Complex Linear Physical-Layer Network Coding (2017)

This paper presents the results of a comprehensive investigation of complex linear physicallayer 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, w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> = αw <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> + βw <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> , 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (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 = αw <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> + βw <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> when the relay attempts 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, w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> , w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</sub> , and w <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> are the elements of a finite field of Gaussian integers, that is, the field of 7G[i]/q, where q is a Gaussian prime. Previous vector formulation, in which wA, wB, and wN were represented by 2-D vectors and α and β were represented by 2 x 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 can be complex, gives us a larger set of signal constellations to achieve different rates at different values of SNR; and 2) we show how to divide the complex plane of η into different Voronoi regions, such that the η within each Voronoi region shares the same optimal PNC mapping (α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">opt</sub> , β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">opt</sub> ). 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 (α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">opt</sub> , β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">opt</sub> ). Overall, the contributions in 1) and 2) yield a toolset for a comprehensive understanding of complex linear PNC in 7G[i]/q. We believe investigation of linear PNC beyond 7G[i]/q can follow the same approach.