## 3 MIMO Detection
### LR-SIC Detector  
The LR-SIC detector represents a nonlinear detector which improves the performance by transforming the channel into a better-conditioned representation and successively canceling the interference from different UEs [23; 24; 26; 27; 43]. Consider systems with \(N\geq K\). We can first apply the QR decomposition of \(\mathbf{H}=\mathbf{Q}\mathbf{R}\) to reduce the computational complexity by reducing dimensions, i.e., we generate first  
\[\mathbf{r}=\mathbf{Q}^{H}\mathbf{y}_{q}=\mathbf{G}\mathbf{x}+\boldsymbol{ \zeta}, \tag{12}\]  
where  
\[\mathbf{G}=\alpha\mathbf{Q}^{H}\mathbf{H}\sqrt{\mathbf{P}}=\alpha\mathbf{R} \sqrt{\mathbf{P}}, \tag{13}\]  
\[\boldsymbol{\zeta}=\mathbf{Q}^{H}(\alpha\mathbf{n}+\mathbf{n}_{q}). \tag{14}\]  
For simplicity, we assume \(\boldsymbol{\zeta}\) as an AWGN with variance  
\[\sigma_{\xi}^{2}=\alpha^{2}\sigma_{n}^{2}+\alpha\beta P_{\mathrm{Rx}}. \tag{15}\]  
We then apply the complex Lenstra-Lenstra-Lovasz (CLLL) algorithm in [25] to the extended channel matrix  
\[\mathbf{\tilde{G}}=[\mathbf{G}^{T}\,\sigma_{\xi}\mathbf{I}_{K}]^{T}. \tag{16}\]  
The LR-SIC receiver then consists of a set of linear detectors based on the lattice-reduced channel matrix  
\[\mathbf{\tilde{G}}=\mathbf{G}\mathbf{T}, \tag{17}\]  
where \(\mathbf{T}\) is a unimodular matrix with \(\det(\mathbf{T})=1\). The received signal is rewritten as  
\[\mathbf{r}=\mathbf{\tilde{G}}\mathbf{z}+\boldsymbol{\zeta}, \tag{18}\]with  
\[\mathbf{z}=\mathbf{T}^{-1}\mathbf{x}. \tag{19}\]  
Therefore, we can detect \(\mathbf{z}\) from \(\mathbf{r}\) first and then use the relationship in (19) to recover \(\mathbf{x}\). Signal detection can then be applied to the signal model (18) in the lattice-reduced domain. The MMSE-SIC algorithm [43] can be implemented using the extended received signal  
\[\mathbf{r}^{\prime}=[\mathbf{r}^{T}\ \ \mathbf{0}_{K}^{T}]^{T} \tag{20}\]  
and extended matrix  
\[\widetilde{\mathbf{G}}^{e}=[\widetilde{\mathbf{G}}^{T}\ \sigma_{5}\mathbf{T}^{T}]^{T}. \tag{21}\]  
The estimate \(\widehat{\mathbf{z}}\) of \(\mathbf{z}\) can be computed successively. Assume a detector order of \(k=1,2,\cdots\). With the already detected data canceled, the received signal is updated as:  
\[\mathbf{G}=\alpha\mathbf{Q}^{H}\mathbf{H}\sqrt{\mathbf{P}}=\alpha\mathbf{R} \sqrt{\mathbf{P}}, \tag{13}\]0  
Accordingly, we define a \(2K\times(K-i+1)\) matrix as  
\[\mathbf{G}=\alpha\mathbf{Q}^{H}\mathbf{H}\sqrt{\mathbf{P}}=\alpha\mathbf{R} \sqrt{\mathbf{P}}, \tag{13}\]1  
In order to generate \(\widehat{z}_{i}\), MMSE estimation is applied to estimate \(z_{i}\) as  
\[\mathbf{G}=\alpha\mathbf{Q}^{H}\mathbf{H}\sqrt{\mathbf{P}}=\alpha\mathbf{R} \sqrt{\mathbf{P}}, \tag{13}\]2  
where  
\[\mathbf{G}=\alpha\mathbf{Q}^{H}\mathbf{H}\sqrt{\mathbf{P}}=\alpha\mathbf{R} \sqrt{\mathbf{P}}, \tag{13}\]3  
Following [44], the estimate \(\widehat{z}_{i}\) of \(z_{i}\) is obtained by shifting and scaling the output from \(\widetilde{z}_{i}\) as  
\[\mathbf{G}=\alpha\mathbf{Q}^{H}\mathbf{H}\sqrt{\mathbf{P}}=\alpha\mathbf{R} \sqrt{\mathbf{P}}, \tag{13}\]4  
where \(\mathbf{t}_{i}^{-1}\) is the \(i\) row of \(\mathbf{T}^{-1}\), \(\left\lfloor\cdot\right\rfloor\) denotes the rounding operator, \(\alpha\) and \(\beta\) denote the scaling and shifting coefficients respectively, and \(\mathbf{1}\) is a vector of \(1\)'s. The LR-SIC can significantly improve the performance as compared with linear detection [23; 24]. Compared with other nonlinear detection algorithms such as the fixed complexity sphere detector (FCSD), LR-SIC can reuse the filters and thus may exhibit a lower average complexity [43]. In the above, a natural ordering \(k=1,2,\cdots\) of the transmitted symbol is assumed. In fact, the order of detection can be optimized to improve performance. One option is to select the symbol that can be detected with the highest reliability at each stage. This can be based on the MMSE criterion.