# A Scheduling Scheme for Improving the Performance and Security of MU-MIMO Systems
## 4 Scheduling Schemes
### Proposed Scheduling Scheme
By the above, the following metric allows us to determine the sum of the random variables \(\zeta_{k}\) for each matrix \(\mathbf{\hat{H}}_{\mathbf{u}_{\ell}}\),  
\[\mathcal{M}(\mathbf{u}_{\ell})=\sum_{k=1}^{U}\frac{1}{[(\mathbf{\hat{H}}_{ \mathbf{u}_{\ell}}^{H}\mathbf{\hat{H}}_{\mathbf{u}_{\ell}})^{-1}]_{k,k}},\ \ \text{for}\ \ell=1,2,...,\binom{U_{t}}{U}. \tag{18}\]  
Then, the MU-MIMO system selects the set of users based on the following criteria  
\[\mathbf{\tilde{u}}=\operatorname*{arg\,max}_{\mathbf{u}_{\ell}}\ \mathcal{M}(\mathbf{u}_{\ell}). \tag{19}\]  
As result, \(\mathbf{\tilde{u}}\) contains the indices of the \(U\) users that must transmit. It is important to notice that the order of the columns for different matrices \(\mathbf{\hat{H}}_{\mathbf{u}_{\ell}}\) does not modify the final decision made by the proposed scheme. For this reason, the number of options evaluated by the scheduler is specifically limited to \(\binom{U_{t}}{U_{t}}\). Algorithm 1 summarizes the process for finding vector \(\mathbf{\tilde{u}}\) as a pseudocode. ```
1:\(U,U_{t},\mathbf{\hat{H}}\)
2:Create matrix \(\mathbf{U}\) with dimensions \(U\times\binom{U_{t}}{U}\) (see as reference eq. (17))
3:for\(\ell=1,2,\ldots,\binom{U_{t}}{U}\)do
4: Take the \(\ell\)-th column from \(\mathbf{U}\), i.e, \(\mathbf{u}_{\ell}\)
5: Take the columns of \(\mathbf{\hat{H}}\) based on the elements of \(\mathbf{u}_{\ell}\) and create matrix \(\mathbf{\hat{H}}_{\mathbf{u}_{\ell}}\)
6: acum = 0
7:for\(k=1,2,\ldots,U\)do
8: acum = acum + \(\left\{[(\mathbf{\hat{H}}_{\mathbf{u}_{\ell}}^{H}\mathbf{\hat{H}}_{\mathbf{u }_{\ell}})^{-1}]_{k,k}\right\}^{-1}\)
9:endfor
10:\(\mathcal{M}(\mathbf{u}_{\ell})=\) acum
11:endfor
12: Select \(\mathbf{u}_{\ell}\) that generates \(\max\{\mathcal{M}(\mathbf{u}_{\ell})\}\) for \(\ell\in\left\{1,2,\ldots,\binom{U_{t}}{U}\right\}\)
13:\(\mathbf{\tilde{u}}=\mathbf{u}_{\ell}\)
14:\(\mathbf{\tilde{u}}\)
```  
**Algorithm 1** Scheduling algorithm pseudocode  
Notice that (18) and (19) allows selecting the \(U\) users with the highest SNR after the multiuser detection process, that is, the \(U\) users with the highest SNR at the ZF detector output. (Selecting the \(U\) users, among the \(U_{t}\) users, whose sum of SNRs is maximum, is equivalent to selecting the \(U\) users with the highest individual SNRs.) From the literature, it is known that a higher SNR implies a higher channel capacity or a lower BER. In this sense, the presented proposal allows the guarantee of better performance for the users in the MIMO system. In addition, the proposed scheduling scheme differs from others in the literature in which the SNR is considered at the receiver input and, therefore, only the channel gains are considered before the processing performed by the multiuser detector, as evidenced in (13) and (14). When the MIMO system is not fully loaded, that is, when \(A>U\), the ZF multiuser detector alone can guarantee a certain diversity order for the MU-MIMO system. In this case, the proposed allocation scheme will further increase the system diversity. However, the presented proposal can be more attractive if \(A=U\), since it improves the system diversity even when the system is fully loaded. From the literature, it is known that when system diversity increases, then performance improves, which implies that lower SNR values or higher modulation orders can be used in the system and still guarantee an adequate performance [36]. These aspects are validated in Section 5. Finally, it could be thought that one or more UTs do not transmit with the proposed scheme. This can be interpreted as unfair to UTs whose channel gains are weaker. At this point, it is important to notice that, according to 3 GPP Release 15, mobile systems under the 5G standard employ the OFDM technique in the uplink [49]. Therefore, if it is considered that the proposed scheme operates with that standard, there are several subcarriers available for users to transmit.