## 2 Conventional QO-DSTMs and Differential Pair-Wise Detections  
The conventional QO-DSTMs [7, 8] with four transmit antennas can be constructed by serially concatenating the coherent QO code with differential encoding, as shown in Figure 1.  
The transmitter of Figure 1 first encodes a \(k\)th input vector \(\mathbf{x}^{k}=[\mathbf{x}_{1}^{k},\mathbf{x}_{2}^{k}]^{T}\) of length 4 through the conventional coherent QO encoder [2, 3] consisting of precoder \(\mathbf{\Theta}\) and Alamouti encoder [10], resulting in:  
\[\mathbf{V}_{1}^{k} =\frac{1}{\sqrt{2}}\Bigg{[} \begin{array}{cc}r_{1,1}^{k}&r_{2,1} ^{k}\\ -\left(r_{2,1}^{k}\right)^{*}&\left(r_{1,1}^{k}\right)^{*}\\ \end{array} \Bigg{]}, \tag{1}\] \[\mathbf{V}_{2}^{k} =\frac{1}{\sqrt{2}}\Bigg{[} \begin{array}{cc}r_{1,2}^{k}&r_{2,2} ^{k}\\ -\left(r_{2,2}^{k}\right)^{*}&\left(r_{1,2}^{k}\right)^{*}\\ \end{array} \Bigg{]} \tag{2}\]  
where \(\mathbf{r}_{i}^{k}=\left[r_{i,1}^{k}r_{i,2}^{k}\right]^{T}=\mathbf{\Theta} \mathbf{x}_{i}^{k}\) with an unitary precoder \(\mathbf{\Theta}\). Notice that the precoder \(\mathbf{\Theta}\) is chosen so that the precoded vector \(\mathbf{r}_{i}^{k}\) has different values for any distinct \(\mathbf{x}_{i}^{k}\)[2, 3], and thus is mapped to \(\mathbf{x}_{i}^{k}\) in a one-to-one relationship. Then, each \(\mathbf{V}_{l}^{k}\) is differentially modulated with an iterative fashion as follows:  
\[\mathbf{S}_{l}^{0}=\mathbf{I}_{2},\ \ \mathbf{S}_{l}^{k}=\frac{\mathbf{V}_{l}^{k} }{a_{l}^{k-1}}\mathbf{S}_{l}^{k-1},k\geq 1 \tag{3}\]  
where \(a_{l}^{k-1}\!=\!\sqrt{\frac{1}{2}\left(\left|r_{1,l}^{k-1}\right|^{2}\!+\! \left|r_{2,l}^{k-1}\right|^{2}\right)}\) is a power-normalization factor to satisfy \(\mathbf{S}_{l}^{k-1}\!\left(\mathbf{S}_{l}^{k-1}\right)^{H}\)\(=\left(a_{l}^{k-1}\right)^{2}\mathbf{I}_{2}\) with an \(2\times 2\) identity matrix \(\mathbf{I}_{2}\). \(\left(\bullet\right)^{H}\) denotes a Hermitian operator. The differentially modulated \(\mathbf{S}_{l}^{k}\) are finally transmitted through four transmit antennas in a time-multiplexed form as shown in Figure 1, and finally arrive at a receiver through \(4\times 1\) independent and identical MIMO fading channels. Figure 1: Block diagram of conventional QO-DSTM [7, 8] with four transmit antennas. Assuming a receive vector \(\mathbf{y}_{l}^{k}=\left[y_{l,1}^{k}y_{l,2}^{k}\right]^{T}\) corresponding to \(\mathbf{S}_{l}^{k}\), [PERSON]'s QO-DSTM [7] presents a near-ML complex pair-wise detection of \(\mathbf{x}_{l}^{k}\), where \(i=1,2\) for an optimal complex precoder \(\mathbf{\Theta}_{\mathrm{C}}=\frac{1}{\sqrt{2}}\left[ \begin{array}{cc}1&e^{ i\pi/4}\\ 1&-e^{j\pi/4}\end{array} \right]\), given as  
\[\mathbf{\hat{x}}_{l}^{k}=\min_{\mathbf{r}_{i}^{k}}\sum_{l=1}^{2}\left[\frac{ \left|\left|\mathbf{y}_{l}^{k-1}\right|\right|^{2}}{2a_{l}^{k-1}}|r_{i,l}^{k}|^ {2}-\left\{\left\langle\mathbf{y}_{l,i}^{k-1},\mathbf{y}_{l,1}^{k}\right\rangle \right\}^{k}_{i,l}\right]^{R}\right] \tag{4}\]  
where \(\mathbf{y}_{l,1}^{k-1}=[(y_{l,1}^{k-1})^{*}y_{l,2}^{k-1}]^{T}\) and \(\mathbf{y}_{l,2}^{k-1}=[(y_{l,2}^{k-1})^{*}-y_{l,1}^{k-1}]^{T}\). \((\bullet)^{T}\) is a transpose operator and \(\left\langle a,b\right\rangle=a^{H}b\). Furthermore, in order to further reduce the decoding complexity of (4), [PERSON]'s QO-DSTM [8] uses a real precoder \(\mathbf{\Theta}_{R}=\left[ \begin{array}{cc}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{array}