## 4 Performance Analysis of the MA-SM
### Performance Analysis of MA-SM with Maximum-Likelihood Receiver
 \tag{12}\]  
Finally, the union bound on the UPEP is obtained by averaging over all the hypotheses of the transmit vectors as follows:  
\[\begin{split} Pr\left[e\right]\leq\frac{1}{2^{M}}\sum_{n_{1}=1}^ {2^{M}}\sum_{n_{2}=1}^{2^{M}}Pr\left[\mathbf{g}_{m_{1}}\rightarrow\mathbf{g}_ {m_{2}}\right]&=\binom{2n_{R}-1}{n_{R}}\frac{\rho^{-n_{R}}}{2^{M} }\sum_{m_{1}=1}^{2^{M}}\sum_{n_{2}=1}^{2^{M}}\left(d_{m_{1},m_{2}(tx)}^{2} \right)^{-n_{R}}\\ &=\binom{2n_{R}-1}{n_{R}}\frac{\rho^{-n_{R}}}{2^{M}}\sum_{i_{1}=1 }^{n_{R}}\sum_{i_{1}=1}^{n_{Norm}}\sum_{i_{1}=1}^{L}\sum_{\hat{1}}\sum_{\hat{1 }}\sum_{\hat{1}}\sum_{\hat{1}}\sum_{\hat{2}}\sum_{\hat{2}}\sum_{\hat{2}}\left( d_{m_{1},m_{2}(tx)}^{2}\right)^{-n_{R}}\\ &=\binom{2n_{R}-1}{n_{R}}  
This case occurs when the combinations \(\gamma_{i}=(i_{1},i_{2})\) and \(\gamma_{j}=(\hat{i}_{1},\hat{i_{2}})\) are equal, which implies that there is no error in the spatial symbol. In this case, the two symbols in the transmitted vector and those in the estimated vector are located at the same index. In other words, the signal symbols in both vectors are transmitted from the same antennas. Accordingly, \((i_{1}=\hat{i_{1}},i_{2}=\hat{i_{2}})\) and \(\theta_{i}=\theta_{j}\). Based on the first and second columns of Table 1, the frequency of this term is equal to 4. The \(\Omega\) term is given by:  
\[\Omega_{1}=\sum_{k_{1}}^{L}\sum_{k_{2}}^{L}\sum_{k_{1}}^{L}\sum_{k_{2}}^{L} \left(\left|s_{k_{1}}-s_{k_{1}}\right|^{2}+\left|s_{k_{2}}-s_{k_{2}}\right|^{2 }\right)^{-n_{R}} \tag{15}\]  
Therefore, the error probability will depend on the Euclidean distance among the signal symbols transmitted from the same antenna. **Case 2:**\(\delta(\xi_{i},\xi_{j})=2\). In this case, one signal symbol from the transmitted vector and one in the estimated vector are located at the same index. The second symbols in the transmitted and estimated vectors are located at different indexes. This corresponds to one of the following equiprobable four cases:  
1. \(i_{1}=\hat{i_{1}},i_{2}\
eq\hat{i_{2}}\), \(\gamma_{1}=(1,3)\leftrightarrow\gamma_{3}=(1,4)\) and \(f_{2}=2\). The corresponding \(\Omega\) term is given by:  
\[\Omega_{2}=\sum_{k_{1}}^{L}\sum_{k_{2}}^{L}\sum_  
\[\theta^{*}=\arg_{0\leq\theta_{1}\leq\pi/2i=1,\cdots,A}\{\Omega_{2}+\Omega_{3}+ \Omega_{4}+\Omega_{5}\} \tag{21}\]  
where \(\theta\) is a vector of length of 4. The optimal angles are obtained using the Matlab optimization tool. First, we analyze the obtained angles in the case of PSK modulation. The general form of the optimal angles obtained using (21) is given by:  
\[|\theta_{1}-\theta_{3}|=|\theta_{2}-\theta_{4}|=\frac{\pi}{L},\ \ \text{with}\ \ \theta_{1}=\theta_{2},\theta_{3}=\theta_{4} \tag{22}\]  
To verify the formula in (22), a three-dimensional graph is generated for four different PSK modulation schemes and illustrated in Figure 1. The value of the cost function depends on the difference between the angles rather than on individual values. As presented in Figure 1, we can conclude that the difference in the value of the cost function in (21) using the optimal angles and non-optimal angles vanishes as the modulation order increases. Table 2 depicts the minimum value of the cost function when the optimal angles are used and the maximum value of the cost function assuming \(\theta_{i}=0\), \(\forall i\) for several modulation schemes. The cost values are obtained using (21) by substituting the corresponding values of the rotation angles. For modulation orders 16 or higher, the value of the cost function becomes independent of the rotation angles. As shown in Figure 1, the change in the function's value (y-axis value) is marginal for the case of 16-PSK. The error performance of the MA-SM, assuming the whole set of \(\Omega\) terms, is considered and analyzed using PSK modulation. \(\Omega_{6}\) is simplified as follows:  
\[\Omega_{6}=L^{4}4^{-n_{R}} \tag{23}\]