## 3 Diversity-Controlled MUST Technique
### Receiver Design at an MUST-Enhanced UE  
Figure 4 shows the overall structure of the proposed receiver at the MUST-enhanced UE. Unlike the signals at the MUST-based UEs, the signal of the MUST-enhanced UE is superposed over \(N\) MUST-based signals since it is repeatedly sent over \(N\) OFDM sub-carriers. Thus, in order for the MUST-enhanced UE to obtain full diversity gain, we need to combine all the signals over \(N\) OFDM sub-carriers and in \(N_{r}\) receive antennas. The received signal at the MUST-enhanced UE is given by  
\[\mathbf{Y}_{E}=\mathbf{H}_{E}\sqrt{\mathcal{E}}diag\left(\mathbf{s}\right)+ \mathbf{W}_{E}, \tag{9}\]  
where \(\mathbf{Y}_{E}\triangleq\left[\mathbf{y}_{E}(1)\quad\mathbf{y}_{E}(2)\quad \cdots\quad\mathbf{y}_{E}(N)\right]\in\mathbb{C}^{N_{r}\times N}\). A vector \(\mathbf{y}_{E}(i)\) denotes the signal vector received at the \(i\)-th OFDM sub-carrier of the MUST-enhanced UE. A matrix \(\mathbf{H}_{E}\triangleq\left[\mathbf{h}_{E}(1)\quad\mathbf{h}_{E}(2)\quad \cdots\quad\mathbf{h}_{E}(N)\right]\in\mathbb{C}^{N_{r}\times N}\) denotes the wireless channel matrix from the BS to the MUST-enhanced UE, where each element follows i.i.d. complex-valued normal Gaussian distribution and \(\mathbf{h}_{E}(i)\) denotes the channel vector of the \(s_{i}\) transmitted over the \(i\)-th OFDM sub-carrier. A matrix \(diag(\mathbf{s})\) indicates the diagonal matrix whose diagonal elements form the vector \(\mathbf{s}\). In addition, \(\mathbf{s}\triangleq\left[s_{1}\quad s_{2}\quad\cdots\quad s_{N}\right]\in \mathbb{C}^{1\times N}\) indicates the transmit signal vector. A matrix \(\mathbf{W}_{E}=\left[\mathbf{w}_{E}(1)\quad\mathbf{w}_{E}(2)\quad\cdots\quad \mathbf{w}_{E}(N)\right]\in\mathbb{C}^{N_{r}\times N}\) represents the additive white Gaussian noise matrix whose element is a zero mean complex Gaussian RV with \(N_{0}\) variance, and \(\mathbf{w}_{E}(i)\) denotes the noise vector in the received signal vector over the \(i\)-th OFDM sub-carrier. Each column of \(\mathbf{Y}_{E}\) can be separated and dealt with independently over \(N\) sub-carriers. Then, the received signal vector over the \(i\)-th sub-carrier at the MUST-enhanced UE is given by  
\[\mathbf{y}_{E}(i)=\mathbf{h}_{E}(i)\sqrt{\mathcal{E}}s_{i}+\mathbf{w}_{E}(i). \tag{10}\]  
Similarly to \(\tilde{y}_{B,i}\), after MRC, the received signal vector in the \(i\)-th sub-carrier at the MUST-enhanced UE is given by  
\[\tilde{y}_{E}(i)=\left\|\mathbf{h}_{E}(i)\right\|^{2}\sqrt{\mathcal{E}}\left( \sqrt{\alpha-1}x_{B,i}+\sqrt{\alpha}x_{E,i}\right)+\mathbf{h}_{E}(i)^{H} \mathbf{w}_{E}(i). \tag{11}\]  
The receiver of the MUST-enhanced UE first detects \(x_{B,i}\) like the receiver operation of the MUST-based UE, and then removes \(\hat{x}_{B,i}\) from the \(\tilde{y}_{E}(i)\), which is called symbol-level SIC operation in the literature. Considering the Gray-mapping function \(G\left(\cdot\right)\), Equation (11) can be rewritten as:  
\[ \begin{split}\tilde{y}_{E}(i)&=\left[\Re\left( \tilde{y}_{E}(i)\right)-\underbrace{\left\|\mathbf{h}_{E}(i)\right\|^{2}\sqrt{ \mathcal{E}}\sqrt{1-\alpha}Sgn\left(\Re\left(\tilde{y}_{E}(i)\right)\right)} _{\text{SIC}}\right]\cdot\underbrace{Sgn\left(\Re\left(\tilde{y}_{E}(i)\right) \right)}_{\text{Gray-demapping}}\\ &\quad+j\left[\Im\left(\tilde{y}_{E}(i)\right)-\underbrace{\left\| \mathbf{h}_{E}(i)\right\|^{2}\sqrt{\mathcal{E}}\sqrt{1-\alpha}Sgn\left(\Im \left(\tilde{y}_{E}(i)\right)\right)}_{\text{SIC}}\right]\cdot\underbrace{Sgn \left(\Im\left(\tilde{y}_{E}(i)\right)\right)}_{\text{Gray-demapping}}.\end{split}