## 5 Performance Evaluation
### The Application of the Proposed Scheme to the Two-Antenna Case  
In this case, we can use closed form solutions for the eigenvalues of \(2\times 2\) matrices to simplify the link level capacity expressions in the previous section. In this case, the simpler form of link level capacity of Equation (12) is described as:  
\[C_{\text{SM}}\leq\sum_{k=1}^{M}\log_{2}\left(1+\frac{\left|\mathbf{R}_{t} \right|\gamma_{0}}{2}\left(\frac{1}{\left|\mathbf{R}_{t}^{11}\right|}+\frac{1 }{\left|\mathbf{R}_{t}^{22}\right|}\right)+\frac{\left|\mathbf{R}_{t}\right|^ {2}\gamma_{0}^{2}}{4\left|\mathbf{R}_{t}^{11}\right|\left|\mathbf{R}_{t}^{22} \right|}\right) \tag{17}\]  
where:  
\[\mathbf{R}_{t}= \begin{cases}p_{m}^{2}\Bigg{|}\begin{matrix}1&\eta_{\text{SEN}m,1,2}\\ \left(\eta_{\text{SEN}m,1,2}\right)^{*}&1\end{matrix}\Bigg{|},&\text{single- node(SENm)}\\ \Bigg{|}\begin{matrix}p_{1}^{2}&0\\ 0&p_{2}^{2}\end{matrix}\Bigg{|},&\text{multi-node.}\end{cases}  \tag{18}\]  
Note that correlation and path gain are the main parameters that determine the SM capacity. MN transmission can avoid spatial correlation, increasing the capacity, while power loss is experienced due to \(p_{2}^{2}(\leq 1)\), decreasing the capacity. Thus, the SM comparison result is significantly dependent upon the path gain ratio and spatial correlation. For the capacity of SN SM, SEN1 SM provides better performance than SEN2 SM when the following inequality holds, and _vice versa_. \[4p_{1}^{2}\left(1-\left|\eta_{\text{SEN}1,1,2}\right|^{2}\right)+\gamma_{0}p_{ 1}^{4}\left(1-\left|\eta_{\text{SEN}1,1,2}\right|^{2}\right)^{2}\geq 4p_{2}^{2} \left(1-\left|\eta_{\text{SEN}2,1,2}\right|^{2}\right)+\gamma_{0}p_{2}^{4} \left(1-\left|\eta_{\text{SEN}2,1,2}\right|^{2}\right)^{2} \tag{19}\]  
which reduces to \(p_{1}^{2}\geq p_{2}^{2}\) when the spatial correlations are the same, _i.e._, \(\eta=\left|\eta_{\text{SEN}1,1,2}\right|=\left|\eta_{\text{SEN}2,1,2}\right|\), as intuitively expected. Assuming SEN1 SM is better than SEN2 SM without loss of generality, the MN SM provides higher capacity than SEN1 SN SM when:  
\[\gamma_{0}p_{1}^{2}\left((1-\eta^{2})^{2}p_{1}^{2}-p_{2}^{2}\right)\leq 2(p_{1} ^{2}+p_{2}^{2})-4(1-\eta^{2})p_{1}^{2}. \tag{20}\]  
This can be simplified using \(p_{1}^{2}=1\) and \(\alpha(p_{2},\eta)=p_{2}^{2}+2\eta^{2}-1\) as:  
\[\gamma_{0}\left(\eta^{4}-\alpha(p_{2},\eta)\right)\leq 2\alpha(p_{2},\eta). \tag{21}\]  
Here, channel environments can be classified into three cases as:  
\[ \begin{split} C1:&\alpha(p_{2},\eta)\leq 0\\ C2:&\alpha(p_{2},\eta)>\eta^{4}>0\\ C3:&\eta^{4}>\alpha(p_{2},\eta)>0.\end{split}  \tag{22}\]The regions for the three cases are depicted in Figure 6. For \(C1\), SN SM is always better than MN SM, irrespective of the SNR. This can be intuitively expected, since \(\alpha(p_{2},\eta)\leq 0\) corresponds to a smaller \(p_{2}\) and \(\eta\), where MN processing cannot obtain its advantage over SN MIMO. On the other hand, MN SM is always better than SN SM for \(C2\), which corresponds to a major portion in Figure 6. Thirdly, a crossing between SN SM and MN SM occurs in \(C3\), where SN SM outperforms MN SM when the SNR is less than the crossing point value, \(\gamma_{MN-SN}\left(=2\alpha(p_{2},\eta)/(\eta^{4}-\alpha(p_{2},\eta))\right)\). We can also calculate SNR crossing points between MN (or SN) SM and SN OSTBC by equating Equations (12) and (16). Although numerical calculation is necessary for a large number of antennas, the SNR crossing point can be easily obtained for two and three antennas.