Exact Conditional and Unconditional Cram\`er-Rao Bounds for Near Field Localization

Abstract

This paper considers the Cram\`er-Rao lower Bound (CRB) for the source localization problem in the near field. More specifically, we use the exact expression of the delay parameter for the CRB derivation and show how this exact CRB can be significantly different from the one given in the literature and based on an approximate time delay expression (usually considered in the Fresnel region). This CRB derivation is then generalized by considering the exact expression of the received power profile (i.e., variable gain case) which, to our best knowledge, has been ignored in the literature. Finally, we exploit the CRB expression to introduce the new concept of Near Field Localization (NFL) region for a target localization performance associated to the application at hand. We illustrate the usefulness of the proposed CRB derivation and its developments as well as the NFL region concept through numerical simulations in different scenarios