# Message Passing Detectors for UAV-Based Uplink Grant-Free NOMA Systems
## 2 System Model and Factor Graph  
### System Model  
A typical GF-NOMA system is considered [14], where the UAV-BS is equipped with a single antenna. \(K\) potential single-antenna UEs under the service area of a UAV-BS perform GF-NOMA random access (RA), sharing the same bandwidth resource and transmitting signals to the UAV-BS, as illustrated in Figure 1. The transmission is assumed to utilize the multi-carrier code division multiple access (MC-CDMA) scheme detailed in [16], which belongs to the NOMA techniques. The MC-CDMA is a hybrid scheme combing code division multiple access (CDMA) and orthogonal frequency-division multiplexing (OFDM), making the MC-CDMA signal tolerant against frequency selective channels, inter-symbol interference (ISI), and inter-carrier interference (ICI) through the use of cyclic prefix and equalization [17]. The uplink transmission of the UEs is multiplexed over \(N\) frequency-divided subcarriers. In each time slot, the ASs of different UEs are assumed to be independent of each other, \(K_{a}\leq K\) of which are active and transmit packets to the BS. The active rate \(p_{a}\) is used to indicate the percentage of active UEs. The active UE \(k\) sends the complex symbol \(b_{k}\) from a predefined complex signal constellation and spreads it over the \(N\) sub-carriers. Meanwhile, the inactive UE is considered to be transmitting a zero symbol \(b_{k}=0\). The received signal \(\mathbf{y}=[y_{1},\cdots,y_{N}]^{\mathrm{T}}\) at the BS for the \(j\)-th time slot can be expressed as  
\[\mathbf{y}=\mathbf{H}\mathbf{b}+\mathbf{w}, \tag{1}\]  
where \(\mathbf{H}=[\mathbf{h_{1}},\cdots,\mathbf{h_{K}}]\in\mathbb{C}^{N\times K}\) denotes the complex channel response matrix, representing the spread codes of \(K\) UEs over \(N\) sub-carriers. \(\mathbf{b}=[b_{1},\cdots,b_{K}]^{\mathrm{T}}\) is the transmitted signal from \(K\) UEs. \(\mathbf{w}=[w_{1},\cdots,w_{N}]^{\mathrm{T}}\sim\mathcal{CN}(\mathbf{0},\lambda^{-1} \mathbf{I})\) denotes the additive white Gaussian noise (AWGN) with the variance \(\lambda^{-1}\). The received signals \(\mathbf{Y}=[\mathbf{y}_{1},\cdots,\mathbf{y}_{J}]\in\mathbb{C}^{N\times J}\) span \(J\) continuous time slots and are represented as a multiple measurement vector (MMV) model, [14]  
\[\mathbf{Y}=\mathbf{H}\mathbf{B}+\mathbf{W}, \tag{2}\]  
where \(\mathbf{B}=[\mathbf{b}_{1},\cdots,\mathbf{b}_{J}]\in\mathbb{C}^{K\times J}\) denotes the transmitted signal matrix and the noise is \(\mathbf{W}=[\mathbf{w}_{1},\cdots,\mathbf{w}_{J}]\in\mathbb{C}^{N\times J}\). In a practical mMTC scenario, the potential UEs keep their AS in some random consecutive time slots within the entire frame [10]. As shown in Figure 1, the AS of the UE exhibits a temporal correlation in the time domain, which promotes the sparse structure of signal \(\mathbf{B}\) in (2). The task of multi-UEs detection is to estimate the sparse transmitted signal \(\mathbf{B}\) from the received signal \(\mathbf{Y}\) with the unknown UE activity value.  
Figure 1: The left part of the figure denotes the uplink transmission from UEs to the base station (BS). The right part of the figure indicates the sparse and temporally correlated structures of the transmit signal \(\mathbf{B}\).