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.
Nonlinear PNC mapping based on Latin square was proposed in @cite_20 . Here, the row of the Latin square corresponds to the symbols of one node, and the column represents the symbols of the other node. Entry @math of the Latin square contains the NC symbol mapped to symbol @math and symbol @math of the two users. The exclusive law of PNC mapping is satisfied by the Latin square's constraint: an NC symbol appears once and only once in each row and in each column. The study of Latin-square nonlinear PNC in @cite_20 focused on low-order @math -PSK (the end nodes transmit @math -PSK signals), and the extension to high-order modulations requires high-order Latin squares. By contrast, as we will show, our Gaussian-integer formulation for linear PNC mapping is scalable with the NC operation with various high-order modulations such as @math -PAM in @cite_25 @cite_29 @cite_28 and complex modulations in this paper. In particular, for higher-order modulations, the Gaussian-integer formulation only requires selecting the optimal coefficients @math among a larger set of non-zero Gaussian integers.
@cite_28 @cite_29 @cite_25 @cite_20
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This paper investigates various subtleties of applying linear physical-layer network coding (PNC) with @math -level pulse amplitude modulation ( @math -PAM) in two-way relay channels. A critical issue is how the PNC system performs when the received powers from the two users at the relay are imbalanced. In particular, how would the PNC system perform under slight power imbalance that is inevitable in practice, even when power control is applied? To answer these questions, this paper presents a comprehensive analysis of @math -PAM PNC. Our contributions are as follows. First, we give a systematic way to obtain the analytical relationship between the minimum distance of the signal constellation induced by the superimposed signals of the two users (a key performance determining factor) and the channel-gain ratio of the two users, for all @math . In particular, we show how the minimum distance changes in a piecewise linear fashion as the channel-gain ratio varies. Second, we show that the performance of @math -PAM PNC is highly sensitive to imbalanced received powers from the two users at the relay, even when the power imbalance is slight (e.g., the residual power imbalance in a power-controlled system). This sensitivity problem is exacerbated as @math increases, calling into question the robustness of high-order modulated PNC. Third, we propose an asynchronized PNC system in which the symbol arrival times of the two users at the relay are deliberately made to be asynchronous. We show that such asynchronized PNC, when operated with a belief propagation decoder, can remove the sensitivity problem, allowing a robust high-order modulated PNC system to be built. The design of a reliable physical-layer network coding (PNC) scheme for practical fading two-way relay channels is a challenging task. This is because the signals transmitted by two users arrive at the relay with varied amplitudes and a relative carrier-phase offset, which will impair the performance of PNC. This paper studies a linear PNC scheme for fading two-way relay channels where the transmitters lack the channel state information. In this scheme, the relay computes and broadcast some finite-set integer combinations of two users' messages. The coefficients for the integer combinations used at the relay are carefully designed to minimize the error probability. This scheme can be viewed as a practical embodiment of the compute-and-forward concept. We develop a new LPNC design criterion called minimum set-distance maximization. Using this criterion, we derive an explicit expression for the optimized integer coefficients that minimizes the error probability of LPNC. The optimized integer coefficients turn out to resemble the fading channel coefficients. We further derive a closed-form expression on the average error probability performance over a complex-valued Rayleigh fading two-way relay channel, which shows that our designed LPNC scheme approaches the optimal error performance at a high SNR. Numerical results show that our designed LPNC outperforms existing schemes by more than 5 dB at a medium-to-high SNR regime. We study a new linear physical-layer network coding (LPNC) scheme for fading two-way relay channels. In the uplink phase, two users transmit simultaneously. The relay selects some integer coefficients and computes a linear combination (in a size-q finite set) of the two users' messages, which is broadcast in the downlink phase. We develop a design criterion for choosing the integer coefficients that minimizes the error probability. Based on that, we derive an asymptotically tight bound, in a closed-form, for the error probability of the LPNC scheme over Rayleigh fading channels. Our analysis shows that the error-rate performance of the LPNC scheme becomes asymptotically optimal at a high SNR, and our designed LPNC scheme significantly outperforms existing schemes in the literature.