## 3 Performance Analysis
### Asymptotic Behavior  
The asymptotic upper bound of the PEP is analyzed to gain deeper insight into how PEP influences the diversity gain of AML detection with one-bit ADCs. Therefore, the asymptotic behavior of the array gain function \(f(\gamma_{ab},\mathbf{x}_{a},\tilde{\mathbf{x}}_{a})\) in Theorem 1 is examined. Subsequently, the diversity gain of PEP is investigated for various combinations of pairwise symbol vectors. The array gain function \(f(\gamma_{ab},\mathbf{x}_{a},\tilde{\mathbf{x}}_{a})\), which governs the behavior of the PEP, exhibits distinct asymptotic characteristics depending on whether \(\mathbf{x}_{a}=-\tilde{\mathbf{x}}_{a}\) or \(\mathbf{x}_{a}\
eq-\tilde{\mathbf{x}}_{a}\). The asymptotic behavior of this array gain function is summarized in the following corollary. **Corollary 1**. : _Let \(\mathbf{x}_{a}\) and \(\tilde{\mathbf{x}}_{a}\) represent two distinct symbol vectors. Subsequently, the asymptotic behavior of the array gain function \(f(\gamma_{ab},\mathbf{x}_{a},\tilde{\mathbf{x}}_{a})\) as \(\gamma_{ab}\rightarrow\infty\) is characterized as_  
* _If_ \(\mathbf{x}_{a}=-\tilde{\mathbf{x}}_{a}\)_, then_ \[f(\gamma_{ab},\mathbf{x}_{a},\tilde{\mathbf{x}}_{a})\rightarrow\frac{2\sqrt{2 N_{a}}}{\|\tilde{\mathbf{x}}_{a}\|}\gamma_{ab}^{-\frac{1}{2}}.\] (29)
* _If_ \(\mathbf{x}_{a}\
eq-\tilde{\mathbf{x}}_{a}\)_, then_ \[f(\gamma_{ab},\mathbf{x}_{a},\tilde{\mathbf{x}}_{a})\rightarrow\frac{2}{\pi} \arctan\biggl{(}\frac{\|\tilde{\mathbf{x}}_{a}+\tilde{\mathbf{x}}_{a}\|}{\| \tilde{\mathbf{x}}_{a}-\tilde{\mathbf{x}}_{a}\|}\biggr{)}.\] (30)  
The corollary states that the asymptotic behavior of the array gain function remains constant when \(\mathbf{x}_{a}\
eq-\tilde{\mathbf{x}}_{a}\). In contrast, the function decays with increasing \(\gamma_{ab}\) at a rate proportional to the order of \(\frac{1}{2}\) when \(\mathbf{x}_{a}=-\tilde{\mathbf{x}}_{a}\). This behavior arises because the two symbol vectors (i.e., \(\mathbf{x}_{a}\) and \(\tilde{\mathbf{x}}_{a}\)) in Rayleigh fading channels become indistinguishable at the receiver when they are not sign-reversed pairs. Using Corollary 1 and Theorem 1, the asymptotic behavior of PEP for AML using one-bit ADCs can be analyzed as  
**Corollary 2**. : _Suppose that \(\mathbf{x}_{S}=-\tilde{\mathbf{x}}_{S}\) and \(\mathbf{x}_{R}=\tilde{\mathbf{x}}_{R}\). In this case, full diversity gain can theoretically be achieved if \(\tilde{\mathbf{x}}_{S}=-\hat{\mathbf{x}}_{R}\). However, under these conditions, it follows that \(\mathbf{x}_{S}=\mathbf{x}_{R}\), which contradicts the assumption \(\mathbf{x}_{S}=-\mathbf{x}_{R}\). Thus, the PEP behaves as_  
\[\mathcal{P}\{\mathbf{x}_{S}\rightarrow\tilde{\mathbf{x}}_{S}\} \to A_{1}\gamma_{SD}^{-N_{D}}, \tag{31}\]  
_where \(A_{1}=\left(\frac{2}{\pi}\arctan\left(\frac{\|\mathbf{x}_{R}+\tilde{\mathbf{x }}_{R}\|}{\|\mathbf{x}_{R}-\tilde{\mathbf{x}}_{R}\|}\right)\right)^{2N_{D}} \times\left(\frac{2}{\pi}\arctan\left(\frac{\|\mathbf{x}_{S}+\tilde{\mathbf{x }}_{R}\|}{\|\tilde{\mathbf{x}}_{S}-\tilde{\mathbf{x}}_{R}\|}\right)\right)^{2 N_{R}}\times\frac{2\sqrt{2N_{S}}}{\|\mathbf{x}_{S}\|}\)._  
In this corollary, the full diversity gain of \(N_{D}+N_{R}\) is not achieved in cooperative DF MIMO systems with one-bit ADCs. This limitation arises because the full diversity gain can only be obtained when the involved symbol vectors in PEP satisfy specific sign-inversion (i.e., \"minus\") conditions. However, these conditions are not satisfied in the presence of source-to-relay detection errors, fundamentally limiting the achievable diversity in these scenarios. **Corollary 3**. : _Under the conditions \(\mathbf{x}_{S}=-\tilde{\mathbf{x}}_{S}\), \(\mathbf{x}_{S}=\mathbf{x}_{R}\), and \(\tilde{\mathbf{x}}_{S}=\tilde{\mathbf{x}}_{R}\), full diversity gain can be achieved because \(\mathbf{x}_{R}=-\tilde{\mathbf{x}}_{R}\).