## 3 Diversity-Controlled MUST Technique
### Performance Analysis of the MUST-Enhanced UE  
The error events of the MUST-enhanced UE happen with the following two cases:  
* Incorrect detection of both the symbol \(x_{E,i}\) and the bit \(b_{B.1}\).
* Accurate detection of the symbol \(x_{E,i}\) but incorrect detection of the bit \(b_{B.1}\).  
Thus, the BER of the MUST-enhanced UE is given as:  
\[\begin{split} P_{b,E}&=\Pr\left\{\left(\hat{b}_{ B,1}^{i},\hat{b}_{B,2}^{i}\right)\
eq\left(b_{B,1}^{i},b_{B,2}^{i}\right),\hat{b}_{E.1}\
eq b_{E.1}\right\}+\Pr\left\{\left(\hat{b}_{B,1}^{i},\hat{b}_{B,2}^{i} \right)=\left(b_{B,1}^{i},b_{B,2}^{i}\right),\hat{b}_{E.1}\
eq b_{E.1}\right\} \\ &=\Pr\left\{\left(\hat{b}_{B,1}^{i},\hat{b}_{B,2}^{i}\right)\
eq \left(b_{B,1}^{i},b_{B,2}^{i}\right)\right\}\Pr\left\{\hat{b}_{E.1}\
eq b_{E.1} \mid\left(\hat{b}_{B,1}^{i},\hat{b}_{B,2}^{i}\right)\
eq\left(b_{B,1}^{i},b_{ B,2}^{i}\right)\right\}\\ &\quad+\Pr\left\{\left(\hat{b}_{B,1}^{i},\hat{b}_{B,2}^{i}\right)= \left(b_{B,1}^{i},b_{B,2}^{i}\right)\right\}\Pr\left\{\hat{b}_{E.1}\
eq b_{E.1} \mid\left(\hat{b}_{B,1}^{i},\hat{b}_{B,2}^{i}\right)=\left(b_{B,1}^{i},b_{B,2} ^{i}\right)\right\}.\end{split} \tag{18}\]  
Since the error probability of decoding \(\left(b_{B,1}^{i},b_{B,2}^{i}\right)\) is close to zero at high SNR values, we can obtain the lower bound of the BER by assuming that the decoding on \(\left(b_{B,1}^{i},b_{B,2}^{i}\right)\) is always correct:  
\[P_{b.E}\geq\Pr\left\{\hat{b}_{E.1}\
eq b_{E.1}\mid\left(\hat{b}_{B,1}^{i},\hat {b}_{B,2}^{i}\right)=\left(b_{B,1}^{i},b_{B,2}^{i}\right)\right\}\triangleq P _{b.E}^{LB}. \tag{19}\]  
Without loss of generality, the lower bound of the BER of the MUST-enhanced UE with this assumption for a given \(\mathbf{H}_{E}\) is obtained as:  
\[\begin{split} P_{b.E}^{LB}&=\Pr\left\{\hat{b}_{E.1 }=0\mid b_{E.1}=1,\left(\hat{b}_{B,1}^{i},\hat{b}_{B,2}^{i}\right)=\left(b_{B,1}^{i},b_{B,2}^{i}\right),\mathbf{H}_{E}\right\}\\ &=\Pr\left\{\Re\left(y_{comb}\right)\leq 0\mid b_{E.1}=1,\left( \hat{b}_{B,1}^{i},\hat{b}_{B,2}^{i}\right)=\left(b_{B,1}^{i},b_{B,2}^{i} \right),\mathbf{H}_{E}\right\}\\ &=\Pr\left\{\Re\left(y_{comb}\right)\leq 0\mid\Re\left(x_{E} \right)=1,\left(\hat{b}_{B,1}^{i},\hat{b}_{B,2}^{i}\right)=\left(b_{B,1}^{i}, b_{B,2}^{i}\right),\mathbf{H}_{E}\right\}\\ &=\int_{-\infty}^{0}\frac{1}{\sqrt{2\pi\left(N_{0}/2\varepsilon a \operatorname{Tr}\left(\mathbf{H}_{E}^{H}\mathbf{H}_{E}\right)\right)}}\exp \left(-\frac{\left(x-1\right)^{2}}{2\left(N_{0}/2\varepsilon a\operatorname{Tr }\left(\mathbf{H}_{E}^{H}\mathbf{H}_{E}\right)\right)}\right)dx\\ &=\int_{-\infty}^{-\sqrt{2\frac{\mathcal{F}}{N_{0}}\alpha \operatorname{Tr}\left(\mathbf{H}_{E}^{H}\mathbf{H}_{E}\right)}}\frac{1}{ \sqrt{2\pi}}\exp\left(-\frac{x^{2}}{2}\right)dx\\ &=Q\left(\sqrt{2\rho\alpha\operatorname{Tr}\left(\mathbf{H}_{E}^{ H}\mathbf{H}_{E}\right)}\right).\end{split} \tag{20}\]  
We define an RV as follows:  
\[Z\triangleq\operatorname{Tr}\left(\mathbf{H}_{E}^{H}\mathbf{H}_{E}\right) \sim\chi_{2N\cdot N_{r}}^{2}. \tag{21}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Number of Multipliers** & **Number of Adders** & **Number of Comparators** \\ \hline MUST-enhanced UE & \(2 NN_{r}\) & \(2N(N_{r}-1)\) & \(N+1\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Computation complexity of the receiver at the MUST-enhanced UE.  
Then, the lower bound of the average BER of the MUST-enhanced UE is given by  
\[\begin{split}\bar{P}_{b.E}^{LB}&=\int_{0}^{\infty}P_{b.E}^{LB}f_{Z}\left(z\right)dz\\ &=\int_{0}^{\infty}\mathcal{Q}\left(\sqrt{2\rho az}\right)\frac{1} {(N_{r}\cdot N-1)!}z^{N_{r}\cdot N-1}\exp(-z)dz\\ &=\left(1-\sqrt{\frac{\rho\alpha}{1+\rho\alpha}}\right)^{N_{r} \cdot N}\sum_{k=0}^{N_{r}\cdot N-1}\binom{N_{r}\cdot N-1+k}{k}\left(\frac{1}{2} \right)^{N_{r}+k}\left(1+\sqrt{\frac{\rho\alpha}{1+\rho\alpha}}\right)^{k}. \end{split} \tag{22}\]