## Appendix A Transmitters
### Proposed Identification Algorithm
One expresses (10) as \[\mathbf{Q}_{\widehat{\kappa}(u^{\prime},\xi),\widehat{\Omega}(u^{\prime},\xi)}^{(u ^{\prime})}=\int\overline{\mathbf{C}}_{\widehat{\kappa}(u^{\prime},\xi),\widehat {\Omega}(u^{\prime},\xi)}^{(u^{\prime})}\mathrm{Pr}\bigg{(}\overline{\mathbf{C }}_{\widehat{\kappa}(u,\xi),\widehat{\Omega}(u^{\prime},\xi)}^{(u^{\prime})} \bigg{|}\mathbf{y}_{u},\widehat{\mathbf{h}}^{(u^{\prime})}(\zeta)\bigg{)}d \overline{\mathbf{C}}_{\widehat{\kappa}(u^{\prime},\xi),\widehat{\Omega}(u^{ \prime},\xi)}^{(u^{\prime})}.\] (11)
* We give a description of the log-likelihood function employed to calculate the revised setting of user \(u\) parameter subset as \[\mathcal{L}=\mathrm{logPr}\bigg{(}\mathbf{y}_{u}\bigg{|}\overline{\mathbf{C}}_ {\widehat{\kappa}(u,\xi),\widehat{\Omega}(u,\xi)}^{(u)}\widehat{\mathbf{h}}^ {(u)}(\xi)\bigg{)}.\] (12) We take into account that \[\mathrm{Pr}\bigg{(}\mathbf{y}_{u}\bigg{|}\overline{\mathbf{C}}_{\widehat{ \kappa}(u,\xi),\widehat{\Omega}(u,\xi)}^{(u)}\widehat{\mathbf{h}}^{(u)}(\zeta )\bigg{)}\propto\exp\bigg{(}\begin{matrix}-\frac{1}{\sigma_{n}^{2}}\bigg{\|} \mathbf{y}_{u}-\overline{\mathbf{C}}_{\widehat{\kappa}(u,\xi),\widehat{\Omega }(u,\xi)}^{(u)}\widehat{\mathbf{h}}^{(u)}(\zeta)\bigg{\|}^{2}\bigg{)},\] (13) where \(\sigma_{n}^{2}\) is the noise variance. * After removing the superfluous components, we translate (12) as \[\mathcal{L}\propto 2\Re\bigg{\{}\mathbf{y}_{u}^{\ddagger}\overline{\mathbf{C}}_ {\widehat{\kappa}(u,\xi),\widehat{\Omega}(u,\xi)}^{(u)}\widehat{\mathbf{h}}^ {(u)}(\zeta)\bigg{\}}-\widehat{\mathbf{h}}^{(u)\ddagger}(\zeta)\overline{ \mathbf{C}}_{\widehat{\kappa}(u,\xi),\widehat{\Omega}(u,\xi)}^{(u)}\overline{ \mathbf{C}}_{\widehat{\kappa}(u,\xi),\widehat{\Omega}(u,\xi)}^{(u)}\widehat{ \mathbf{h}}^{(u)}(\zeta),\] (14) where the superscript \(\ddagger\) denotes the conjugate transpose of a vector and \(\Re(\cdot)\) is the real part of a complex number. * The SAGE algorithm's expectation phase employs the current estimates to derive the average value of \(\mathcal{L}\) with respect to information symbols, as demonstrated in (15): \[\mathbb{T}\bigg{(}\alpha(u),\Omega(u),\overline{\mathbf{h}}^{(u)}\bigg{|} \widehat{\alpha}(u,\zeta),\widehat{\Omega}(u,\zeta),\widehat{\overline{\mathbf{ h}}}^{(u)}(\zeta)\bigg{)}\propto\] (15) \[2\Re\bigg{\{}\mathbf{y}_{u}^{\ddagger}\mathbf{Q}_{\widehat{\kappa}(u,\xi), \widehat{\Omega}(u,\xi)}^{(u)}\widehat{\mathbf{h}}^{(u)}(\zeta)\bigg{\}}- \widehat{\mathbf{h}}^{(u)\ddagger}(\zeta)\mathbf{Q}_{\widehat{\kappa}(u,\xi), \widehat{\Omega}(u,\xi)}^{(u)}\mathbf{Q}_{\widehat{\kappa}(u,\xi),\widehat{ \Omega}(u,\xi)}^{(u)}\widehat{\mathbf{h}}^{(u)}(\zeta).\]
* The parameters of user \(u\) are modified during the maximization step of the SAGE algorithm in the following fashion: \[ \begin{split}\bigg{[}\widehat{\alpha}(u,\zeta+1),\widehat{ \Omega}(u,\zeta+1),\widehat{\overline{\mathbf{h}}}^{(u)}(\zeta+1)\bigg{]}= \underset{\alpha(u),\Omega(u),\overline{\mathbf{h}}^{(u)}}{\arg\max}\\ \mathbb{T}\bigg{(}\alpha(u),\Omega(u),\overline{\mathbf{h}}^{(u)} \bigg{|}\widehat{\alpha}(u,\zeta),\widehat{\Omega}(u,\zeta),\widehat{ \overline{\mathbf{h}}}^{(u)}(\zeta)\bigg{)}.\end{split} \] (16) Here, we illustrate how to partition the coupled problem represented by (16) into simpler one-dimensional search tasks. Specifically, for every single couple of \((\alpha(u),\Omega(u))\), the revised value of \(\overline{\mathbf{h}}^{(u)}\) is obtained by placing the function's gradient in (15) to zero a \[\widehat{\overline{\mathbf{h}}}^{(u)}(\alpha(u),\Omega(u),\zeta+1) = \Big{(}\mathbf{Q}_{a(u),\Omega(u)}^{(u)\ddagger}\mathbf{Q}_{a(u), \Omega(u)}^{(u)}\mathbf{\Omega}_{a(u),\Omega(u)}^{-1}\mathbf{Q}_{a(u),\Omega(u )}^{(u)}\mathbf{y}_{u}.\] (17)
* After incorporating (17) into (15), the revised values of \(\alpha(u)\) and \(\Omega(u)\) are recalculated as \[