## 2 Uplink GF-NOMA Network System Model  
A typical uplink GF-NOMA network is shown in Figure 1. In this network, users come on and off for transmission purposes in a random manner, without needing to apply for transmission grant permission. The network is made up of a single base station (BS) that the users in the networks can connect with. The active users are the nodes that are transmitting signals to the BS, and the users corresponding to nodes that are silent or are not transmitting signals to the BS in any given time slot are referred to as inactive users. Since the activities of users in networks are sporadic, the BS has the enormous task of identifying the users that are active, and simultaneously recovering the transmitted signals of the users.  
The system model is assumed to comprise \(U\) total users. These users are distributed in the network randomly, connecting to the BS in a GF-free transmission mode. It is further assumed that each of the user nodes that are connected to the BS possesses a single antenna. Supposing that modulation and channel coding procedures have been completed, the symbol of an active user's transmitted information can be represented as \(\mathbf{x}_{u}\). This symbol is assumed to be from a signal constellation set that is complex, such as M-phase shift keying (M-PSK). In this network, each of the users is allocated their corresponding spreading sequence. The unique spreading sequence allocated to the \(uth\) user symbol \(\mathbf{x}_{u}\) is given by an \(N\)-dimensional complex-valued vector, \(\mathbf{c}_{u}\). One of the qualities of the NOMA multiuser  
Figure 1: Typical Uplink GF-NOMA network with a small number of active users among multiple inactive users.  
transmission scheme, in comparison with its OMA counterpart, is the capacity to allow overloading in the system. In such a scenario, the available number of resources (e.g., subcarriers) is less than the number of users \(U\), i.e., \(N<U\). The signals of all of the active users in the networks are superimposed, and they all, at the same time, propagate over \(N\) subcarriers via the wireless fading channels. In the course of a random access scenario, the _uth_ user transmits over \(T\) continuous time slots.  
By following a signal model based on a single measurement vector (SMV) that is similar to that followed in most of the reviewed papers, on the _nth_ subcarrier at the BS, the signal received at the receiver \(\mathbf{y}_{n}\) is given as:  
\[\mathbf{y}_{n}=\sum_{u}^{U}g_{n,u}c_{n,u}\mathbf{x}_{u}+w_{n}, \tag{1}\]  
where parameter \(g_{n,u}\) represents the fading channel's gain of the _uth_ user at the _nth_ subcarrier to the BS. The gains \(g_{n,u}\) are independently complex distributions with mean zero and unit variance, \(\mathbb{CN}(0,\ 1)\). Furthermore, in (1), \(c_{n,u}\) represents the _nth_ element of the spreading vector \(c_{u}\), and \(w_{n}\) is the _nth_ subcarrier component of the noise, which is complex and has a Gaussian distribution \(\mathbb{CN}\big{(}0,\ \sigma^{2}\big{)}\). If all \(\mathbf{y}_{n},\mathbf{x}_{u}\), and \(w_{n}\) are arranged in their corresponding vectors \(\mathbf{y}=[\mathbf{y}_{1},\ \mathbf{y}_{2},\ \ldots,\ \mathbf{y}_{N}]^{Tr}\), \(\mathbf{x}=[\mathbf{x}_{1},\ \mathbf{x}_{2},\ \ldots,\ \mathbf{x}_{U}]^{Tr}\), and \(\mathbf{w}=[w_{1},\ w_{2},\ \ldots,\ w_{N}]^{Tr}\), respectively, then Equation (1) can be re-written as:  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{w}. \tag{2}\]  
The parameter \(\mathbf{H}\) in Equation (2) is a matrix of size \(N\times U\) with \(h_{n,u}=g_{n,u}c_{n,u}\), the matrix's element, situated on the _uth_ column and _nth_ row. The operator \([.]^{Tr}\) is the transpose.