## 4 Simulations  
The numerical results are obtained via Monte Carlo simulations, averaging over \(10^{6}\) independent realizations of the channel matrices and additive noise. Perfect channel state information is assumed at each node to demonstrate the accuracy of the proposed analytical results. Binary phase-shift keying (BPSK) modulation is employed throughout the simulations to focus on the PEP behavior because it facilitates the generation of sign-inverted symbol pairs relevant to the analysis. Therefore, the number of transmit antennas at the source, relay, and destination nodes is fixed at one, whereas the number of receive antennas is varied across simulation scenarios \((N_{S},N_{R},N_{D})\). Each transmission block consists of 256 data symbols, and a rate-\(\frac{1}{2}\) turbo code is adopted for channel coding. The turbo code is based on a parallel concatenated structure using feedforward and feedback polynomials \((15,13)\) in the octal notation. In addition, it is assumed that a source-to-relay link is available at the destination node for detection. The block error rate (BLER) and PEP are evaluated to provide the practical system performance and validation of the analytical findings. Key abbreviations and system parameters used in the simulations are listed in Table 1.  
The BLER performance of various detection schemes is compared with that of the AML detection method in Figure 2. The BLER of direct communication (DC) is included to highlight the benefits of incorporating a relay. Conventional MRC cooperation exhibits poorer BLER performance than other cooperative methods such as AML and soft decode-and-forward (SDF). Moreover, the slope of the MRC curve significantly differs from those of AML, SDF, and JML, indicating its inability to achieve a diversity gain owing to a lack of robustness against relay detection errors. In contrast, the AML, SDF, and JML schemes exhibit similar BLER slopes, demonstrating their effectiveness in mitigating the error propagation inherent in DF relaying. As a summary, the performance and complexity are provided in Table 2, as shown below:  
Figure 3 presents the PEP of AML detection with one-bit ADCs under various transmit/receive antenna configurations when \(\mathbf{x}_{S}=-\bar{\mathbf{x}}_{S}\) and \(\mathbf{x}_{R}=\bar{\mathbf{x}}_{R}\). To validate Corollary 2, the symbol pairs are selected as \(\left[\mathbf{x}_{S}^{T},\bar{\mathbf{x}}_{S}^{T}\right]^{T}=\left[1,-1\right] ^{T}\) and \(\left[\mathbf{x}_{R}^{T},\bar{\mathbf{x}}_{R}^{T}\right]=\left[1,1\right]^{T}\), thereby satisfying \(\mathbf{x}_{S}=-\bar{\mathbf{x}}_{S}\) and \(\mathbf{x}_{R}=\bar{\mathbf{x}}_{R}\). This setup corresponds to a scenario in which decoding errors occur at the relay. Further, the asymptotic upper bound derived in Corollary 2 is plotted for comparison. The asymptotic behavior is analyzed in terms of the slope of the PEP curves, which reflects the diversity gain. The slope of the simulated PEP curves closely matches that of the analytical bound for all antenna configurations. Given that the slope of \begin{table}
\begin{tabular}{c c} \hline
**Abbreviations and System Parameters** & **Descriptions** \\ \hline DC & Direct Communication \\ MRC & Maximum Ratio Combining \\ SDF & Selective Decode and Forward \\ AML & Approximate Maximum Likelihood \\ JML & Join Maximum Likelihood \\ \(N_{S}\) & Number of Transmit Antennas at Source Node \\ \(N_{R}\) & Number of Transmit Antennas at Relay Node \\ \(N_{D}\) & Number of Transmit Antennas at Destination Node \\ \hline \end{tabular}
\end{table} 
Table 1: Key abbreviations and system parameters in simulation. \begin{table}
\begin{tabular}{c c c c c} \hline  & **MRC** & **SDF** & **JML** & **AML** \\ \hline Performance & Poor & Good & Good & Good \\ Complexity & Low & Low & High & Low \\ \hline \end{tabular}
\end{table} 
Table 2: Performance and complexity comparison for different detections. Figure 2: BLER of AML detection with different detection schemes when \((N_{S},N_{R},N_{D})=(1,8,8)\).