## 1 Introduction  
As users and devices explosively grow in the next generation mobile communication systems and Internet of Things (IoT), it is more difficult to provide ubiquitous and reliable massive access with conventional terrestrial networks [1]. Since satellite communication has greatly developed over the years, it is considered a promising solution as a complement for terrestrial communications with its extensive coverage and continuous service [2]. However, it is still not enough to meet the demands with satellites alone; new multiple access (MA) technologies are also called for [3]. In long-term evolution (LTE), orthogonal frequency division multiple access (OFDMA) and singe carrier frequency division multiple access (SC-FDMA) are adopted to boost the capacity in downlink and uplink, respectively. By assigning users orthogonal sub-carriers or discrete Fourier transform (DFT) precoders, multiple access among them becomes possible. However, the orthogonality requirement limits their capacity. The massive machine type communication (mMTC) in 5G and 6G cannot be supported by them. Several non-orthogonal multiple access (NOMA) technologies are proposed to further improve the spectral efficiency. Among them, power domain NOMA (PD-NOMA) has been extensively studied over the years. Different transmission powers are allocated based on users' channel conditions. By utilizing the successive interference cancellation (SIC) technique, multiple users could share the same frequency band and time slot, resulting in capacity boost. Nonetheless, it requires user ordering and significant channel discrepancy for better performance. The increase of users and devices also makes it more difficult and complex for user paring, ordering and power allocation, causing performance deterioration. Recently, a novel rate splitting multiple access (RSMA) has drawn a lot attention. It divides user's messages into a public part and a private part, assigning them corresponding precoders. Users also share the same frequency and time resources. But it does not need user paring and ordering or great channel differences to perform well. It has higher capacity compared with OFDMA, SC-FDMA, and PD-NOMA with relatively low receiver complexity [4; 5]. The idea of rate splitting was first presented in a single input single output (SISO) system [6]. Over the years, it has been extended for multiple input single output (MISO) and multiple input multiple output (MIMO) systems [7; 8]. In the groundbreaking works of rate splitting (RS), the authors thoroughly study this method in terms of sum-rate theoretically; besides the obvious larger achievable rate region, the results prove that RS not only significantly decrease the complexity of precoder design and user scheduling but also exhibits robustness with imperfect channel state information (CSI) compared with conventional MA schemes [7; 8]. For the classical weighted sum-rate (WSR) maximization in RSMA, usually this non-convex problem is converted to a convex weighted minimum mean square error (WMMSE) to solve [9]. But recently, Ref. [10] has adopted a new deep learning approach called deep unfolding to simplify and improve the WMMSE algorithm in an integrated satellite terrestrial network (ISTN), while [11] proposes to use the block-coordinate descent algorithm for an intelligent reflecting surface (IRS) aided system, both of which could achieve a better performance. Another hot topic of RSMA is the max-min fairness in a multi-group or multi-beam scenario. Reference [12] uses semi-definite relaxation and a convex-concave procedure to attain better fairness. Another reference [13] introduces LogSumExp approximation and a generalized power iteration framework for beamforming. Lately, the application of RSMA for simultaneous wireless information and power transfer (SWIPT) has gained more attention due to exponential energy consumption growth. In a multigroup multicast system, the authors of [14] uses the successive convex approximation (SCA) algorithm to minimize the transmission power.