## 3 The Proposed Detector for Uplink GF-NOMA Networks: FROMPUS  
In this section, the proposed Forward-Reverse OMP-Union-SP (FROMPUS)-based MUD is developed. As previously mentioned, in any uplink GF-NOMA system, the precise number of active users at any given time will be significantly lower than the total number of users in the coverage area of the BS in a particular network. In terms of massive internet of things (IoT), as is the case in mMTC-based 5G wireless systems, with the associated massive connectivity (with the presence of large numbers of users), in a particular time slot, there are only a few users that are transmitting information data. In this type of uplink GF mode of transmission, signal processing in terms of multiuser detection becomes what is known as a sparse signal recovery problem. Moreover, given that users can be sporadically active and inactive in the networks in practical communication scenarios, in adjacent time slots, some users will be present in the networks that will be transmitting their information data with a significantly high probability. Accordingly, in several continuous time slots in the systems, active user sets will temporarily correlate with each other. In constructing the proposed FROMPUS-based MUD, a dynamic model is assumed for uplink GF-NOMA systems, where it operates in a continuous time slot. At the _t-th_ time index, the transmitted signal can be denoted as \(\mathbf{x}^{[t]}\), then the collective transmitted signals \(\mathbf{X}\) can be represented as \(\mathbf{X}=\left[\mathbf{x}^{[1]},\ \mathbf{x}^{[2]},\ \ldots,\ \mathbf{x}^{[T]}\right]\). From the signal received at the BS, \(\mathbf{Y}=\left[\mathbf{y}^{[1]},\ \mathbf{y}^{[2]},\ \ldots,\ \mathbf{y}^{[T]}\right]\), in the \(T\) continuous time slots, where \(t=1,\ 2,\ \ldots,\ T\), the signal \(\mathbf{X}\) is to be recovered. At the _t-th_ time slot, the signal received at the BS can be expressed as:  
\[\mathbf{y}^{[t]}=\mathbf{H}^{[t]}\mathbf{x}^{[t]}+\mathbf{w}^{[t]},\ \text{for all}\ t\, \tag{3}\]  
with \(\mathbf{H}^{[t]}\) standing for the corresponding measurement channel matrix, which is complex, at the _t-th_ time index. The Gaussian noise vector is represented by the parameter \(\mathbf{w}^{[t]}\) in Equation (1) at the \(t\)-\(th\) time index. By representing sparsity's level of \(\mathbf{x}^{[t]}\) as \(\Im\), the support set, \(\mathbf{\zeta}^{[t]}\) of active users can be written as:  
\[\mathbf{\zeta}^{[t]}=\Big{\{}u:\mathbf{x}^{[t]}_{u}\
eq 0,\;u\in\{1,\,2,\;\ldots,\;U\} \Big{\}}. \tag{4}\]  
In Equation (4), \(\mathbf{x}^{[t]}_{u}\) is the transmitted signal of the \(uth\) user from among all of the transmitted signals' vector \(\mathbf{x}^{[t]}\) of all users. The complement of Equation (4) is given as:  
\[\mathbf{\zeta}^{[t]}=\Big{\{}u:\mathbf{x}^{[t]}_{u}=0,\;u\in\{1,\,2,\;\ldots,\;U\} \Big{\}}. \tag{5}\]  
With the idea of the support set, the active component of \(\mathbf{x}^{[t]}\) can be denoted as \(\mathbf{x}^{[t]}_{\mathbf{\zeta}^{[t]}}\), referring to the selection of non-zero components of \(\mathbf{x}^{[t]}\), active users. As a result, \(\mathbf{x}^{[t]}_{\mathbf{\zeta}^{[t]}}=0\). In this paper, the idea of an initial support set \(\mathbf{\zeta}_{ini}\), serving as the initial knowledge employed by a reconstruction algorithm to enhance the detection process, is introduced. Since users can randomly be active and inactive in the system in practical communication scenarios, in the adjacent time slots, some users will be transmitting data with a significantly high likelihood, leading to active user sets within several continuous time slots in the network being temporarily correlated with each other. Consequently, by sharing support set information among these active users, estimation of some parts of the a priori support set can be obtained and then employed as an initialization parameter to enhance detection processing.