This paper presents the results of a comprehensive investigation of complex linear physical-layer network (PNC) in two-way relay channels (TWRC). A critical question at relay R is as follows: "Given channel gain ratio @math , where @math and @math are the complex channel gains from nodes A and B to relay R, respectively, what is the optimal coefficients @math that minimizes the symbol error rate (SER) of @math when we attempt to detect @math in the presence of noise?" Our contributions with respect to this question are as follows: (1) We put forth a general Gaussian-integer formulation for complex linear PNC in which @math , and @math are elements of a finite field of Gaussian integers, that is, the field of @math where @math is a Gaussian prime. Previous vector formulation, in which @math , @math , and @math were represented by @math -dimensional vectors and @math and @math were represented by @math matrices, corresponds to a subcase of our Gaussian-integer formulation where @math is real prime only. Extension to Gaussian prime @math , where @math can be complex, gives us a larger set of signal constellations to achieve different rates at different SNR. (2) We show how to divide the complex plane of @math into different Voronoi regions such that the @math within each Voronoi region share the same optimal PNC mapping @math . We uncover the structure of the Voronoi regions that allows us to compute a minimum-distance metric that characterizes the SER of @math under optimal PNC mapping @math . Overall, the contributions in (1) and (2) yield a toolset for a comprehensive understanding of complex linear PNC in @math . We believe investigation of linear PNC beyond @math can follow the same approach.
We investigate modulation schemes optimized for two-way wireless relaying systems, for which network coding is employed at the physical layer. We consider network coding based on denoise-and-forward (DNF) protocol, which consists of two stages: multiple access (MA) stage, where two terminals transmit simultaneously towards a relay, and broadcast (BC) stage, where the relay transmits towards the both terminals. We introduce a design principle of modulation and network coding, considering the superposed constellations during the MA stage. For the case of QPSK modulations at the MA stage, we show that QPSK constellations with an exclusive-or (XOR) network coding do not always offer the best transmission for the BC stage, and that there are several channel conditions in which unconventional 5-ary constellations lead to a better throughput performance. Through the use of sphere packing, we optimize the constellation for such an irregular network coding. We further discuss the design issue of the modulation in the case when the relay exploits diversity receptions such as multiple-antenna diversity and path diversity in frequency-selective fading. In addition, we apply our design strategy to a relaying system using higher-level modulations of 16QAM in the MA stage. Performance evaluations confirm that the proposed scheme can significantly improve end-to-end throughput for two-way relaying systems.