# A Scalable Spatial-Temporal Correlated Non-Stationary Channel Fading Generation Method
## 3 Scalable and Real-Time Hardware Generation Method
### Overview of Hardware Generation Structure
The SoFM method is used to generate the Gaussian random variable \(\hat{\sigma}_{n}(\xi)\), which can be expressed as  
\[\hat{\sigma}_{n}(\xi)=\tfrac{1}{\sqrt{M}}\sum\limits_{m=1}^{M}\exp\left\{j(2\pi \sum\limits_{k=0}^{\xi}T_{s}\hat{f}_{n,m}[k]+\hat{\varphi}_{n,m})\right\}= \tfrac{1}{\sqrt{M}}\sum\limits_{m=1}^{M}\exp\{j(\hat{\theta}_{n,m}(\xi))\} \tag{13}\]  
where \(\xi\) represents the time index of the discrete domain, \(M\) denotes the number of frequency modulation, \(T_{s}\) represents the clock period of hardware, and \(\hat{\theta}_{n,m}\), \(\hat{f}_{n,m}\), and \(\hat{\varphi}_{n,m}\) denote the time-varying phase, discrete Doppler frequency, and initial phase, respectively. The initial phase is uniformly distributed over \((0,2\pi]\). The discrete Doppler frequency \(\hat{f}_{n,m}\) can be characterized by the DPSD \(\hat{S}_{\sigma_{n}\sigma_{n}}(f)\). We first divide the DPSD into \(M\) parts and determine \(\hat{f}_{n,m}\) with the MEA method [24], which can be expressed as  
where \(f_{0}\) denotes the spectrum offset, and \(f_{k}\) represents the discrete frequency stepping. Assuming that the inverse function \(\hat{I}_{n,m}^{-1}(f_{k})\) exists, the discrete Doppler frequency \(\hat{f}_{n,m}\) can be further expressed as \(\hat{f}_{n,m}=\hat{I}_{n,m}^{-1}(0)\). Rayleigh fading can be expressed as the complex sum of two sets of mutually orthogonal Gaussian random sequences  
\[\hat{r}_{n}^{\text{Ray}}(\xi)=|\hat{\sigma}_{n,1}(\xi)+j\hat{\sigma}_{n,2}(\xi)| \tag{16}\]  
where \(\sigma_{n,1}(\xi)\) and \(\sigma_{n,2}(\xi)\) represent the in-phase component and the quadrature component of the Gaussian random sequence, respectively. Rice fading can be simulated by superimposing the line-of-sight path on the basis of Rayleigh fading, which can be expressed as  
\[\hat{r}_{n}^{\text{R}ic}(\xi)=\left|\rho_{m}\cdot\exp(j(\sum_{k=0}^{\xi}T_{s} \hat{f}_{LoS}[k]+\hat{\sigma}_{LoS}))+\hat{r}_{n}^{\text{Ray}}(\xi)\right| \tag{17}\]  
where \(\rho_{m}\) represents the amplitude, \(\hat{f}_{LoS}\) and \(\hat{\sigma}_{LoS}\) denote the Doppler frequency and initial phase of LoS component, respectively. The Formula (17) can be transformed into Rayleigh fading When \(\rho_{m}=0\). Nakagami fading can be simulated by several independent Gaussian random sequences, which can be further expressed as  
\[\hat{r}_{n}^{\text{Nak}}(\xi)=\sqrt{\zeta_{n}^{Sha}/2m}\cdot\sqrt{\left[\hat{ \sigma}_{n,1}(\xi)\right]^{2}+\left[\hat{\sigma}_{n,2}(\xi)\right]^{2}+\cdots+ \left[\hat{\sigma}_{n,2m}(\xi)\right]^{2}} \tag{18}\]  
where \(\zeta_{n}^{Sha}\) represents the average power of each cluster signal fading. According to the measured data, Weibull fading can be generated by two Gaussian random sequences with zero mean and orthogonal to each other, which can be further expressed as  
\[\hat{r}_{n}^{\text{Web}}(\xi)=\exp(w_{m}/2\cdot\ln(\hat{\sigma}_{n,1}(\xi)+j \hat{\sigma}_{n,2}(\xi))=\exp(w_{m}/2\cdot\ln(\hat{r}_{n}^{\text{Ray}}(\xi))) \tag{19}\]  
where \(w_{m}\) represents Weibull form factor, when \(w_{m}=2\), (19) degenerates to a Rayleigh distribution. Furthermore, shadow fading can usually be modeled as a Lognormal distribution, which can be further expressed as  
\[\hat{r}_{n}^{\text{Sha}}(\xi)=\exp\bigl{(}\xi_{n,\beta}\hat{\sigma}_{n,0}(\xi) +\alpha_{n,\beta}\bigr{)} \tag{20}\]  
where \(\xi_{n,\beta}\) and \(\alpha_{n,\beta}\) denote the standard deviation and mean of shadow fading, respectively. Based on the above, the types of channel fading that can be implemented by this hardware architecture are summarized in Table 1.