## 3 Performance Analysis
### PEP Analysis
Applying this bound to (19) with \(s=1/2\), the PEP in (19) can be further upper bounded as  
\[\mathbb{P}\{\mathbf{x}_{S}\rightarrow\mathbf{\tilde{x}}_{S}| \mathbf{H}_{SRD}\} \leq\sum_{\mathbf{q}_{SRD}\in\{-1,1\}^{4N_{D}+2N_{R}}}\mathbb{P} \{\mathbf{q}_{SRD}|\mathbf{H}_{SRD},\mathbf{x}_{S}\}\left(\frac{\mathbb{P}\{ \mathbf{q}_{SRD}|\mathbf{H}_{SRD},\mathbf{\tilde{x}}_{S}\}}{\mathbb{P}\{ \mathbf{q}_{RD}|\mathbf{H}_{SRD},\mathbf{x}_{S}\}}\right)^{\frac{1}{2}}\] \[=\sum_{\mathbf{q}_{SRD}\in\{-1,1\}^{4N_{D}+2N_{R}}}\sqrt{\mathbb{ P}\{\mathbf{q}_{SRD}|\mathbf{H}_{SRD},\mathbf{x}_{S}\}\mathbb{P}\{\mathbf{q}_{ SRD}|\mathbf{H}_{SRD},\mathbf{\tilde{x}}_{S}\}}. \tag{20}\]  
The likelihood probability \(\mathbb{P}\{\mathbf{q}_{SRD}|\mathbf{H}_{SRD},\mathbf{x}_{S}\}\) of the AML is further approximated using the max-log and PEP approximations, following the approach in (15), which yields  
\[\mathbb{P}\{\mathbf{x}_{S}\rightarrow\mathbf{\tilde{x}}_{S}| \mathbf{H}_{SRD}\}\] \[\leq\sum_{\mathbf{q}_{SD}\in\{-1,1\}^{2N_{D}}}\sum_{\mathbf{q}_ {RD}\in\{-1,1\}^{2N_{D}}}\sqrt{\mathbb{P}\{\mathbf{q}_{SD}|\mathbf{H}_{SD}, \mathbf{x}_{S}\}\mathbb{P}\{\mathbf{x}_{S}\rightarrow\mathbf{x}_{R}\}\mathbb{P }\{\mathbf{q}_{RD}|\mathbf{H}_{RD},\mathbf{x}_{R}\}}\] \[\times\sqrt{\mathbb{P}\{\mathbf{q}_{SD}|\mathbf{H}_{SD},\mathbf{ \tilde{x}}_{S}\}\mathbb{P}\{\mathbf{\tilde{x}}_{S}\rightarrow\mathbf{\tilde{x} }_{R}\}\mathbb{P}\{\mathbf{q}_{RD}|\mathbf{H}_{RD},\mathbf{\tilde{x}}_{R}\}}. \tag{21}\]Given that \(\sum_{i=1}^{n}\sqrt{a_{i}b_{i}c_{i}}\leq\sum_{i=1}^{n}\sqrt{a_{i}}\sum_{i=1}^{n} \sqrt{b_{i}}\sum_{i=1}^{n}\sqrt{c_{i}}\) when \(0\leq a_{i},b_{i},c_{i}\leq 1,\forall i\), the inequality in (21) can be upper-bounded accordingly as  
\[\mathbb{P}\{\mathbf{x}_{S}\rightarrow\tilde{\mathbf{x}}_{S}| \mathbf{H}_{SRD}\}\leq\sum_{\mathbf{q}_{SD}\in\{-1,1\}^{2N_{D}}}\sqrt{\mathbb{ P}\{\mathbf{q}_{SD}|\mathbf{H}_{SD},\mathbf{x}_{S}\}\mathbb{P}\{\mathbf{q}_{SD}| \mathbf{H}_{SD},\tilde{\mathbf{x}}_{S}\}}\] \[\times\sum_{\mathbf{q}_{RD}\in\{-1,1\}^{2N_{D}}}\sqrt{\mathbb{P} \{\mathbf{q}_{RD}|\mathbf{H}_{RD},\mathbf{x}_{R}\}\mathbb{P}\{\mathbf{q}_{RD}| \mathbf{H}_{RD},\tilde{\mathbf{x}}_{R}\}}\times\sqrt{\mathbb{P}\{\mathbf{x}_{S }\rightarrow\mathbf{x}_{R}\}\mathbb{P}\{\tilde{\mathbf{x}}_{S}\rightarrow \tilde{\mathbf{x}}_{R}\}}. \tag{22}\]  
Subsequently, the PEP is obtained by taking the expectation over channel realizations, which results in  
\[\mathbb{P}\{\mathbf{x}_{S}\rightarrow\mathbf{\tilde{x}}_{S}| \mathbf{H}_{SRD}\} =\sum_{\mathbf{q}_{SRD}\in\{-1,1\}^{4N_{D}+2N_{R}}}\mathbb{P}\{ \mathbf{q}_{SRD}|\mathbf{H}_{SRD},\mathbf{x}_{S}\}\]0 \[\mathbb{P}\{\mathbf{x}_{S}\rightarrow\mathbf{\tilde{x}}_{S}| \mathbf{H}_{SRD}\} =\sum_{\mathbf{q}_{SRD}\in\{-1,1\}^{4N_{D}+2N_{R}}}\mathbb{P}\{ \mathbf{q}_{SRD}|\mathbf{H}_{SRD},\mathbf{x}_{S}\}\]1 \[\mathbb{P}\{\mathbf{x}_{S}\rightarrow\mathbf{\tilde{x}}_{S}| \mathbf{H}_{SRD}\} =\sum_{\mathbf{q}_{SRD}\in\{-1,1\}^{4N_{D}+2N_{R}}}\mathbb{P}\{ \mathbf{q}_{SRD}|\mathbf{H}_{SRD},\mathbf{x}_{S}\}\]2  
where the detections of \(A1\) and \(A2\) at the destination and relay are independent. Terms \(A1\) and \(A2\) in (23) can be derived by using the following lemma  
\[\mathbb{P}\{\mathbf{x}_{S}\rightarrow\mathbf{\tilde{x}}_{S}| \mathbf{H}_{SRD}\} =\sum_{\mathbf{q}_{SRD}\in\{-1,1\}^{4N_{D}+2N_{R}}}\mathbb{P}\{ \mathbf{q}_{SRD}|\mathbf{H}_{SRD},\mathbf{x}_{S}\}\]3  
where \(f(\gamma_{ab},\mathbf{x}_{a},\tilde{\mathbf{x}}_{a})=\Big{(}1-\frac{2}{\pi} \arctan\Big{(}\frac{\|\mathbf{x}_{a}+\tilde{\mathbf{x}}_{a}\|}{\|\mathbf{x}_{ a}-\tilde{\mathbf{x}}_{a}\|}\Big{)}\Big{)}\Big{(}\sqrt{\frac{2N_{a}}{2N_{a}+\gamma_{ab}\| \mathbf{x}_{a}\|^{2}}}+\sqrt{\frac{2N_{a}}{2N_{a}+\gamma_{ab}\|\tilde{\mathbf{ x}}_{a}\|^{2}}}\Big{)}\)  
Proof. : See Appendix A. In addition, term \(B\) in (23) can be obtained using the following lemma.