## 3 Proposed SAGE Estimation Algorithm  
The primary duty of the destination is to make use of the \(\overline{\mathbf{y}}_{D}^{(1)}\) and \(\overline{\mathbf{y}}_{D}^{(2)}\) observations in order to recover the data that were sent by different users. Hence, it is required to determine the IQM parameters of each user, relay, and destination, as well as the radio links connecting the different nodes. Using (5), (9), into (11), we express \(\overline{\mathbf{y}}_{D}^{(1)}\) and \(\overline{\mathbf{y}}_{D}^{(2)}\) as  
\[\overline{\mathbf{y}}_{D}^{(1)}=\sum_{k=0}^{K-1}\left(\overline{\mathbf{s}}^{ (k)}\otimes\mathcal{H}_{D,1}^{(k)}+\overline{\mathbf{s}}^{(k)\dagger}\otimes \mathcal{H}_{D,2}^{(k)}\right)+\overline{\mathbf{n}}_{D}^{(1)}, \tag{14}\]  
Figure 2: The basic structure of the sources, relay, and destination nodes. \[\overline{\mathbf{y}}_{D}^{(2)}=\sum_{k=0}^{K-1}\left(\overline{\mathbf{s}}^{(k)} \otimes\mathcal{H}_{D3}^{(k)}+\overline{\mathbf{s}}^{(k)\dagger}\otimes \mathcal{H}_{D4}^{(k)}\right)+\overline{\mathbf{n}}_{D}^{(2)}, \tag{15}\]  
where  
\[\mathcal{H}_{D,1}^{(k)}=\eta_{D}\eta^{(k)}\mathbf{h}_{D}^{(k)}+\mu_{D}\mu^{(k) \dagger}\mathbf{h}_{D}^{(k)\dagger}, \tag{16a}\] \[\mathcal{H}_{D,2}^{(k)}=\eta_{D}\mu^{(k)}\mathbf{h}_{D}^{(k)}+\mu_{D}\eta^{(k) \dagger}\mathbf{h}_{D}^{(k)\dagger}, \tag{16b}\]  
\[\mathcal{H}_{D,3}^{(k)}=\Delta\eta_{D}\eta_{K}\Big{(}\eta^{(k)} \mathbf{h}_{D}^{(k)}\otimes\mathbf{h}_{RD}+\mu^{(k)\dagger}\mathbf{h}_{D}^{(k) \dagger}\otimes\mathbf{h}_{RD}\Big{)}\] (16c) \[\quad+\Delta^{\dagger}\mu_{D}\mu_{K}^{\dagger}\Big{(}\mu^{(k) \dagger}\mathbf{h}_{D}  
where \(\Pr\left(\circ|\triangleleft\right)\) describes the probability density function of \(\diamond\) given \(\triangleleft\), and \(\triangleleft\) corresponds to the predicted value of \(\diamond\). Given that \(\overline{\mathbf{y}}_{D}^{(1)}\) and \(\overline{\mathbf{y}}_{D}^{(2)}\) are independent sequences, we express  
\[ \begin{array}{c}\Pr\left(\overline{\mathbf{y}}_{D}^{(1)},\overline{\mathbf{y }}_{D}^{(2)}\Big{|}\mathrm{S}^{(0)},\cdots,\mathrm{S}^{(K-1)},\mathrm{H}_{1}^ {(0)},\mathrm{H}_{2}^{(0)},\cdots,\mathrm{H}_{1}^{(K-1)},\mathrm{H}_{2}^{(K-1)} \right)=\\ \Pr\left(\overline{\mathbf{y}}_{D}^{(1)}\Big{|}\mathrm{S}^{(0)},\cdots, \mathrm{S}^{(K-1)},\mathrm{H}_{1}^{(0)},\cdots,\mathrm{H}_{1}^{(K-1)}\right) \\ \times\Pr\left(\overline{\mathbf{y}}_{D}^{(2)}\Big{|}\mathrm{S}^{(0)},\cdots, \mathrm{S}^{(K-1)},\mathrm{H}_{2}^{(0)},\cdots,\mathrm{H}_{2}^{(K-1)}\right) \end{array}  \tag{21}\]  
where  
\[\Pr\left(\overline{\mathbf{y}}_{D}^{(1)}\Big{|}\mathrm{S}^{(0)},\cdots, \mathrm{S}^{(K-1)},\mathrm{H}_{1}^{(0)},\cdots,\mathrm{H}_{1}^{(K-1)}\right) \propto\exp\left(\frac{-1}{\sigma_{n}^{2}}\bigg{\|}\overline{\mathbf{y}}_{D}^ {(1)}-\sum_{k=0}^{K-1}\mathrm{S}^{(k)}\mathrm{H}_{1}^{(k)}\bigg{\|}^{2}\right), \tag{22}\]  
and  
\[\Pr\left(\overline{\mathbf{y}}_{D}^{(2)}\Big{|}\mathrm{S}^{(0)},\cdots, \mathrm{S}^{(K-1)},\mathrm{H}_{2}^{(0)},\cdots,\mathrm{H}_{2}^{(K-1)}\right) \propto\exp\left(\frac{-1}{\sigma_{n}^{2}}\bigg{\|}\overline{\mathbf{y}}_{D}^ {(2)}-\sum_{k=0}^{K-1}\mathrm{S}^{(k)}\mathrm{H}_{2}^{(k)}\bigg{\|}^{2}\right). \tag{23}\]  
Here, \(\sigma_{n}^{2}\) describes the variance of the additive white Gaussian noise. Equations (20)-(23) reveal that knowing the broadcast matrices \(\mathrm{S}^{(0)},\cdots,\mathrm{S}^{(K-1)}\) in advance is required for the precise ML solution, although this information is typically unattainable at the destination. Direct computations of maximum likelihood value predictions are problematic in a number of signal processing scenarios owing to the complexity of the likelihood functions or the coupling created by the smoothness penalties or priors. In such situations, the expectation maximization process is often used to provide parameter estimates with the greatest likelihood in an iterative manner [61; 62; 63].