# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
## 2 Materials and Methods
### DMD/PiMD/npeDMD  
#### 2.5.1 Dynamic Mode Decomposition  
DMD is a powerful data-driven technique that has found applications in various fields, ranging from fluid dynamics to neuroscience and beyond [28]. At its core, DMD seeks to capture the underlying dynamic behavior of a system from time-series data, offering insights into its modes of oscillation, growth rates, and spatial structures [8]. DMD operates on the premise that complex dynamics can be approximated by a set of coherent modes that evolve over time [8]. Given a sequence of data snapshots, DMD extracts these modes and associated frequencies to create a compact representation of the system's behavior. This is particularly valuable in scenarios where the underlying dynamics are complex and difficult to model analytically. DMD has demonstrated its efficacy in extracting relevant features from complex and high-dimensional data, making it particularly suitable for applications in which understanding system dynamics is crucial. In the context of CF-MMIMO wireless systems, DMD offers a novel approach to Channel Estimation by capturing the spatial-temporal evolution of the wireless channel's behavior. By leveraging DMD, it becomes possible to adaptively track and predict the changing channel characteristics, thereby enhancing the efficiency and reliability of communication in dynamic environments [8]. The DMD algorithm is as follows. \[y_{k+1}\approx Ay_{k}, \tag{13}\]  
\[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]  
which has the solution  
\[A\approx y_{k+1}y_{k}^{\dagger}=y_{k+1}W\Sigma V^{*}. \tag{15}\]  
where \(\dagger\) indicates the Moore-Penrose pseudoinverse. In (15),  
\[W\in\mathbb{R}^{N\times r}\quad V\in\mathbb{R}^{(m-1)\times r}, \tag{16}\]  
\[y_{k}^{\dagger}=V\Sigma^{-1}W^{T}. \tag{17}\]  
\[\hat{A}=y_{k+1}V\Sigma^{-1}W^{T}\in\mathbb{R}^{N\times N}, \tag{18}\]  
\[W^{T}AW=Wy_{k+1}V\Sigma^{-1}W^{T}W\in\mathbb{R}^{N\times N}, \tag{19}\]  
\[\hat{A}=U_{r}^{*}AV_{r}=W_{r}^{\dagger}y_{k+1}V_{r}\Sigma_{-1}^{-1}\in(R)^{r \times r}. \tag{20}\]  
Figure 5: The formation of the \(\Phi_{k}\) matrix involves \(n\) pilots and \(k\) consecutive samples. (**A**) Time-frequency snapshot Number 1. Pilots are show as red squares. (**B**) \(k\) time frequency snapshots. Each column of matrix \(\Phi_{k}\) is formed by \(k\) consecutive pilot samples. It is now feasible to compute the eigendecomposition of \(\hat{A}\). The eigenvectors of A can be approximated from the reduced eigenvectors \(\Psi\) by (21):  
\[\hat{A}\Psi=\Psi\hat{\Lambda}\ \Longrightarrow\ \hat{A}=X\Lambda X^{-1}. \tag{21}\]  
If we start in state \(u_{0}\), \(u_{k}\) can be found using \(A\), as shown next:  
\[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]0 \[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]1 (23) \[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]2 (24) \[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]3 (25) \[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]4 (26) \[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]5 (27) \[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]6 \[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]7 \[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]8 (28) \[\min_{A\in\mathbb{R}^{N\times N}}||y_{k+1}-Ay_{k}||_{F}. \tag{14}\]9 (29) \[A\approx y_{k+1}y_{k}^{\dagger}=y_{k+1}W\Sigma V^{*}. \tag{15}\]0 (30) \[A\approx y_{k+1}y_{k}^{\dagger}=y_{k+1}W\Sigma V^{*}. \tag{15}\]1 (31) \[A\approx y_{k+1}y_{k}^{\dagger}=y_{k+1}W\Sigma V^{*}. \tag{15}\]2 (32) \[A\approx y_{k+1}y_{k}^{\dagger}=y_{k+1}W\Sigma V^{*}. \tag{15}\]3 (33) \[A\approx y_{k+1}y_{k}^{\dagger}=y_{k+1}W\Sigma V^{*}. \tag{15}\]4 (34) \[A\approx y_{k+1}y_{k}^{\dagger}=y_{k+1}W\Sigma V^{*}. \tag{15}\]5  
Equation (37) is a method to compute \(b\); alternative approaches to compute b can be found in [28, 29, 30].