# A Scheduling Scheme for Improving the Performance and Security of MU-MIMO Systems
## 5 Numerical Results and Discussions
### Secrecy Outage Probability  
Finally, in this subsection, it is verified how the proposed scheme can help reduce the SOP in case of presence of a spy external receiver (eavesdropper) that tries to decode the information sent by the UTs in the MU-MIMO system. Thus, the analysis performed considers \(U_{t}\) users who try to connect to a trusted BS, of which only \(U\) connect to it according to the proposed scheduling algorithm. In addition, there is also an eavesdropper BS that tries to decode the information of the \(U\) users in the MIMO system. In particular, the SOP is defined as the probability that the secrecy capacity is below a given threshold value [32; 52], where the secrecy capacity is defined as the channel capacity obtained with the SNR of the trusted BS minus the channel capacity obtained with the untrusted receiver's SNR. Thus, the lower the SOP, the greater the protection of the information. From [53] and (11), the achievable rate of the BS decoding the signal of the \(k\)-th user can be obtained as  
\[C_{B,k}\triangleq\log_{2}\left(1+\frac{E_{b}}{N_{0}}\frac{1}{[(\mathbf{\hat{H }}_{B}^{H}\mathbf{\hat{H}}_{E})^{-1}]_{k,k}}\log_{2}M\right), \tag{21}\]  
where \(\mathbf{\hat{H}}_{B}\) is an \(A\times U\) matrix containing the estimated channels gains for the links between the UTs and the BS after the scheduling process. The entries of \(\mathbf{\hat{H}}_{B}\) have mean power \(P\), given by (4). Similarly, the achievable rate of the eavesdropper decoding the signal of the \(k\)-th UT can be obtained as  
\[C_{E,k}\triangleq\log_{2}\left(1+\frac{E_{b}}{N_{0}}\frac{1}{[(\mathbf{\hat{H }}_{E}^{H}\mathbf{\hat{H}}_{E})^{-1}]_{k,k}}\log_{2}M\right), \tag{22}\]  
where \(\mathbf{\hat{H}}_{E}\) is an \(A_{E}\times U\) matrix containing the estimated channels gains for the links between the UTs and the eavesdropper. Thus, notice that the number of receiving antennas at the trusted BS and at the eavesdropper can be different. The entries of \(\mathbf{\hat{H}}_{E}\) have mean power \(P_{E}=cP\), where \(0<c\leq 1\) modifies the mean power of the eavesdropper channel gains. Thus, a small value of \(c\) can be interpreted as the distance between the UTs and the eavesdropper being much larger than the distance between the UTs and the trusted BS. At the other extreme, \(c=1\) implies that the BS and the eavesdropper are at the same distance from the UTs. Moreover, no scheduling process based on the elements of \(\mathbf{\hat{H}}_{E}\) is performed, and the elements of \(\mathbf{\hat{H}}_{E}\) are independent from the elements of \(\mathbf{\hat{H}}_{B}\). The non-negative secrecy capacity for the \(k\)-th UT can be obtained as  
\[\mathcal{C}_{k}=[C_{B,k}-C_{E,k}]^{+}, \tag{23}\]  
where \([x]^{+}=\max(x,0)\). Then, the SOP for the \(k\)-th UT is given by  
\[\mathrm{SOP}_{k}=\mathrm{P}(\mathcal{C}_{k}\leq R), \tag{24}\]  
where \(R\) denotes the secrecy target rate to guarantee the security of the \(k\)-th UT in the MIMO system. By considering the above results, Figures 9 and 10 show the SOP as a function of \(c\), parameterized by \(U_{t}\), considering \(M=4\), \(A_{E}=2\), \(U=2\), \(E_{b}/N_{0}=15\), and \(e=0.1\) in a Rician fading channel with \(K=2\). For the SOP calculation, \(R=1\) bps is employed and, without loss of generality, it is considered that the UT \(k=1\) for the analysis. Figure 9 considers \(A=2\) antennas at the trusted BS and Figure 10 considers \(A=3\) antennas at the trusted BS. In both figures, the SOP increases as \(c\) increases, since the mean power of the fading channel for the paths between the eavesdropper and the UTs increases and consequently, its achievable rate increases. In other words, the eavesdropper can decode the information sent by the users more easily. On the other hand, for small values of \(c\), or equivalently, when the eavesdropper is away from the UTs, the SOP decreases. For all scenarios, it is observed that when \(U_{t}\) increases, then the SOP decreases.