# A Scheduling Scheme for Improving the Performance and Security of MU-MIMO Systems
## 4 Scheduling Schemes
### Computational Complexity Analysis of the Proposed Scheduling Scheme  
Now, the computational complexity of the proposed scheduling scheme is analyzed based on the number of complex operations performed. For this, let **A**, **B**, and **C** be arbitrary complex matrices with dimensions \(m\times u\), \(n\times p\) and \(m\times m\), respectively. Thus, the product **AB** has \(mp(n-1)\) complex additions and \(mmp\) complex multiplications. In addition, the inversion of **C** requires \(\frac{1}{3}m^{3}+\frac{1}{2}m^{2}-\frac{5}{6}m\) complex additions and subtractions and \(\frac{1}{3}m^{3}+m^{2}-\frac{1}{3}m\) complex multiplications and divisions considering that Gaussian elimination with reverse substitution is used [51]. In the literature, there are several techniques to reduce the complexity of matrix operations, however, in the computational complexity analysis carried out in this work these techniques have not been considered.  
With the above result, and knowing that \(\hat{\textbf{H}}\) has dimensions \(A\times U\) and that the vector **y** has dimensions \(A\times 1\), by considering (9), the number of complex additions and subtractions required by the ZF detector to calculate the product \(\hat{\textbf{H}}^{\textbf{t}}\textbf{y}=(\hat{\textbf{H}}^{H}\hat{\textbf{H}} )^{-1}\hat{\textbf{H}}^{H}\textbf{y}\), when \(A>U\), is \(\frac{1}{3}U^{3}+(2A-\frac{1}{2})U^{2}-\frac{11}{6}U\). In addition, the number of complex multiplications and divisions required to perform such an operation is \(\frac{1}{3}U^{3}+(2A+1)U^{2}+(A-\frac{1}{3})U\). On the other hand, if \(\hat{\textbf{H}}\) is a square matrix, the ZF detector performs the operation \(\hat{\textbf{H}}^{-1}\textbf{y}\). In this case, the detector performs \(\frac{1}{3}U^{3}+\frac{3}{2}U^{2}-\frac{11}{6}U\) complex additions and subtractions and \(\frac{1}{3}U^{3}+2U^{2}-\frac{1}{3}U\) complex multiplications and divisions.  
Next, the computational complexity of the scheduling criterion defined in (18) and (19) is analyzed. The number of complex additions and subtractions needed to calculate \(\hat{\textbf{H}}^{H}_{\textbf{u}_{\ell}}\hat{\textbf{H}}_{\textbf{u}_{\ell}}\) and its inverse are \((A-1)U^{2}\) and \(\frac{1}{3}U^{3}+\frac{1}{2}U^{2}-\frac{5}{6}U\), respectively, and the number of complex multiplications and divisions required are \(AU^{2}\) and \(\frac{1}{3}U^{3}+U^{2}-\frac{1}{3}U\), respectively. Also, notice that (18) requires \(U\) divisions and \(U\) sums for each \(\ell\). Therefore, (19) requires \(\binom{U_{t}}{U^{2}}\left[\frac{1}{3}U^{3}+(A-\frac{1}{2})U^{2}+\frac{1}{6}U\right]\) complex additions and subtractions and \(\binom{U_{t}}{U^{2}}\left[\frac{1}{3}U^{3}+(A+1)U^{2}+\frac{2}{3}U\right]\) complex multiplications and divisions. Thus, the total number of complex operations required by the proposed algorithm is  
\[\text{O}_{\text{P}}=\binom{U_{t}}{U}\left[\frac{2}{3}U^{3}+\left(2A+\frac{1}{ 2}\right)U^{2}+\frac{5}{6}U\right]. \tag{20}\]  
In the above expression, notice that the binomial coefficient \(\binom{U_{t}}{U^{2}}\) determines the number of operations performed by the algorithm  
important to remember that the presented algorithm is an interesting proposal in scenarios in which the system is fully loaded. Thus, its use is expected to be performed in scenarios where the difference between \(U_{t}\) and \(U\) is not high and, therefore, the number of operations carried out by the scheduling algorithm can be executed in practical systems. Despite the small difference between \(U_{t}\) and \(U\), the MIMO system performance can be improved considerably, as discussed in the next section.