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This paper derives the maximum achievable sum-rate and presents the optimal encoding/decoding framework for rate-diverse wireless network coding (RD-WNC) over broadband channels consisting of multiple parallel subchannels. RD-WNC applies to a communication scenario in which a base station wants to deliver two different messages with different rates to two users. The base station combines the two separate messages into one network-coded message and broadcasts the network-coded message to both users. Each user then extracts its desired message from the network-coded message by subtracting from it the other message, which we assume to be side information available to the user. Deriving the maximum achievable sum-rate for RD-WNC is challenging when the channel consists of multiple parallel subchannels with different channel coefficients (e.g., the subcarrier channels of OFDM systems), since apart from the rate allocation between the two users, optimal power allocation among multiple subchannels needs to be identified. The first contribution of this paper is a new “mountain-leveling” power allocation algorithm to achieve the maximum sum-rate. With the resulting power allocation, we can then achieve the corresponding optimal sum-rate by having a separate encoding/decoding mechanism for each subchannels, but doing so is cumbersome and complex when the number of subchannels is large. The second contribution of this paper is a practical encoding/decoding framework using only one encoder-decoder mechanism for all subchannels without sacrificing sum-rate optimality. We provide numerical results to corroborate our theoretical findings and to demonstrate the benefits of our encoding/decoding framework. [ABSTRACT FROM AUTHOR]

This paper presents the results of a comprehensive investigation of complex linear physical-layer network coding (PNC) in two-way relay channels. In this system, two nodes A and B communicate with each other via a relay R. Nodes A and B send complex symbols, wA and wB , simultaneously to relay R. Based on the simultaneously received signals, relay R computes a linear combination of the symbols, wN=\alpha wA+\beta wB , as a network-coded symbol and then broadcasts wN to nodes A and B. Node A then obtains wB from wN and its self-information wA by wB=\beta ^{-1}(wN-\alpha wA) . Node B obtains wB in a similar way. A critical question at relay R is as follows: “given channel gain ratio \eta = hA/hB , where hA and hB are the complex channel gains from nodes A and B to relay R, respectively, what is the optimal coefficients (\alpha ,\beta ) that minimizes the symbol error rate (SER) of \alpha ,\beta ,w_{A}, w_{B} , and wN are the elements of a finite field of Gaussian integers, that is, the field of \mathbb {Z}[i]/q , where q is a Gaussian prime. Previous vector formulation, in which \alpha and \beta were represented by 2\times 2 matrices, corresponds to a subcase of our Gaussian-integer formulation, where q is real prime only. Extension to the Gaussian prime q , where q . We uncover the structure of the Voronoi regions that allows us to compute a minimum-distance metric that characterizes the SER of (\alpha \mathrm{ opt},\beta \mathrm{ opt}) . Overall, the contributions in 1) and 2) yield a toolset for a comprehensive understanding of complex linear PNC in \mathbb {Z}[i]/q . We believe investigation of linear PNC beyond \mathbb Z[i]/q can follow the same approach. [ABSTRACT FROM AUTHOR]

This paper proposes an encoding/decoding framework for achieving the optimal channel capacities of the two-user broadcast channel where each user (receiver) has the message targeted for the other user (receiver) as side information. Since the link qualities of the channels from the base station to the two users are different, their respective single-user non-broadcast channel capacities are also different. A goal is to simultaneously achieve/approach the single-user non-broadcast channel capacities of the two users with a single broadcast transmission by applying network coding. This is referred to as the rate-diverse wireless network coding problem. For this problem, this paper presents a capacity-achieving framework based on linear-structured nested lattice codes. The significance of the proposed framework, besides its theoretical optimality, is that it suggests a general design principle for linear rate-diverse wireless network coding going beyond the use of lattice codes. We refer to this design principle as the principle of virtual single-user channels. Guided by this design principle, we propose two implementations of our encoding/decoding framework using practical linear codes amenable to decoding with affordable complexities: the first implementation is based on Low Density Lattice Codes (LDLC) and the second implementation is based on Bit-interleaved Coded Modulation (BICM). These two implementations demonstrate the validity and performance advantage of our framework. [ABSTRACT FROM PUBLISHER]

