# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
## 3 Channel Estimation
### Channel Aging  
This section delves into the advanced methodologies for predicting channels in CF-MIMO systems. Conventional channel-prediction methods often rely on theoretical models, which might not fully encapsulate the intricacies of real-world scenarios. The relative movement between User Equipments (UEs) and Access Points (APs) leads to temporal fluctuations in the propagation environment, impacting channel coefficients within a single resource block. The channel realization \(h_{R}[n]\) is defined by its initial state \(h_{R}[0]\) and an innovation term [32]. Within this framework, we introduce and assess DMD/PiDMD-based predictors, benchmarking them against current techniques. Historically, channel estimation and prediction in massive MIMO systems have employed methods such as Wiener or Kalman filtering, assuming model-based analytical  
Figure 14: The matrix \(\mathbf{R}\) is presented with varying numbers of pilots to highlight its effect on the phenomenon known as _favorable propagation_. mpeDMD is used. Figure 13: The matrix \(\mathbf{R}\) matrix is presented with varying numbers of antennas to highlight its effect on the phenomenon known as _channel hardening_. PiDMD is used. channels [33, 34, 35]. For example, [PERSON] et al. [19] employed a simple first-order Gauss-Markov process for channel prediction, while more sophisticated autoregressive (AR) and autoregressive moving average (ARMA) models--linear stochastic models for correlated random processes--were explored in [35, 36]. Despite their effectiveness, these approaches rely on simplified analytical models for long-term channel statistics, such as using a rectangular power spectrum to model temporal variations in channels, as discussed in [36], which may not fully encapsulate real-world conditions. Previous prediction methods have also utilized autoregressive models derived from Yule-Walker equations [19], with additional research investigating Machine Learning (ML) approaches [37]. An ML-based predictor with noise pre-processing using linear Minimum Mean Square Error (LMMSE) has also been proposed. Numerical analyses reveal that PiDMD/mpeDMD channel predictors offer a significant improvement in prediction accuracy and data rate compared to conventional channel models. While the ML-based predictor initially exhibits greater computational complexity than the DMD-based predictors, its operational complexity markedly decreases post-training, eventually surpassing the efficiency of the DMD/PiDMD-based methods. The Kalman prediction model used for comparison is detailed in [35]. The correlation matrices employed for LMMSE Channel Estimation are empirically derived as follows:  
\[C_{h_{u}y_{n}} =\mathbb{E}[h_{n}y_{n}^{H}],\] \[C_{y_{n}} =\mathbb{E}[y_{n}y_{n}^{H}],\] \[C_{h_{n}} =\mathbb{E}[h_{n}h_{n}^{H}].\]  
The base station acquires the sampled auto-covariance \(C_{y_{n}}\) through the following expression:  
\[C_{y_{n}}=\frac{1}{N_{s}}\sum_{i=1}^{N_{s}}y_{i}y_{i}^{H} \tag{57}\]  
The LMMSE estimate \(g_{n}\) of \(y_{n}\) is computed as  
\[g_{n} =C_{h_{u}y_{n}}C_{y_{n}}^{-1}y_{n} \tag{58}\] \[=C_{h_{n}}\Psi_{n}^{H}(\Psi_{n}C_{h_{n}}\Psi_{n}^{H}+I)^{-1}y_{n}. \tag{59}\]  
Here, \(\Psi_{n}\) is defined as  
\[\Psi_{li}^{j}=\Bigg{(}\sum_{j,j\in P}p_{li}\tau_{p}\mathbb{R}_{li}^{j}+\sigma_ {lUL}^{2}\mathbb{I}_{M}^{j}\Bigg{)}^{-1}. \tag{60}\]  
Following the algorithm proposed in this study, after obtaining empirical covariance matrices, PiDMD/mpeDMD is applied. The accuracy of the estimate diminishes as the time between pilot transmission and the considered channel realization increases. In DMD channel prediction, once matrix \(A\) is computed as \(R_{k+1}\approx AR_{k}\), the prediction for \(R_{k+2}\) becomes \(R_{k+2}\approx AR_{k+1}\). Matrix \(R\) encapsulates the primary eigenstructure of the system and is reduced in order, thereby decreasing computational complexity. Numerical results highlight the superior performance of DMD/PiDMD compared to Kalman and Machine Learning (MLP) prediction models. The MLP model described can be found in [35].