  Reconfigurable intelligent surfaces (RISs) are able to provide passive
beamforming gain via low-cost reflecting elements and hence improve wireless
link quality. This work considers two-way passive beamforming design in
RIS-aided frequency division duplexing (FDD) systems where the RIS reflection
coefficients are the same for downlink and uplink and should be optimized for
both directions simultaneously. We formulate a joint optimization of the
transmit/receive beamformers at the base station (BS) and the RIS reflection
coefficients. The objective is to maximize the weighted sum of the downlink and
uplink rates, where the weighting parameter is adjustable to obtain different
achievable downlink-uplink rate pairs. We develop an efficient manifold
optimization algorithm to obtain a stationary solution. For comparison, we also
introduce two heuristic designs based on one-way optimization, namely,
time-sharing and phase-averaging. Simulation results show that the proposed
manifold-based two-way optimization design significantly enlarges the
achievable downlink-uplink rate region compared with the two heuristic designs.
It is also shown that phase-averaging is superior to time-sharing when the
number of RIS elements is large.
