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The rising concept of reconfigurable intelligent surface (RIS) has promisingpotential for Beyond 5G localization applications. We herein investigatedifferent phase profile designs at a reflective RIS, which enablenon-line-of-sight positioning in nearfield from downlink single antennatransmissions. We first derive the closed-form expressions of the correspondingFisher information matrix (FIM) and position error bound (PEB). Accordingly, wethen propose a new localization-optimal phase profile design, assuming priorknowledge of the user equipment location. Numerical simulations in a canonicalscenario show that our proposal outperforms conventional RIS random anddirectional beam codebook designs in terms of PEB. We also illustrate the fourbeams allocated at the RIS (i.e., one directional beam, along with itsderivatives with respect to space dimensions) and show how their relativeweights according to the optimal solution can be practically implementedthrough time sharing (i.e., considering feasible beams sequentially).

Constrained RIS Phase Profile Optimization and Time Sharing for Near-field Localization