑡 }, perfectly, the maximum likelihood (ML) decoding is given by ˆm = arg max m ln ∑ u,v: m=u⊕v exp (−𝑊mac (𝒞1(u), 𝒞2(v))) , (2) where 𝑊mac(b, c) is the decoding metric, which is given in (3) with 𝛾mac,𝑘 = 𝐸 𝑘/𝜎2 R, 𝑘 = 1, 2. Then, ˆm is encoded by a 1This is not necessary to derive decoding algorithms, but we consider this assumption to see the diversity gain over fading channels. convolutional encoder 𝒞R to form the coded sequence d = 𝒞R( ˆm). The code rate of 𝒞R is 𝑟′ = 1/𝑁′ , where 𝑁′ is a positive integer. Note that the length of d is 𝐿𝑁′ . In the BC phase consisting of 𝐿𝑁′ -symbol duration, relay node R broadcasts the coded signals to both the source nodes. In particular, at 𝑡 = (𝑙 − 1)𝑁′ + 𝑗, 𝑙 = 1, 2, . . ., 𝐿 and 𝑗 = 1, 2, . . ., 𝑁′ , in the BC phase, S 𝑘, 𝑘 = 1, 2, receives the following signal: 𝑦 (𝑘) 𝑡 = 𝑔 (𝑘) 𝑡 √ 𝐸R (1 − 2𝑑𝑙,𝑗) + 𝑛 (𝑘) 𝑡 , 𝑡 = 1, 2, . . . , 𝐿𝑁′ , (4) where 𝐸R and {𝑔 (𝑘) 𝑡 } are the transmission power and the sequence of channel gains from relay node R to source node S 𝑘, respectively, and {𝑛 (𝑘) 𝑡 } is the background noise sequence at S 𝑘 which is an iid sequence of CSCG random variables, i.e., 𝑛 (𝑘) 𝑙,𝑗 ∼ 𝒞𝒩(0, 𝜎2 𝑘). For the AWGN channel, it is assumed to be 𝑔 (𝑘) 𝑡 = 1 ∀𝑡, while {𝑔 (𝑘) 𝑡 } is iid for fading channels. With known {𝑔 (𝑘) 𝑡 }, ˆm is decoded by S 𝑘 from the received signals, {𝑦 (𝑘) 𝑡 }, as ˇm(𝑘) = arg min ˆm 𝑊 (𝑘) bc (𝒞R( ˆm)) , (5) where 𝑊 (𝑘) bc (d) = 𝐿𝑁′ ∑ 𝑡=1 𝑦 (𝑘) 𝑡 𝜎 𝑘 − 𝑔 (𝑘) 𝑡 √ 𝛾bc,𝑘 (1 − 2𝑑𝑙,𝑖) 2 with 𝛾bc,𝑘 = 𝐸R/𝜎2 𝑘. Finally, S1 decodes v, the message sent by S2, by taking the XOR operation of u and ˇm(1) , i.e., ˇv = u ⊕ ˇm(1) . Similarly, u is found at S2 as ˇu = v ⊕ ˇm(2) . Since for given u (resp. v), the mapping from v (resp. u) to m is a one-to-one mapping, ˇv and ˇu are the ML decoding results.
  • 3. 2726 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 𝑀𝑙 (𝑠𝑙; 𝑢𝑙, 𝑣𝑙) = 𝑁∑ 𝑖=1 𝑟(𝑙−1)𝑁+𝑖 𝜎R − ℎ (1) (𝑙−1)𝑁+𝑖 √ 𝛾mac,1 (1 − 2𝑏𝑙,𝑖) − ℎ (2) (𝑙−1)𝑁+𝑖 √ 𝛾mac,2 (1 − 2𝑐𝑙,𝑖) 2 . (8) III. DECODING WITH FULL-STATE TRELLIS FOR XORED MESSAGES IN MAC PHASE A. Two-User Decoding using Full-Trellis In the BC phase, VA can be directly used for decoding at the two source nodes. Note that the complexity of a VA- based decoder depends on the trellis structure, where the main parameters are the number of states and the number of transitions from a state. On the other hand, in the MAC phase, the role of relay node is to decode the XORed message sequence of two binary message sequences from the two source nodes. Certainly, based on (2), the XORed message sequence can be optimally decoded by an exhaustive search, which would have prohibitively high complexity. Since the relay node may have a limited computing power, it is desirable to have a decoding algorithm of reasonably low complexity. In the original decoding rule at relay node R for MAC decoding shown in (2), since the decoding metric is the logarithm of a sum of exponential functions, VA cannot be applied to decoding. Fortunately, if the original decoding met- ric is modified using its log-max approximation, the decoding process