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.
: In nonlinear PNC systems, the NC mapping at the relay cannot be expressed as a linear weighted sum of the symbols transmitted from the end nodes. A representative work on nonlinear PNC is @cite_22 . Based on an exclusive law to avoid ambiguity in the decoding of NC symbols at the relay, @cite_22 made use of the closest-neighbor clustering principle (corresponding to mapping constellation points of two superimposed symbols separated by @math to the same NC symbol in this paper) to map the superimposed symbols of two QPSK symbols of two users to NC symbols in @math QAM constellation at the relay.
@cite_22
31453327 1575671689 2158058261 2293741284
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. We propose and design a practical modulation-coded (MC) physical-layer network coding (PNC) scheme to approach the capacity limits of Gaussian and fading two-way relay channels (TWRCs). In the proposed scheme, an irregular repeat–accumulate (IRA) MC over @math with the same random coset is employed at two users, which directly maps the message sequences into coded PAM or QAM symbol sequences. The relay chooses appropriate network coding coefficients and computes the associated finite-field linear combinations of the two users' message sequences using an iterative belief propagation algorithm. For a symmetric Gaussian TWRC, we show that, by introducing the same random coset vector at the two users and a time-varying accumulator in the IRA code, the MC-PNC scheme exhibits symmetry and permutation-invariant properties for the soft information distribution of the network-coded message sequence (NCMS). We explore these properties in analyzing the convergence behavior of the scheme and optimizing the MC to approach the capacity limit of a TWRC. For a block fading TWRC, we present a new MC linear PNC scheme and an algorithm used at the relay for computing the NCMS. We demonstrate that our developed schemes achieve near-capacity performance in both Gaussian and Rayleigh fading TWRCs. For example, our designed codes over GF(7) and GF(3) with a code rate of 3 4 are within 1 and 1.2 dB of the TWRC capacity, respectively. Our method can be regarded as a practical embodiment of the notion of compute-and-forward with a good nested lattice code, and it can be applied to a wide range of network configurations. We study the application of convolutional codes to two-way relay networks (TWRNs) with physical-layer network 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. Abstract This paper presents the first implementation of a two-way relay network based on the principle of physical-layer network coding (PNC). To date, only a simplified version of PNC, called analog network coding (ANC), has been successfully implemented. The advantage of ANC is that it is simple to implement; the disadvantage, on the other hand, is that the relay amplifies the noise along with the signal before forwarding the signal. PNC systems in which the relay performs XOR or other denoising PNC mappings of the received signal have the potential for significantly better performance. However, the implementation of such PNC systems poses many challenges. For example, the relay in a PNC system must be able to deal with symbol and carrier-phase asynchronies of the simultaneous signals received from multiple nodes, and the relay must perform channel estimation before detecting the signals. We investigate a PNC implementation in the frequency domain, referred to as FPNC, to tackle these challenges. FPNC is based on OFDM. In FPNC, XOR mapping is performed on the OFDM samples in each subcarrier rather than on the samples in the time domain. We implement FPNC on the universal soft radio peripheral (USRP) platform. Our implementation requires only moderate modifications of the packet preamble design of 802.11a g OFDM PHY. With the help of the cyclic prefix (CP) in OFDM, symbol asynchrony and the multi-path fading effects can be dealt with simultaneously in a similar fashion. Our experimental results show that symbol-synchronous and symbol-asynchronous FPNC have essentially the same BER performance, for both channel-coded and non-channel-coded FPNC systems.