# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
## 6 Results
### Evaluation of the Applicability of DMD for Channel Estimation in CF-MMIMO Wireless Systems  
In this section, we evaluate the applicability of Dynamic Mode Decomposition (DMD) for Channel Estimation in CF-MMIMO wireless systems through several experiments. Standard metrics, such as the Cumulative Distribution Function (CDF) of the per-UE uplink/downlink Spectral Efficiency (SE), are used for performance evaluation. These CDFs are applied to various methods, including MRC, MRC-DMD, FZF-DMD, MMSE, MMSE-DMD, PFZF, and PFZF-DMD. To assess adaptability, we employ a highly dynamic scenario as described in Section [referenced section]. Furthermore, PiDMD and mpeDMD are compared with traditional predictors, such as Machine Learning (ML) algorithms and the Kalman Filter (KF). The impact of the number of time-frequency snapshots, denoted as \(\tau_{p}\), is also investigated. In wireless networks, algorithm efficiency is commonly evaluated by calculating Spectral Efficiency. Figure 15 presents the CDF of the per-user downlink SE for different numbers of BS antennas using various combining schemes: MRC, MMSE, FZF, PFZF, and their DMD-enhanced versions--MRC-piDMD, MMSE-piDMD, FZF-piDMD, and PFZF-piDMD. The parameters used are \(L=100\), \(K=16\), \(N=8\), \(\tau_{p}=6\), \(\tau_{5}=40\), \(p_{k}=100\) mW, and \(50\leq v\leq 150\) for each UE. There is a notable performance disparity between MR-based combinations and ZF-based schemes, particularly for UEs with high channel gains, largely due to inter-user interference. FZF and PiFZF schemes effectively mitigate this interference, with FZF combining leading to a 16% improvement in average SE. Moreover, FZF-PiDMD outperforms FZF, and PFZF-piDMD achieves a 33%-likely SE improvement over PFZF, primarily by offering enhanced protection to weaker UEs with lower channel gains. The increase in the number of antennas per base station (BS) from 4 to 64 results in an average improvement in algorithm performance of 28%. Figure 16 compares the UL CDF of the SEs with different estimators, MR, MR-DMD, ZF, ZF-DMD, MMSE, MMSE-DMD, and PZF, with PZF-DMD combining the scheme with \(L=128\), \(K=128\), \(N=64\), \(\tau_{p}=16\) and \(p_{max}=100\) mW with different number of UEs, \(K=10\)[10, 40]. As the number of UEs decreases while maintaining the same pilot length, both interference and pilot contamination are reduced, resulting in higher SEs for each UE. In this case, the density of UEs increased raises the number of pilots. At first glance, it is surprising that MR-mpeDMD under these conditions presents very good results. While MR is optimal in the presence of noise, it is not as effective with both noise and interference. This reveals another important characteristic of MR-mpeDMD: by intrinsically separating stronger signals from weaker ones and treating the weaker signals as noise, it proves to be extremely efficient. MMSE-mpe, ZF-mpeDMD, and PZF-mpeDMD also benefit from the separation of strong and weak signals using mpeDMD. Figure 17 depicts the average uplink Spectral Efficiency (SE) per User Equipment (UE) plotted as a function of the pilot sequence length, \(\tau_{p}\). The simulation parameters include \(L=100\) Access Points (APs), \(N=4\) antennas per AP, \(K=40\) UEs, a pilot sequence length (\(\tau_{p}\)) of 40, and spatially correlated Rayleigh fading with an Angular Spread Distribution (ASD) of \(\sigma_{\phi}=\sigma_{\theta}=15^{\circ}\) and \(50<v<150\). FZF-mpeDMD shows great superiority over the other algorithms primarily due to the minimal number of pilots required for its operation, which allows for the optimization of data transmission. On the other hand, MMSE-mpeDMD presents almost constant behavior. The hypothesis is that MMSE is already optimal in environments with high interference, so adding DMD does not provide a significant improvement in SE. FZF-PiDMD also shows high performance around a length of 20 \(\tau_{p}\), but then it decreases because the payload starts being affected by the increase in pilots.