## 3 Proposed Biased-Power Allocation (Bi-PA) for SM-LDM Systems  
In contrast to the Preassigned-Power Allocation (Pre-PA) methods [12; 13; 15], this research proposes a biased-PA (Bi-PA) for the SM-LDM systems. To improve the fairness between the layers in terms of the BER performance, the ratios in the Bi-PA method are produced such that the SINRs of the _UL_ and _LL_ are almost identical. As explained previously, in LDM systems, the _LL_ employs high-order modulation, and accordingly, it has a small SMED between the constellation points. The _UL_, on the other hand, utilizes low-order modulation with powerful SMED to accomplish strong transmission. This difference yields a significant BER performance gap between the layers. In broadcasting systems, the _LL_ fixed receiver has a better SNR threshold than the _UL_ mobile receiver, but it employs a higher modulation constellation which conversely deteriorates the BER performance. Additionally, this higher modulation affects the _UL_ performance and becomes interference limited at the high SNR values and yields to an error floor problem. Accordingly, the power ratios need to be adaptively assigned. Due to the importance and significance of the SINRs and the SMED factor, the suggested Bi-PA distributes the power ratios to obtain equal SINRs, and these SINRs take the SMED factor into mind. The _UL_ receiver uses the ML detector in the SM-LDM receiver model described in Section 2 to estimate its information bits that are mapped to the modulation and spatial symbols, as shown in (2). The _UL_ signal includes interference from the _LL_ signal in addition to the AWGN with zero mean and variance \(\sigma_{0}^{2}=1/\gamma\). Accordingly, the _UL_ SINR is given by \(\rho_{U}/(\rho_{L}+1/\gamma)=\rho_{U}\gamma/(\rho_{L}\gamma+1)\). On the other hand, the _LL_ receiver applies the SIC detector to first estimate and cancel the _UL_ signal as in (3). Then, it applies the ML detector to estimate its information bits as in (4). Hence, the SINR at the _LL_ receiver is given by \(\beta_{L}\rho_{L}/(1/\gamma)=\beta_{L}\rho_{L}\gamma\) assuming that the _UL_ signal has been perfectly canceled. Now, we add the SMED factor to the numerator of both SINRs to include its effect. Large SMED as in _UL_ constellations (i.e., QPSK is an example) yields higher SINR and better system performance. Low SMED as in _LL_ constellations (i.e., 64 QAM is an example) yields low SINR and bad performance. Thus, the SINRs become \(\rho_{U}\gamma\rho_{U}/(\rho_{L}\gamma+1)\) and \(\beta_{L}\rho_{L}\gamma D_{L}\) for _UL_ and _LL_, respectively. By equating both SINRs, the closed form of power ratios can be given as  
\[\rho_{L}=\frac{\sqrt{\left(\beta_{L}D_{L}+D_{U}\right)^{2}+4\beta_{L}D_{L}D_{U} \gamma}-\left(\beta_{L}D_{L}+D_{U}\right)}{2\beta_{L}D_{L}\gamma},\rho_{U}=1- \rho_{L}. \tag{5}\]  
Figure 1: Transceiver model of a typical SM-LDM system. From (5), the power ratios are generated such that \(\rho_{L}<\rho_{U}\) and both ratios should equal 1 (\(\rho_{L}+\rho_{U}=1\)). Hence, the adapted ratios for _UL_ and _LL_, respectively, will be \(\frac{1}{2}<\rho_{U}<1\) and \(0<\rho_{L}<1/2\). It can be noticed from (5) that the power ratios are produced in response to the modulation schemes (\(D_{L}\) and \(D_{U}\)), SNR (\(\gamma\)), and the SNR threshold difference ratio between the layers (\(\beta_{L}\)). In other words, the power ratios \(\rho_{L}\) and \(\rho_{U}\) are dynamically changed according to the mentioned parameters. The adaptive change of the power ratios has positive advantages for both layers. The _LL_'s power ratio \(\rho_{L}\) increases if the SMED between _LL_ symbols (\(D_{L}\)) is low and in cases of less \(\beta_{L}\) and low SNR values. Hence, at a low SNR region, the _LL_ layer obtains a high ratio (relative to the case of Pre-PA). Because the _LL_ applies the SIC process and becomes a noise-limited layer, the relatively high _LL_ ratio is expected to improve its possibility of correct detection and consequently enhance its performance.