  Reconfigurable intelligent surface (RIS) is a promising technological enabler
for the 6th generation (6G) of wireless systems with applications in
localization and communication. In this paper, we consider the problem of
positioning a single-antenna user in 3D space based on the received signal from
a single-antenna base station and reflected signal from an RIS by taking into
account the mobility of the user and spatial-wideband (WB) effects. To do so,
we first derive the spatial-WB channel model under the far-field assumption,
for orthogonal frequency-division multiplexing signal transmission with the
user having a constant velocity. We derive the Cramer Rao bounds to serve as a
benchmark. Furthermore, we devise a low-complexity estimator that attains the
bounds in high signal-to-noise ratios. Our estimator neglects the spatial-WB
effects and deals with the user mobility by estimating the radial velocities
and compensating for their effects in an iterative fashion. We show that the
spatial-WB effects can degrade the localization accuracy for large RIS sizes
and large signal bandwidths as the direction of arrival or departure deviate
from the RIS normal. In particular, for a 64X64 RIS, the proposed estimator is
resilient against the spatial-WB effects up to 140 MHz bandwidth. Regarding
user mobility, our results suggest that the velocity of the user influences
neither the bounds nor the accuracy of our estimator. Specifically, we observe
that the state of the user with a high speed (42 m/s) can be estimated
virtually with the same accuracy as a static user.
