Directional beamforming will play a paramount role in 5G and beyond networks in order to combat the higher path losses incurred at millimeter wave bands. Appropriate modeling and analysis of the angles and distances between transmitters and receivers in these networks are thus essential to understand performance and limiting factors. Most existing literature considers either infinite and uniform networks, where nodes are drawn according to a Poisson point process, or finite networks with the reference receiver placed at the origin of a disk. Under either of these assumptions, the distance and azimuth angle between transmitter and receiver are independent, and the angle follows a uniform distribution between $0$ and $2\pi$. Here, we consider a more realistic case of finite networks where the reference node is placed at any arbitrary location. We obtain the joint distribution between the distance and azimuth angle and demonstrate that these random variables do exhibit certain correlation, which depends on the shape of the region and the location of the reference node. To conduct the analysis, we present a general mathematical framework which is specialized to exemplify the case of a rectangular region. We then also derive the statistics for the 3D case where, considering antenna heights, the joint distribution of distance, azimuth and zenith angles is obtained. Finally, we describe some immediate applications of the present work, including the analysis of directional beamforming, the design of analog codebooks and wireless routing algorithms.

翻译：定向波束成形将在5G及未来网络中发挥关键作用，以应对毫米波频段更高的路径损耗。因此，在这些网络中对发射机与接收机之间的角度与距离进行适当建模与分析，对于理解性能及其限制因素至关重要。现有文献主要考虑两种场景：一是节点服从泊松点过程的无限均匀网络，二是参考接收机位于圆盘中心的有界网络。在这两种假设下，发射机与接收机之间的距离和方位角相互独立，且角度在$0$到$2\pi$之间服从均匀分布。本文考虑了一种更实际的场景——参考节点可置于任意位置的有界网络。我们推导了距离与方位角的联合分布，并证明这些随机变量确实存在一定相关性，其相关程度取决于区域形状及参考节点位置。为完成该分析，我们提出一个通用数学框架，并具体应用于矩形区域案例。随后，我们进一步导出了三维情况下的统计特性——考虑天线高度后，获得了距离、方位角与天顶角的联合分布。最后，我们阐述了本研究的若干直接应用，包括定向波束成形分析、模拟码本设计及无线路由算法等。