## 2 MIMO System Model  
In this paper, we consider a MIMO system employing \(N\) transmit and \(M\) receive antennas as shown in Figure 1, where the different transmit antennas transmit different data streams in order to attain a multiplexing gain [24]. The channel is considered to be flat fading, where the channels between the different transmit and different receive antennas are considered to be spatially uncorrelated and are independent and identically distributed.  
Let \(\mathbf{x}^{c}\) denote the transmitted complex symbol vector of size \(N\times 1\), where \(\mathbf{x}^{c}=[x_{1}^{c},x_{2}^{c},...,x_{N}^{c}]^{T}\) such that \(x_{i}^{c}\in X^{c}\) is drawn from a complex constellation of P-Quadrature Amplitude Modulation (P-QAM), where P is the constellation size. Furthermore, the channel can be described by a complex matrix \(\mathbf{H}^{c}\) of size \(M\times N\), where \(\mathbf{H}^{c}\) changes independently from one frame to another. Therefore, the received signal \(\mathbf{y}\) can be expressed as:  
\[\mathbf{y}^{c}=\mathbf{H}^{c}\mathbf{x}^{c}+\mathbf{n}^{c}, \tag{1}\]  
where \(\mathbf{y}^{c}=[y_{1}^{c},y_{2}^{c},...,y_{M}^{c}]^{T}\) is the received complex signal vector of size \(M\times 1\) and \(\mathbf{n}^{c}=[n_{1}^{c},n_{2}^{c},...,n_{M}^{c}]^{T}\) represents the complex Additive White Gaussian Noise (AWGN) vector of size \(M\times 1\) with zero mean and variance \(N_{0}\). Given that there are different symbols transmitted from the different transmit antennas at the same time, these symbols will interfere with each other at the receiver side. Hence, the detector at the receiver should retrieve the transmitted vector \(\mathbf{x}^{c}\) from the received signal \(\mathbf{y}^{c}\), which combines all the transmitted symbols.  
In the previous paragraph, we have used the superscript \(c\) in the notations representing the transmitted symbol vector, received symbol vector, channel and noise in order to denote that these are complex valued. The complex model of (1) can be represented in an equivalent real model as follows:  
Figure 1: MIMO system block diagram.  
\[\begin{bmatrix}\Re[\mathbf{y}^{c}]\\ \Im[\mathbf{y}^{c}]\end{bmatrix}=\begin{bmatrix}\Re[\mathbf{H}^{c}],-\Im[ \mathbf{H}^{c}]\\ \Im[\mathbf{H}^{c}],\ \Re[\mathbf{H}^{c}]\end{bmatrix}\begin{bmatrix}\Re[\mathbf{x}^{c}]\\ \Im[\mathbf{x}^{c}]\end{bmatrix}+\begin{bmatrix}\Re[\mathbf{n}^{c}]\\ \Im[\mathbf{n}^{c}]\end{bmatrix}, \tag{2}\]  
where \(\Re\) and \(\Im\) represent the real and imaginary parts of a complex number, respectively. Additionally, (2) can be represented in the following form:  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}. \tag{3}\]  
The equivalent real model of (3) has \(\mathbf{y}=[y_{1},y_{2},...,y_{2M}]^{T}\) and \(\mathbf{n}=[n_{1},n_{2},...,n_{2M}]^{T}\) both of size \(2M\times 1\), while \(\mathbf{H}\) has a size of \(2M\times 2N\), and \(\mathbf{x}=[x_{1},x_{2},...,x_{2N}]^{T}\) of size \(2N\times 1\). Using the equivalent real model, each \(x_{i}\in X\) is drawn from the real constellation set of P-QAM as \(\{-\sqrt{P}+1\), \(-\sqrt{P}+3,...,\sqrt{P}-1\}\). For example the equivalent real model of a constellation \(x^{c}=[1+j,3-j]^{T}\) is given by \(x=[1,3,1,-1]^{T}\). In what follows, we use the equivalent real model for explaining the different detectors.