## 3 New Pair-Wise Detections
Moreover, for a real \(\mathbf{\Theta}\), the mean-based estimation can be applied to \(\hat{\mu}_{i,l}=\sqrt[]{\frac{1}{2}\left(|(r^{k}_{i,l})^{R}|^{2}+|(r^{k}_{i,l })^{I}|^{2}+1\right)+\left(a^{k-1}_{l}\right)^{2}}\) in (13) one more time for the separate decoding of \(\left(\mathbf{x}^{k}_{i}\right)^{R}\) and \(\left(\mathbf{x}^{k}_{i}\right)^{I}\), i.e., \(E\Big{\{}|(r^{k}_{i,l})^{I}|^{2}\Big{\}}=\frac{1}{2},^{\forall}l\) when decoding \(\left(\mathbf{x}^{k}_{i}\right)^{R}\) and also \(E\Big{\{}|(r^{k}_{i,l})^{R}|^{2}\Big{\}}=\frac{1}{2},^{\forall}l\) when decoding \(\left(\mathbf{x}^{k}_{i}\right)^{I}\), resulting in:  
\[\left(\mathbf{\hat{x}}^{k}_{i}\right)^{R}=\min_{\left(\mathbf{x}^{k}_{i} \right)^{R}}\left\|\mathbf{\hat{A}}^{R}_{i}\left[\left(\mathbf{z}^{k}_{i} \right)^{R}-\rho^{k-1}\mathbf{\Theta}\left(\mathbf{x}^{k}_{i}\right)^{R} \right]\right\|^{2},i=1,2 \tag{14}\] \[\left(\mathbf{\hat{x}}^{k}_{i}\right)^{I}=\min_{\left(\mathbf{x}^ {k}_{i}\right)^{I}}\left\|\mathbf{\hat{A}}^{I}_{i}\left[\left(\mathbf{z}^{k}_ {i}\right)^{I}-\rho^{k-1}\mathbf{\Theta}\left(\mathbf{x}^{k}_{i}\right)^{I} \right]\right\|^{2},i=1,2 \tag{15}\]  
where \(\mathbf{\hat{A}}^{R}_{i}=\text{diag}\Big{(}1/\hat{\mu}^{R}_{i,1},1/\hat{\mu}^ {R}_{i,2}\Big{)}\) with \(\hat{\mu}^{R}_{i,l}=\sqrt{\frac{1}{2}\left(|(r^{k}_{i,l})^{R}|^{2}+\frac{3}{2 }\right)+(a^{k-1}_{l})^{2}}\) and \(\mathbf{\hat{A}}^{I}_{i}=\text{diag}\Big{(}1/\hat{\mu}^{I}_{i,1},1/\hat{\mu}^ {I}_{i,2}\Big{)}\) with \(\hat{\mu}^{I}_{i,l}=\sqrt{\frac{1}{2}\left(|(r^{k}_{i,l})^{I}|^{2}+\frac{3}{2 }\right)+(a^{k-1}_{l})^{2}}\). Obviously, the new pair-wise detections of (13)-(15) are not equal to the ML decoding of (12), but exhibit performances within only about 0.5 dB compared to the ML receiver for all simulation cases, which will be shown in the following simulation results. Notice that with some manipulations, the conventional detection of (4) can be shown to be equal to the new one of (13) setting \(\hat{\mu}_{i,l}\) to be \(\sqrt{a^{k-1}_{l}}\), i.e., \(\mathbf{\hat{A}}_{i}=\text{diag}\bigg{(}\sqrt{a^{k-1}_{1}},\sqrt{a^{k-1}_{2 }}\bigg{)}\), given as  
\[\min_{\mathbf{x}^{k}_{i}}\left\|\mathbf{\hat{A}}_{i}\left(\mathbf{ z}^{k}_{i}-\rho^{k-1}\mathbf{\Theta}\mathbf{x}^{k}_{i}\right)\right\|^{2}\] \[=\min_{\mathbf{r}^{k}_{i}}\left(\left\|\mathbf{\hat{A}}_{i}\rho^{ k-1}\mathbf{r}^{k}_{i}\right\|^{2}-2\Big{\{}\Big{\langle}\mathbf{\hat{A}}_{i} \mathbf{z}^{k}_{i},\mathbf{\hat{A}}_{i}\rho^{k-1}\mathbf{r}^{k}_{i}\Big{\rangle} \Big{\}}^{R}\right) \tag{16}\] \[=\min_{\mathbf{r}^{k}_{i}}\sum_{l=1}^{2}\left[\frac{(\rho^{k-1}_{ i})^{2}}{a^{k-1}_{l}}|r^{k}_{i,l}|^{2}-2\Big{\{}\Big{\langle}\mathbf{y}^{k-1}_{l,l },\mathbf{y}^{k}_{l,1}\Big{\rangle}r^{k}_{i,l}\Big{\}}^{R}\right] \tag{17}\]  
where the equality of (16) is from that \(\mathbf{\hat{A}}_{i}\) is absolutely irrelevant to the current input signals \(\mathbf{x}^{k}_{i}\) or \(\mathbf{r}^{k}_{i}\). Following the same derivations, (5) and (6) can also be proven to be equal to the new ones of (14) and (15), respectively, setting \(\hat{\mu}^{R(I)}_{i,l}\) to be \(\sqrt{a^{k-1}_{l}}\). This means that the conventional detections can be seen as the ones to cut off the dependency between two \(\mathbf{x}^{k}_{i}\) or four \(\left(\mathbf{x}^{k}_{i}\right)^{R(I)}\) from each other by estimating \(\mu_{l}\) to be simply \(\sqrt{a^{k-1}_{l}}\). Moreover, since the proposed methods are the same symbol by symbol decoding as the conventional methods except for the \(m_{l}\) computation, performance improvement can be expected without increasing the same decoding complexity. Defining \(\varepsilon(\hat{\mu}_{l})=E\Big{\{}(\mu_{l}-\hat{\mu}_{l})^{2}\Big{\}}\), Table 1 compares the accuracies of \(\hat{\mu}_{l}\)s used in the conventional and new detections for the QO-DSTMs [7; 8] with perfect knowledge of \(a^{k-1}_{l}\). From Table 1, the estimation errors of \(\hat{\mu}_{l}\) in the new detections are shown to be greatly lower than those of the convention detections.