Physical Layer Network Coding Schemes over Finite and Infinite Fields Shengli Zhang, Soung Chang Liew, Lu Lu The Department of Information Engineering The Chinese University of Hong Kong, Hong Kong, China Email: {slzhang5, soung, ll007}@ie.cuhk.edu.hk Abstract: Direct application of network coding at the physical layer - physical layer network coding (PNC) - is a promising technique for two-way relay wireless networks. In a two-way relay network, relay nodes are used to relay two-way information flows between pairs of end nodes. This paper proposes a precise definition for PNC. Specifically, in PNC, a relay node does not decode the source information from the two ends separately, but rather directly maps the combined signals received simultaneously to a signal to be relayed. Based on this definition, PNC can be further sub-classed into two categories - PNCF (PNC over finite field) and PNCI (PNC over infinite field) - according to whether the network-code field (or groups, rings) adopted is finite or infinite. For each of PNCF and PNCI, we consider two specific estimation techniques for dealing with noise in the mapping process. The performance of the four schemes is investigated by means of analysis and simulation, assuming symbol-level time synchronization only. I. INTRODUCTION Network coding, first proposed in [1], is a promising technique for achieving max-flow min-cut capacity in multicast transmission. Linear network coding has been shown to be powerful enough to achieve this capacity [2]. This paper focuses on the use of linear network coding in wireless networks. Although the original investigation of network coding was in the context of wired networks, its potential to boost performance in wireless networks could be even more significant thanks to the broadcast nature of the wireless medium [3]. Ref. [3] shows that direct application of network coding at the physical layer in a wireless relay network could double the capacity of bi-directional point-to-point communication. A similar idea was independently presented in [4]. The analog network coding proposed in [5] is essentially another variation of physical layer network coding (PNC). Several other PNC schemes have also been proposed for the wireless two-way relay channel (TWRC). For example, [6] proposed a PNC scheme based on Tomlinson-Harashima precoding. In [7], a number of memoryless relay functions, including the BER optimal function, were identified and analyzed assuming phase synchronization between signals of the transmitters. Under the general definition for PNC given in our paper here, there is a one-to-one correspondence between a relay function and a specific PNC scheme. We give a precise definition for PNC to distinguish it from the traditional straightforward network coding (SNC). Additionally, we classify PNC schemes into two categories - This work was supported by the Competitive Earmarked Research Grant (Project Number 414507) established under the University Grant Committee of the Hong Kong Special Administrative Region, China. PNCF (PNC over finite field) and PNCI (PNC over infinite field) - according to whether the network-code field (or groups, rings) adopted is finite or infinite. With the definition and classification, the construction of a PNC scheme can be regarded as consisting of two parts: (i) determination of the network code to be used at the relay node; (ii) computation of the information to be relayed at the relay node based on the signals received from the two end nodes. In this paper, we investigate several well-known signal estimation techniques for (ii) under PNCF and PNCI. Among the resulting schemes, one is ANC, one is a novel scheme and the other two are the generalizations of the PNC schemes in [3, 7]. For all the four PNC schemes, only symbol-level time synchronization is assumed. Throughout this paper, PNC is only considered as a modulation-demodulation technique, and channel coding is not taken into account. The investigation of PNC with channel coding can be found in [8], where channel code is involved during the PNC transformation at the relay node. II. SYSTEM MODEL AND DEFINITIONS A. System model 1 N 2 N 3 N Figure 1. Two way relay channel We consider the two-way relay channel as shown in Fig.1, in which nodes N 1 and N 2 exchange information with the help of relay node N 3 . We assume that all nodes are half-duplex, i.e., a node can not receive and transmit simultaneously. This is an assumption arising from practical considerations because it is difficult for the wireless nodes to remove the strong interference of its own transmitting signal from the received signal. We also assume that there is no direct link between N 1 and N 2 . An example in practice is a satellite communication system in which the two end nodes on the earth can only communicate with each other via the relay satellite. In this paper, W i denotes the un-coded packet of N i ; X i denotes the corresponding transmitted packet after channel coding and modulation; and Y i denotes the received base-band packet at N i . A lowercase letter, w, x, or y, denotes one symbol within the corresponding packet. Because of half-duplexity and the lack of a direct link between N 1 and N 2, the transmission must consist two phases: the uplink and downlink phases. Under PNC, for the uplink phase, N 1 and N 2 transmit to N 3 at the same time. Therefore, N 3 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings. 978-1-4244-2324-8/08/$25.00 © 2008 IEEE. 1 Authorized licensed use limited to: Chinese University of Hong Kong. Downloaded on August 26, 2009 at 23:02 from IEEE Xplore. Restrictions apply.