can be decomposed into two steps, and one of them is actually joint decoding, namely two-user decoding, for MAC, where VA can be applied to two-user decoding. Note that, while two users can transmit at arbitrary rates in general MACs, the two source nodes should have the same rate in the TWRNs with PNC. Using the log-max approximation (i.e., ln ∑ 𝑖 exp (𝑧𝑖) ≈ max𝑖 𝑧𝑖), the modified decoding is given by ˆm = arg min m min u,v: m=u⊕v 𝑊mac (𝒞1(u), 𝒞2(v)) . (6) We observe that the decoding process in (6) can be carried out by the following two steps: i) the two message sequences are jointly decoded as {ˆu, ˆv} = arg min {u,v} 𝑊mac (𝒞1(u), 𝒞2(v)); (7) and ii) the XORed message sequence is obtained by ˆm = ˆu ⊕ ˆv. The first step is the two-user decoding for MAC. In this step, VA can be applied based on the following two- user trellis structure. Provided that u and v are transmit- ted, the state at the 𝑙th stage is represented by 𝑠𝑙 = {𝑢𝑙−1, . . . , 𝑢𝑙−𝜈1 , 𝑣𝑙−1, . . . , 𝑣𝑙−𝜈2 }, where 𝜈 𝑘, 𝑘 = 1, 2, is the memory length of 𝒞 𝑘. From 𝑠𝑙, if the joint input is (𝑢𝑙, 𝑣𝑙), the next state is 𝑠𝑙+1 = {𝑢𝑙, . . . , 𝑢𝑙−𝜈1+1, 𝑣𝑙, . . . , 𝑣𝑙−𝜈2+1}. The transition from 𝑠𝑙 to 𝑠𝑙+1 produces a joint output se- quence of length 𝑁 given by {(𝑏𝑙,1, 𝑐𝑙,1, ) , . . . , (𝑏𝑙,𝑁 , 𝑐𝑙,𝑁 )}. For convenience, denote by 𝒮 the state alphabet. Its size is given by ∣𝒮∣ = 2 𝜈1+𝜈2 . Since {(𝑏𝑙,1, 𝑐𝑙,1, ) , . . . , (𝑏𝑙,𝑁 , 𝑐𝑙,𝑁 )} depends only on the current state 𝑠𝑙 and input (𝑢𝑙, 𝑣𝑙), the branch metric is given in (8). Therefore, VA can be used in searching the optimal path over all possible paths, each path goes through a state sequence. Given a state sequence S = {𝑠1, 𝑠2, . . . , 𝑠 𝐿+1}, its path metric is 𝑉mac ({𝑠𝑙}) = 𝐿∑ 𝑙=1 𝑀𝑙 (𝑠𝑙; 𝑢𝑙, 𝑣𝑙), (9) which is actually the metric 𝑊mac (𝒞1(u), 𝒞2(v)) in (7). Here, u and v are the two message sequences associated with S. This decoding method is referred to as the full-state decoding, while a low complexity approximation, called the reduced- state decoding, will be described in the next section. Note that, at each stage, there are 2 𝜈1+𝜈2 possible states. Therefore, the decoder needs to keep track of 2 𝜈1+𝜈2 survival paths at each stage. Thus, the complexity of the full-state decoding increases exponentially with the sum of memory lengths of the two convolutional encoders of the two source nodes. B. Hamming Distance Analysis for Full-State Trellis Denote by 𝑃mac and 𝑃bc the message error probabilities (or frame error probabilities (FERs)) for MAC and BC phases, respectively. If an XORed message sequence is incorrectly decoded in both the MAC and BC phases, the probability of that the decision in overall end-to-end transmission returns to be correct is 1/(2 𝐿 − 1). Therefore, the end-to-end FER is given by 𝑃EtoE = 𝑃mac (1 − 𝑃bc) + 𝑃bc (1 − 𝑃mac) + 2 𝐿 − 2 2 𝐿 − 1 𝑃mac 𝑃bc = 𝑃mac + 𝑃bc − 2 𝐿 2 𝐿 − 1 𝑃mac 𝑃bc ≈ 𝑃mac + 𝑃bc − 𝑃mac 𝑃bc (10) for sufficiently large 𝐿. It can be seen that 𝑃EtoE is lower- and upper-bounded by max {𝑃mac, 𝑃bc} and (𝑃mac + 𝑃bc), respectively. Therefore, the diversity of end-to-end transmis- sion is