## Appendix A Transmitters
### Proposed Identification Algorithm  
The matrix version of (3) is utilized here for the purpose of mathematical comfortability as  
\[\mathbf{r}=\sum_{u=0}^{U-1}\sum_{p=0}^{P(k)-1}\mathbf{C}_{\alpha(u),\Omega(u)} ^{(u,p)}\mathbf{h}^{(u,p)}+\mathbf{w}, \tag{4}\]  
and \(\mathbf{C}_{\alpha(u),\Omega(u)}^{(u,p)}\) is an \(\left((\mathcal{N}+\
u)N^{(u)}+L-1\right)\times(L-1)\) matrix formed as  
\[\mathbf{C}_{\alpha(u),\Omega(u)}^{(u,p)}(v_{1},v_{2})=\left\{ \begin{array}{ ll}\mathfrak{E}_{\alpha(u),\Omega(u)}^{(u,p)}(v_{1}-v_{2})\\ \text{for }v_{1}=0,\cdots,(\mathcal{N}+\
u)N^{(u)}+L-1,\\ \text{and }v_{2}=0,\cdots,L-1,\,v_{1}\geq v_{2}\\ 0\qquad\qquad\qquad\text{otherwise},\end{array} \right. \tag{5}\]  
where \(\mathbf{C}_{\alpha(u),\Omega(u)}^{(u,p)}(v_{1},v_{2})\) is the component situated at row \(v_{1}\), and column \(v_{2}\) of matrix \(\mathbf{C}_{\alpha(u),\Omega(u)}^{(u,p)}\) and \(\mathfrak{E}_{\alpha(u),\Omega(u)}^{(u,p)}(v_{1}-v_{2})\) is the \((v_{1}-v_{2})\)th component of vector \(\mathfrak{E}_{\alpha(u),\Omega(u)}^{(u,p)}\). Restating (4) in its condensed form yields  
\[\mathbf{r}=\sum_{u=0}^{U-1}\overline{\mathbf{C}}_{\alpha(u),\Omega(u)}^{(u)} \overline{\mathbf{h}}^{(u)}+\mathbf{w}, \tag{6}\]  
where \(\overline{\mathbf{C}}_{\alpha(u),\Omega(u)}^{(u)}=\left[\mathbf{C}_{\alpha(u ),\Omega(u)}^{(u,0)},\cdots,\mathbf{C}_{\alpha(u),\Omega(u)}^{(u,P(k)-1)} \right]\) and \(\overline{\mathbf{h}}^{(u)}=\left[\mathbf{h}^{(u,0)\dagger},\cdots,\mathbf{ h}^{(u,P(k)-1)\dagger}\right]^{\dagger}\). Here, \(\dagger\) refers to the transpose action of a vector. Optimizing the log-likelihood function with regard to the unknown parameters results in the ML strategy, as  
\[ \begin{split}\left[\widehat{\alpha}(0),\widehat{\Omega}(0), \cdots,\widehat{\alpha}(U-1),\widehat{\Omega}(U-1)\right]=\arg\max_{\widetilde {\alpha}(0),\widetilde{\Omega}(0),\cdots,\widetilde{\alpha}(U-1),\widetilde{ \Omega}(U-1)}\\ \Big{\{}\log\Pr\Big{(}\mathbf{r}|\widehat{\alpha}(0),\widehat{ \Omega}(0),\cdots,\widetilde{\alpha}(U-1),\widehat{\Omega}(U-1)\Big{)} \Big{\}},\end{split}  \tag{7}\]\(u=0,\cdots,U-1\), \(\delta\) and \(\widetilde{\circ}\) are the measured and trail quantities of \(\circ\), respectively, and \(\Pr(\circ|\triangledown)\) is the probability density function of \(\circ\) given \(\triangledown\). The probability density function of the observations given the unknown parameters is stated as  
\[ \begin{array}{l}\Pr\Big{(}\mathbf{r}|\widetilde{\alpha}(0),\widetilde{ \Omega}(0),\cdots,\widetilde{\alpha}(U-1),\widetilde{\Omega}(U-1)\Big{)}= \int\cdots\int\\ \\ \Pr\Big{(}\mathbf{r}|\overline{\widetilde{\alpha}}^{(0)}_{\widetilde{\alpha}( 0),\widetilde{\Omega}(0)},\overline{\mathbf{h}}^{(0)},\cdots,\overline{ \mathbf{C}}^{(U-1)}_{\widetilde{\alpha}(U-1),\widetilde{\Omega}(U-1)}, \overline{\mathbf{h}}^{(U-1)}\Big{)}\times\\ \\ \Pr\Big{(}\overline{\widetilde{\alpha}}^{(0)}_{\widetilde{\alpha}(0), \widetilde{\Omega}(0)}\Big{)}\times\cdots\times\Pr\Big{(}\overline{ \widetilde{\mathbf{C}}}^{(U-1)}_{\widetilde{\alpha}(U-1),\widetilde{\Omega}( U-1)}\Big{)}\times\\ \\ d\overline{\widetilde{\alpha}}^{(0)}_{\widetilde{\alpha}(0),\widetilde{\Omega}( 0)}\times\cdots\times d\overline{\widetilde{\mathbf{C}}}^{(U-1)}_{\widetilde{ \alpha}(U-1),\widetilde{\Omega}(U-1)}\times d\overline{\mathbf{h}}^{(0)} \times\cdots\times d\overline{\mathbf{h}}^{(U-1)}.\end{array}  \tag{8}\]  
It is important to note that (8) indirectly takes an average of all potential channel paths and information symbols broadcast for arbitrary choices of \(\widetilde{\alpha}(u)\) and \(\widehat{\Omega}(u)\), \(u=0,\cdots,U-1\). This is because the ML recognizer is blind to the actual values of these factors. As a direct consequence, determining (8) in a real-world context is incredibly difficult. In this scenario, the SAGE process is beneficial, since it offers a smaller method to predict the ML solution in spite of the existence of nuisance parameters.