# A Scheduling Scheme for Improving the Performance and Security of MU-MIMO Systems
## 4 Scheduling Schemes
### Proposed Scheduling Scheme  
A MU-MIMO system that employs the ZF detector at the receiver can serve a number of users less than or equal to the number of receiving antennas, as shown in Figure 1. With the ZF detector, the system diversity is \(\mathcal{D}=A-U+1\)[35; 8]. In particular, the greater the diversity, the better the system performance. Thus, if the number of users is equal to the number of antennas in the receiver, the detector eliminates the interference but loses the spatial diversity, namely, the performance for all users is similar to the performance of a system in which there is only one transmitting and one receiving antenna [8]. Now, consider a scenario where the total number of users \(U_{t}\) is greater than the number of antennas \(A\), as in the example shown in Figure 1, where \(A=3\), \(U_{t}=4\) and \(U=3\). In this case, the system can select a group of \(U\) users who will transmit simultaneously over the same radio resource. Thus, a scheduling scheme is proposed below to select \(U\) users among the \(U_{t}\) users competing to transmit. In addition, this channel allocation scheme along with the ZF detector is intended to eliminate interference, but also to guarantee some diversity in order to improve system performance. Consider as an example Figure 2, where four different user selection scenarios appear. The proposed scheme considers the wireless communication channel for all users at once, that is, it employs the complete estimated channel matrix \(\mathbf{\hat{H}}\) to decide which users should transmit, and consequently, considers the instantaneous SNR of each user, which was obtained in (11). This approach differs from other works in which the channel gains of each user are considered separately [26], and not the whole matrix \(\mathbf{\hat{H}}\) altogether. Let \(\mathbf{U}\) be a matrix in which each column is one of the combinations of \(U\) different integer numbers taken from the subset \(\mathcal{I}=\{1,2,\ldots,U_{t}\}\). Therefore, \(\mathbf{U}\) has \(U\) rows and \(\binom{U_{t}}{U}\) columns, where \(\binom{x}{y}=\frac{x!}{(x-y)!y!}\). Moreover, let \(\mathbf{u}_{\ell}\) be the \(\ell\)-th column of \(\mathbf{U}\). Associating each of the \(U_{t}\) candidate users with each integer in the subset \(\mathcal{I}\), \(\mathbf{\hat{H}}_{\mathbf{u}_{\ell}}\) can be defined as a matrix similar to \(\mathbf{\hat{H}}\) but containing only the channel gains for the users defined by the vector \(\mathbf{u}_{\ell}\). For a better understanding, consider again as an example the scenario with \(A=3,U_{t}=4\) and \(U=3\). The matrices \(\mathbf{U}\) and \(\mathbf{\hat{H}}_{\mathbf{u}_{2}}\), respectively, are given by  
\[\mathbf{U}= \begin{pmatrix}1&1&1&2\\ 2&2&3&3\\ 3&4&4&4\end{pmatrix} ,\ \mathbf{\hat{H}}_{\mathbf{u}_{2}}= \begin{pmatrix}\hat{h}_{1,1}&\hat{h}_{1,2}&\hat{h}_{1,4}\\ \hat{h}_{2,1}&\hat{h}_{2,2}&\hat{h}_{2,4}\\ \hat{h}_{3,1}&\hat{h}_{3,2}&\hat{h}_{3,4}\end{pmatrix} , \tag{17}\]  
where \(\hat{h}_{a,k}\) is the estimated channel gain at the receiver for the \(a\)-th antenna and the \(k\)-th user. The SNR for the \(k\)-th user is proportional to \(\zeta_{k}\), defined in (12). Therefore, every possible matrix \(\mathbf{\hat{H}}_{\mathbf{u}_{\ell}}\) must be taken for \(\ell=1,2,\ldots(\frac{U_{t}}{U_{t}})\), and the related random variables \(\zeta_{k}\)  
Figure 1: MU-MIMO system with \(A=3\) antennas in the BS, in which \(U=3\) users are served simultaneously out of a total of \(U_{t}=4\) users. Figure 2: Channel assignment scenarios with the proposed scheduling scheme considering \(U_{t}=4\) and \(U=3\) for \(A=3\) receiving antennas. must be determined. Then, the scenario in which the instantaneous SNR of the \(U\) users is maximized is the one in which the sum of the variables \(\zeta_{k}\) is the greatest, assuming that the \(E_{b}/N_{0}\) ratio is the same for all users. (This assumption is valid, since current cellular systems perform almost perfect power control on the uplink [48] and therefore, signals from different users arrive with similar power at the BS.)