## 2 System Description  
As indicated in Figure 1, we investigated a radio upstream multiple users SC-FDMA scheme with \(K\) operating users in the context of dual-hop broadcasts. The relay employed an AF routing protocol and operated in a half-duplex mode, which meant that emissions and receptions are separated in time. There were \(K\) distinct subsets of the total number of subcarriers \(N\). Each subset contained \(M=N/K\) separate subcarriers, each of which was used by one user at a time. Quantitatively, the subset of subcarriers allotted for user \(k\), \(Q^{(k)}\), fulfilled \(\bigcup\limits_{k=0}^{K-1}Q^{(k)}=\{0,1,\cdots,N-1\}\) and \(Q^{(k)}\bigcap Q^{(k^{\prime})}=\phi\) for \(k\
eq k^{\prime}\), with \(\phi\) denoting the empty set. Figure 2 shows the basic structure of the users' transmitters, relay, and destination nodes. A channel-control encoder operating at rate \(r^{(k)}\) appends redundancy bits to a stream of binary data belonging to user \(k\) in order to compensate for faults introduced during the broadcast. The encoded bits are interleaved and then transferred to complex-valued information symbols drawn from a specified signal pattern \(\Omega^{(k)}\) of unit energy. Here, the modulator, encoder, and interleaver settings used by each user are completely unrestricted. In order to kick off the estimation procedure, a small number of known symbols are inserted into information symbols. Each of the \(G^{(k)}\) vectors in the resultant chain contains \(M\) symbols. We denote \(\mathbf{d}_{S}^{(k)}=\left[d_{S}^{(k)}(0),\cdots,d_{S}^{(k)}(M-1)\right]\) as the \(g\)th vector, with \(d_{S}^{(k)}(m)\) being its \(m\)th component. Using the \(M\)-point fast Fourier-transform (FFT) algorithm, the frequency-domain observations \(\mathbf{f}_{g}^{(k)}=\left[f_{S}^{(k)}(0),\cdots,f_{S}^{(k)}(M-1)\right]\) are described as [57,58]  
\[f_{g}^{(k)}(m_{1})=\sum_{m_{2}=0}^{M-1}d_{S}^{(k)}(m_{2})\exp(-j2\pi m_{1}m_{2 }/M), \tag{1}\]  
where \(f_{S}^{(k)}(m_{1})\) is the \(m_{1}\)th sample of \(\mathbf{f}_{S}^{(k)}\) and \(j=\sqrt{-1}\). The components of \(\mathbf{f}_{S}^{(k)}\) activate \(M\) subcarriers using an interleaved mapper, which is described as  
\[\mathbf{\hat{x}}_{S}^{(k)}(m_{1})=\left\{ \begin{array}{ll}f_{S}^{(k)}(m_{2} )&m_{1}=k+m_{2}M\\ 0&\text{otherwise,}\end{array} \right. \tag{2}\]  
Figure 1: Upstream dual-hop scheme. The solid lines represent the transmission during the first time interval, and the dashed ones depict the broadcast during the following time interval. and a localized mapper specified as  
\[\bar{x}_{g}^{(k)}(m_{1})=\left\{ \begin{array}{ll}f_{g}^{(k)}(m_{2})&m_{1}=m_{2} +kM\\ 0&\text{otherwise},\end{array} \right. \tag{3}\]  
where \(0\leq k\leq K-1\). Using an \(N\)-point inverse FFT (IFFT) with incorporating a cyclic prefix of length \(\
u\), we transform \(\mathbf{f}_{g}^{(k)}\) into an SC-FDMA symbol, \(\mathbf{s}_{g}^{(k)}=\left[s_{g}^{(k)}(-\
u),\cdots,s_{g}^{(k)}(N-1)\right]\), as  
\[s_{g}^{(k)}(n_{1})=\sum_{n_{2}=-\
u}^{N-1}\bar{x}_{g}^{(k)}(n_{2})\exp(j2\pi n_ {1}n_{2}/N), \tag{4}\]  
where \(s_{g}^{(k)}(n_{1})\) is the \(n_{1}\)th sample of the \(g\)th SC-FDMA symbol, \(\mathbf{s}_{g}^{(k)}\), of user \(k\). Therefore, the entire sequence sent by user \(k\) is expressed as \(\mathbf{\bar{s}}^{(k)}=\left[s_{0}^{(k)},\cdots,s_{g_{-1}}^{(k)}\right]^{\dagger}\), where \(\dagger\) is the complex-conjugate action. We represent \(\theta^{(k)}\) and \(\rho^{(k)}\) as the phase and amplitude mismatches between I and Q routes at the transmitter of user \(k\). The signal damaged by the IQM at user \(k\) is described as [59]  
\[\overrightarrow{\mathbf{s}}^{(k)}=\eta^{(k)}\mathbf{\bar{s}}^{(k)}+\mu^{(k)} \mathbf{\bar{s}}^{(k)\dagger}, \tag{5}\]  
where \(\eta^{(k)}\) and \(\mu^{(k)}\) are defined as  
\[\eta^{(k)}=\cos\Bigl{(}\theta^{(k)}\Bigr{)}+j\rho^{(k)}\sin\Bigl{(}\theta^{(k )}\Bigr{)}, \tag{6}\]  
\[\mu^{(k)}=\rho^{(k)}\cos\Bigl{(}\theta^{(k)}\Bigr{)}+j\sin\Bigl{(}\theta^{(k) }\Bigr{)}. \tag{7}\]  
The following is a breakdown of the broadcast schedule.