## 4 Proposed SAS-Based SM-LDM System Model  
In the typical SM-LDM system outlined in Section 2, each layer makes its own independent choice about which active transmit antenna to use. It increases SE but requires two RF chains (considering two layers _UL_ and _LL_), which lowers EE and adds hardware complexity. It makes it harder for receivers to identify each layer's active antenna. Additionally, it requires SIC for the _LL_ signal and increases ILI considering that every layer transmits a modulation symbol. To overcome these problems, we propose a new technique called shared antenna selection (SAS) to activate only one antenna per transmission time which yields to transmit a unique modulation symbol from both layers. The decision of choosing which antenna is active is determined cooperatively via _UL_ and _LL_. More specifically, the whole data of the _UL_ are combined with the AS bits of the _LL_ layer to determine the active antenna as shown in Figure 2. Hence, \(b_{AS}^{U}+b_{AS}^{L}=\log_{2}(N_{t})\), and \(b_{SS}^{L}=\log_{2}(M_{L})\). It can be noticed here that one active antenna requires a single RF chain which improves the EE and reduces the hardware complexity. This will be more incredible for multiple layers of SM-LDM, especially with many antennas at the transmitter. In addition, the _UL_ delivers its service completely through implicit modulation of the spatial domain; there is no modulation symbol transmitted. Hence, it conveys an SSK signal which offers several features. Firstly, there is no need for the SIC process at the _LL_ receiver which reduces the complexity of the whole system and alleviates the problem of SIC propagation error. Secondly, the total power is allocated to the _LL_ signal. Accordingly, it is free from optimizing the power allocation and finding suitable ratios. Additionally, assigning the full power to the transmitted symbol of the _LL_ signal yields reinforcement of the possibility of successful detection of both _LL_ and _UL_ bits. Thirdly, there is no ILI which is expected to yield better system performance. An additional advantage of the SAS technique comes from the flexibility to deliver the \(LL\) service through both spatial (i.e., AS bits) and modulation symbols (i.e., SS bits). More specifically, sharing the decision of antenna selection provides more freedom to convey the services via AS bits. This allows utilizing the case of a higher number of antennas more efficiently; increasing \(N_{t}\) leads to more AS bits for the \(LL\) which decreases the SS bits, and consequently there is less modulation constellation size. Therefore, it relieves the interference and consequently provides better BER performance. Assume that a \(c\)th common antenna is activated through the combined AS bits from both layers' data and the \(LL\)th modulation symbol \(s_{l}\) is selected by the SS bits (\(b_{SS}^{L}\)). Hence, the received signal at the \(UL\) and \(LL\) receivers are given as follows:  
\[ \begin{array}{l}\boldsymbol{y}_{U}=\boldsymbol{h}_{c}^{U}s_{l}+\boldsymbol{ w}_{U}.\\ \boldsymbol{y}_{L}=\sqrt{\beta_{L}}\boldsymbol{h}_{c}^{L}s_{l}+\boldsymbol{ w}_{L}.\end{array}  \tag{6}\]  
where the channel vector between the shared active \(c\)th antenna and the \(UL\)th and \(LL\)th receivers is denoted by \(h_{c}^{U}\) and \(\boldsymbol{h}_{c}^{L}\), respectively. All elements of the vector \(\boldsymbol{h}_{c}^{U}\) and \(\boldsymbol{h}_{c}^{L}\) are i.i.d. complex Gaussian random variables with zero mean and unit variance. As shown in Figure 2, the \(UL\)th receiver applies the ML detector to estimate the index of the active transmit antenna as follows:  
\[\hat{c}=\underbrace{\arg\min}_{1\leq c\leq N_{t},1\leq l\leq M_{L}}\left\| \boldsymbol{y}_{U}-\boldsymbol{h}_{c}^{L}s_{l}\right\|^{2}. \tag{7}\]  
From the estimated index \(\hat{c}\) the AS bits of \(UL\) are obtained.