Joint and marginal probabilities for time of arrival and angle of arrival using ellipsoidal model

In this paper, time of arrival and angle of arrival statistics for wireless propagation environments are evaluated in closed form using three dimensional ellipsoidal model. Scatterers are assumed to be uniformly distributed inside the ellipsoid whose foci collocate with the transmitter and the receiver. This model finds numerous applications in micro- and pico-cell outdoor as well as for indoor environments where scatterers significantly impact the transmitter and receiver characteristics. Joint and marginal probability density functions of time of arrival and angles (both azimuth and elevation) of arrival at the receiver side are evaluated in closed form. Numerical simulations are used to validate the analytical derivations.