  In this paper, we introduce the concept of partially-connected Receiving
Reconfigurable Intelligent Surfaces (R-RISs), which refers to metasurfaces
designed to efficiently sense electromagnetic waveforms impinging on them, and
perform localization of the users emitting them. The presented R-RIS hardware
architecture comprises subarrays of meta-atoms, with each of them incorporating
a waveguide assigned to direct the waveforms reaching its meta-atoms to a
reception Radio-Frequency (RF) chain, enabling signal/channel parameter
estimation. We particularly focus on the scenarios where the user is located in
the 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-RIS
subarray. For the AoA estimation, which relies on spatially sampled versions of
the received signals via each subarray's phase configuration of meta-atoms, we
devise an off-grid atomic norm minimization approach, which is followed by
subspace-based root MUltiple SIgnal Classification (MUSIC). The AoA estimates
are finally combined via a least-squared line intersection method to obtain the
position coordinates of a user emitting synchronized localization pilots. Our
derived theoretical Cram\'er Rao Lower Bounds (CRLBs) on the estimation
parameters, which are compared with extensive computer simulation results of
our localization approach, verify the effectiveness of the proposed
R-RIS-empowered 3D localization system, which is showcased to offer cm-level
positioning accuracy. Our comprehensive performance evaluations also
demonstrate the impact of various system parameters on the localization
performance, namely the training overhead and the distance between the R-RIS
and the user, as well as the spacing among the R-RIS's subarrays and its
partitioning patterns.
