## 4 Numerical Results  
This section presents the evaluation of the performance of the proposed multiuser MIMO FD relay downlink algorithm using numerical simulation. The CVX toolbox was employed, which is a Matlab-based software package for solving convex optimization problems [21]. For simplicity, we considered a system with two sources and two destinations. The extension to more than two source-destination pairs is straightforward. We assumed that the sources and destinations were each equipped with two antennas, and the relay was equipped with four receive antennas and four transmit antennas. The simulation parameters are given in Table 2.  
As in the related literature, flat-fading MIMO channels were considered. It was assumed that the elements of \(\mathbf{H}_{k}\), \(\mathbf{G}_{k}\), and \(\mathbf{R}_{Ll}\) were i.i.d. complex Gaussian random variables with a zero mean and unit variance. Moreover, all noise terms were i.i.d. complex, circularly symmetric Gaussian random variables with a zero mean and unit variance. According to [9], the residual LI can be 0 dB to 20 dB larger than the channel noise. Therefore, LI levels of 0 dB, 5 dB and 10 dB were employed here. In all cases, the results are given for an average of 1000 independent channel realizations. Note that the HD system mentioned in this paper is the same as the proposed FD system except that the residual LI term is zero and the achievable rate of the HD system is half of the FD system since two time slots are required for transmission between a source and destination. Thus, the same optimization procedure is employed. Figure 2 presents the SMSE for the HD and FD systems with different levels of residual LI. The signal-to-noise ration (SNR) between the sources and relay varies from 0 dB to 30 dB, and the SNR between the relay and destinations is fixed at 30 dB. These results indicate that the HD system provides the best SMSE as the FD performance is degraded as the LI increases. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(\mathbf{s}_{k}\) & Signal vector for the \(k\)th source \\ \(\mathbf{\Theta}_{k}\) & Linear transceiver matrix at the source for the \(k\)th user \\ \(\mathbf{H}_{k}\) & Channel between source \(k\) and the relay node \\ \(\mathbf{R}_{Ll}\) & Loop interference (LI) channels \\ \(\mathbf{n}_{r}\) & i.i.d. additive white Gaussian noise (AWGN) matrix \\ \(\mathbf{M}_{k}\) & Linear combiner at the destination \\ \(\mathbf{T}\) & Transceiver matrix at the relay for the \(k\)th user \\ \(\mathbf{M}_{k}\) & Linear combiners at the destinations for the \(k\)th user \\ R & Achievable rate \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbol descriptions. The achievable rate for the system in (5) can be obtained using an approach similar to that in [8], and can be written as  
\[R=\sum_{k=1}^{K}R_{k}, \tag{31}\]  
where  
\[ \begin{split} R_{k}&=\log_{2}\det[\mathbf{I}_{N_{d}}+( \mathbf{G}_{k}\mathbf{\Gamma}\mathbf{H}_{k}\mathbf{\Theta}_{k})(\mathbf{G}_{k} \mathbf{\Gamma}\mathbf{H}_{k}\mathbf{\Theta}_{k})^{H}\\ &\times(\sigma_{t}^{2}\mathbf{G}_{k}\mathbf{\Gamma}\mathbf{R}_{L }\mathbf{R}_{L}\mathbf{I}^{H}\mathbf{\Gamma}^{H}\mathbf{G}_{k}^{H}+\mathbf{G} _{k}\mathbf{\Gamma}\mathbf{\Gamma}^{H}\mathbf{G}_{k}^{H}+\mathbf{I}_{N_{d}})^{ -1}].\end{split}  \tag{32}\]  
Figure 3 gives the achievable rate for the HD and FD systems. In this figure, the SNR between the sources and relay varies from 0 dB to 30 dB while the SNR between the relay and destinations is fixed at 30 dB. The HD system corresponds to the case when the residual LI is zero and two time slots are required for transmission from sources to destinations. Therefore, the FD achievable rate is twice the HD achievable rate if the LI is cancelled completely. Further, the achievable rate with the FD system is higher when the residual LI is 0 dB to 10 dB. The achievable rate of the FD system is degraded as the residual LI increases in the high-SNR region since the residual LI is greater than the multiuser interference.