## 2 System Model  
Consider a MIMO full-duplex (FD) relay system with \(K\) source-destination pairs communicating simultaneously with the aid of a relay. The source-to-destination links are assumed to be inconsequential due to the large-scale fading and the large distances between them. Figure 1 presents the system model with the \(k\)th source-destination pairs equipped with \(N_{s_{k}}\) and \(N_{d_{k}}\) antennas, respectively. The FD relay has \(N_{r}\) and \(N_{t}\) antennas to simultaneously receive and transmit signals, respectively. Therefore, communication between the source-destination pairs is accomplished in one time slot compared to an HD system that requires two time slots. Let \(\mathbf{s}_{k}[n]\) represent the length \(d\) signal vector at time \(n\) for the \(k\)th source. A linear transceiver matrix \(\mathbf{\Theta}_{k}[n]\) is applied to \(\mathbf{s}_{k}[n]\) before transmission. The received signal at the relay can then be expressed as  
\[\mathbf{y}_{r}[n]=\sum_{k=1}^{K}\mathbf{H}_{k}[n]\mathbf{\Theta}_{k}[n] \mathbf{s}_{k}[n]+\mathbf{R}_{LI}[n]\mathbf{t}[n]+\mathbf{n}_{r}[n], \tag{1}\]  
where \(\mathbf{H}_{k}[n]\in\mathbb{C}^{N_{k}\times N_{k}}\) is the channel between source \(k\) and the relay node, \(\mathbf{R}_{LI}[n]\in\mathbb{C}^{N_{r}\times N_{k}}\) is the loop interference (LI) channels, and \(\mathbf{n}_{r}[n]\in\mathbb{C}^{N_{r}\times 1}\) is an independent and identically distributed (i.i.d.) additive white Gaussian noise (AWGN) matrix. As the received signal \(\mathbf{y}_{r}[n]\) is corrupted by residual LI, from [12], it can be written as  
\[\mathbf{y}_{r}[n]=\sum_{k=1}^{K}\mathbf{H}_{k}[n]\mathbf{\Theta}_{k}[n] \mathbf{s}_{k}[n]+\mathbf{R}_{LI}[n]\Delta\mathbf{t}[n]+\mathbf{n}_{r}[n], \tag{2}\]  
where \(\Delta\mathbf{t}[n]=\mathbf{t}[n]-\tilde{\mathbf{t}}[n]\), and \(\mathbf{R}_{LI}[n]\Delta\mathbf{t}[n]\) is the residual LI. At time \(n+1\), the relay amplifies the received signal with a relay transceiver matrix \(\mathbf{\Gamma}[n+1]\in\mathbb{C}^{N_{k}\times N}\) and forwards the resulting signal to the destinations. The received signal at the \(k\)th destination is then  
\[\mathbf{y}_{k}[n+1]=\mathbf{G}_{k}[n+1]\mathbf{\Gamma}[n+1]\mathbf{y}_{r}[n] +\mathbf{n}_{d_{k}}[n+1], \tag{3}\]  
where \(\mathbf{G}_{k}[n+1]\in\mathbb{C}^{N_{k}\times N_{k}}\) is the channel matrix between the relay and destination \(k\), and \(\mathbf{n}_{d_{k}}[n+1]\in\mathbb{C}^{N_{k}\times 1}\) is an i.i.d. AWGN vector with a zero mean and unit variance. As in [6,7,12], channel state information (CSI) is assumed to be available at all nodes. In addition, channel variations during the transceiver matrix update interval are assumed to be relatively small [19] and so do not influence the transceiver design [12]. Thus, the received signal at the \(k\)th destination can be expressed as  
\[\mathbf{y}_{k}=\mathbf{G}_{k}\mathbf{\Gamma}\sum_{k=1}^{K}\mathbf{H}_{k} \mathbf{\Theta}_{k}\mathbf{s}_{k}+\mathbf{n}_{k}, \tag{4}\]  
where  
\[\mathbf{n}_{k}=\mathbf{G}_{k}\mathbf{\Gamma}\mathbf{R}_{LI}\Delta\mathbf{t}+ \mathbf{G}_{k}\mathbf{\Gamma}\mathbf{n}_{r}+\mathbf{n}_{d_{k}}. \tag{5}\]  
As in [18,20], \(\Delta\mathbf{t}\) is modeled as white Gaussian noise which is independent of \(\mathbf{n}_{r}\) and \(\mathbf{n}_{d_{k}}\), and has variance \(\sigma_{t}^{2}\). Consequently, the covariance matrix of \(\mathbf{n}_{k}\) is  
\[\mathbf{C}_{\mathbf{n}_{k}}=\mathbb{E}[\mathbf{n}_{k}\mathbf{n}_{k}^{H}]. \tag{6}\]  
Figure 1: The multiple-user MIMO full-duplex (FD) relay system. A linear combiner \(\mathbf{M}_{k}\in\mathbb{C}^{N_{k}\times N_{b}}\) is used at the \(k\)th destination to obtain the signal estimate  
\[\mathbf{\hat{s}}_{k}=\mathbf{M}_{k}^{H}\mathbf{y}_{k}. \tag{7}\]  
The mean square error (MSE) at the \(k\)th user is defined as  
\[