Wireless systems can be designed and analyzed at different levels of abstraction. At a high level the network can be seen as a set of random points, representing clients that are covered by
a set of discs, representing base stations broadcast domains.
Following this representation, in the first part of the talk we present a generalization of the mathematical theory of Continuum Percolation that describes the covering process and we show
how, using such extension, we can prove the existence of a phase transition in the network connectivity graph.
Observing that a practical way to achieve clients coverage is to let an algorithm place a minimum number of base stations on a regular lattice, we then present a theorem in geometry that
shows how the lattice density influences the quality of the coverage.
Finally, we turn to the question of existence of an optimal covering of points on the infinite plane using a minimal density of discs, and conjecture on the existence of a phase transition
behavior in its computability.
In the second part of the talk we consider a lower level of abstraction, where the abstract platonic world of discs, connected graphs, and points, is replaced by the laws of electromagnetic
(EM) wave propagation described by Maxwell equations.
In particular, we look at the problem of determining the propagation loss of a wireless channel, which is essential for link budget calculations in wireless network planning.
We introduce a new model of propagation in urban environments, based on random walks, that is analytically tractable. This leads to a path loss formula that is validated by comparison
with experimental data and with previously proposed empirical formulas.
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