Multi-dimensional bounds on time-of-arrival statistics in random scattering channels

Bounds on Gaussian and hyperbolic time-of-arrival (ToA) probability density functions (pdfs) in multi-path mobile environments are derived. The bounds are expressed as functions of the coverage distance D between a base station (BS) and a user equipment (UE), and the propagation time ratio Ï„n = Ï„/Ï„0 where Ï„ is the general time delay and Ï„0 is the nominal time delay between the BS and the UE via the direct line-of-sight path. Detailed discussions on the variation of the ToA pdfs in 1-D and 3-D are given which are useful for system modelling in mobile environments.