Complex Linear Physical-Layer Network Coding

Abstract

This paper presents the results of a comprehensive investigation of complex linear physical-layer network (PNC) in two-way relay channels (TWRC). 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 = αwA + βwB, 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 = β −1 (wN − αwA). Node B obtains wB in a similar way. A critical question at relay R is as follows: "Given channel gain ratio η = 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 (α, β) that minimizes the symbol error rate (SER) of wN = αwA + βwB when we attempt to detect wN 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 α, β, wA, wB, and wN are elements of a finite field of Gaussian integers, that is, the field of Z[i]/q where q is a Gaussian prime. Previous vector formulation, in which wA, wB, and wN were represented by 2-dimensional vectors and α and β were represented by 2 × 2 matrices, corresponds to a subcase of our Gaussian-integer formulation where q is real prime only. Extension to Gaussian prime q, where q 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 η into different Voronoi regions such that the η within each Voronoi region share the same optimal PNC mapping (αopt, βopt). We uncover the structure of the Voronoi regions that allows us to compute a minimum-distance metric that characterizes the SER of wN under optimal PNC mapping (αopt, βopt). Overall, the contributions in (1) and (2) yield a toolset for a comprehensive understanding of complex linear PNC in Z[i]/q. We believe investigation of linear PNC beyond Z[i]/q can follow the same approach.

Implementation of uplink network‐coded modulation for two‐hop networks

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“…"> 2.The PNC mapping function must be well designed to resolve all singular fade states (SFSs). 3.The hub must be able to restore both UE messages based on the two NCS streams from both APs. Instead of using the PNC approach in the work of Shi and Liew, where the linear combinations are performed on complex symbols, we design the PNC mapping directly on the message bits; thus, the AP decodes the linear network‐coded function over . We define as the bit‐level linear network‐coded function of the j th AP and as the network‐coded vector (NCV) consisting two linear network‐coded bits, ie, where w denotes the 1 × 4 joint UE message binary vector, M j denotes the 2 × 4 binary mapping matrix, and ⋆ denotes the modulo‐2 matrix multiplication.…”

Physical-Layer Network Coding: An Efficient Technique for Wireless Communications

“…In this way, we can obtain a unique NC symbol derived from the superimposed symbol which is generated even by multiple transmission pairs. The choice of decoding coefficients has been intensively studied in [6].…”

“…Thus, it is significant to investigate K-user PNC communication scenarios, K ≥ 3. How to characterize the decoding behavior for K-user PNC based on the constellation minimum distance [6] is still a challenging task. 3) Design of Channel Codes for PNC: To approach the PNC channel capacity at high SNR regime, prior works have designed irregular repeat-accumulate codes and irregular LDPC codes for XOR-CD and NC-CD, respectively.…”

Low-Complexity Estimation for Spatially Modulated Physical-Layer Network Coding Systems

“…Recently, two-way relay systems have received much attention [1][2][3][4] as they not only can extend system coverage but also increase transmission efficiency. Among various two-way relay schemes, physical-layer network coding (PNC) [1][2][3][4][5] is known as an effective scheme as it allows the two terminal nodes to transmit at the same time and same frequency during multiple access phase, thus reducing the number of exchange phases to two. As a consequence, throughput and spectral efficiency of the PNC system are higher than those of the traditional network coding (NC) [6].…”