## 3 System Performance Analysis
### Outage Probability at the Coupled Relays \(\mathcal{R}\)  
**Theorem 1**. : _The outage event at a relay in the coupled relays \(\mathcal{R}\) occurs when the relay cannot successfully decode at least a data symbol \(x_{i}\in\{x_{N},\ldots,x_{1}\}\) from the best received signal \(\underset{[A_{1}\times A_{2}]}{\max}\left\{\left|\mathbf{Y}_{R_{1}}^{\left( \tau_{1}^{\left(\text{ideal}\right)}\right)}\right|\right\}\) for an odd transmission block or \(\underset{[A_{1}\times A_{2}]}{\max}\left\{\left|\mathbf{Y}_{R_{2}}^{\left( \tau_{2}^{\left(\text{even}\right)}\right)}\right|\right\}\) for an even transmission block, which is the best signal after TAS. Therefore, the coupled relays \(\mathcal{R}\) receive the best superimposed signal \(\underset{[A_{1}\times A_{2}]}{\max}\left\{\left|\mathbf{Y}_{\mathcal{R}}^{ \left(\tau_{2}^{\left(\text{eq}\right)}\right)}\right|\right\}\) by applying a TAS protocol to select the best signal from the pre-coding channel matrix \(\mathbf{H_{0}}\) for an odd transmission block or \(\mathbf{G_{0}}\) for an even transmission block to maximize the SINRs \(\gamma_{\mathcal{R}-x_{i}}^{\left(\tau_{1}^{\left(\text{eq}\right)}\right)}\), as given by (8), (9), (21), and (21), and maximize the instantaneous bit-rate threshold \(\underset{[A_{0}\times A_{1}]}{\max}\left\{\underset{[A_{0}\times A_{1}]}{ \mathbf{R}_{\mathcal{R}-x_{i}}^{\left(\tau_{i}^{\left(\text{eq}\right)} \right)}}\right\}\), as given by (10) or (23). Maximization of the instantaneous bit rate threshold \(\underset{[A_{0}\times A_{1}]}{\max}\left\{\underset{[A_{0}\times A_{1}]}{ \mathbf{R}_{\mathcal{R}-x_{i}}^{\left(\tau_{i}^{\left(\text{eq}\right)} \right)}}\right\}\) is then compared to a device's predefined bit-rate threshold \(R_{i}^{*}\), where \(i=\{N,\ldots,1\}\). If maximization of the instantaneous bit-rate threshold \(\underset{[A_{0}\times A_{1}]}{\max}\left\{\underset{[A_{0}\times A_{1}]}{ \mathbf{R}_{\mathcal{R}-x_{i}}^{\left(\tau_{i}^{\left(\text{eq}\right)} \right)}}\right\}\) is less than a device's predefined bit-rate threshold \(R_{i}^{*}\), an outage event will occur, i.e, the OP at coupled relays \(\mathcal{R}\) is expressed as follows:_  
\[ \begin{split} OP_{\mathcal{R}}^{\left(\tau_{1}^{\left(\text{eq} \right)}\right)}&=\text{Pr}\Bigg{\{}\underset{[A_{0}\times A_{1}]}{ \max}\Bigg{\{}\mathbf{R}_{\mathcal{R}-x_{N}}^{\left(\tau_{1}^{\left(\text{eq }\right)}\right)}}<R_{N}^{*}\Bigg{\}}\\ &+\text{Pr}\Bigg{\{}\underset{[A_{0}\times A_{1}]}{\max}\Bigg{\{} \mathbf{R}_{\mathcal{R}-x_{N}}^{\left(\tau_{1}^{\left(\text{eq}\right)} \right)}}\Bigg{\}}\geq R_{N^{*}}^{*}\underset{[A_{0}\times A_{1}]}{\max} \Bigg{\{}\mathbf{R}_{\mathcal{R}-x_{N-1}}^{\left(\tau_{1}^{\left(\text{eq} \right)}\right)}\Bigg{\}}<R_{N-1}^{*}\Bigg{\}}\\ &+\\ &\vdots\\ &+\text{Pr}\Bigg{\{}\underset{[A_{0}\times A_{1}]}{\max}\Bigg{\{} \mathbf{R}_{\mathcal{R}-x_{N}}^{\left(\tau_{1}^{\left(\text{eq}\right)} \right)}\Bigg{\}}\geq R_{N^{*}}^{*}\underset{[A_{0}\times A_{1}]}{\max} \Bigg{\{}\mathbf{R}_{\mathcal{R}-x_{N-1}}^{\left(\text{eq}\right)}\Bigg{\}} \geq R_{N-1}^{*}\text{,}\ldots,\\ &\underset{[A_{0}\times A_{1}]}{\max}\Bigg{\{}\mathbf{R}_{ \mathcal{R}-x_{2}}^{\left(\text{eq}\right)}\Bigg{\}}\geq R_{2^{*}}^{*} \underset{[A_{0}\times A_{1}]}{\max}\Bigg{\{}\mathbf{R}_{\mathcal{R}-x_{1}}^ {\left(\text{eq}\right)}\Bigg{\}}<R_{1}^{*}\end{split}  \tag{30}\]  
_We obtain the OP at coupled relays \(\mathcal{R}=\{R_{1},R_{2}\}\) in novel closed-form as follows:_\[ \begin{split} OP_{\mathcal{R}}^{\left(\Upsilon_{1}^{(\theta)}\right)}& =\sum_{i=N}^{1}\left(\prod_{a_{0}=0}^{A_{0}}\prod_{a_{1}=1}^{A_{1}} \left(1-\exp\left(-\frac{\gamma_{i}^{*}}{(1-\lambda)\beta_{i}\rho_{S}\sigma_{ \mathcal{R}}^{2}}\right)\right)\right.\\ &\times\underbrace{\prod_{k=N}^{i+1}\left(1-\prod_{a_{0}=1}^{A_{1 }}\prod_{a_{1}=1}^{A_{1}}\left(1-\exp\left(-\frac{\gamma_{k}^{*}}{(1-\lambda) \left(\alpha_{k}-\gamma_{k}^{*}\sum\limits_{j=1}^{k-1}\alpha_{j}\right)\rho_{S }\sigma_{\mathcal{R}}^{2}}\right)\right)\right)}_{\text{for $i<N$}},\end{split}