decided by the minimum diversity of MAC and BC. Since the diversity order of BC is given by the free distance of 𝒞ℛ, in this subsection, we only focus on the diversity order of MAC. Since the mapping from a pair of binary message sequences to its XORed message sequence is a many-to-one mapping, it is possible that the XORed message sequence can be correctly decoded for incorrectly decoded individual message sequences. However, the error probability of XORed message decoding for given a set of channel realizations is still upper- bounded by a sum of probabilities, each of them is the probability of that an incorrect pair of message sequences is decoded. Such kind of probabilities is referred to as the pair-wise error probability (PEP). Note that each PEP is the probability that an incorrect path is decoded and each path is labeled by a distinct joint output sequence. Therefore, the diversity order of MAC phase is the minimum two-user Hamming distance.
  • 4. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 2727 {0,0,0,1} {0,0,1,0} {0,0,1,1} {0,1,0,0} {0,1,0,1} {0,1,1,0} {0,1,1,1} {1,0,0,0} {1,0,0,1} {1,0,1,0} {1,0,1,1} {1,1,0,0} {1,1,0,1} {1,1,1,0} {1,1,1,1} {0,0,0,0} {(1,1),(0,0)},{(1,0),(0,1)},{(0,1),(1,0)},{(0,0).(1,1)} {(1,0),(1,0)},{(1,1),(1,1)},{(0,0),(0,0)},{(0,1).(0,1)} {(1,0),(1,1)},{(1,1),(1,0)},{(0,0),(0,1)},{(0,1).(0,0)} {(1,1),(1,1)},{(1,0),(1,0)},{(0,1),(0,1)},{(0,0).(0,0)} {(1,1),(1,0)},{(1,0),(1,1)},{(0,1),(0,0)},{(0,0).(0,1)} {(0,0),(1,0)},{(0,1),(1,1)},{(1,0),(0,0)},{(1,1).(0,1)} {(0,0),(1,1)},{(0,1),(1,0)},{(1,0),(0,1)},{(1,1).(0,0)} {(0,1),(1,1)},{(0,0),(1,0)},{(1,1),(0,1)},{(1,0).(0,0)} {(0,1),(1,0)},{(0,0),(1,1)},{(1,1),(0,0)},{(1,0).(0,1)} {(1,0),(0,0)},{(1,1),(0,1)},{(0,0),(1,0)},{(0,1).(1,1)} {(1,0),(0,1)},{(1,1),(0,0)},{(0,0),(1,1)},{(0,1).(1,0)} {(1,1),(0,1)},{(1,0),(0,0)},{(0,1),(1,1)},{(0,0).(1,0)} {(0,0),(0,0)},{(0,1),(0,1)},{(1,0),(1,0)},{(1,1).(1,1)} {(0,0),(0,1)},{(0,1),(0,0)},{(1,0),(1,1)},{(1,1).(1,0)} {(0,1),(0,1)},{(0,0),(0,0)},{(1,1),(1,1)},{(1,0).(1,0)} {(0,1),(0,0)},{(0,0),(0,1)},{(1,1),(1,0)},{(1,0).(1,1)} Fig. 2. Full-state trellis for the pair of (5,7) and (6,7) convolutional codes. Definition 1: Let a = {𝑎𝑙,𝑖} and a′ = { 𝑎′ 𝑙,𝑖 } be two different joint output sequences of length 𝐿𝑁, where their elements are the pairs of coded bits, i.e., 𝑎𝑙,𝑖 = (𝑏𝑙,𝑖, 𝑐𝑙,𝑖), 𝑎′ 𝑙,𝑖 = (𝑏′ 𝑙,𝑖, 𝑐′ 𝑙,𝑖). The two-user Hamming distance between a and a′ is the number of the places where the two correspond- ing elements of a and a′ are different from each other. Lemma 1: Let 𝑑free,𝑘 be the free distance of 𝒞 𝑘. Let 𝑑min be the minimum two-user Hamming distance over all the pairs of joint output sequences from 𝒞1 and 𝒞2. We have 𝑑min = min {𝑑free,1, 𝑑free,2} . (11) The outline of proof of Lemma 1 is given as follows: First, we need to verify that two paths of Hamming distance less than min {𝑑free,1, 𝑑free,2} do not exist, which can be proved by the properties of given convolutional codes. Second, we can show that the combination of the all-zero sequence and a sequence of minimum Hamming weight of the weaker