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Complex Linear Physical-Layer Network Coding

机译：复杂线性物理层网络编码

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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, w_{A} and w_{B} , simultaneously to relay R. Based on the simultaneously received signals, relay R computes a linear combination of the symbols, w_{N}=alpha w_{A}+beta w_{B} , as a network-coded symbol and then broadcasts w_{N} to nodes A and B. Node A then obtains w_{B} from w_{N} and its self-information w_{A} by w_{B}=beta ^{-1}(w_{N}-alpha w_{A}) . Node B obtains w_{B} in a similar way. A critical question at relay R is as follows: “given channel gain ratio eta = h_{A}/h_{B} , where h_{A} and h_{B} 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 w_{N}=alpha w_{A}+beta w_{B} when the relay attempts to detect w_{N} 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 alpha ,beta ,w_{A}, w_{B} , and w_{N} 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 w_{A} , w_{B} , and w_{N} were represented by 2-D vectors and alpha and beta were represented by 2times 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 eta into different Voronoi regions, such that the eta within each Voronoi region shares the same optimal PNC mapping (alpha _{mathrm{ opt}},beta _{mathrm{ opt}}) . We uncover the structure of the Voronoi regions that allows us to compute a minimum-distance metric that characterizes the SER of w_{N} under optimal PNC mapping (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.
机译：本文介绍了对双向中继信道中的复杂线性物理层网络编码（PNC）进行全面研究的结果。在该系统中，两个节点A和B通过中继R彼此通信。节点A和B同时向中继R发送复杂符号w_ {A}和w_ {B}。基于同时接收到的信号，中继R计算符号w_ {N} = alpha w_ {A} + beta w_ {B}的线性组合作为网络编码符号，然后将w_ {N}广播到节点A和B。然后，节点A获得w_ {来自w_ {N}的B}及其自身信息w_ {A}，其中w_ {B} = beta ^ {-1}（w_ {N} -alpha w_ {A}）。节点B以类似的方式获得w_ {B}。继电器R上的一个关键问题如下：“给定通道增益比eta = h_ {A} / h_ {B}，其中h_ {A}和h_ {B}是从节点A和B到继电器R的复数通道增益分别使中继器尝试检测w_ {N} =αw_ {A} + beta w_ {B}的符号错误率（SER）最小的最佳系数（alpha，beta）是多少？有噪音吗？”关于这个问题，我们的贡献如下：1）针对α，β，w_ {A}，w_ {B}和w_ {N}是元素的复杂线性PNC，我们提出了一个一般的高斯整数公式。有限域的高斯整数，即mathbb {Z} [i] / q的域，其中q是高斯素数。以前的矢量公式，其中w_ {A}，w_ {B}和w_ {N}用2维矢量表示，而alpha和beta用2 x 2矩阵表示，对应于我们的高斯整数公式的一个子情形，其中q仅是实质数。扩展到高斯素数q，其中q可能很复杂，这为我们提供了更多的信号星座图集，以在不同的SNR值下实现不同的速率。和2）我们展示了如何将eta的复平面划分为不同的Voronoi区域，以使每个Voronoi区域内的eta共享相同的最佳PNC映射（α ）。我们发现了Voronoi区域的结构，该结构使我们能够计算最小距离度量，该度量表征在最佳PNC映射下的w_ {N}的SER（alpha _ {mathrm {opt}}，beta _ {mathrm {opt}}） 。总体而言，在1）和2）中的贡献产生了一个工具集，可用于全面理解mathbb {Z} [i] / q中的复杂线性PNC。我们相信对超出mathbb {Z} [i] / q的线性PNC的研究可以遵循相同的方法。

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