A simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) assisted downlink (DL) active and uplink (UL) backscatter communication (BackCom) framework is proposed. More particularly, a full-duplex (FD) base station (BS) communicates with the DL users via the STAR-RIS's transmission link, while exciting and receiving the information from the UL BackCom devices with the aid of the STAR-RIS's reflection link. Non-orthogonal multiple access (NOMA) is exploited in both DL and UL communications for improving the spectrum efficiency. The system weighted sum rate maximization problem is formulated for jointly optimizing the FD BS active receive and transmit beamforming, the STAR- RIS passive beamforming, and the DL NOMA decoding orders, subject to the DL user's individual rate constraint. To tackle this challenging non-convex problem, we propose an alternating optimization (AO) based algorithm for the joint active and passive beamforming design with a given DL NOMA decoding order. To address the potential high computational complexity required for exhaustive searching all the NOMA decoding orders, an efficient NOMA user ordering scheme is further developed. Finally, numerical results demonstrate that: i) compared with the baseline schemes employing conventional RISs or space division multiple access, the proposed scheme achieves higher performance gains; and ii) higher UL rate gain is obtained at a cost of DL performance degradation, as a remedy, a more flexible performance tradeoff can be achieved by introducing the STAR-RIS.

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