# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
## 2 Materials and Methods
### DMD/PiMD/npeDMD
#### 2.5.2 Physics-Informed Dynamic Mode Decomposition (piDMD)  
Several variants of Dynamic Mode Decomposition (DMD) have been developed. When analyzing a system from a data-driven perspective, it is common to have partial knowledge of the system's underlying physics. The initial step in physics-informed DMD (piDMD) requires the user to summarize the known or suspected physical properties of the system [9].  
Once the physical principles to be enforced are determined, they must be translated into the matrix manifold to which the linear model should be constrained [9]. This step involves understanding the data collection procedure, such as the spatial grid or the energy inner product.  
With a target matrix manifold defined, the relevant Procrustes problem [9] can be solved. Equation (46) is commonly known as the Procrustes problem in the literature. By appropriately modifying the optimization routine, it can be ensured that the resulting model adheres to the previously identified physical principles.  
The final step of piDMD involves extracting physical information from the learned model [9]. This could include analyzing the spectrum or modes, computing the resolvent modes [31], making predictions, or exploring other diagnostics. When the matrix is Toeplitz, the algorithm takes the form of Equation (47). A Toeplitz matrix can be reduced to a circulant matrix [9].  
\[\operatorname*{arg\,min}_{R_{k+1}\in M}||R_{k+1}-AR_{k}||_{F}, \tag{46}\]  
where the \(A\) matrix is given by (44) or (45) if the reduced form is used. If matrix R is restricted to Toeplitz. then the problem can be written as (47). A key point to emphasize is that PiDMD leverages empirical data while simultaneously incorporating prior knowledge. Specifically, it enforces the condition that the matrix \(R\) exhibits a Toeplitz structure, thereby ensuring that this constraint is satisfied throughout the process.  
\[\operatorname*{arg\,min}_{R_{k+1}\text{ is Toeplitz}}||R_{k+L}-AR_{k}||_{F} \tag{47}\]  
\[R=\mathcal{F}\operatorname*{diag}(\hat{r})\mathcal{F}^{-1} \tag{48}\]  
Circulant matrices possess very useful properties, one of the most important being their ability to be diagonalized using the Fourier transform. In (48), \(\mathcal{F}_{jk}=e^{2\pi i(j-2)(k-1)/n}/\sqrt{2}\) and \(\mathcal{F}^{-1}=\mathcal{F}^{*}\), with \(\hat{r}_{j}\) as the unknown eigenvalues and \(R_{j,k}=r_{(j-k)\mod n}\). Once \(R\) is diagonalized, it can be written as  
\[R=\begin{bmatrix}r_{0}&r_{-1}&\ldots&r_{-(n-1)}\\ r_{1}&r_{0}&\ddots&\vdots\\ \vdots&\ddots&\ddots&r_{-1}\\ r_{n-1}&\ldots&r_{1}&r_{0}\end{bmatrix}. \tag{49}\]  
A Toeplitz matrix can be straightforwardly transformed into a circulant matrix form. Similarly to the circulant case, a Toeplitz matrix has elements that satisfy the relation \(R_{i,j}=r_{i-j}\), with the distinction that in this case, the indices are not taken modulo \(n\). As demonstrated in Equation (50), a Toeplitz matrix can be embedded within a circulant matrix. Once this transformation is achieved, the resulting circulant matrix can be diagonalized, as outlined above. Denoting the circulant matrix as \(C\), this matrix is constructed using an auxiliary matrix, denoted \(\tilde{R}\), whose entries are known a priori. The explicit forms of both matrices are illustrated in Equation (50).  
\[\begin{array}{c}{\subset}{\begin{bmatrix}R&\tilde{R}\\ \tilde{R}&R\end{bmatrix}}\\ \\ \tilde{R}{=}{\begin{bmatrix}0&r_{n-1}&\dots&r_{1}\\ r_{-(n-1)}&0&\ddots&\vdots\\ \vdots&\ddots&\ddots&r_{-(n-1)}\\ r_{-1}&\dots&r_{-(n-1)}&0\end{bmatrix}}\end{array} \tag{50}\]  
Finally, if the matrix is circulant, the process for solving Equation (47) becomes significantly simplified, as detailed in [9]. The procedure is straightforward: first, a Toeplitz matrix is constructed, and then, for mathematical simplification, a circulant matrix is formed.