# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
## Appendix A Delay Correlation  
Delay correlation describes the relationship between the channel's response to different delayed versions of the transmitted signal. It plays a crucial role in understanding how multipath propagation influences signal reception. The delay correlation function, denoted as \(R_{h}(\tau)\), quantifies this relationship and is defined as  
\[R_{h}(\tau)=\mathbb{E}[h(t,\tau)h^{*}(t,\tau+\Delta\tau)],\]  
where \(h(t,\tau)\) is the channel's impulse response at time \(t\) with delay \(\tau\), \(h^{*}(t,\tau+\Delta\tau)\) is the complex conjugate of the channel's response at a delay offset \(\Delta\tau\), and \(\mathbb{E}[\cdot]\) denotes the expectation operator. This function indicates how the channel's response to one multipath component correlates with another separated by a delay \(\Delta\tau\).  
Understanding delay correlation is essential for designing effective equalization and diversity techniques, as it directly impacts the channel's frequency selectivity and inter-symbol interference (ISI). The accurate modeling of delay correlation helps improve system performance, optimize receiver design, and enhance the overall reliability of communication systems in multipath environments.  
In the context of spatially correlated Rayleigh fading channels, the channel response from \(UE_{k}\) in cell \(l\) to the BS in cell \(j\) is represented by \(h^{j}_{lk}\in\mathbb{C}^{M_{j}}\), where each component of \(h^{j}_{lk}\) is sampled from a circularly symmetric complex Gaussian distribution:  
\[h^{j}_{lk}\sim\mathcal{N}(0,R^{j}_{lk}), \tag{1}\]  
with \(R^{j}_{lk}\) being the positive semi-definite covariance matrix capturing spatial correlation among non-line-of-sight (NLoS) elements. The variance \(\overline{h}^{j}_{lk}\) accounts for factors like path attenuation, shadowing, and antenna characteristics.  
The average channel gain from one of the antennas at BS \(j\) to \(UE_{k}\) in cell \(l\) is given by the normalized trace:  
\[\beta^{j}_{lk}=\frac{1}{M_{j}}\text{tr}(R^{j}_{lk}), \tag{2}\]  
where \(M_{j}\) denotes the number of antennas at BS \(j\).  
The reception in large antenna arrays involves spatial correlation among signals received by individual antennas, which translates into correlated delays. Assuming a separation of \(\lambda/2\) between antennas (equivalent to half a wavelength), the received vector adopts the structure described in Equation (1), where \(g_{n}\in\mathbb{C}\) represents gain and phase rotation for a particular path, and \(d_{H}\) is the antenna spacing (in wavelengths).  
\[a_{n}=g_{n}\Big{[}1\quad e^{2\pi jd_{H}\sin(\hat{\phi}_{n})}\quad\cdots\quad e ^{2\pi jd_{H}(M-1)\sin(\hat{\phi}_{n})}\Big{]}^{T}. \tag{3}\]When considering \(N\) paths from reflected trajectories, the channel's response \(h\) is the summation of these components, as shown in Equation (A4). The resulting correlation matrix \(R\) is given by Equation (A5), which is directly related to delay correlation:  
\[h=\sum_{n=1}^{N_{\text{path}}}a_{n},\] (A4)  
\[[R]_{l,m}=\sum_{n=1}^{N_{\text{path}}}\mathbb{E}\{|g_{n}|^{2}\}\mathbb{E}\{e^{2 \pi jd_{H}(l-1)\sin(\hat{\phi})}e^{2\pi jd_{H}(m-1)\sin(\hat{\phi})}\}.\] (A5)  
Here, \(E\{|g_{n}|^{2}\}\) represents the expectation of the squared gain, and \(\sin(\hat{\phi})\) accounts for the delay correlation in terms of angles.  
In practice, the angular standard deviation (ASD) often follows a jointly Gaussian distribution [5], leading to a spatial correlation matrix represented by Equation (A6):  
\[[R]_{m,l}=\beta\int\int e^{j\pi(m-l)\sin(\hat{\phi})\cos(\hat{\theta})}f(\hat{ \phi},\hat{\theta})d\hat{\phi}d\hat{\theta}.\] (A6)  
Here, \(f(\phi,\theta)\) denotes the joint Gaussian distribution function.