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.
For link-by-link channel-coded PNC, the relay is aware of the channel coding employed by the two end nodes (specifically, the relay knows the codebooks used by the two end nodes), and the relay can exploit the correlations among the symbols within each of the channel-coded packets to further improve the accuracy of PNC decoding mapping. The study of channel-coded PNC systems also originated from low-order modulations such as BPSK @cite_24 @cite_19 , and then evolved to high-order modulations in search of higher throughput in the high SNR regime @cite_6 @cite_14 @cite_27 @cite_26 @cite_10 @cite_17 @cite_15 .
@cite_14 @cite_26 @cite_6 @cite_24 @cite_19 @cite_27 @cite_15 @cite_10 @cite_17
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This paper investigates link-by-link channel-coded PNC (physical layer network coding), in which a critical process at the relay is to transform the superimposed channel-coded packets received from the two end nodes (plus noise), Y3 = X1+ X2+W3, to the network-coded combination of the source packets, S1 oplus S2. This is in contrast to the traditional multiple-access problem, in which the goal is to obtain both S1 and S2 explicitly at the relay node. Trying to obtain S1 and S2 explicitly is an overkill if we are only interested in S1oplusS2. In this paper, we refer to the transformation Y3 rarr S1 oplus S2 as the channel-decoding- network-coding process (CNC) in that it involves both channel decoding and network coding operations. This paper shows that if we adopt the repeat accumulate (RA) channel code at the two end nodes, then there is a compatible decoder at the relay that can perform the transformation Y3 rarr S1oplusS2 efficiently. Specifically, we redesign the belief propagation decoding algorithm of the RA code for traditional point-to-point channel to suit the need of the PNC multiple-access channel. Simulation results show that our new scheme outperforms the previously proposed schemes significantly in terms of BER without added complexity. 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. 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. We study relay selection with the physical-layer network coding (PNC), which is a major decode-and-forward (DF)-based bidirectional protocol. We consider a network consists of two different end-sources and multiple relays, where each relay adopts XOR-encoding to combine two received symbols from the two end-sources. For this network, we propose a relay selection scheme by modifying the well-known selection cooperation (SC) to use in the PNC protocol, and it is referred to as SC-PNC. The SC-PNC consists of two phases: a multiple access channel (MAC) phase and a broadcast channel (BC) phase. In the MAC phase, a set of relays that correctly decode two received symbols from the two end-sources is determined; in the BC phase, among the relays in the determined set, a single best relay is selected such that the minimum mutual information of the two links from each relay to the two end-sources is maximized. Finally, we derive the exact outage probability of the SC-PNC in closed-form.