## 3 System Model  
In this paper, we concentrate on the BPSK transmission system in the \((M\times N,\ d)^{K}\) MIMO interference channel as shown in Figure 1. Admittedly, many other types of modulation schemes such as quadrature phase shift keying (QPSK), offset QPSK (O-QPSK) and minimum phase shift keying (MSK) can be applied to the system. Figure 1: The BPSK transmission system in the \(K\)-user MIMO interference channel. The discussion on the selection of modulation schemes is beyond the scope of this paper, and the readers can refer to [34] for more information. Besides, perfect carrier recovery and symbol synchronization are assumed to be achieved. Therefore, the bandpass waveforms can be transformed to the equivalent baseband signals [35]. As shown in Figure 1, the binary bit vector \(\mathbf{m}_{k}\in\mathbf{R}^{d\times 1}\) of the \(k\)-th user is first modulated by the waveform function \(g(t)\), and the modulated waveform vector \(\mathbf{s}(t)\in\mathbf{R}^{d\times 1}\) can be expressed as:  
\[\mathbf{s}_{k}(t)=\mathbf{m}_{k}g_{k}(t),\ 0\leq t\leq T_{b},\ k=1,2,...,K \tag{1}\]  
where \(T_{b}\) represents the bit period and:  
\[\mathbf{m}_{k}= \left[ \begin{array}{cc}m_{k,1},m_{k,2},...,m_{k,d}\end{array}  \right]^{\top},\ m_{k,l}\in\left\{+1,-1\right\},\,l=1,\,2,\,...,\,d \tag{2}\] \[\mathbf{s}_{k}(t)= \left[ \begin{array}{cc}s_{k,1}(t),s_{k,2}(t),...,s_{k,d}(t) \end{array} \right]^{\top} \tag{3}\]  
Then the modulated waveform vector is precoded by the precoding matrix \(\mathbf{V}_{k}\in\mathbf{C}^{\text{th}\times d}\) to produce the transmitted waveform vector, _i.e._,  
\[\mathbf{X}_{k}(t)=\mathbf{V}_{k}\mathbf{s}_{k}(t) \tag{4}\]  
where:  
\[\mathbf{V}_{k}= \left[\mathbf{V}_{k,1},\mathbf{V}_{k,2},...,\mathbf{V}_{k,d}\right] \tag{5}\] \[\mathbf{V}_{k,l}\in\mathbf{C}^{\text{th}\times 1},\,l=1,\,2,\,...,\,d\] (6) \[\parallel\mathbf{V}_{k,l}\parallel= 1 \tag{7}\]  
The precoded waveforms will be sent across the interference channel, and the received waveform vector \(\mathbf{Y}_{k}(t)\in\mathbf{C}^{\text{N}\times 1}\) at the \(k\)-th receiver can be expressed as:  
\[\mathbf{Y}_{k}(t)= \sum_{i=1}^{K}\gamma_{ik}\mathbf{H}_{ik}\mathbf{X}_{i}(t)+ \mathbf{n}_{k}(t) \tag{8}\] \[= \sum_{i=1}^{K}\gamma_{ik}\mathbf{H}_{ik}\mathbf{V}_{i}\mathbf{s} _{i}(t)+\mathbf{n}_{k}(t),\ 0\leq t\leq T_{b}\]  
where \(\mathbf{n}(t)\in\mathbf{C}^{\text{N}\times 1}\) denotes the received noise vector at the \(k\)-th receiver, each element of which is an independent zero-mean additive white Gaussian noise (AWGN) process with two-sided power spectral density of \(N_{0}/2\) W/Hz; \(\mathbf{H}_{ki}\in\mathbf{C}^{\text{N}\times M}\) represents the small scale fading channel gain matrix from the \(i\)-th transmitter to the \(k\)-th receiver, each entity of which is an independent random variable with the distribution of \(\mathbf{CN}\)(0,1); \(\gamma_{ki}\) denotes the large-scale channel attenuation between the \(i\)-th transmitter and the \(k\)-th receiver. Most of the previous works have focused on the symmetric networks, where the users are located in the proximity of each other and share the same path-loss, _i.e._, \(\gamma_{ki}=1\) for all users. However, in practical communication systems, the users might scatter around geographically. Therefore, we will also consider the asymmetric networks, where the large-scale fading is associated with the distance and is given by [13]:  
\[\gamma_{k}= \sqrt{r_{ki}^{-\alpha}} \tag{9}\]The notations \(\alpha\) and \(\gamma_{ki}\) represent the path-loss exponent and the distance between the transmitter \(i\) and receiver \(k\), respectively. At the \(k\)-th receiver, a decoding matrix \(\mathbf{U}_{k}\in\mathbf{C}^{\mathrm{N}\times d}\) is applied to suppress the interference from the unintended users, and the reconstructed signal waveform vector can be formulated as:  
\[