code is the joint output sequence of min {𝑑free,1, 𝑑free,2} Hamming distance, which confirms (11). Although FER is considered so far, the same diversity can also be observed by bit error rate (BER). Fig. 2 illustrates a trellis diagram for a joint encoder with 𝒞1 and 𝒞2 of generators (5,7) and (6,7), respectively, both are of memory length 2. On the left of each state is the list of joint output sequences resulting from transitions triggered by the pairs of message bits, (0, 0),(0, 1),(1, 0), and (1, 1), respectively. The two paths that are merged after 4 stages shown by thicker lines represent the two joint output sequences corresponding to the sequence of pairs of zeros and {(0, 1), (0, 1), (0, 0), (0, 0), (0, 1), (0, 0), (0, 0), (0, 1), . . .}. The resulting Hamming distance is 4, which is the free distance of the (6,7) convolutional code. On the selection for a pair of convolutional codes, which are constrained on the memory lengths of codes, the optimal choice for convolutional codes at source nodes can be easily found from Lemma 1. That is, the two source nodes should choose the same convolutional code, which has the largest free distance over all the convolutional codes for a given memory
  • 5. 2728 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 {0,0} {0,1} {1,0} {1,1} {{(0,0),(0,0)},{(0,0),(1,1)},{(1,1),(0,0)},{(1,1),(1,1)}} {{(0,1),(0,1)},{(0,1),(1,0)},{(1,0),(0,1)},{(1,0),(1,0)}} {{(0,0),(0,0)},{(0,0),(1,1)},{(1,1),(0,0)},{(1,1),(1,1)}} {{(0,1),(0,1)},{(0,1),(1,0)},{(1,0),(0,1)},{(1,0),(1,0)}} {{(0,0),(0,1)},{(0,0),(1,0)},{(1,1),(0,1)},{(1,1),(1,0)}} {{(0,1),(0,0)},{(0,1),(1,1)},{(1,0),(0,0)},{(1,0),(1,1)}} {{(0,1),(0,0)},{(0,1),(1,1)},{(1,0),(0,0)},{(1,0),(1,1)}} {{(0,0),(0,1)},{(0,0),(1,0)},{(1,1),(0,1)},{(1,1),(1,0)}} Fig. 3. Reduced-state trellis for two (5,7) convolutional codes. length constraint. For example, if 𝜈1, 𝜈2 ≤ 2, the convolutional code of (5, 7) is optimal for two source nodes. IV. REDUCED-STATE TRELLIS STRUCTURE FOR XORED MESSAGES DECODING IN MAC PHASE Since the relay node only needs to decode the XORed message sequence rather than two original message sequences, say u and v, there might be efficient decoding schemes that only decode the XORed message sequence. In this section, we study such a decoding scheme by exploiting a reduced-state trellis. Our proposed reduced-state trellis is constructed by merging every disjoint group of states in the full-state trellis into a single merged state. For convenience, we assume that two constituent convolutional encoders have the same memory length, i.e., 𝜈1 = 𝜈2 = 𝜈. (If their memory lengths are different, the shorter memory can be extended and the ex- tension does not affect the outputs of encoder.) Let ˜𝒮 denote the state alphabet for 𝒞 𝑘 (due to the same memory length, ˜𝒮 is identical for both codes). The full-state state alphabet for joint two-user decoding is 𝒮 = ˜𝒮 × ˜𝒮. Any group of full-state states 𝑠𝑙 = {𝑢𝑙−1, . . . , 𝑢𝑙−𝜈, 𝑣𝑙−1, . . . , 𝑣𝑙−𝜈 } that have the same {𝑢𝑙−1 ⊕ 𝑣𝑙−1, . . . , 𝑢𝑙−𝜈 ⊕ 𝑣𝑙−𝜈 } is merged into a single merged state defined