## 5 Spectral Efficiency (SE) Analysis for SAS-Based SM-LDM
### SE of Lower Layer  
Now shifting to the \(LL\), \(I(\mathcal{H}_{L},\mathcal{S}_{L};\mathbf{y}_{L})\) is the achievable rate (mutual information) between the continuous received signal and discrete symbols (spatial and modulation) of the \(LL\) which is given by [28; 29]:  
\[I(\mathcal{H}_{L},\mathcal{S}_{L};\mathbf{y}_{L})=\sum_{g=1}^{\mathsf{G}_{L}}\sum_ {l=1}^{\mathsf{M}_{L}}p\left(\mathbf{h}_{g}^{L},s_{l}^{L}\right)\Bigg{[}\int p\left( \mathbf{y}_{L}\Big{|}s_{l}^{L},\mathbf{h}_{g}^{L}\right)\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\  
\[I(\mathcal{H}_{L},\mathcal{S}_{L};\mathbf{y}_{L})=\frac{1}{G_{L}M_{L}}\sum_{g-1}^{G_{ L}}\sum_{l=1}^{M_{L}}\int p\left(\mathbf{y}_{L}\Big{|}s_{l}^{L},\mathbf{h}_{g}^{L} \right)\log_{2}\frac{p\left(\mathbf{y}_{L}\Big{|}s_{l}^{L},\mathbf{h}_{g}^{L}\right)}{ \frac{1}{G_{L}M_{L}}\sum_{m=1}^{G_{L}}\sum_{m=1}^{M_{L}}p\left(\mathbf{y}_{L} \Big{|}s_{h}^{L},\mathbf{h}_{m}^{L}\right)}dy_{L}, \tag{20}\]  
It is also challenging to find a closed formula for (20); thus, we use MC integration to obtain:  
\[I(\mathcal{H}_{L},\mathcal{S}_{L};\mathbf{y}_{L})\approx\log_{2}\left(G_{L}M_{L} \right)+\frac{1}{G_{L}M_{L}}\sum_{g=1}^{G_{L}}\sum_{l=1}^{M_{L}}\sum_{s=1}^{S} log_{2}\frac{p\left(\mathbf{y}_{L}^{s}\Big{|}s_{l}^{L},\mathbf{h}_{g}^{L}\right)}{ \sum_{m=1}^{G_{L}}\sum_{m=1}^{M_{L}}p\left(\mathbf{y}_{L}^{s}\Big{|}s_{h}^{L},\bm {h}_{m}^{L}\right)}, \tag{21}\]  
Rearranging (21), it yields:  
\[I(\mathcal{H}_{L},\mathcal{S}_{L};\mathbf{y}_{L})\approx\log_{2}\left(G_{L}M_{L} \right)-\frac{1}{G_{L}M_{L}}\sum_{g=1}^{G_{L}}\sum_{l=1}^{M_{L}}\sum_{s=1}^{S} \log_{2}\left[1+\sum_{\begin{subarray}{c}m=1\\ \
eq g\end{subarray}}^{G_{L}}\sum_{n=1}^{M_{L}}\frac{p\left(\mathbf{y}_{L}^{s} \Big{|}s_{h}^{L},\mathbf{h}_{m}^{L}\right)}{p\left(\mathbf{y}_{L}^{s}\Big{|}s_{l}^{L}, \mathbf{h}_{g}^{L}\right)}\right] \tag{22}\]  
The PDF \(p\left(\mathbf{y}_{L}^{s}\Big{|}s_{l}^{L},\mathbf{h}_{g}^{L}\right)\) can be rewritten by taking into consideration the \(i\)th index of the TA among the \(g\)th group antennas, and it can be given as follows:  
\[p\left(\mathbf{y}_{L}^{s}\Big{|}s_{l}^{L},\mathbf{h}_{g}^{L}\right)=\frac{1}{l_{k}} \sum_{i=1}^{l_{k}}p\left(\mathbf{y}_{k}\Big{|}s_{l}^{L},\mathbf{h}_{g,i}^{L}\right) \tag{23}\]  
Hence, (17) can be reformulated as  
\[I(\mathcal{H}_{L},\mathcal{S}_{L};\mathbf{y}_{L})\approx log_{2}(G_{L}M_{L})-\frac {1}{G_{L}MS}\sum_{g=1}^{G_{L}}\sum_{l=1}^{M_{L}}\sum_{s=1}^{S}log_{2}\left[1+ \sum_{\begin{subarray}{c}m=1\\ \
eq g\end{subarray}}^{G_{L}}\sum_{i=1}^{L_{k}}\sum_{n=1}^{M_{L}}\frac{p\left( \mathbf{y}_{L}^{s}\Big{|}s_{n}^{L},\mathbf{h}_{m,i}^{L}\right)}{\sum_{i=1}^{l_{k}}p \left(\mathbf{y}_{L}^{s}\Big{|}s_{l}^{L},\mathbf{h}_{g,i}^{L}\right)}\right] \tag{24}\]  
In (24), On the right-hand side, the first term represents the highest rate that may be achieved using \(LL\). The conditional PDF \(p\left(\mathbf{y}_{L}^{s}\Big{|}s_{l}^{L},\mathbf{h}_{g,i}^{L}\right)\) is expressed as  
\[p\left(\mathbf{y}_{L}^{s}\Big{|}s_{l}^{L},\mathbf{h}_{g,i}^{L}\right)=\frac{\pi^{-N_{L} }}{|\mathbf{\psi}_{L}|}\exp\biggl{\{}-\left(\mathbf{y}_{L}^{s}-\sqrt{\beta_{L}}s_{l}^{ L}\mathbf{h}_{g,i}^{L}\right)^{H}(\mathbf{\psi}_{L})^{-1}\left(\mathbf{y}_{L}^{s}-\sqrt{\beta_{L }}s_{l}^{L}\mathbf{h}_{g,i}^{L}\right)\biggr{\}} \tag{25}\]  
where \(\mathbf{\psi}_{L}=\beta_{L}\mathbf{h}_{g,i}^{L}\Big{(}\mathbf{h}_{g,i}^{L}\Big{)}^{H}+\sigma _{0}^{2}\mathbf{I}\) represents the covariance matrix.