Symbol misalignment is inevitable in asynchronous physical-layer network coding (PNC) systems. It is paramount that such symbol misalignment is taken into account in PNC decoding for good performance. Thus, accurate estimation of symbol misalignment is crucial. This paper argues that, when Nyquist pulses (i.e., intersymbol-interference (ISI)-free pulses) are adopted, signal samples only need to be collected at baud rate for optimal symbol misalignment estimation. Based on this principle, we propose a highly accurate symbol misalignment estimation method with low complexity. Our method makes use of the constant amplitude zero autocorrelation sequence (Zadoff–Chu sequence (ZC sequence)). We derive a maximum-likelihood (ML) estimator for symbol misalignment based on the cross-correlation result of the ZC sequence. Unlike previous methods that employ oversampling, our estimation method requires only baud-rate sampling, thus having much lower complexity. Extensive simulations show that our method can accurately estimate both integral and fractional symbol misalignments using sinc pulse and raised-cosine (RC) pulse. The root-mean-square error (RMSE) of the estimation is below 10−2 (in unit of symbol duration) when the SNR is above 15, 18, and 21 dB for 127-, 63-, and 31-bit-length ZC sequences, respectively. Furthermore, our method, being an ML estimation method, has no error floor in the high-SNR regime, whereas the prior methods exhibit an error floor. [ABSTRACT FROM PUBLISHER]

This paper presents several soft decision iterative decoding schemes for physical-layer network coding (PNC) operated with coded modulation (CM) and bit-interleaved coded modulation (BICM). With respect to PNC operated with CM, we consider network coding-based channel decoding (NC-CD) and multi-user complete decoding (MUD-NC) for PNC decoding at the relay. Their BICM counterparts are XOR-based channel decoding (XOR-CD) and MUD-XOR, respectively. First, we show that, when the decoding is non-iterative, there is a gap between the BICM capacities of both XOR-CD and MUD-XOR under Gray mapping and the capacities of their CM counterparts, NC-CD, and MUD-NC. This is in contrast to the conventional point-to-point communication system, for which the BICM capacity with Gray mapping is known to be very close to the CM capacity, without the need for iterative decoding. Second, we investigate the error performance of iteratively decoded BICM XOR-CD and MUD-XOR. Extrinsic information transfer chart analysis and simulation results indicate that for these Gray-mapped BICM PNC systems, iterative decoding can achieve considerable gains over non-iterative decoding. Again, this is in contrast to the Gray-mapped BICM point-to-point communication system, for which iterative decoding provides little gain over non-iterative decoding. We further show that Gray mapping gives rise to best PNC rate for MUD-XOR and XOR-CD systems among several bits-to-symbol mappings under study. Overall, our results indicate that BICM PNC systems exhibit different decoding behavior from conventional BICM point-to-point systems. This paper serves as a first foray into the investigation of this issue. [ABSTRACT FROM PUBLISHER]

The growing demand for high-speed data, quality of service (QoS) assurance, and energy efficiency has triggered the evolution of fourth-generation (4G) Long-Term Evolution-Advanced (LTE-A) networks to fifth generation (5G) and beyond. Interference is still a major performance bottleneck. This paper studies the application of physical-layer network coding (PNC), which is a technique that exploits interference, in heterogeneous cellular networks. In particular, we propose a rate-maximizing relay selection algorithm for a single cell with multiple relays assuming the decode-and-forward (DF) strategy. With nodes transmitting at different powers, the proposed algorithm adapts the resource allocation according to the differing link rates, and we prove theoretically that the optimization problem is log-concave. The proposed technique is shown to perform significantly better than the widely studied selection-cooperation technique. We then undertake an experimental study—on a software radio platform—of the decoding performance of PNC with unbalanced signal-to-noise ratios (SNRs) in the multiple-access transmissions. This problem is inherent in cellular networks, and it is shown that, with channel coding and decoders based on multiuser detection and successive interference cancellation, the performance is better with power imbalance. This paper paves the way for further research on multicell PNC, resource allocation, and the implementation of PNC with higher order modulations and advanced coding techniques. [ABSTRACT FROM PUBLISHER]

This paper investigates various subtleties of applying linear physical-layer network coding (PNC) with $q$ -level pulse amplitude modulation ( $q$ -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 $q$ -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 $q$ . 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 $q$ -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 $q$ 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. [ABSTRACT FROM AUTHOR]

