Simultaneously Transmitting and Reflecting (STAR) RIS assisted over-the-air computation systems

The performance of over-the-air computation (Air-Comp) systems degrades due to the hostile channel conditions of wireless devices (WDs), which can be significantly improved by the employment of reconfigurable intelligent surfaces (RISs). However, the conventional RISs require that the WDs have to be located in the half-plane of the reflection space, which restricts their potential benefits. To address this issue, the novel family of simultaneously transmitting and reflecting reconfigurable intelligent surfaces (STAR-RIS) is considered in AirComp systems to improve the computation accuracy across a wide coverage area. To minimize the computation mean-squared-error (MSE) in STAR-RIS assisted AirComp systems, we propose a joint beamforming design for optimizing both the transmit power at the WDs, as well as the passive reflect and transmit beamforming matrices at the STAR-RIS, and the receive beamforming vector at the fusion center (FC). Specifically, in the updates of the passive reflect and transmit beamforming matrices, closed-form solutions are derived by introducing an auxiliary variable and exploiting the coupled binary phase-shift conditions. Moreover, by assuming that the number of antennas at the FC and that of elements at the STAR-RIS/RIS are sufficiently high, we theoretically prove that the STAR-RIS assisted AirComp systems provide higher computation accuracy than the conventional RIS assisted systems. Our numerical results show that the proposed beamforming design outperforms the benchmark schemes relying on random phase-shift constraints and the deployment of conventional RIS. Moreover, its performance is close to the lower bound achieved by the beamforming design based on the STAR-RIS dispensing with coupled phase-shift constraints