  Since reconfigurable intelligent surface (RIS) is considered to be a passive
reflector for rate performance enhancement, a RIS-aided amplify-and-forward
(AF) relay network is presented. By jointly optimizing the beamforming matrix
at AF relay and the phase shifts matrices at RIS, two schemes are put forward
to address a maximizing signal-to-noise ratio (SNR) problem. Firstly, aiming at
achieving a high rate, a high-performance alternating optimization (AO) method
based on Charnes-Cooper transformation and semidefinite programming (CCT-SDP)
is proposed, where the optimization problem is decomposed to three subproblems
solved by CCT-SDP and rank-one solutions can be recovered by Gaussian
randomization. While the optimization variables in CCT-SDP method are matrices,
which leads to extremely high complexity. In order to reduce the complexity, a
low-complexity AO scheme based on Dinkelbachs transformation and successive
convex approximation (DT-SCA) is put forward, where matrices variables are
transformed to vector variables and three decoupled subproblems are solved by
DT-SCA. Simulation results verify that compared to two benchmarks (i.e. a
RIS-assisted AF relay network with random phase and a AF relay network without
RIS), the proposed CCT-SDP and DT-SCA schemes can harvest better rate
performance. Furthermore, it is revealed that the rate of the low-complexity
DT-SCA method is close to that of CCT-SDP method.
