## 4 Performance Analysis of the MA-SM
### Performance Analysis of MA-SM with Maximum-Likelihood Receiver
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Modulation** & **QPSK** & **8-PSK** & **16-PSK** & **32-PSK** \\ \hline \(\theta\) & optimal & \(\theta_{i}=0\) & optimal & \(\theta_{i}=0\) & optimal & \(\theta_{i}=0\) & optimal & \(\theta_{i}=0\) \\
**Cost** & 32.8 & 41.2 & 590.2 & 592.2 & 9459.3 & 9459.3 & \(1.5135\times 10^{5}\) & \(1.5135\times 10^{5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The value of the cost function in (21) for optimal angles and \(\theta_{i}=0\) for several PSK modulation schemes and \(n_{R}=3\). Figure 1: Three-dimensional graphs of the cost function in (21) as a function of \(\theta_{1}\) and \(\theta_{3}\) for BPSK, QPSK, 8-PSK, and 16-PSK modulation schemes and \(n_{R}=3\). Figure 2 depicts the weighted values of \(\Omega_{2}-\Omega_{5}\), weighted by their corresponding frequency \(f\), on the left \(y\)-axis and the value of \(\Omega_{1}\) weighted by \(f_{1}\) on the right \(y\)-axis. The \(x\)-axis represents the iteration number of the optimization process given in (21). The 8-PSK modulation is used for the generation of both graphs. The number of receive antennas is equal to 3 and 5 in the left and right sub-figure, respectively. Based on (13), the upper bound on the pairwise error probability depends on the weighted sum of the six \(\Omega\) terms. When the optimization algorithm converges to the optimal solution for \(n_{R}=3\), \(f_{1}\Omega_{1}\) is about 6094 (\(f_{1}\)=4, \(\Omega_{1}=1523.5\), from (15)) and \(f_{6}\Omega_{6}=256\) (\(f_{6}=4\) and \(\Omega_{6}=64\), from (23)), and the sum of the terms that depend on the rotation angles is about 1180. Note that the values of \(\Omega\) are obtained by substituting the values of the optimal rotation angles in (15)-(20) for any modulation set. Therefore, the error probability will be dominated by the terms that do not depend on the rotation angles. Further dominance is remarked as \(n_{R}\) increases, where  
\[\frac{f_{1}\Omega_{1}+f_{6}\Omega_{6}}{\sum_{i=2}^{5}f_{i}\Omega_{i}}\approx 75.4,\]  
From the previous analysis and for \(n_{R}=5\), we conclude that:  
* For a given modulation order, the effect of the rotation angle on the upper bound on the pairwise error probability vanishes as \(n_{R}\) increases. * For a given \(n_{R}\), the effect of the rotation angle on the upper bound on the pairwise error probability vanishes as the modulation order increases. It is therefore expected that the change in the rotation angles would have minimal impact, if any, on the error performance of the MA-SM system. The error performance is verified in Section 5 using simulations. The optimization of the rotation angles is analyzed for the case when both signal symbols are drawn from the same L-QAM constellation with \(L>4\). Based on the result of the optimization process, the optimal rotation angle is given as:  
\[|\theta_{1}-\theta_{3}|=|\theta_{2}-\theta_{4}|=\frac{\pi}{4},\ \ \text{with}\ \ \theta_{1}=\theta_{2},\theta_{3}=\theta_{4} \tag{24}\]  
Equation (22) is satisfied only for \(n_{R}\)= 2 or 3. For \(n_{R}\) = 4 and 6, we remarked that \(|\theta_{1}-\theta_{3}|=|\theta_{2}-\theta_{4}|\approx 65^{\circ}\) leads to a better performance than values in (22). Several other sets of angles that are not symmetric achieve similar performances. Figure 3 depicts the weighted \(\Omega\) terms that depend on the rotation angles on the left y-axis, and the weighted \(\Omega_{1}\) on the right \(y\)-axis, versus the iteration number of the multi-objective optimization process. Figure 2: The value of the weighted \(\Omega\) terms that are affected by the rotation angle (left y-axis) and the value of the rotation-independent \(\Omega_{1}\) (right y-axis) for \(n_{R}=3\) (**left sub-figure**) and \(n_{R}=5\) (**right sub-figure**), assuming 8-PSK. The value of \(\Omega_{6}\) is neglected as it is much smaller than \(\Omega_{1}\). For instance, \(\Omega_{1}/\Omega_{6}\) approximately equals 72 for 64-QAM.