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

干扰对准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建立的，因此一个给定的资源一次只能由一个用户使用。通过干扰对准，所有用户同时进入信道，每个用户获得一半的无干扰率，从而实现与用户数量线性扩展的网络吞吐量。两种最常研究的干涉对准方法是线性和非线性格式。在前一种情况下，对准问题可以表述为求解一个关于未知子集的超定方程组的问题，并且可以转换为熟悉的向量空间语言。后者需要数论中关于用有理数逼近无理数的深刻成果。目前还没有一个统一的框架囊括这两种方法。本研究使用现代代数中的模（残环或多项式环上的模）理论作为干涉对准的统一框架。首先介绍了该框架，并展示了通过选择合适的环来减少众所周知的线性和非线性干涉对准。然后，根据子模块分解对通道进行分类，并对通道的自由度进行评估。最后，通过一种新颖的方法设计了实际的代码，其中目的地将晶格代码解码为接收信号的壁橱（在嵌套晶格链中）。后一项任务与最近提出的计算转发中继方法有关。本文还探讨了两者之间的联系。所提出的工作的中心影响源于其对干涉对准的统一理论处理。目前还没有这种统一的框架或方法。尽管人们经常观察到经典的线性代数工具不足以解决网络问题，但到目前为止，模块理论还没有成为解释、分析和设计干扰网络代码的工具。本研究通过探索干扰对准的新基础，将为社区提供一个极其丰富的工具集来设计和分析代码，预计将影响和受益于其他网络问题，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信息处理的范围，并为巩固分布式、分散、干扰网络的综合网络科学理论迈出重要一步。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提出，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.