## 3 New Pair-Wise Detections
\right]\right)\right\|^{2}\] (11) \[=\left\|\mathbf{\Lambda}\Big{(}\mathbf{z}_{1}^{k}-\rho^{k-1} \mathbf{\Theta x}_{1}^{k}\Big{)}\right\|^{2}+\left\|\mathbf{\Lambda}\Big{(} \mathbf{z}_{2}^{k}-\rho^{k-1}\mathbf{\Theta x}_{2}^{k}\Big{)}\right\|^{2} \tag{12}\]  
where diagonal matrix \(\mathbf{\Lambda}\mathbf{=diag(1/\mu_{y}1/\mu_{2})}\), \(\mathbf{z}_{i}^{k}\mathbf{=}\left[z_{i,1}^{k}z_{i,2}^{k}\right]^{T}\) with \(z_{i,l}^{k}\mathbf{=}\left\{a_{l}^{k-1}\Big{(}\mathbf{y}_{l,l}^{k-1}\mathbf{ y}_{l,1}^{k}\Big{)}\right\}\)/\(\rho_{l}^{k-1}\) and \(\rho^{k-1}\mathbf{=diag}\Big{(}\rho_{1}^{k-1},\rho_{2}^{k-1}\Big{)}\). The equality of (10) uses an energy conserving property of the unitary matrix \(\mathbf{B}^{H}\). From (12), we can see that if all \(\mu_{l}\) in \(\mathbf{\Lambda}\) are independent of \(\mathbf{x}^{k}\), each \(\mathbf{x}_{i}^{k}\) can be separately decoded by minimizing \(\left\|\mathbf{\Lambda}\Big{(}\mathbf{z}_{i}^{k}-\rho^{k-1}\mathbf{\Theta x}_ {i}^{k}\Big{)}\right\|^{2}\). Unfortunately, the values \(\mu_{l}=\sqrt{\frac{1}{2}\left(\left|r_{1,l}^{k}\right|^{2}+\left|r_{2,l}^{k} \right|^{2}\right)+\left(a_{l}^{k-1}\right)^{2}}\) in \(\mathbf{\Lambda}\) are variant with respect to \(r_{i,l}^{k}\) or all the input QAM signals \(\mathbf{x}^{k}\), implying the unfeasibility of pair-wise ML detection at the receiver. Specifically, in order to decode \(\mathbf{x}_{1}^{k}\), we need to know the two exact values of \(\left|r_{2,l}^{k}\right|^{2}\) and \(\left|r_{2,2}^{k}\right|^{2}\) in \(\mu_{l}\) containing the other signals \(\mathbf{x}_{2}^{k}\). Conversely, for decoding \(\mathbf{x}_{2}^{k}\), the two values \(\left|r_{1,1}^{k}\right|^{2}\) and \(\left|r_{1,2}^{k}\right|^{2}\) corresponding to the other \(\mathbf{x}_{1}^{k}\) need to be known. Considering that \(\mu_{l}\) are normalizing terms including \(a_{l}^{k}\) and \(a_{l}^{k-1}\), we conclude that the dependency between two symbol pairs \(\mathbf{x}_{1}^{k}\) and \(\mathbf{x}_{2}^{k}\) indeed originates from the power-normalization with the constraint of total transmit power in (3), thus implying that this dependency will be not avoidable for the general QAM. Hence, in order to make separate detections of \(\mathbf{x}_{i}^{k}\) possible, it should be done to break up the dependency relationship between \(\mathbf{x}_{1}^{k}\) and \(\mathbf{x}_{2}^{k}\) in \(\mu_{l}\). For this goal, when decoding \(\mathbf{x}_{1}^{k}\), we simply replace or estimate the values of the other \(\left|r_{2,1}^{k}\right|^{2}\) by their mean values, i.e., \(E\Big{(}\left|r_{2,1}^{k}\right|^{2}\Big{)}=E\Big{(}\left|r_{2,2}^{k}\right| ^{2}\Big{)}=1\), and also when decoding \(\mathbf{x}_{2}^{k}\), \(E\Big{(}\left|r_{1,1}^{k}\right|^{2}\Big{)}=E\Big{(}\left|r_{1,2}^{k}\right|^{2} \Big{)}=1\). In this way, the decoding of \(\mathbf{x}_{1}^{k}\) can be performed only by using \(r_{1,l}^{k}\) elements containing \(\mathbf{x}_{1}^{k}\) and the same can be done for decoding \(\mathbf{x}_{2}^{k}\) only by using \(r_{2,l}^{k}\) elements. Hence, this simple mean-based estimation produces a new complex pair-wise decoding of \(\mathbf{x}_{i}^{k}\), given as  
\[\hat{\mathbf{x}}_{i}^{k}=\min_{\mathbf{x}_{i}^{k}}\left\|\hat{\mathbf{\Lambda}}_{i} \Big{(}\mathbf{z}_{i}^{k}-\rho^{k-1}\mathbf{\Theta x}_{i}^{k}\Big{)}\right\|^{2},i=1,2 \tag{13}\]where \(\mathbf{\hat{A}}_{i}=\text{diag}(1/\hat{\mu}_{i,1},1/\hat{\mu}_{i,2})\) with \(\hat{\mu}_{i,l}=\sqrt[]{\frac{1}{2}\left(|r^{k}_{i,l}|^{2}+1\right)+\left(a^{k-1 }_{l}\right)^{2}}\).