by ˜𝑠𝑙 = {𝑢𝑙−1 ⊕ 𝑣𝑙−1, . . . , 𝑢𝑙−𝜈 ⊕ 𝑣𝑙−𝜈} ∈ ℛ, where ℛ denotes the state alphabet of the reduced-state trellis. Clearly, ℛ = ˜𝒮. From now on, we refer the states in the full-state and reduced- state trellis to as type-A and type-B states, respectively. In addition, multiple parallel transitions from a group to another of type-A states, where the elements in each group are merged into a type-B state, are also merged into a single transition in the reduced-state trellis. While there is a one- to-one correspondence between transitions and their labels (a label is the joint output sequence triggered by a transition) in the full-state trellis, each transition in the reduced-state trellis is associated with multiple labels. It is clear that any path in the reduced-state trellis uniquely corresponds to an XORed message sequence. Therefore, if a type-B state sequence, denoted by ˜S, is recovered from the reduced-state trellis, the corresponding XORed message is immediately decoded, which is what we want at R with a PNC protocol. For decoding, VA can be applied to the reduced-state decoding with the following branch metric: 𝑀𝑙 (˜𝑠𝑙; 𝑚𝑙) = max 𝑠 𝑙,𝑢 𝑙,𝑣 𝑙: 𝑠 𝑙∈𝒫(˜𝑠 𝑙) 𝑢 𝑙⊕𝑣 𝑙=𝑚 𝑙 𝑀𝑙 (𝑠𝑙; 𝑢𝑙, 𝑣𝑙), (12) where 𝒫(˜𝑠𝑙) = {𝑠𝑙 ∣ ˜𝑠𝑙 = {𝑢𝑙−1 ⊕ 𝑣𝑙−1, . . . , 𝑢𝑙−𝜈 ⊕ 𝑣𝑙−𝜈 }}. Although we need additional multiple binary comparators and the number of comparators is decided by the number of distinct labels on each reduced-state transition (for the case of 𝒞1 = 𝒞2 the number of binary comparators is 2 𝑁 − 1), it is only required to track 2 𝜈 survival paths in each stage. Thus, the complexity of reduced-state decoding is approximately a square root of that of the full-state decoding. Definition 2: The free distance of a reduced-state trellis is the minimum two-user Hamming distance over all the pairs of paths in this trellis. Lemma 2: If 𝒞1 = 𝒞2 = 𝒞 and denote by 𝑑free the free distance of these codes, the free distance of the reduced-state trellis is equal to the free distance of 𝒞, i.e., 𝑑free. Proof: Consider the following joint output sequence: a = {(𝑏1,1, 𝑐1,1) , . . . , (𝑏 𝐿,𝑁, 𝑐 𝐿,𝑁 )} , which is associated with a state sequence ˜S from the reduced- state trellis, where b, c ∈ 𝒞. Let ˜a = b ⊕ c, which is called the XORed-coded version of a. Let u = 𝒞−1 (b) and v = 𝒞−1 (c). Due to the linear property of convolutional codes, ˜a = 𝒞(u ⊕ v). This shows that the XORed-coded versions of all joint output sequences associated with ˜S are identical to ˜a. Now, we regard the XORed-coded sequence associated with each path in the reduced-state trellis as the secondary label, which is unique. The minimum Hamming distance between two secondary labels associated with two different paths is equal to 𝑑free. Since (b, c) ∕= (b′ , c′ ) if b ⊕ c ∕= b′ ⊕ c′ , we can conclude that the free distance of the reduced-state trellis is equal to 𝑑free. The result in Lemma 2 shows that the diversity gain of decoding based on a reduced-trellis is identical to that based on a full-trellis over fading channels. Thus, the use of a reduced-trellis in decoding can efficiently