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In this paper, we introduce the concept of partially-connected ReceivingReconfigurable Intelligent Surfaces (R-RISs), which refers to metasurfacesdesigned to efficiently sense electromagnetic waveforms impinging on them, andperform localization of the users emitting them. The presented R-RIS hardwarearchitecture comprises subarrays of meta-atoms, with each of them incorporatinga waveguide assigned to direct the waveforms reaching its meta-atoms to areception Radio-Frequency (RF) chain, enabling signal/channel parameterestimation. We particularly focus on the scenarios where the user is located inthe far-field of all the R-RIS subarrays, and present a three-Dimensional (3D)localization method which is based on narrowband signaling and Angle of Arrival(AoA) estimates of the impinging signals at each single-receive-RF R-RISsubarray. For the AoA estimation, which relies on spatially sampled versions ofthe received signals via each subarray's phase configuration of meta-atoms, wedevise an off-grid atomic norm minimization approach, which is followed bysubspace-based root MUltiple SIgnal Classification (MUSIC). The AoA estimatesare finally combined via a least-squared line intersection method to obtain theposition coordinates of a user emitting synchronized localization pilots. Ourderived theoretical Cram\'er Rao Lower Bounds (CRLBs) on the estimationparameters, which are compared with extensive computer simulation results ofour localization approach, verify the effectiveness of the proposedR-RIS-empowered 3D localization system, which is showcased to offer cm-levelpositioning accuracy. Our comprehensive performance evaluations alsodemonstrate the impact of various system parameters on the localizationperformance, namely the training overhead and the distance between the R-RISand the user, as well as the spacing among the R-RIS's subarrays and itspartitioning patterns.

3D Localization with a Single Partially-Connected Receiving RIS: Positioning Error Analysis and Algorithmic Design