This paper studies the transceiver design of the Gaussian two-pair two-way relay channel (TWRC), where two pairs of users exchange information through a common relay in a pairwise manner. Our main contribution is to show that the capacity of the Gaussian two-pair TWRC is achievable to within $\frac{1}{ 2}$ bit for arbitrary channel conditions. For the outer bound, we derive a genie-aided bound of the Gaussian two-pair TWRC, which is tighter than the cut-set bound. For the inner bound, we develop a hybrid coding scheme involving Gaussian random coding, nested lattice coding, superposition coding, and network-coded decoding. We further present a message-reassembling strategy to decouple the coding design for the user-to-relay and relay-to-user links, so as to provide flexibility to fully exploit the channel randomness. We show that judicious power allocation at the users and at the relay is necessary to approach the channel capacity under various channel conditions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized channel decoding scheme for nonbinary physical-layer network coding (CD-NC) in two-way relay channels (TWRCs), where two source nodes A and B exchange their nonbinary symbols via a relay. The two sources use the same nonbinary low-density parity-check (LDPC) channel code over the integer ring $\mathbb {Z}_M$ and $M$ -pulse-amplitude modulation, respectively. The existing channel decoding schemes for nonbinary network coding suffer severe rate loss compared with the cut-set bound of TWRC, especially in the low-to-medium signal-to-noise ratio regime. The proposed CD-NC can decrease the rate loss. Our contributions are as follows: 1) We develop a generalized nonbinary sum product algorithm (G-SPA) for CD-NC according to the principle of virtual encoding of the superimposed symbols. Simulation results show that our CD-NC can achieve significant performance gains over the conventional nonbinary network coding for both additive white Gaussian noise and fading channels; and 2) We exploit two-dimensional fast-Fourier-transform-based belief propagation (2-D-FFT-BP) and extended min-sum (EMS) decoding algorithms to reduce the decoding complexity of G-SPA. Simulation results show that the 2-D-FFT-BP has the same performance as G-SPA, while EMS can greatly reduce the decoding complexity of G-SPA at the cost of slight performance degradation. [ABSTRACT FROM AUTHOR]

This paper studies and analyzes a multi-pair MIMO two-way relay (TWR) system from a principal-angle perspective. In this system, users are grouped into pairs and two users in each pair exchange information through a common relay. The transmission scheme consists of two phases. In the uplink phase, the relay’s receive signal space is divided into two subspaces for each pair: in one subspace, the signals of the two users are near-parallel, and physical-layer network coding (PNC) decoding is applied to retrieve network-coded messages; in the other subspace, the two user signals are near-orthogonal, and complete decoding (CD) is applied to retrieve individual user messages. In the downlink phase, the relay’s transmit signal space is split into orthogonal subspaces for each pair, and in each subspace a common message is broadcasted to the two users. Our main contribution is to establish a critical link between the transmission scheme and the concept of principal angle in linear algebra. In particular, principal angle quantizes the degree of orthogonality between the user signals and helps determining how to partition the relay’s receive signal space into PNC decoding subspace and CD decoding subspace. Building on that, we discuss the optimal design of user/relay precoders for both uplink and downlink phases to maximize the asymptotic sum-rate of the proposed scheme at high signal-to-noise ratio (SNR). The analysis and numerical results indicate that our scheme performs close to the cut-set outer bound and significantly outperforms the existing schemes in the literature. [ABSTRACT FROM AUTHOR]

We present an optimal decoder for physical-layer network coding (PNC) in multipath fading channels. Previous studies on PNC have largely focused on the single-path case. For PNC, multipath introduces not only intersymbol interference (ISI) but cross-symbol interference (Cross-SI) between signals simultaneously transmitted by multiple users as well. To overcome these problems, the decoder at the relay of our PNC design makes use of a belief propagation (BP) algorithm to decode the multipath-distorted signals received from multiple users into a network-coded packet. We refer to our multipath decoding algorithm as MP-PNC. Our simulation results show that, benchmarked against synchronous PNC over a one-path channel, the bit-error-rate (BER) performance penalty of MP-PNC under a two-tap International Telecommunication Union (ITU) channel model can be kept within 0.5 dB. Moreover, it outperforms a multiuser detection (MUD)– xor algorithm by 3 dB (MUD– xor decodes the individual information from both users explicitly before performing the xor network-coding mapping). Although the framework of fading-channel PNC presented in this paper is demonstrated based on two- and three-path channel models, our algorithm can be extended to cases with more than three paths. [ABSTRACT FROM PUBLISHER]