Wyner-Ziv Lattice Coding for Two-Way Relay Channel Sinda Smirani, Mohamed Kamoun, Mireille Sarkiss CEA, LIST, Communicating Systems Laboratory BC 94, Gif Sur Yvette, F91191 - France {sinda.smirani, mohamed.kamoun, mireille.sarkiss}@cea.fr Abdellatif Zaidi Universit´ e Paris-Est Marne La Vall´ ee Champs-sur-Marne, F77454 - France abdellatif.zaidi@univ-mlv.fr Pierre Duhamel CNRS/LSS, Supelec Gif Sur Yvette, F91192 - France pierre.duhamel@lss.supelec.fr Abstract—A Two-Way Relay Channel (TWRC) in which duplex transmission between two users via a relay station is considered. A physical layer network coding strategy based on compress- and-forward relaying scheme for the TWRC is proposed. In the underlying coding strategy, we use nested lattices for Wyner-Ziv coding and decoding. The relay uses the weaker side information available at the receivers from the first transmission phase to broadcast a common quantized version of its received signal. We characterize the achievable rate region of the presented scheme. Then we show that lattice codes can achieve random coding rates. I. I NTRODUCTION The two way relaying problem where two communicating nodes want to exchange information via a relay station is encountered in various wireless communication scenarios: ad-hoc networks, range extension for cellular and local networks ... While network level routing is the classical solution to this problem it has been shown that network coding (NC) strategies provide better performance by leveraging the side information that is available in each node. In fact, NC allows to improve the rates by combining raw bits or packets at the network layer. The capacity of the system can be further improved when NC is applied to the physical layer. It takes advantage of the broadcast and multiple-access properties of the radio link that are considered usually as an interference nuisance to the system [1]. In this context, we consider a physical network coding (PNC) strategy where the overall communication takes two phases, namely a multiple Access (MAC) phase and a Broadcast (BC) phase. Among various strategies that can be used in the TWRC, three relaying protocols can be employed: Amplify and Forward (AF), Decode and Forward (DF), and Compress and Forward (CF). The AF scheme is a linear relaying protocol where the relay only scales the received signal to meet its output power constraint. In the DF scheme, the relay decodes separately both messages then re-encodes them before broadcasting the resulting codeword. Finally, CF scheme, introduced first in [2], has been recently extended to TWRC [3]. In this protocol, the relay station sends a quantized version of the received signal. Performance bounds of these schemes were investigated in [4], [5], [3]. Independently, achievable rate regions of CF relaying have been investigated in [6], [7]. It has been shown that for specific channel conditions, specially symmetric channels, CF outperforms DF for high SNR regimes and AF at all SNR regimes. Besides, the authors in [6] proved that CF relaying scheme achieves rates within one half bit of the capacity region in the Gaussian case for symmetric noise variances. In the aforementioned references, the derivation of the achievable rate regions has employed random coding tools. Structured codes, on the other hand, have been found to be more advantageous in a practical settings thanks to their reduced complexity in encoding and decoding [8]. Lattice codes represent an important class of structured codes that can be used in many wireless systems. It has been shown in [9] that for an Additive White Gaussian Noise (AWGN) channel, lattice based codes can achieve the Shannon capacity for Gaussian point-to-point communication. Based on this result, lattice coding and decoding schemes have been suggested for TWRC scenario in [10]. In this scheme, the transmitters employ nested lattices as codebooks, and the relay decodes a modulo-lattice sum of the transmitted codewords from the received signal. In addition, all the nodes should transmit with the same power. In [11], this scheme has been extended for different power constraints, however no channel conditions were considered. In our work, we investigate a CF strategy at the relay for block fading channels with different transmit powers at the transmitting nodes. In the MAC phase, these nodes send simultaneously their messages and the relay receives a mixture of the transmitted signals. This received signal represents the relay source to be encoded. Then, the relay schemes a message which can be decoded by both receivers using the available side information. When only the decoder has access to side information about the source, this setting is equivalent to Wyner-Ziv (WZ) with lossy source compression [12] or Slepian-Wolf with lossless source compression [13]. Nested lattice-code-based lossy source quantization is introduced in [14] for Gaussian sources in point-to-point communication. In this paper, we extend this scheme to the TWRC where a common source is compressed and broadcasted to the receivers. The proposed strategy provides a practical PNC scheme for TWRC. The rest of the paper is organized as follows. In section II, we introduce our system model. In section III, we outline the fundamentals of nested lattice codes and the lattice based WZ scheme for the TWRC. In section VI, achievable rate region of the considered model will be provided and then compared