# A Scalable Spatial-Temporal Correlated Non-Stationary Channel Fading Generation Method
## 3 Scalable and Real-Time Hardware Generation Method
### Temporal Correlated Fading Generation Based on Modified CORDIC Method
We can convert the input angle \(z_{1}^{HR}=\theta\) into the form of \(z_{1}^{HR}=Q\cdot\ln 2+R\), where \(Q\) is the rounded value of \(\theta/\ln 2\), and R is the value of the remainder of \(\theta/\ln 2\). The corresponding exponential output can be converted to the form of \(e^{z_{1}^{HR}}=2^{Q}\cdot e^{R}\), and the extent of the input angle convergence region increases with the value of \(Q\). Figure 3a shows the relative error of the e-exponential function calculation after expanding the range of convergence region. It can be seen from the figure that when the range of the extended convergence domain is \([-150,150]\), the relative error range of the e-exponential function calculation is maintained within the order of \([-1.21,+1.21]\times 10^{-4}\), which meets the data accuracy requirements of the hardware system. As shown in Figure 2c, the hyperbolic coordinate rotation vector mode in CORDIC is used to calculate the logarithmic function and square root function. Different from the rotation mode, the rotation factor \(d_{i}^{HV}\) in (21) is determined by \(y_{i}^{HV}\),  
\[d_{i}^{HV}= \begin{cases}+1,y_{i}^{HV}\leq 0\\ -1,y_{i}^{HV}>0\end{cases}  \tag{25}\]  
the last iteration output value in vector mode can be expressed as  
\[ \begin{split} x_{n}^{HV}&=A_{n}^{HV}\sqrt{x_{1}^{HV} {}^{2}-y_{1}^{HV}{}^{2}}\\ y_{n}^{HV}&=0\\ z_{n}^{HV}&=z_{1}^{HV}+\tanh^{-1}(y_{1}^{HV}/x_{1}^{HV})\end{split}  \tag{26}\]  
Set the initial value \(x_{1}^{HV}=\theta+1,y_{1}^{HV}=\theta-1,z_{1}^{HV}=0\); then, Equation (26) can be further expressed as  
\[ \begin{split} x_{n}^{HV}&=2A_{n}^{HV}\sqrt{\theta}\\ y_{n}^{HV}&=0\\ z_{n}^{HV}&=\tanh^{-1}(\theta-1/\theta+1)\end{split}  \tag{27}\]  
With \(\ln\theta=2\cdot tan^{-1}h(\theta-1/\theta+1)\), we obtain \(z_{1}^{HV}=\ln\theta/2\). To extend the domain of convergence, we can convert the input angle \(z_{1}^{HV}=\theta\) into the form \(\theta=\alpha\cdot 4^{Q}\), and the corresponding logarithmic function output becomes \(\ln\theta=\ln\alpha+Q\cdot 2\ln 2\), where \(\alpha\in[0.1069,9.3573]\), and \(Q\) is an integer. The range of input \(\theta\) is related to the value range of \(Q\). If the convergence area continues to expand, we need to change the value range of \(Q\). For hardware implementation, \(2^{Q}\) can be implemented by shifting, which facilitates a fixed point of the input angle. Figure 3b,c shows the relative error calculated by the  
Figure 3: The relative errors of (**a**) e–exponent function, (**b**) logarithm function, and (**c**) square root function in the extended region. logarithmic function and the square root function after expanding the range of convergence. It can be seen from the figure that when the range of the extended convergence domain is \([10,400]\), the relative error range of the logarithmic function is within the order of \([-6.66,+7.72]\times 10^{-5}\) and the relative error range of the square root function is within the order of \([+0.01,+7.23]\times 10^{-9}\), both of which meet the requirements of the hardware system for data accuracy.