## 1 Introduction  
Non-terrestrial networks (NTNs), which are aimed at achieving seamless global coverage and supporting high data rate services, will be an important part of the Beyond-5G (B5G) and 6G ecosystems. Within the NTN, low-Earth-orbit (LEO) giant constellations have attracted significant interest in recent years [1]. Compared to geostationary Earth orbit (GEO) satellites, LEO satellites can effectively reduce the long propagation delays and high path losses and then provide a more favorable communication and networking environment [2]. In satellite-to-ground communication systems, the received power is often weak when the ground users (GUs) receive signals directly from the satellites. To increase the received power, one can choose to increase the transmit power or gain of a single satellite or to use multiple satellites cooperatively. Multi-satellite cooperation should be a better solution to cope with the problem of cost and flexibility [3]. However, due to the mobility of LEO, spectrum management has become an urgent challenge, and cell-free (CF)communication systems have also been identified as a potential source of improved SE in ongoing works [4]. Despite the challenges posed by the escalated costs of hardware equipment and the intricate system complexity of CF systems, contemporary mobile communications continue to exhibit a growing predilection towards this architecture. CF massive multiple-input multiple-output (CF-mMIMO) is a distributed massive multiple-input multiple-output (mMIMO) technique. The model was first proposed in [5]. CF-mMIMO networks deploy a large number of access points (APs) to serve GUs, which can provide ubiquitous connectivity with high spectral efficiency (SE) and energy efficiency (EE). Thus, it can be one of the promising technologies for B5G and 6G wireless networks [6; 7; 8]. Due to the high density of LEO constellations, it is highly feasible to use CF-mMIMO architectures to improve the performance of satellite networks [2]. The author of [9] proposed an integrated space-terrestrial architecture that combines ultra-dense terrestrial deployment with the extensive coverage of LEO satellites, aiming at maximizing the minimum per-user rate in the coverage area. Subsequently, they proposed an integrated network architecture combining a CF-mMIMO terrestrial network layout with LEO satellites to manage terrestrial-to-satellite user transfers using scalable techniques In [10]. The great potential of LEO satellites in supporting and expanding CF-mMIMO terrestrial coverage is revealed.A user demand-based quantum activation strategy was proposed in [11] for LEO satellites in CF-mMIMO communications. The authors in [12] proposed a scalable distributed mMIMO system for LEO satellite networks based on user-centric dynamic clustering, which can improve the system's SE and minimize the serving cluster (SC) size for each GU. However, since satellites use multi-beam transmission techniques, full-frequency multiplexing is often employed, resulting in inter-beam interference. Therefore, linear precoding is used at the transmitter to mitigate the interference [13]. Due to the limited complexity and favorable performance, precoding architectures based on hybrid analog/digital precoding were widely applied in satellite communication systems [14]. The hybrid precoding scheme is capable of reducing the hardware cost and power consumption of the system. When integrated with suitable precoding algorithms, this approach can significantly enhance the overall system performance. A hybrid analog/digital precoding technique was developed in [13] in an mMIMO LEO satellite communication (SATCOM) downlink using a dual-resolution phase-shift network. The authors in [14] proposed a hybrid precoding architecture based on dynamic subarrays and low-resolution phase shifters. A hybrid transmitting precoding scheme under spectrum-sharing constraints in a backhaul system was proposed in [15].