Nonmatrix closed-form expressions of the Cramér-Rao Bounds for near-field localization parameters

Near-field source localization problem by a passive antenna array makes the assumption that the time-varying sources are located near the antenna. In this situation, the far-field assumption (planar wavefront) is no longer valid and we have to consider a more complicated model parameterized by the bearing (as in the far-field case) and by the distance, named range, between the source and a reference sensor. We can find a plethora of estimation schemes in the literature but the ultimate performance has not been fully investigated. In this paper, we derive and analyze the Cramer-Rao Bound (CRB) for a single time-varying source. In this case, we obtain nonmatrix closed-form expressions. Our approach has two advantages: (i) the computational cost for a large number of snapshots of a matrix-based CRB can be high while our approach is cheap and (ii) some useful informations can be deduced from the behavior of the bound. In particular, we show that closer is the source from the array and/or higher is the carrier frequency, better is the estimation of the range.