# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
### Paper Structure  
This paper is organized as follows.  
Section 2, Materials and Methods, begins with an overview of the System Model, detailing the application of DMD for Channel Estimation in CF-MMIMO systems. It discusses the incorporation of channel correlation and introduces a novel proposed algorithm aimed at improving performance metrics.  
Section 3, Channel Estimation, delves deeper into the methodologies employed for Channel Estimation using DMD techniques. It explores channel prediction and addresses the challenges posed by channel aging within CF-MMIMO environments.  
Section 4, Uplink Transmission, focuses on the crucial components of uplink communication in CF-MMIMO systems. Topics covered include uplink training methods, Uplink Data Transmission strategies, and the evaluation of uplink Spectral Efficiency. Additionally, it examines key transmission techniques such as Maximum Ratio Combining (MRC), Minimum Mean Squared Error (MMSE) Channel Estimation, Full Zero Forcing (FZF), Partial Zero Forcing (PZF), and Optimal Least Squares Frequency Domain (LSFD) weights.  
Section 5, Downlink Transmission, shifts attention to the complexities of Downlink Transmission. It contrasts Coherent and Non-Coherent Downlink transmission strategies, highlighting their respective advantages and challenges within the CF-MMIMO framework.  
Section 6, Results, presents and analyzes empirical findings derived from the simulations and experiments conducted to validate the proposed methodologies and algorithms.  
Section 7, Discussion, expands on the implications and insights drawn from the results, exploring broader implications and potential avenues for future research.  
Section 8, Conclusion, offers a concise summary of the paper's contributions, emphasizing how the adoption of DMD techniques in CF-MMIMO systems enhances Channel Estimation accuracy and transmission efficiency.  
The CF-MMIMO architecture is illustrated in Figure 1. Several APs are randomly distributed within a circular area of radius \(D\), accommodating \(N\) UEs. In principle, each UE communicates with every AP. A central CPU is capable of collecting data from each AP. This setup presents several challenges. For instance, the volume of data transmitted from the APs to the CPU grows substantially with the increasing number of UEs and APs. A centralized transmission strategy becomes impractical, and Channel Estimation becomes complex due to the large amount of data to be processed.  
It is easy to realize that strategies that allow dealing with a huge amount of information while producing satisfactory results for good performance are highly desirable. This is precisely where DMD/PiDMD proves suitable for two main tasks: it optimally reduces the dimensions of the systems, and it leverages all available information, particularly the structure of the correlation matrix.  
DMD methods are purely data-driven techniques. In general, data-driven methods in wireless networks lack accuracy because the number of samples is always small, especially for the estimation of the correlation matrix \(R\), which is used for Channel Estimation in most methods. DMD is able, with a relatively small number of samples, to capture the spatial and temporal correlations of the wireless channel [8].  
We assume that the base station (\(BS_{k}\)) and User Equipment (\(UE_{s}\)) are perfectly synchronized and operate using a pilot-based protocol. Each coherence block contains \(\tau_{p}\) channel uses, with \(K\) designated for pilots. There are \(K\) orthogonal unit-norm pilot sequences, which are reused across cells. The pilot for UE \(k\) in cell \(j\) is represented by \(\phi_{jk}\in\mathbb{C}^{K}\) and satisfies \(\left\|\phi_{jk}\right\|^{2}=1\)[20].  
Figure 1: CF-MMIMO system [19]. Each color represents a dynamically formed user-centric cell.