## 4 Simulation Results  
All simulations are done based on an independent data frame consisting of 128 blocks. Each block has four QAM symbols. For propagation channel models, it is assumed that the four MIMO channel gains have independent and identical Rayleigh distributions, and also are constant during each frame with independent distribution. Furthermore, in all decodings, we use \(a_{1}^{k-1}\) and \(a_{2}^{k-1}\) calculated from previously detected symbols.  
Figure 2 shows average bit error rates (BERs) of the new and conventional complex pair-wise detections for the [PERSON]'s QO-DSTM [7] using \(\Theta_{C}\). For the comparison of performances, the ML results of (12) are also included. Firstly, the new pair-wise detection is shown to achieve more improved performance compared to the conventional one for all simulation cases. The performance gain is much larger especially as the modulation order increases. Specifically, for 16, 64 and 256 QAMs, the respective SNR gains at BER = \(10^{-4}\) are about 0.8, 1.0, 1.2 dBs. This is mainly because the new detection is performed based on more accurately estimated \(\hat{\mu}_{I}\) compared to the conventional one, especially for a higher modulation order, as shown in Table 1. Furthermore, we note that the proposed detection shows an SNR loss of less than only 0.2 dB compared to the ML decoding for all cases.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mod.**} & \multicolumn{2}{c}{\(\Theta_{C}\)[7]} & \multicolumn{2}{c}{\(\Theta_{R}\)[8]} \\ \cline{2-5}  & \(\epsilon\left(\sqrt{a_{I}^{k-1}}\right)\), (4) & \(\epsilon\left(\hat{\mu}_{I}^{i}\right)\) (13) & \(\epsilon\left(\sqrt{a_{I}^{k-1}}\right)\) (5), (6) & \(\epsilon\left(\hat{\mu}_{I,R}^{i}\right)\) (14), (15) \\ \hline
16-QAM & 7.462 & 0.195 & 2.679 & 0.359 \\ \hline
64-QAM & 44.593 & 0.866 & 6.558 & 0.754 \\ \hline
256-QAM & 263.204 & 2.898 & 16.042 & 1.381 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparisons of \(\epsilon(\hat{\mu}_{I})\) for the conventional and new detections.  
Figure 2: Average BERs for the QO-DSTM [7] with \(\Theta_{C}\).  
Figure 3 shows the average BERs of the new and conventional real pair-wise detections for the [PERSON]'s QO-DSTM [8] using \(\Theta_{R}\). Here, the ML results are also included for comparing performances. First the performance trends in Figure 3 are almost the same as those of Figure 2 with the identical reasons as in Table 1. Specifically, the respective SNR gains for 16, 64 and 256 QAMs at BER = \(10^{-4}\) are about 0.3, 0.6, 0.9 dBs. Furthermore, the proposed detection shows an SNR loss of less than 0.5 dB compared to the ML decoding for all cases.  
Comparing the simulation results in Figures 2 and 3, we can see that the performance gain comparing to the conventional system is more significant when using the complex precoder in Figure 2. This is mainly due to the difference in estimation errors between the proposed and conventional methods listed in Table 1. Namely, when using the complex precoder, the gap of estimation errors for \(\mu_{l}\) between these two methods are larger than those when using the real precoder for all modulation cases, as shown in Table 1.