reduce the complexity without a significant performance loss. Note that the number of binary comparisons is 2 𝑁 − 1 in computing each branch metric if 𝒞1 = 𝒞2 as implied by the proof of Lemma 2. We illustrate the reduced-state trellis from two identi- cal (5,7) convolutional codes in Fig. 3. At the right of each state, the list of labels on the transitions caused by 𝑢𝑙 ⊕ 𝑣𝑙 = 0 and 𝑢𝑙 ⊕ 𝑣𝑙 = 1 are given in the first and second rows, respectively. For instance, the transition from the (0,0) to (1,0) type-B state is constructed by merging a set of type-A transitions, each is from an element of state
  • 6. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 9, SEPTEMBER 2010 2729 4 6 8 10 12 14 16 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Eb/No [dB] BER (5,7) & (5,7) convolutional codes, full−state decoding (5,7) & (6,7) convolutional codes, full−state decoding (5,7) & (5,7) convolutional codes, reduced−state decoding (23,35) & (23,35) convolutional codes, full−state decoding (23,35) & (23,35) convolutional codes, reduced−state decoding Fig. 4. Simulation results for full-state decoding using the same and different convolutional codes and reduced-state decoding using the same convolutional codes, Rayleigh fading channels. group {(0,0,0,0),(0,1,0,1),(1,0,1,0),(1,1,1,1)} to an element of the group {(0,0,1,0),(0,1,1,1),(1,0,0,0),(1,1,0,1)}. The dis- tinct labels for such transition are {(0,1),(0,1)}, {(0,1),(1,0)}, {(1,0),(0,1)}, and {(1,0),(1,0)}. In the depicted reduced-state trellis, the free distance is 5, the same as that of the (5,7) convolutional code. V. SIMULATION RESULTS In all simulations, we assume that {ℎ (𝑘) 𝑡 } and {𝑔 (𝑘) 𝑡 }, 𝑘 = 1, 2, are independent. Furthermore, each element in these sequences of channel gains is a CSCG random variable with zero mean and unit variance (Rayleigh fading channels). All channel codes have the same code rate such that 𝑟 = 𝑟′ = 1/2. For a given memory length, we choose the convolutional code with maximum Hamming distance in the BC phase for all simulations, i.e., if memory length is 2, the (5,7) convolutional code is chosen, while (23,35) convolutional code is used for the memory length 4 constraint. The performance index is the average end-to-end BER over two communication flows. As shown in Fig. 4, when the two source nodes use the same (5,7) convolutional code and the relay node uses the full-state decoding, the BER slope is 2 dB/decade. For a comparison purpose, we consider another case, where two different channel codes, (5,7) and (6,7) convolutional codes, are used. This case provides the 2.5 dB/decade BER slope, which confirms that the diversity order of the system is equal to the free distance of the worst code. This two supporting results also confirm the principle of choosing codes for a given memory length constraint. As mentioned, the diversity order is retained with reduced- state trellis decoding provided that the same code is used at both the source nodes. It is verified in Fig. 4 for two cases, where the same (5,7) convolutional code is used in the first case and