Transmit Power Minimization for RIS-Assisted CF-NOMA in Space–Ground Integrated Networks

Low Earth Orbit (LEO) satellite communications have emerged as a promising paradigm for achieving ubiquitous coverage, driving the evolution of space-ground integrated networks (SGINs). The cell-free (CF) architecture has attracted significant attention in SGINs as the terrestrial segment for its potential to enhance capacity and connectivity. However, deploying CF necessitates numerous access points (APs), resulting in a prohibitive cost. To this end, we propose reconfigurable intelligent surface (RIS)- and simultaneous transmitting and reflecting (STAR)-RIS-assisted CF systems for SGINs, where part of the APs is replaced with cost-efficient RISs and STAR-RISs. Non-orthogonal multiple access (NOMA) is incorporated to improve connectivity under limited spectrum. We formulate transmit power minimization problems for both RIS- and STAR-RIS-assisted CF-NOMA in SGINs, jointly optimizing the active beamforming vectors of the satellite and APs, as well as the discrete passive beamforming (DPB) vectors of RISs/STAR-RISs. For the RIS-assisted scenario, a semi-definite programming (SDP)-based method is proposed to optimize the active beamforming vectors, while an enhanced integer linear programming (ILP) method is proposed to obtain the optimal DPB of RISs. To reduce complexity, we develop a low-complexity penalty-based SDP (PB-SDP) algorithm that achieves near-optimal DPB solutions. For the STAR-RIS-assisted scheme, both independent and coupled DPB for transmission and reflection are optimized alone with the active beamforming vectors. Numerical results demonstrate that: 1) The proposed systems outperform cell-based systems and heuristic optimization algorithms in terms of transmit power consumption; 2) The proposed PB-SDP algorithm achieves near-optimal performance with reduced complexity; 3) It is shown that DBP with 3 quantization bits achieves performance comparable to continuous passive beamforming (CPB) in both RIS- and STAR-RIS-assisted systems; 4) Also, it is shown that beyond a certain number of APs, further increasing the APs yields only limited transmit power consumption gains under a fixed total number of antennas.