## 2 System Model  
### MIMO Channel Model  
We consider a flat fading MIMO system with \(N_{t}\) transmit antennas at the transmitter and \(N_{r}\) receive antennas at the receiver \((N_{t}\leq N_{r})\) as shown in Figure 1. For the flat fading channels, the fading coefficients are assumed to be constant during a frame modeled as independent zero-mean complex Gaussian random variables with a variance of \(1/2\) per dimension. The perfect channel knowledge is assumed to be known at the receiver.  
The received signal vector \(\mathbf{y}\in\mathbb{C}^{N_{t}}\) can be expressed as  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}, \tag{1}\]  
where \(\mathbf{H}\in\mathbb{C}^{N_{r}\times N_{t}}\) is the Rayleigh fading channel matrix. Channel element, \(h_{ij}\), denotes the complex gain between \(j\)th transmit antenna and \(i\)th receive antenna and is an independent zero mean circularly symmetric complex Gaussian random variable with unit variance. \(\mathbf{n}\in\mathbb{C}^{N_{t}\times 1}\) represents the additive white Gaussian noise (AWGN) vector in which the entries are the independent and identically distributed (i.i.d) complex Gaussian random variables with zero mean and a variance \(\sigma^{2}\). The transmitted signal vector is \(\mathbf{x}\in\mathbf{S}^{N_{t}}\), where the element of \(\mathbf{x}\) is drawn from a finite constellation alphabet, \(\mathbf{S}\), of cardinality \(M\). The constellation alphabets that we consider in this paper are 64-QAM, 256-QAM, and 1024-QAM.  
Figure 1: MIMO system.  
For computational reasons, we shall replace the complex-valued model (1) by its real-valued equivalent (2) in the usual way, where \(\Re(\mathbf{y})\) and \(\Im(\mathbf{y})\) denote the real and imaginary parts of \(\mathbf{y}\), respectively.  
\[\mathbf{\tilde{y}}=\begin{bmatrix}\Re(\mathbf{y})\\ \Im(\mathbf{y})\end{bmatrix} =\begin{bmatrix}\Re(\mathbf{H})&-\Im(\mathbf{H})\\ \Im(\mathbf{H})&\Re(\mathbf{H})\end{bmatrix}\bullet\begin{bmatrix}\Re( \mathbf{x})\\ \Im(\mathbf{x})\end{bmatrix}+\begin{bmatrix}\Re(\mathbf{n})\\ \Im(\mathbf{n})\end{bmatrix}\] \[=\tilde{\mathbf{H}}\tilde{\mathbf{x}}+\mathbf{\tilde{n}}. \tag{2}\]  
The basic task of MIMO detection is to estimate the input vector \(\tilde{\mathbf{x}}\) relying on the knowledge of the received signal \(\mathbf{\tilde{y}}\) and the channel matrix \(\tilde{\mathbf{H}}\). In other words, the objective function of MIMO detection is to find the closest lattice point \(\tilde{\mathbf{x}}^{\prime}\) to the received signal \(\mathbf{\tilde{y}}\) in a lattice as follows  
\[\mathbf{\tilde{x}}^{\prime}=\operatorname*{arg\,min}_{\mathbf{x}\in\mathbf{ \tilde{S}}}d(\mathbf{\tilde{x}}), \tag{3}\]  
where the input vector \(\mathbf{\tilde{x}}\) belong to an \(2N_{l}\)-dimensional lattice \(\mathbf{\tilde{S}}\) of size \(|\mathbf{\tilde{S}}|=M^{2N_{l}}\) and \(d(\mathbf{\tilde{x}})\) is an Euclidean distance metric defined by [28]  
\[d(\mathbf{\tilde{x}})\triangleq\lVert\mathbf{\tilde{y}}-\tilde{\mathbf{H}} \mathbf{\tilde{x}}\rVert^{2}. \tag{4}\]  
In order to estimate the optimal \(\mathbf{\tilde{x}}\), all \(64^{2N_{l}}\), \(256^{2N_{l}}\), and \(1024^{2N_{l}}\)\(2N_{l}\)-dimensional lattice included in \(\mathbf{\tilde{S}}\) should be searched for 64-QAM, 256-QAM, and 1024-QAM, respectively, according to \(N_{l}\), which can consider one to thousands. The number of the lattice that should be considered for optimal MIMO detection increases exponentially with \(N_{l}\) and \(M\). However, it is practically impossible to implement an ML method that finds the optimal \(\mathbf{\tilde{x}}\) by considering the lattice in all cases because it requires too much computational complexity. Therefore, meta-heuristic algorithms [20; 27] conduct broad and intensive searches with hundreds to thousands of lattices for \(\mathbf{\tilde{S}}\), where an optimal solution is likely to exist for estimating suboptimal \(\mathbf{\tilde{x}}\), which enable low-complexity MIMO detectoion while providing ML-like performance.