CIF: Small: Modules as a Framework for Interference Alignment in Networks

干扰对齐是无线网络中多个用户共享相同信道资源的突破性范例。今天的商用网络是为了避免干扰而构建的，使得给定的资源一次仅由一个用户使用。通过干扰对齐，所有用户同时接入信道，每个用户获得一半的无干扰速率，从而实现与用户数量线性缩放的网络吞吐量。两种最常研究的干扰对齐方法是线性和非线性方案。在前一种情况下，对准问题可以用公式表示为求解关于未知数子集的超定方程组，并且可以转换为熟悉的向量空间语言。后者要求数论中关于用有理数近似无理数的深入结果。目前还没有一个统一的框架包含这两种方法。 本研究使用现代代数中的模理论（关于剩余环或多项式环）作为干涉对齐的统一框架。首先，该框架被引入，并示出减少众所周知的线性和非线性干扰对齐通过选择适当的环。然后，渠道进行分类，根据他们的子模块分解和他们的自由度评估。最后，实际的代码设计通过一种新的方法，其中一个目的地解码的格码是最接近（在嵌套格链）的接收信号。后一项任务与最近提出的计算和转发中继方法有关。两者之间的联系也进行了探讨。所提出的工作的中心影响源于其统一的理论处理的干扰对齐。目前还不存在这样的统一框架或方法。虽然人们经常观察到，经典的线性代数工具是不够的网络问题，模块的理论还没有出现到目前为止，作为一个工具来解释，分析和设计代码的干扰网络。本研究通过探索干扰对齐的新基础，将为社区提供极其丰富的工具集来设计和分析代码，这有望影响和有益于其他网络问题，如网络编码和计算转发中继。从技术角度来看，本研究提出的新范式将扩大干扰网络中的信息处理范围，并提供了一个重要的一步，为分布式，分散，干扰网络的网络科学的综合理论的巩固。这项研究的结果将在主要的国家和国际专业场所，在信息理论和通信网络社区，预计将立即用于工业部门。

Interference Alignment is a breakthrough paradigm for wireless networks where several users share the same channel resources. Today's commercially available networks are built so as to avoid interference, so that a given resource is used by only one user at a time. With interference alignment, all users access the channel simultaneously and each user obtains half the interference-free rate, thereby achieving a network throughput that scales linearly with the number of users. The two most commonly studied approaches to interference alignment are the linear and non-linear schemes. In the former case the alignment problem can be formulated as that of solving an overdetermined system of equations with respect to a subset of the unknowns, and can be cast into the familiar language of vector spaces. The latter requires deep results in number theory on approximating an irrational number by a rational one. At present no unifying framework exists that encompasses both approaches. This research uses the theory of modules (on residue rings or polynomial rings) from modern algebra as a unifying framework for interference alignment. First the framework is introduced and shown to reduce to well known linear and non-linear interference alignment by choosing proper rings. Then, channels are classified in terms of their submodule decomposition and their degrees of freedom evaluated. Finally, actual codes are designed through a novel method where a destination decodes the lattice code that is the closet (in a nested lattice chain) to the received signal. This latter task has connections with the recently proposed compute-and-forward method of relaying. Connections among the two are also explored.The central impact of the proposed work stems from its unified theoretical treatment of interference alignment. No such unified framework or methodology exists at present. Although it has been often observed that classical linear algebra tools are insufficient for network problems, the theory of modules has not emerged so far as a tool to explain, analyze and design codes for interference networks. This research, by exploring novel foundations of interference alignment, will provide the community with an extremely rich toolset to design and analyze codes, which is expected to impact and benefit other network problem such as network coding and compute-and-forward relaying.From a technical perspective the new paradigm proposed in this research will enlarge the scope of information processing in interference networks, and provide an important step forward into the consolidation of a comprehensive theory of network science for distributed, decentralized, interfering networks. The results of this research will be presented at major national and international professional venues, in the information theory and communication networks communities and are expected to be of immediate use to the industrial sector.