# Performance Evaluation of CF-MMIMO Wireless Systems Using Dynamic Mode Decomposition
## 2 Materials and Methods
### Impact of Channel Correlation  
Small-scale fading arises from the presence of multiple propagation paths, resulting in signal replicas with varying phase shifts [23]. Within a local area, the variation in these phase shifts across different locations is influenced by the angle between the propagation paths and the line connecting those locations. This concept is illustrated in Figure 4, which depicts the line connecting two locations and a propagation path at an angle \(\theta\) relative to it. The change in path length between the two locations is the projection of this line in the direction of the path [24]. Since this projection depends on \(\phi\), a key aspect of characterizing small-scale fading is a function that describes the angular distribution of paths, known as the Angular Spread Distribution (ASD). The ASD represents the received signal power as a function of the angle when acting as a receiver, or conversely, it represents the transmitted signal power reaching the receiver at the other end of the link as a function of the angle of departure. The ASD is a continuous function of angle, normalized to be interpreted as a probability density function (PDF), consistent with the unit-power normalization of small-scale fading. The standard deviation of the ASD, when viewed as a distribution, provides the root-mean-square (RMS) angle spread [24]. While uncorrelated Rayleigh fading is a common model due to its simplicity, it does not always accurately represent practical scenarios [24]. This paper proposes a Doppler-delay correlation model where \(R_{\text{delay}}\) is the delay correlation matrix and \(R_{\text{Doppler}}\) is the Doppler correlation matrix. Incorporating delay and Doppler correlation into the model makes it more realistic, as these variables persist for longer periods, as demonstrated in Orthogonal Time-Frequency Space (OTFS) [25]. Figure 4: Propagation illustration and the multi-path sources. To begin, consider an uncorrelated scattering environment. The frequency-domain correlation \(R_{h}(. )\) and the Power Delay Profile (PDP) can be related through a Fourier transform [24]. The distribution of received power as a function of delay, normalized to be interpreted as a PDF, is termed the Power Delay Profile (PDP) and is defined by \(S_{h}(\tau)=E\big{\{}|h(\tau)|^{2}\big{\}}\)[26]. \[R_{h}(\Delta_{\
u}) =\mathbb{E}[h(\
u)h^{*}(\
u+\Delta_{\
u})] \tag{4}\] \[=\mathbb{E}\bigg{[}\int_{-\infty}^{\infty}h(\tau_{0})e^{-j2\pi \
u\tau_{0}}d\tau_{0}\int_{-\infty}^{\infty}h(\tau_{1})e^{-j2\pi(\
u+\Delta_{ \
u})\tau_{1}}d\tau_{1}\bigg{]}\] (5) \[=\int_{-\infty}^{\infty}\mathbb{E}[|h(\tau)|^{2}]e^{j2\pi\Delta_{ \
u\tau}}d\tau. \tag{6}\]  
\[R_{h}(\Delta_{\
u})=\int_{-\infty}^{\infty}\mathbb{E}\Big{[}|h(\tau)|^{2} \Big{]}e^{j2\pi\Delta_{\
u\tau}}d\tau, \tag{7}\]  
\[=\int_{-\infty}^{\infty}S_{h}(\tau)e^{j2\pi\Delta_{\
u\tau}}d\tau. \tag{8}\]  
Assuming \(\mathbb{E}[h(\tau_{0})h(\tau_{1})]=0\) for \(\tau_{0}\
eq\tau_{1}\), the condition outlined in (7) ensures that \(\mathbb{E}[h(\tau_{0})h(\tau_{1})]=0\) for \(\tau_{0}\
eq\tau_{1}\). This condition, known as the uncorrelated scattering condition and specified in (8), is based on the idea that propagation mechanisms generating paths with different delays are inherently independent. When combined with the assumption of local stationarity, this condition forms the basis of traditional small-scale fading modeling within the wide-sense stationary uncorrelated scattering (WSSUS) framework [27]. Additionally, the uncorrelated scattering condition in the delay domain ensures that \(R_{h}(. )\) depends only on \(\Delta\
u\) rather than \(\
u\), making the fading wide-sense stationary in the frequency domain as well. Due to the Fourier relationship between \(R_{h}(. )\) and \(S_{h}(. )\), it follows that \(B_{c}\propto\frac{1}{T_{d}}\), with the exact proportionality determined by the characteristics of \(R_{h}(. )\), \(S_{h}(. )\), and the level of decorrelation represented by \(B_{c}\)[24]. These conditions are often not met in practical scenarios [24].