the same (23,35) convolutional code is used in the second case. However, there are about 2 dB performance degradation in both the cases. The performance loss can be seen as the cost resulting from the reduced decoding complexity. VI. CONCLUDING REMARKS In this letter, we studied the use of convolutional codes for TWRNs with PNC protocol. It was shown that a nearly- optimal decoding for element-wise XORed messages can be achieved at the relay node using a VA based decoder over a two-user trellis where the number of states is the product of numbers of states for individual codes. This decoding scheme provided the same diversity order as the free distance of the worst code for fading channels. Consequently, it was shown that the optimal selection for two source nodes’ convolutional codes is the pair of the same code that has the largest free distance for a given memory length constraint. In addition, due to the nature of PNC protocols, where only XORed messages need to be decoded rather than individual messages at the relay node, we proposed the use of a reduced-state trellis for decoding to reduce decoding complexity. It was shown that the complexity of the reduced-state decoding is an approximately square root of that of the full-state decoding, while there is no loss of diversity gain for fading channels. Simulation results showed that, however, there is a performance gap of approximately 2 dB SNR paying for the complexity reduction. REFERENCES [1] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network infor- mation flow,” IEEE Trans. Inf. Theory, vol. 46, pp. 1204-1216, July 2000. [2] S. Zhang, S. C. Liew, and P. Lam, “Hot topic: physical-layer network coding,” in Proc. ACM MOBICOM ’06, Sep. 2006, pp. 358-365. [3] R. Rankov and A. Wittneben, “Spectral efficient protocols for half- duplex fading relay channels,” IEEE J. Sel. Areas Commun., vol. 25, pp. 379-389, Feb. 2007. [4] S. Zhang, S.-C. Liew, and L. Lu, “Physical-layer network coding schemes over finite and infinite fields,” in Proc. IEEE GLOBECOM’08, Nov.-Dec. 2008, pp. 3784-3789. [5] J. Liu, M. Tao, Y. Xu, and X. Wang, “Superimposed XOR: a new physical layer network coding scheme for two-way relay channels,” in Proc. IEEE GLOBECOM’09, Nov. 2009. [6] T. Koike-Akino, P. Popovski, and V. Tarokh, “Optimized constellations for two-way wireless relaying with physical network coding,” IEEE J. Sel. Areas Commun., vol. 27, pp. 773-787, June 2009. [7] S. Zhang and S.-C. Liew, “Channel coding and decoding in a relay system operated with physical-layer network coding,” IEEE J. Sel. Areas Commun., vol. 27, pp. 788-796, June 2009. [8] A. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm,” IEEE Trans. Inf. Theory, vol. 13, pp. 260- 269, Apr. 1967. [9] R. Peterson and D. J. Costello Jr., “Binary convolutional codes for a multiple-access channel,” IEEE Trans. Inf. Theory, vol. 25, pp. 101-105, Jan. 1979. [10] M. V. Eyuboglu and S. U. H. Qureshi, “Reduced-state sequence estima- tion with set partitioning and decision feedback,” IEEE Trans. Commun., vol. 36, pp. 13-20, Jan. 1988.