  In this paper, we present an analytical framework for the joint
characterization of the 3D direction of arrival (DoA), i.e., azimuth and
elevation components, and time of arrival (ToA) in multipath environments. The
analytical framework is based on the use of nearly frequency-invariant
beamformers (FIB) formed by toric arrays. The frequency response of the toric
array is expanded as a series of phase modes, which leads to azimuth-time and
elevation-time diagrams from which the 3D DoA and the ToA of the incoming waves
can be extracted over a wide bandwidth. Firstly, we discuss some practical
considerations, advantages and limitations of using the analytical method.
Subsequently, we perform a parametric study to analyze the influence of the
method parameters on the quality of the estimation. The method is tested in
single-path and multipath mm-wave environments over a large bandwidth. The
results show that the proposed method improves the quality of the estimation,
i.e., decreases the level of the artifacts, compared to other state-of-art FIB
approaches based on the use of single/concentric circular and elliptical
arrays.
