## 5 Spectral Efficiency (SE) Analysis for SAS-Based SM-LDM
### SE of Upper Layer  
For _UL_, the MI is measured between the discrete channel input and continuous output which can be given as follows [29]:  
\[I(\mathcal{H}_{UL};\mathbf{y}_{U})=E_{\mathbf{h}_{g}^{U},\mathbf{y}_{U}}\Big{[}\log_{2}p \Big{(}\mathbf{y}_{U}\Big{|}\mathbf{h}_{g}^{U}\Big{)}/p(\mathbf{y}_{U})\Big{]}=\sum_{g=1} ^{G_{UL}}\int p(\mathbf{h}_{g}^{U})p\Big{(}\mathbf{y}_{U}\Big{|}\mathbf{h}_{g}^{U}\Big{)} \log_{2}\frac{p\Big{(}\mathbf{y}_{U}\Big{|}\mathbf{h}_{g}^{U}\Big{)}}{p(\mathbf{y}_{U})} \mathbf{dy}_{U}, \tag{11}\]  
where \(p\Big{(}\mathbf{h}_{g}^{U}\Big{)}=1/G_{UL}\) reflects with equal probability the spatial symbol \(\mathbf{h}_{g}^{U}\) supplied by _UL_. Additionally, in (9), \(p(\mathbf{y}_{U})\) is the probability density function (PDF) of the received vector \(\mathbf{y}_{U}\), and \(p\Big{(}\mathbf{y}_{U}\Big{|}\mathbf{h}_{g}^{U}\Big{)}\) is the conditional PDF, assuming that _UL_ selects the spatial symbol \(\mathbf{h}_{g}^{U}\). Accordingly, (11) can be rewritten as  
\[I(\mathcal{H}_{UL};\mathbf{y}_{U})=\frac{1}{G_{UL}}\sum_{g=1}^{G_{UL}}\int p\Big{(} \mathbf{y}_{U}\Big{|}\mathbf{h}_{g}^{U}\Big{)}\log_{2}\frac{p\Big{(}\mathbf{y}_{U}\Big{|} \mathbf{h}_{g}^{U}\Big{)}}{\frac{1}{G_{UL}}\sum_{m=1}^{G_{UL}}p\Big{(}\mathbf{y}_{U} \Big{|}\bm  
Because a closed-form formula for \(I(\mathcal{H}_{U};\mathbf{y}_{U})\) is difficult to identify, the Monte Carlo (MC) integration is utilized to compute it in Equation (13) [30]. The expression \(I(\mathcal{H}_{U};\mathbf{y}_{U})\) can be represented as follows in the form of expectations:  
\[I(\mathcal{H}_{U};\mathbf{y}_{U})\approx\log_{2}G_{U}-\frac{1}{G_{U}}\sum_{g=1}^{G _{U}}E_{y_{U}|\mathcal{H}_{U}}\Bigg{[}\log_{2}\Bigg{(}1+\frac{\sum_{m=1, \
eq g}^{G_{U}}p\left(\mathbf{y}_{U}\Big{|}\mathbf{h}_{m}^{U}\Big{)}}{p\left(\mathbf{y}_{U} \Big{|}\mathbf{h}_{g}^{U}\right)}\Bigg{)}\Bigg{]}. \tag{14}\]  
A significant number (let us say \(\mathsf{S}\)) of independent random samples is collected from the signal that was received (9) to evaluate (14) [31]. This approach is used for all \(\mathsf{S}\) transmission channels, and the estimated values obtained are averaged to provide the resultant value:  
\[I(\mathcal{H}_{U};\mathbf{y}_{U})\approx\log_{2}G_{U}-\frac{1}{G_{U}}\sum_{g=1}^{ G_{U}}\sum_{s=1}^{\mathsf{S}}\log_{2}\Bigg{[}1+\frac{\sum_{m=1,\
eq g}^{G_{U}}p \left(\mathbf{y}_{U}\Big{|}\mathbf{h}_{m}^{U}\right)}{p\left(\mathbf{y}_{U}\Big{|}\mathbf{h}_{ g}^{U}\right)}\Bigg{]}. \tag{15}\]  
In Equation (15), \(\mathbf{y}_{U}^{s}\) is a unique identifier for the received signal \(\mathbf{y}_{U}\), which is described in (9). \(p\left(\mathbf{y}_{U}^{s}\Big{|}\mathbf{h}_{g}^{U}\right)\) also signifies that the \(g\)th group was active and the _UL_ signal was transmitted from any of the \(i\)th antennas in the \(g\)th group. This probability is related to all transmission probabilities from other antennas in other groups via \(p\left(\mathbf{y}_{U}^{s}\Big{|}\mathbf{h}_{m}^{U}\right)\), where \(m\
eq n\). Consequently, \(p\left(\mathbf{y}_{U}^{s}\Big{|}\mathbf{h}_{g}^{U}\right)\) can be rewritten as:  
\[p\left(\mathbf{y}_{U}^{s}\Big{|}\mathbf{h}_{g}^{U}\right)=\frac{1}{I_{U}}\sum_{i=1}^{ I_{U}}p\left(\mathbf{y}_{U}^{s}\Big{|}\mathbf{h}_{g,i}^{U}\right), \tag{16}\]  
Therefore, (15) can be rearranged to  
\[I(\mathcal{H}_{U};y_{U})\approx\log_{2}G_{U}-\frac{1}{G_{U}}\sum_{g=1}^{ \mathsf{S}}\sum_{s=1}^{\mathsf{S}}\log_{2}\Bigg{[}1+\frac{\sum_{m=1}^{G_{U}} \sum_{i=1}^{\mathsf{S}_{U}}p\left(\mathbf{y}_{U}^{s}\Big{|}\mathbf{h}_{m,i}^{U}\right) }{\sum_{i=1}^{I_{U}}p\left(\mathbf{y}_{U}^{s}\Big{|}\mathbf{h}_{g,i}^{U}\right)}\Bigg{]} \tag{17}\]  
Additionally, \(p\left(\mathbf{y}_{U}^{s}\Big{|}\mathbf{h}_{g,i}^{U}\right)\) is given by:  
\[p(y_{U}^{s}|\mathbf{h}_{g,i}^{U})=\frac{1}{\pi^{N_{t}}|\mathbf{I}_{U}|}\exp\Big{\{}- \mathbf{y}_{U}^{H}\mathbf{\Gamma}_{U}^{-1}\mathbf{y}_{U}\Big{\}}, \tag{18}\]  
where \(\mathbf{\Gamma}_{U}=\mathbf{h}_{g,i}^{U}\left(\mathbf{h}_{g,i}^{U}\right)^{H}+\sigma_{0}^ {2}\mathbf{I}\) represents the non-singular covariance matrix.