# Message Passing Detectors for UAV-Based Uplink Grant-Free NOMA Systems
## 1 Introduction  
Recently, unmanned aerial vehicle (UAV) assisted communications, such as UAV base stations (UAV-BSs), have garnered significant research interest for their ability to provide rapid and widespread connectivity [1, 2, 3]. The UAV-BS system can offer a cost-effective and innovative solution to managing fluctuating traffic demands, especially in scenarios where traditional infrastructure-based base stations, which are expensive and slow to deploy, are not financially viable. The UAV-BS can be a data collector for sensor networks in remote or restricted areas [4]. Additionally, it can function as an emergency BS when existing ones fail due to natural disasters or become overwhelmed by sudden surges in traffic [5, 6]. However, the large number of potential user equipments (UEs) makes UAV-BS communication difficult and complicated. Currently, grant-free non-orthogonal multiple access (GF-NOMA) is a promising candidate to support UAV-BS communications. The non-orthogonal multiple access (NOMA) technique enables the system resource block to simultaneously serve multiple UEs [7, 8]. At the same time, the grant-free (GF) scheme allows UEs to transmit data to the UAV-BS directly without waiting for its permission, which can decrease the access delay and reduce the control overhead for coordination [9]. In the case of applying GF-NOMA to uplink,recovering the transmitted signal through designing efficient multi-UEs detection methods is challenging, not only due to the large number of UEs present in the cell but also because they are randomly accessing the network in each access slot. However, UEs are only sporadically and randomly activated in practical massive machine-type communications (mMTC) systems, which brings sparsity of the transmitted slots in the time-frequency resource grid [10], which can be changed across different slots within a single frame [11; 12]. Furthermore, once a UE is activated for performing the data transmission, it typically requires several consecutive time slots. Then, this UE will be deactivated for a certain period until the next transmission. This activation and deactivation is denoted as activity state (AS), and it exhibits a temporal correlation [13]. Consequently, to enhance the performance of the detectors in a GF-NOMA system, the existing methods incorporate not only the exploitation of sparsity but also the temporal correlation information provided by AS. According to the literature of GF-NOMA, detectors based on message passing (MP) algorithms [12; 13; 14] have a better performance compared to conventional compressed sensing (CS) methods. In [12], the authors exploited the temporal correlation of the active UE and modeled it by a Markov process. The authors in [13] utilized the sparse Basianyes learing (SBL) and pattern coupled sparse Basianyes learing (PCSBL) models to formulate the slow change of the UEs AS, then proposed the SBL-based multi-UEs detection algorithms, outperforming CS methods. Then, [14] further embedded the generalized approximate message passing (GAMP) algorithm with the SBL and PCSBL models, and proposed the GAMP sparse Basianyes learing (GAMP-SBL) and the GAMP pattern coupled sparse Basianyes learing (GAMP-PCSBL) detector algorithms, which can reduce the complexity. However, the SBL and PCSBL models mainly consider the sparse structure of the UE activity. Motivated by [12], we find that the Markov chain (MC) better captures the slow change of the AS features with successive time slots and the active UE sparsity simultaneously. Furthermore, differing from [12], the hybrid message passing (HMP) rule [15] is used to update the parameters of the MC on the factor graph (FG) to improve the convergence speed. This paper focuses on designing a high-performance MP algorithm, resulting in the GAMP-BG-MC detector algorithm for the UAV-BS-based uplink GF-NOMA system. The main contributions of this paper are summarized as follows. 1.