## 3 New Pair-Wise Detections  
Before deriving new pair-wise decoding, we first rearrange the exact ML decoding ([3], Equation (33)) in a vector form as follows:  
\[\mathbf{\hat{x}}^{k} =\min_{\mathbf{x}^{k}}\sum_{l=1}^{2}\left( \begin{array}{c}\frac {\sigma^{2}+\left(a_{l}^{k-1}\right)^{2}}{\left(a_{l}^{l}\right)}||\mathbf{y} _{l}^{k}||^{2}+\frac{\sigma^{2}+\left(a_{l}^{k}\right)^{2}}{q_{l}}||\mathbf{y }_{l}^{k-1}||^{2}\\ -\frac{2a_{l}^{k-1}}{q_{l}}\left[\left(\mathbf{y}_{l}^{k-1}\right)^{H}\mathbf{ V}_{l}^{k}\mathbf{y}_{l}^{k}\right]^{R}+2\sigma^{2}\ln\left(q_{l}\right) \end{array} \right) \tag{7}\] \[\approx \min_{\mathbf{x}^{k}}\left(\left\|\frac{a_{l}^{k-1}\mathbf{y}_{ l}^{k}}{\mu_{1}}-\frac{\mathbf{V}_{1}^{k}\mathbf{y}_{1}^{k-1}}{\mu_{1}} \right\|^{2}+\left\|\frac{a_{2}^{k-1}\mathbf{y}_{2}^{k}}{\mu_{2}}-\frac{ \mathbf{V}_{2}^{k}\mathbf{y}_{2}^{k-1}}{\mu_{2}}\right\|^{2}\right)\] (8) \[=\min_{\mathbf{x}^{k}}\left\|\left[ \begin{array}{c}a_{l}^{k-1} \mathbf{y}_{l,1}^{k}/\mu_{1}\\ a_{l}^{k-1}\left(y_{1,2}^{k}\right)^{*}/\mu_{1}\\ a_{2}^{k-1}y_{2,1}^{k}/\mu_{2}\\ a_{2}^{k-1}\left(y_{2,2}^{k}\right)^{*}/\mu_{2}\end{array} \right]-\left[ \begin{array}{c}\frac{\mathbf{Y}_{1}^{k-1}}{\mu_{1}}&\mathbf{0}_{2}\\ \mathbf{0}_{2}&\frac{\mathbf{Y}_{2}^{k-1}}{\mu_{2}}\end{array} \right]\left[ \begin{array}{c}r_{1,1}^{k}\\ r_{2,1}^{k}\\ r_{2,2}^{k}\end{array} \right]\right\|^{2} \tag{9}\]  
where \(q_{l}=\sigma^{2}+\left(a_{l}^{k-1}\right)^{2}+\left(a_{l}^{k}\right)^{2}\) with \(\sigma^{2}=1/\text{SNR}\), \(\mu_{l}=\sqrt{\left(a_{l}^{k}\right)^{2}+\left(a_{l}^{k-1}\right)^{2}}\) and \(\mathbf{Y}_{l}^{k-1}=\left[ \begin{array}{c}y_{l,1}^{k-1}&y_{l,2}^{k-1}\\ (y_{l,2}^{k-1})^{*}&-(y_{l,1}^{k-1})^{*}\end{array} \right]\). The approximation of (8) uses an assumption of high SNR, i.e., \(\sigma^{2}\approx 0\)[7, 8]. Furthermore, the equality of (9) comes from no change of magnitude of any conjugated signal. Here, let us define a \(4\times 4\) unitary matrix \(\mathbf{B}=\left[ \begin{array}{cc}\mathbf{Y}_{1}^{k-1}/\rho_{1}^{k-1}&\mathbf{0 }_{2}\\ \mathbf{0}_{2}&\mathbf{Y}_{2}^{k-1}/\rho_{2}^{k-1}\end{array} \right]\) with \(\rho_{l}^{k-1}=\|\mathbf{y}_{l}^{k-1}\|\) to satisfy \(\mathbf{B}^{H}\mathbf{B}=\mathbf{I}_{4}\). Then, by multiplying the left of (9) with \(\mathbf{B}^{H}\), the ML metric can be written as the summation of two equations, including each \(\mathbf{x}_{i}^{k}\) as follows:  
\[\left\|\mathbf{B}^{H}\left[ \begin{array}{cc}a_{1}^{k-1}\big{(} y_{1,1}^{k}/\mu_{1}\\ a_{1}^{k-1}\big{(}y_{1,2}^{k}\big{)}^{*}/\mu_{1}\\ a_{2}^{k-1}\big{(}y_{2,1}^{k}/\mu_{2}\\ a_{2}^{k-1}\big{(}y_{2,2}^{k}\big{)}^{*}/\mu_{2}\end{array} \right]-\mathbf{B}^ {H}\left[ \begin{array}{cc}\frac{\mathbf{x}_{1}^{k-1}}{\mu_{1}}&\mathbf{0}_{2 }\\ \mathbf{0}_{2}&\frac{\mathbf{x}_{2}^{k-1}}{\mu_{2}}\end{array} \right]\left[ \begin{array}{c}r_{1,1}^{k}\\ r_{2,1}^{k}\\ r_{2,2}^{k}\end{array} \right]\right\|^{2}\] \[=\left\|\left[ \begin{array}{ccc}1/\mu_{1}&0&0&0\\ 0&1/\mu_{1}&0&0\\ 0&0&1/\mu_{2}&0\\ 0&0&0&1/\mu_{2}\end{array} \right]\left(\left[ \begin{array}{ccc}z_{1,1}^{k} \\ z_{2,1}^{k}\\ z_{2,2}^{k}\end{array} \right]-\left[ \begin{array}{ccc}\rho_{1}^{k-1}&0&0&0\\ 0&\rho_{1}^{k-1}&0&0\\ 0&0&\rho_{2}^{k-1}&0\\ 0&0&0&\rho_{2}^{k-1}\end{array} \right]\left[ \begin{array}{c}r_{1,1}^{k}\\ r_{2,1}^{k}\\ r_{2,2}^{k}\end{array} \right]\right)\right\|^{2} \tag{10}\] \[=\left\|\left[ \begin{array}{ccc}1/\mu_{1}&0\\ 0&1/\mu_{2}\end{array} \right]\left(\left[ \begin{array}{ccc}z_{1,1}^{k}\\ z_{2,1}^{k}\\ z_{2,2}^{k}\end{array} \right]-\left[ \begin{array}{ccc}\rho_{1}^{k-1}&0\\ 0&\rho_{2}^{k-1}\end{array} \right]\left[ \begin{array}{c}r_{1,1}^{k}\\ r_{1,2}^{k}\\ r_{2,2}^{k}\end{array}