# A Scalable Spatial-Temporal Correlated Non-Stationary Channel Fading Generation Method
## 2 Correlated Channel Fading Model  
Wireless signals propagate along various paths with different delays, powers, and angles. This causes unpredictable signal attenuation known as channel fading \(p_{n}^{i}(t)\), which can be expressed as  
\[p_{n}^{i}(t)=r_{n}^{i}(t)\cdot\exp\Bigl{\{}j\varphi_{n}^{i}(t)\Bigr{\}},n=1,2, \cdots,N \tag{1}\]  
where \(r_{n}^{i}(t)\) denotes the envelope of channel fading, \(i\in\{\)Ray,Ric,Nak,Web,Sha\(\}\), \(\varphi_{n}^{i}(t)\) denotes the random phase of channel fading and obeys the uniform distribution of \(U\sim(0,2\pi]\), and \(N\) denotes the number of multipath clusters. Different types of channel fading occur due to various channel propagation scenarios. In the non-line of sight (NLoS) propagation environment, the PDF of the Rayleigh fading envelope can be expressed as  
\[f_{r}^{\text{Ray}}(x)=\frac{x}{\sigma_{0}^{2}}\exp(\frac{-x^{2}}{2\sigma_{0}^{2 }}) \tag{2}\]  
where \(\sigma_{0}\) is the standard deviation. In the line of sight (LoS) propagation environment, the PDF of the Rician fading envelope can be expressed as  
\[f_{r}^{\text{Ric}}(x)=\frac{x}{\sigma_{0}^{2}}\cdot\exp\left(\frac{x^{2}+A_{0} ^{2}}{2\sigma_{0}^{2}}\right)\cdot J_{0}(\frac{xA_{0}}{\sigma_{0}^{2}}) \tag{3}\]  
where \(A_{0}\) represents the signal amplitude, and \(J_{0}(\cdot)\) represents the zero-order modified Bessel function of the first kind. The multipath fading due to the superposition of NLoS components can be modeled by the Nakagami distribution, and its envelope PDF can be expressed as  
\[f_{r}^{\text{Nak}}(x)=\frac{2}{\Gamma(m_{k})}\left(\frac{m_{k}}{\Omega_{k}} \right)\cdot\overset{m_{k}}{x}\overset{m_{k}}{x}\overset{1}{\sim}\exp\left(- \frac{m_{k}}{\Omega_{k}}x^{2}\right) \tag{4}\]  
where \(\Gamma(m_{k})\) denotes the gamma function, \(m_{k}\) represents the shape factor which can evaluate the severity of channel fading, and \(\Omega_{k}\) represents the average power of each cluster. Furthermore, Weibull fading is more consistent with channel fading in indoor and outdoor environments. The PDF of the Weibull fading amplitude can be expressed as  
\[f_{r}^{\text{Web}}(x)=\frac{m_{\gamma}}{\beta_{w}}\biggl{(}\frac{x}{\beta_{w} }\biggr{)}^{m_{\gamma}-1}\cdot\exp\left(\frac{x}{\beta_{w}}\right)^{m_{\gamma}} \tag{5}\]  
where \(m_{\gamma}\) determines the shape of the magnitude PDF, and \(\beta_{w}\) represents the scale parameter. In addition, the PDF of the shadow fading envelope can be further expressed as  
\[f_{r}^{\text{Sha}}(x)=\frac{1}{\sqrt{2\pi}\sigma_{0}x}\exp\left[-\frac{\left( \ln x-m_{\beta}\right)^{2}}{2\sigma_{0}^{2}}\right] \tag{6}\]  
where the measured data show that \(\sigma_{0}\in[1.5,7]\) dB, and \(m_{\beta}\) represents the mean value. The TCF is a crucial tool in wireless channel modeling, providing information about the temporal characteristics of the channel. It helps estimate the coherence time and the severity of the channel fading. In a MIMO communication system, a single cluster signal will be affected by obstacles, and there is an autocorrelation characteristic between different time delays. It can be expressed as  
\[A_{n}^{p^{j}}(\tau)=\frac{\mathrm{E}\left[p_{n}^{i}(t)p_{n}^{i}(t+\tau)\right] -\mathrm{E}\left[p_{n}^{i}(t)\right]\mathrm{E}\left[p_{n}^{i}(t+\tau)\right]}{ \mathrm{var}\left[p_{n}^{i}(t+\tau)\right]} \tag{7}\]where \(\tau\) represents the time delay, \(n\in\{1,2,\cdots,N\}\), and \(\mathrm{E}[\cdot]\) and \(\mathrm{var}[\cdot]\) denote the expectation and variance, respectively.