The Inclusion-Exclusion Principle is a counting technique used in combinatorics and probability theory to calculate the number of elements in the union of multiple sets. It provides a systematic way to account for the overlaps or intersections between sets when counting their combined elements. The inclusion-exclusion principle is typically stated for two or more sets, but it can be extended to handle more sets.


Multi-user Beam Alignment for Millimeter Wave Systems in Multi-path Environments

Directional transmission patterns (a.k.a. narrow beams) are the key to wireless communications in millimeter wave (mmWave) frequency bands which suffer from high path loss, severe shadowing, and intense blockage. In addition, the propagation channel in mmWave frequencies incorporates only a few number of spatial clusters requiring a procedure, called beam alignment (BA), to align the corresponding narrow beams with the angle of departure (AoD) of the channel clusters. In addition, BA enables beamforming gains to compensate path loss and shadowing or diversity gains to combat the blockage. Most of the prior analytical studies have considered strong simplifying assumptions such as i) having a single-user scenario and ii) having a single dominant path channel model for theoretical tractability. In this study, we relax such constraints and provide a theoretical framework to design and analyze optimized multiuser BA schemes in multi-path environments. Such BA schemes not only reduce the BA overhead and provide beamforming gains to compensate path loss and shadowing, but also provide diversity gains to mitigate the impact of blockage in practical mmWave systems.