# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
## 6 Results
### Performance Evaluation of DMD in a System Considering Delay-Doppler Correlated Channel  
In this section, several experiments are conducted to demonstrate the impact of a system with delay-Doppler correlation and the performance of DMD-based algorithms in this specified scenario. Experiments aim to show Spectral Efficiency under conditions of a correlated channel, analyzing effects for both uplink and downlink transmissions. Correlation always exists in practice and is highly dependent on every communication scenario, primarily generated by ASD, which is quantified in this section. Different Doppler values in the channel are also studied. Some studies suggest mitigating the correlation by increasing the number of pilots \(\tau_{p}\); therefore, the number of pilots is tested to determine this relationship among DMD algorithms. Additionally, the impact of channel correlation on coherent and non-coherent downlink transmissions is analyzed. Channel correlation is largely influenced by ASD, making it insightful to investigate the relationship between Spectral Efficiency (SE) and ASD. Figure 20 depicts the average uplink SE per UE as a function of ASD for azimuth and elevation angles. In this graph, we use \(\sigma_{\phi}=\sigma_{\theta}\). We consider different operation modes of CF-MMIMO with \(L=100\), \(N=4\), \(K=40\), and \(\tau_{p}=10\). Results for uncorrelated Rayleigh fading are shown as a reference. Surprisingly, mpeDMD can leverage correlated channels and outperform the case without correlation. Figure 21 illustrates the 95%-likely per-user non-coherent downlink Spectral Efficiency (SE) against the value of the length of the frames \(\tau_{s}\). The simulation parameters are set to \(L=100\), \(K=20\), \(M=2\), and ASD = \(30^{\circ}\). Simulations show that asymptotically, DMD-based methods converge faster as the delay increases. PiDMD and mpeDMD exhibit better performance than MR and FZF. MR and ZF have good performance when \(f_{D}T_{s}=0\) but degrade rapidly as the Doppler effect increases. DMD models also degrade with a higher Doppler effect but much less than non-DMD schemes. The hypothesis for why this happens is that DMD methods separate strong signals from weaker ones. Weaker signals are treated as noise, and the stronger signals, which are just a few, are easily aligned, allowing good signal alignment at the receiver. Figure 20: The average uplink Spectral Efficiency (SE) per User Equipment (UE) as a function of the angular spread (ASD) for azimuth and elevation angles, where \(\sigma_{\phi}=\sigma_{\theta}\), is analyzed for different operations of CF-MMIMO and small-cell systems. We consider \(L=100\), \(M=4\), \(K=40\), and \(\tau_{p}=10\). The results for uncorrelated Rayleigh fading are included as a reference. Figure 22 depicts the uplink (UL) transmission and its dependence on the number of pilot symbols (\(\tau_{p}\)) with \(L=100\), \(K=20\), \(N=2\), and ASD = \(30^{\circ}\), considering a delay-Doppler correlated channel. The delay is uniformly distributed in \([0,0.01]\) ms, and the normalized Doppler is \(f_{D}T_{s}=0.002\). The graph shows that piDMD and mpeDMD initially grow very rapidly, and then their growth becomes nearly negligible. MR, PZFZ, and ZF exhibit linear growth. This clearly indicates that DMD-based methods require fewer pilots; increasing the number of pilots beyond a certain limit does not substantially increase the SE. Figure 23 illustrates the CDF of per-user uplink Spectral Efficiency (SE) with full power. Figure 23B shows the effect of normalized Doppler shifts of \([f_{d}T_{s}=0,f_{d}T_{s}=0.004]\)  
Figure 21: Varying data frame length and non-coherent transmission (\(L=100\), \(K=20\), \(N=2\), ASD = \(30^{\circ}\)). Figure 22: Average UL SE vs. number of \(\tau_{p}\), centralized LSFD (\(L=100\), \(K=20\), \(N=2\), ASD = \(30^{\circ}\)). on uplink SE. It is evident that DMD systems outperform non-DMD systems at both CDF and 95%-likely Uplink SE with Doppler effect.