# Enabling Intelligent 6G Communications: A Scalable Deep Learning Framework for MIMO Detection
## 3 MIMO Detection: A System Model  
We consider a MIMO communication system comprising \(P_{t}\) transmit and \(Q_{u}\) receive antennas. The received signal model in the complex domain is expressed as follows:  
\[\bar{\mathbf{r}}=\tilde{\mathbf{G}}\tilde{\mathbf{s}}+\bar{\mathbf{w}}, \tag{1}\]  
where \(\bar{\mathbf{r}}\in\mathbb{C}^{Q_{u}\times 1}\) denotes the complex-valued received signal vector, \(\tilde{\mathbf{s}}\in\mathbb{C}^{P_{t}\times 1}\) is the transmitted symbol vector, \(\tilde{\mathbf{G}}\in\mathbb{C}^{Q_{u}\times P_{t}}\) is the Rayleigh fading channel matrix, and \(\bar{\mathbf{w}}\in\mathbb{C}^{Q_{u}\times 1}\) represents the additive white Gaussian noise (AWGN) vector with zero mean and variance \(\sigma^{2}\). The noise vector \(\bar{\mathbf{w}}\) is modeled as a complex Gaussian random vector:  
\[\bar{\mathbf{w}}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}\mathbf{I}_{Q_{u}}), \tag{2}\]  
where \(\mathcal{CN}(\cdot)\) denotes the complex normal distribution, \(\sigma^{2}\) is the noise variance per complex dimension, and \(\mathbf{I}_{Q_{u}}\) is the \(Q_{u}\times Q_{u}\) identity matrix. After transforming to the real-valued domain, the noise vector becomes \(\mathbf{w}\in\mathbb{R}^{2Q_{u}\times 1}\) and is distributed as  
\[\mathbf{w}\sim\mathcal{N}(\mathbf{0},\frac{\sigma^{2}}{2}\mathbf{I}_{2Q_{u}}), \tag{3}\]  
where \(\mathcal{N}(\cdot)\) represents the multivariate real Gaussian distribution. The factor \(\frac{1}{2}\) arises due to the decomposition of each complex dimension into two real dimensions, each having half the variance. We assume perfect channel state information (CSI) at the receiver. Solving the complex-valued MIMO channel model directly is challenging. The real and imaginary components of the transmitted and received symbol vectors are separated to provide an equivalent real-valued channel model without losing generality [31]. The conversion of a complex-valued model to a real-valued one is given as follows:  
\[\begin{bmatrix}\mathrm{Re}(\bar{\mathbf{r}})\\ \mathrm{Im}(\bar{\mathbf{r}})\end{bmatrix}=\begin{bmatrix}\mathrm{Re}(\hat{ \mathbf{G}})&-\mathrm{Im}(\hat{\mathbf{G}})\\ \mathrm{Im}(\hat{\mathbf{G}})&\mathrm{Re}(\hat{\mathbf{G}})\end{bmatrix} \begin{bmatrix}\mathrm{Re}(\hat{\mathbf{s}})\\ \mathrm{Im}(\hat{\mathbf{s}})\end{bmatrix}+\begin{bmatrix}\mathrm{Re}(\hat{ \mathbf{w}})\\ \mathrm{Im}(\hat{\mathbf{w}})\end{bmatrix}. \tag{4}\]  
Defining  
\[\mathbf{r} =\begin{bmatrix}\mathrm{Re}(\bar{\mathbf{r}})\\ \mathrm{Im}(\bar{\mathbf{r}})\end{bmatrix}\in\mathbb{R}^{2Q_{u}\times 1}, \tag{5}\] \[\mathbf{s} =\begin{bmatrix}\mathrm{Re}(\hat{\mathbf{s}})\\ \mathrm{Im}(\hat{\mathbf{s}})\end{bmatrix}\in\mathbb{R}^{2P_{t}\times 1},\] \[\mathbf{G} =\begin{bmatrix}\mathrm{Re}(\hat{\mathbf{G}})&-\mathrm{Im}(\hat{ \mathbf{G}})\\ \mathrm{Im}(\hat{\mathbf{G}})&\mathrm{Re}(\hat{\mathbf{G}})\end{bmatrix}\in \mathbb{R}^{2Q_{u}\times 2P_{t}},\]  
We obtain the canonical real-valued MIMO system model:  
\[\mathbf{r}=\mathbf{Gs}+\mathbf{w}, \tag{6}\]  
For clarity, the system can also be expressed in an expanded matrix form:\[\begin{bmatrix}r_{1}\\ r_{2}\\ \vdots\\ r_{Q_{u}}\end{bmatrix}=\begin{bmatrix}g_{1,1}&g_{1,2}&\cdots&g_{1,P_{t}}\\ g_{2,1}&g_{2,2}&\cdots&g_{2,P_{t}}\\ \vdots&\vdots&\ddots&\vdots\\ g_{Q_{u},1}&g_{Q_{u},2}&\cdots&g_{Q_{u},P_{t}}\end{bmatrix}\begin{bmatrix}s_{1} \\ s_{2}\\ \vdots\\ s_{P_{t}}\end{bmatrix}+\begin{bmatrix}w_{1}\\ w_{2}\\ \vdots\\ w_{Q_{u}}\end{bmatrix} \tag{7}\]