## 5 Spectral Efficiency (SE) Analysis for SAS-Based SM-LDM  
This section introduces the SE analysis of both layers in the SM-LDM system model that applies the proposed SAS technique. Finite alphabet inputs (modulation and spatial symbols) are employed. Assume perfect channel estimation at the receivers and that the modulation symbol constellation of the \(LL\) receiver is known for both receivers. For the analysis, we define the spatial constellation spaces for \(UL\) and \(LL\), respectively, as \(\mathcal{H}_{U}=\left\{\boldsymbol{h}_{1}^{U},\boldsymbol{h}_{2}^{U}, \boldsymbol{h}_{c}^{U},\ldots,\boldsymbol{h}_{N_{t}}^{U}\right\}\) and \(\mathcal{H}_{L}=\left\{\boldsymbol{h}_{1}^{L},\boldsymbol{h}_{2}^{L}, \boldsymbol{h}_{c}^{L},\ldots,\boldsymbol{h}_{N_{t}}^{L}\right\}\) with \(N_{t}\) spatial symbols. Likewise, the signal space for \(LL\) with \(M_{L}\) signal symbols is denoted as \(\mathcal{S}_{L}=\left\{s_{1}^{L},s_{2}^{L},s_{l}^{L},\ldots,s_{M_{L}}^{L}\right\}\). The SE for each layer can be calculated by determining the mutual information (MI) ob  
Figure 2: Transceiver model for SAS-based SM-LDM system.  
tained between the received signal, which is given in Equation (6), and the inputs from both the spatial domain for the _UL_ (i.e., \(I(\mathcal{H}_{UL};\mathbf{y}_{l})\)) and the signal domain as well as the spatial domain for the _LL_ (i.e., \(I(\mathcal{H}_{L},\mathcal{S}_{L};\mathbf{y}_{L})\)) [27; 28].  
Within the context of the proposed SAS-based SM-LDM, each layer contributes to the process of selecting the common TA in which the AS bits are mapped to a shared spatial symbol. As a result, the MI produced by spatial symbols for each layer is determined by the layer's AS bit allocation. These bits split the \(N_{t}\) antennas into groups \(G_{U}=2^{b_{AS}^{U}}\) and \(G_{L}=2^{b_{AS}^{L}}\); each group includes several indices equal to \(I_{U}=N_{t}/G_{U}\) and \(I_{L}=N_{t}/G_{L}\). An example of this AS bits mapping is provided in Appendix A. The spatial constellation of the _UL_ becomes \(\mathcal{H}_{U}=\left\{\mathbf{h}_{1,I}^{U},\mathbf{h}_{g(U),i(U)}^{U},\dots,\mathbf{h}_{G_{ U},I_{U}}^{U}\right\}\) where \(g(U)\) is the group index (i.e.,\(g(U)\in\{1,\dots,G_{U}\}\)) and \(i(U)\) is the antenna index per group (i.e., \(i(U)\in\{1,\dots,I_{U}\}\)). Accordingly, \(\mathbf{h}_{g(U),i(U)}^{U}\) represents the spatial symbol transmitted by _UL_. Similarly, \(\mathbf{h}_{g(U),i(U)}^{L}\) is the spatial symbol transmitted by the _LL_. At specific transmission time, the AS bits of the _UL_ yield a group index \(g\), whereas the AS bits of the _LL_ specify the index \(i\). Hence, the received signals shown in (6) can be reformed as follows:  
\[\mathbf{y}_{U}=\mathbf{h}_{g,i}^{U}\mathbf{s}_{l}+\mathbf{w}_{U}. \tag{9}\] \[\mathbf{y}_{L}=\sqrt{\beta_{L}}\mathbf{h}_{g,i}^{L}\mathbf{s}_{l}+\mathbf{w}_{L}.\]  
The sum-rate of the SAS-based SM-LDM system can be evaluated as follows:  
\[R_{Sum}=I(\mathcal{H}_{UL};\mathbf{y}_{U})+I(\mathcal{H}_{L},\mathcal{S}_{L};\mathbf{ y}_{L}) \tag{10}\]  
where \(I(\mathcal{H}_{UL};\mathbf{y}_{U})\) is the MI gained from the _UL_ (SSK layer). The second term \(I(\mathcal{H}_{L},\mathcal{S}_{L};\mathbf{y}_{L})\) is the joint MI realized from both spatial and modulation symbols transmitted by _LL_.