# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
###### Abstract
Developing and deploying the necessary hardware to support such a large and complex system needs advanced electronic components capable of operating in a distributed and large-scale environment. The costs associated with installing and maintaining such an extensive and complex infrastructure require significant investments and careful planning to ensure the economic viability of the system. Ensuring secure communications and maintaining user privacy in a widely dispersed environment necessitates the implementation of robust security protocols and efficient encryption mechanisms. These technical challenges underscore the complexity of implementing CF-MMIMO systems but also highlight the opportunities for technological innovations and improvements in wireless communications. Traditional approaches to Channel Estimation often rely only on orthogonal pilot sequences to estimate the channel response [4]. However, these methods are susceptible to pilot contamination, leading to inaccurate channel estimates, especially in scenarios with a high density of antennas [7]. Some works suggest increasing the number of pilots to harness this issue, but the overhead incurred by pilot signaling escalates with the number of antennas, hampering the efficiency gains of massive MIMO. In this paper, we propose a paradigm shift in Channel Estimation for CF-MMIMO systems by leveraging the power of DMD and two of its variants, in particular, Physical Informed Dynamic Mode Decomposition (PiDMD) and measure-preserving extended DMD (mpeDMD). DMD/PiDMD/mpeDMD are purely data-driven techniques that extract dynamic modes from time-series data, revealing the underlying dynamics of complex systems. Our approach emphasizes the ability of DMD/PiDMD/mpeDMD to capture the spatial-temporal evolution of the channel, offering a novel solution to the challenges posed by rapid variations in CF-MMIMO channels. Dynamic Mode Decomposition (DMD) is a data-driven technique utilized to decompose complex dynamical systems into spatial-temporal modes [8], effectively capturing the underlying patterns and dynamics from time-resolved data. PiDMD (Physics-informed DMD) [9] enhances this approach by incorporating prior knowledge of the governing physical laws, thereby improving the accuracy of the dynamic modes, particularly when data are limited or noisy. mpeDMD (Multi-parameter Extended DMD) further extends DMD by accommodating dynamical systems influenced by multiple parameters, allowing for a comprehensive analysis of how these parameters affect the observed dynamics [10]. This work also introduces a novel model for channel correlation [11], which is crucial in practical MIMO systems where correlation consistently exists. This model relies on delay-Doppler characteristics and draws inspiration from Orthogonal Time-Frequency Space (OTFS) [12]. To demonstrate the potential of DMD/PiDMD/mpeDMD in wireless networks, we will consider scenarios involving User Equipment (UE) with high mobility, which introduces the concept of channel aging [13]. In the remainder of this work, \"DMD\" will generally refer to DMD itself and its variants, including PiDMD and mpeDMD, unless otherwise specified. For notational simplicity, we define a set of diagonal matrices \(D_{kl}\in\mathbb{C}^{N\times N}\), where \(k=1,\ldots,K\) and \(l=1,\ldots,L\) represent the communication links between Access Points (APs) and User Equipment (UE). This set models link breakage as a uniformly distributed random process. Specifically, \(D_{kl}\) is the identity matrix \(\mathbf{I}_{N}\) if AP \(l\) is permitted to transmit to and receive signals from UE \(k\) and otherwise \(\mathbf{0}_{N}\):  
\[D_{kl}= \begin{cases}\mathbf{I}_{N}&\text{if }l\in\mathcal{M}_{k},\\ \mathbf{0}_{N}&\text{if }l\
otin\mathcal{M}_{k}.\end{cases} \]  
The elements of \(\sqrt{\eta_{k}}\phi_{t_{k}}\) are transmitted as the signal \(s_{k}\) over \(\tau_{p}\) consecutive samples.