  Reconfigurable Intelligent Surfaces (RIS) are an emerging technology that can
be used to reconfigure the propagation environment to improve cellular
communication link rates. RIS, which are thin metasurfaces composed of discrete
elements, passively manipulate incident electromagnetic waves through
controlled reflective phase tuning. In this paper, we investigate co-design of
the multiantenna basestation beamforming vector and multielement RIS phase
shifts. The downlink narrowband transmission uses sub-6 GHz frequency bands,
and the user equipment has a single antenna. Subject to the non-convex
constraints due to the RIS phase shifts, we maximize the spectral efficiency or
equivalent channel power as a proxy. Our contributions in improving RIS-aided
links include (1) design of gradient ascent codesign algorithms, and (2)
comparison of seven codesign algorithms in spectral efficiency vs.
computational complexity. In simulation, the best spectral efficiency vs.
computational complexity tradeoffs are shown by two of our proposed gradient
ascent algorithms.
