## 3 LR-Aided MIMO Detector Design
### LLL-Aided Detectors  
We first present the LLL-aided detectors, where the LLL LR algorithm is used to obtain the \(\mathbf{T}\) matrix [11], which will result in a new channel matrix \(\mathbf{\widetilde{H}}=\mathbf{H}\mathbf{T}\). Then, the new model in (4) is used in the detection process, where we first explain how the LLL is combined with the linear ZF detector and then we explain how the LLL can be combined with the K-Best detector.  
In the LLL-aided ZF detector, the \(\mathbf{T}\) matrix is evaluated and then the new channel matrix \(\mathbf{\widetilde{H}}\) is computed. Then, the equalisation matrix \(\mathbf{W}=\mathbf{\widetilde{H}}^{-1}\) is obtained. Afterwards, \(\mathbf{W}\) is multiplied by the received signal using the transformed MIMO model presented in (4). The output of the LLL-aided ZF is \(\mathbf{\widehat{z}}=\mathbf{z}+\mathbf{\widetilde{H}}^{-1}\mathbf{n}\) and since \(\mathbf{z}=(\mathbf{T}^{-1}\mathbf{x})\in\mathcal{Z}\), the original constellation symbols \(\mathbf{x}\) can be recovered by multiplying with \(\mathbf{T}\) after shifting, scaling and rounding as follows [10]:\[\widehat{\mathbf{x}}=2\mathbf{T}\lceil(\widehat{\mathbf{z}}-\mathbf{T}^{-1}\mathbf{1 }_{2N\times 1})/2\rfloor-\mathbf{1}_{2N\times 1}, \tag{5}\]  
where \(\mathbf{1}_{2N\times 1}\) is a \((2N\times 1)\) vector of all 1 entries.  
On the other hand, in order to attain near-ML performance, ref. [18] showed that the performance of the K-Best detector can be improved by performing a LR preprocessing of the channel matrix prior to the K-Best detection. After obtaining the \(\mathbf{T}\) matrix using the LLL algorithm, the new channel matrix \(\widetilde{\mathbf{H}}=\mathbf{H}\mathbf{T}\) can be attained. Then, the K-Best detection is applied to the new model in (4), where the QR decomposition is performed for the new channel matrix as \(\widetilde{\mathbf{H}}=\widetilde{\mathbf{Q}}\widetilde{\mathbf{R}}\), with \(\widetilde{\mathbf{R}}\) being an upper triangular matrix and \(\widetilde{\mathbf{Q}}\) a unitary matrix (A unitary matrix has the following property \(\widetilde{\mathbf{Q}}^{H}=\widetilde{\mathbf{Q}}^{-1}\), where \({}^{H}\) represented the Hermitian transpose).  
After the QR decomposition, (4) can be reshaped from \(\mathbf{y}=\widetilde{\mathbf{H}}\mathbf{z}+\mathbf{n}\) to \(\mathbf{y}=\widetilde{\mathbf{Q}}\widetilde{\mathbf{R}}\mathbf{z}+\mathbf{n}\). Then, the ML detection problem for \(\widehat{\mathbf{z}}\) can be expressed as:  
\[\widehat{\mathbf{z}}=arg\min_{\mathbf{z}\in\mathcal{Z}^{2N}}\parallel\tilde{ \mathbf{y}}-\widetilde{\mathbf{R}}\mathbf{z}\parallel^{2}, \tag{6}\]  
where \(\tilde{\mathbf{y}}=\widetilde{\mathbf{Q}}^{H}\widetilde{\mathbf{y}}\) and \(\mathbf{z}=(\mathbf{T}^{-1}\mathbf{x})\). Note that \(\widetilde{\mathbf{y}}=(\mathbf{y}-\mathbf{H}\mathbf{1}_{2N\times 1})/2\), which is a shifted and scaled version of \(\mathbf{y}\)[27]. From (6), the K-Best detector requires performing a tree search to detect \(\mathbf{z}\) and then recovers the original symbols by multiplying with \(\mathbf{T}\) after rescaling and re-shifting \(\widehat{\mathbf{z}}\) as \(\widehat{\mathbf{x}}=2\mathbf{T}(\widehat{\mathbf{z}})+\mathbf{1}_{2N\times 1}\).  
A further improvement to the LLL-aided K-Best detector has been proposed in [27] by combining with the Minimum Mean Square Error (MMSE) regularization, where the channel matrix \(\mathbf{H}\) is replaced with the extended \(\overline{\mathbf{H}}\) matrix and the received signal vector \(\mathbf{y}\) is replaced with the extended \(\overline{\mathbf{y}}\) as follows:  
\[\overline{\mathbf{H}}=\left[\sqrt{\frac{\sigma^{2}}{p}}\mathbf{I}_{2N}\right],\ \overline{\mathbf{y}}=\left[\begin{matrix}\mathbf{y}\\ \mathbf{0}_{2N\times 1}\end{matrix}\right], \tag{7}\]  
and then the same K-Best detection process described above is applied. This is referred to as LLL-aided MMSE K-Best detector and has an improved performance compared to the LLL-aided K-Best detector.