## 3 The Proposed Detector for Uplink GF-NOMA Networks: FROMPUS
### Node C: Reverse-Select  
This node is mainly for reliability testing. If there are errors in the \(\boldsymbol{\zeta}_{ini}\) used in the Fused Modified OMP-Modified SP node, these errors will likely propagate to the Fused Modified OMP-Modified SP node's output, \(\boldsymbol{\zeta}_{0}\). Hence, there is a need to execute an ordering procedure that will help to arrange the corresponding residual vectors appropriately, which is essential for reliability testing. This is the purpose of this node, which is given as Algorithm 3. The finally computed support set, \(\boldsymbol{\zeta}_{fin}^{[t]}\), in step 3 is employed by the proposed FROMPUS-based MUD to compute the active users' transmitted signals. ```
Input:\(Y=\left[\boldsymbol{y}^{[1]},\;\boldsymbol{y}^{[2]},\;\ldots,\;\boldsymbol{y}^{[T]} \right],\boldsymbol{H}=\left[\boldsymbol{H}^{[1]},\;\boldsymbol{H}^{[2]},\; \ldots,\;\boldsymbol{H}^{[T]}\right],\boldsymbol{\zeta}_{j+1}^{[t]},\;T,\;j\) Output:\(\boldsymbol{r}_{fin}^{[t]},\boldsymbol{\zeta}_{fin}^{[t]}\)
1.for\(t=1\)to\(T\)do
2.\(\widetilde{\boldsymbol{x}}_{\phi_{j+1}^{[t]}}\leftarrow\left(\boldsymbol{H}^{[t] }\right)^{\dagger}\boldsymbol{\zeta}_{j+1}^{[t]},\;\boldsymbol{\widetilde{ \boldsymbol{x}}}_{\phi_{j+1}^{[t]}}\leftarrow\boldsymbol{0},\,(. )^{\dagger}\) is matrix pseudo-inverse
3.\(\boldsymbol{\zeta}_{fin}^{[t]}\gets supp\bigg{(}\widetilde{\boldsymbol{x}}_ {\phi_{j+1}^{[t]}},\;j\bigg{)}\), Final support set for the estimation of the active user's signal
4.Reside:
5.endfor
6.return\(\boldsymbol{r}_{fin}^{[t]},\boldsymbol{\zeta}_{fin}^{[t]}\)
```  
**Algorithm 3** Reverse-Select  
The full proposed FROMPUS-based MUD is represented in Algorithm 4, noting that \(\boldsymbol{r}_{j}^{[t]},\boldsymbol{r}_{j+2}^{[t]}\), and \(\boldsymbol{r}_{fin}^{[t]}\) are the initially estimated, intermediate estimated, and finally estimated temporary residual signals, respectively. Similarly, \(\boldsymbol{\zeta}_{j}^{[t]},\boldsymbol{\zeta}_{j+1}^{[t]},\;\) and \(\boldsymbol{\zeta}_{fin}^{[t]}\) are the initially estimated, intermediate estimated, and finally estimated active user support sets, respectively. The active users' signals are detected using the LS-based procedure in step 18 and are returned in step 20 of the proposed MUD. The full proposed FROMPUS-based MUD algorithm is illustrated in a block diagram in Figure A1 of Appendix A. ``` Input:\(\mathbf{Y}=\left[\mathbf{y}^{[1]},\ \mathbf{y}^{[2]},\ \ldots,\ \mathbf{y}^{[T]}\right],\mathbf{H}=\left[\mathbf{H}^{[1]},\ \mathbf{H}^{[2]},\ \ldots,\ \mathbf{H}^{[T]}\right],\ T,\mathbf{\xi}_{ini},\ \Im,\Im_{si},\ j\) Output:\(\mathbf{X}=\left[\mathbf{x}^{[1]},\ \mathbf{x}^{[2]},\ \ldots,\ \mathbf{x}^{[T]}\right]\) Node A: Initialization:
1. \((\xi_{0},r_{0})\)\(\leftarrow\)Fused Modified OMP-Modified SP ( \(\mathbf{Y}\), \(\mathbf{H}\), \(T\), \(\mathbf{\xi}_{ini}\), \(\Im_{i}\), \(\Im_{i}\))
2. \(\mathbf{\zeta}_{j}^{[t]}\)\(\leftarrow\)\(\mathbf{\zeta}_{0}\), \(\mathbf{r}_{j}



comparative performances of the proposed FROMPUS-based MUD and the other MUDs are documented in terms of the percentage overloading factor, expressed as:

\[\text{Percentage overloading factor}=\Big{(}U\Big{/}N\times 100\Big{)}\%. \tag{7}\]

This overloading factor is obtained by increasing the total number of users \(U\) in the subsequent simulations from the initially set value of 50 up to a maximum of 400. Furthermore, in the computer simulation, the channel matrix \(H\) is assumed to be known, and satisfies the restricted isometry property (RIP) condition with a high probability, which allows for accurate recovery of signals that are sparse [38]. The various results presented in this section are aimed at identifying the impacts of variations of system parameters on the performances of both the FROMPUS-based MUD and the other MUDs. The first set of results displayed in Figure 2 shows the achievable bit error rate (BER) versus signal-to-noise ratio (SNR) performance of the proposed FROMPUS-based MUD in an uplink GF-NOMA system in comparison with the other five MUDs.