Joint Distribution of Distance and Angles in Finite Wireless Networks

Directional beamforming will play a paramount role in 5G and beyond networks 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 that is specialized to exemplify the case of a rectangular region. We 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 design of analog codebooks, wireless routing algorithms, and the analysis of directional beamforming, which is illustrated by analyzing the coverage probability of an indoor scenario considering misaligned beams.