A Generalized 3-D Scattering Model for a Macrocell Environment With a Directional Antenna at the BS

In this paper, a generalized 3-D scattering model is presented for macrocellular land mobile radio cellular systems with a mobile station (MS) located at the center of a 3-D scattering semispheroid and a base station (BS) employing a directional antenna located outside of the semispheroid. The effect of the directional antenna is thoroughly observed on spatial and temporal characteristics of the proposed model. Closed-form expressions for joint and marginal probability density functions (pdf´s) of the angle of arrival (AoA) seen at the MS and the BS in correspondence with azimuth and elevation angles are derived. Furthermore, closed-form expressions for propagation path delays and trivariate joint pdf´s of the time of arrival (ToA) seen at the MS and the BS in correspondence with azimuth and elevation angles are derived. Moreover, the theoretical results, along with observations, are shown that illustrate the effect of the directional antenna on the spatiotemporal statistics of the proposed 3-D model. The proposed 3-D scattering model is shown to deduce all previously proposed 2-D and 3-D models that assume a uniform distribution of scatters with directional or omnidirectional antennas found in the literature for a macrocell environment. The obtained theoretical results are compared with some notable 2-D and 3-D scattering models to validate the generalization of the proposed model.