## 6 Complexity Analysis  
The computational complexity analysis is the last intriguing result worth highlighting. Floating-points operations (FLOPs) [40] are used to assess the complexity of the algorithms proposed in this paper. Figure 9 compares the FLOPs versus the number of APs, the number of \(N_{r}\), and the number of UEs for the uplink cell-free mm-Wave massive MIMO system. Because DSAS executes more iterations to acquire the antenna subset selection solution, it has higher complexity than CMDAS and CMFASs. In contrast, the proposed matching scheme overcomes DSAS and CMDAS and attains an approximate number of FLOPs. Figure 9a shows the FLOPs versus the number of APs, and it is obvious that the proposed matching scheme can overcome DSAS and CMDAS by achieving 193.51% and 199.177% complexity reduction ratio compared to CMDAS and DSAS, respectively. Figure 9b,c also present the FLOPs against the number \(N_{r}\) of antennas and the number \(K\) of UEs in the cell-free network, respectively. It can be seen that the proposed matching scheme achieves lower computational complexity concerning the number of FLOPs versus both \(N_{r}\) and \(K\) compared to CMDAS and DSAS. It is also noteworthy that although our proposed matching scheme based on the Hungarian algorithm has a slightly higher complexity compared to the CMFAS, it is evident that the proposed matching scheme has better EE with all investigated scenarios of uplink cell-free mm-Wave massive MIMO systems. Moreover, the computational complexities of the proposed schemes can be affected by the number of required iterations to obtain the optimal number of selected antennas for each RF chain at each AP, which results in obtaining the maximum \(\text{SE}_{m}\), and the number of iterations is affected by the number \(N_{r}\) of antennas and the number of APs in the coverage area. Furthermore, the number of iterations is given by \(\imath=\sum_{m=1}^{M}\sum_{i=K+1}^{N_{r}K}(i)_{m}\) for DSAS, \(\imath=\sum_{m=1}^{M}(N_{r})_{m}\) for CMDAS, and \(\imath=M\) for both the matching strategy for antenna selection based on the Hungarian algorithm and CMFAS. Thus, the computational complexity to obtain the total SE for all APs in the cell-free systems based on (17) is \(\mathcal{O}(\imath N_{r}K^{2})\). It is noticeable that DSAS requires many iterations, especially when the \(M\) APs and the \(N_{r}\) equipped for each AP is vast. For example, if \(N_{r}=64\), \(M=32\) APs, and \(K=8\) UEs, the required total number of iterations to obtain the maximum total SE is around 16,380 for all APs. While CMDAS needs 2048 iterations to reach the maximum total SE. In  
Figure 8: Total power consumption versus the number of UEs, where \(M=80\) APs, \(N_{Q}=8\), \(\rho=23\) dBm, and \(N_{r}=48\). addition, the channel magnitude with the predefined value of selected antennas for each AP based on the channel condition strategy has the lowest number of iterations, which is equal to the number of APs inside the coverage area. Regarding the proposed matching algorithm based on the Hungarian method strategy, if \(N_{r}K^{2}>K^{3}\) (i.e., \(N_{r}\geq K\)), then its computational complexity is \(\mathcal{O}(t(N_{r}K^{2}+N_{r}K+K^{3}))=\mathcal{O}(t(N_{r}K^{2}))\), and the required number of iterations to obtain the maximum total spectral efficiency is also equal to the number of APs, which is considered to be similar to the channel magnitude with predefined value of the selected antennas based on the channel condition scheme. In addition, if \(N_{r}K^{2}<K^{3}\) (i.e., \(N_{r}<K\)), then the computational complexity of the proposed matching algorithm based on the Hungarian method strategy is \(\mathcal{O}(t(K^{3}))\)[41], which is similar to the Hungarian algorithm complexity analysis multiplied by \(\iota\), which is equal to the number of APs in the cell-free network, and in this case, the proposed matching scheme for antenna selection has higher complexity than the channel magnitude with the predefined value of the selected antennas based on the channel condition scheme.