# 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  
In various communication scenarios, real-time calculations of the complex exponential, e-exponent, natural logarithm, and square root functions are required for generating channel fading. To date, the above functions can be implemented through the LUT and CORDIC methods. However, the classical CORDIC method has too many iterations, poor real-time performance, and limited input convergence domain, making it unsuitable for the real-time operation of numerous fixed-point data. Figure 2 depicts the hardware implementation architecture of the CORDIC algorithm, where the above function values can be computed using rotation and vector modes in different coordinate systems. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fading Type** & **Fading Symbol** & **Communication Scenario** \\ \hline Rayleigh & \(\hat{r}_{n}^{\text{Ray}}(\xi)\) & Urban high-rise environments without LoS \\ Rice & \(\hat{r}_{n}^{\text{R}ic}(\xi)\) & Urban and suburban areas with LoS \\ Nakagami & \(\hat{r}_{n}^{\text{Nak}}(\xi)\) & High-rise buildings and rural areas \\ Weibull & \(\hat{r}_{n}^{\text{Web}}(\xi)\) & Urban areas \\ Shadow & \(\hat{r}_{n}^{\text{Sha}}(\xi)\) & Tall buildings and terrain obstacles \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The fading types supported by the hardware architecture. In the rotation mode of the circular system, using the improved CORDIC method can reduce the hardware delay and consumption of the real-time operation of the complex exponential, as shown in Figure 2a. For a hyperbolic coordinate system, the target angle \(\theta\) can be defined as a set of angles \(\{1\leq i\leq L-1|\theta^{i}=\tanh^{-1}(2^{-i})\}\) that satisfy the approximate relation \(\tan\theta^{i}\approx 2^{-i}\), where \(L\) denotes the fixed-point data bit width length, and \(i\) denotes the number of current iterations. As shown in Figure 2b, the iterative process of calculating the value of the hyperbolic sine-cosine function can be expressed as  
\[\left[ \begin{array}{c}x_{i=1}^{HR}\\ y_{i=1}^{HR}\\ z_{i+1}^{HR}\end{array} \right]=\left[ \begin{array}{ccc}1&d_{i}^{HR}\cdot 2 ^{-i}&0\\ d_{i}^{HR}\cdot 2^{-i}&1&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}x_{i}^{HR}\\ y_{i}^{HR}\\ z_{i}^{HR}\end{array} \right]+\left[ \begin{array}{c}0\\ 0\\ -d_{i}^{HR}\cdot\tanh^{-1}2^{-i}\end{array} \right] \tag{21}\]  
\[d_{i}^{HR}=\left\{ \begin{array}{c}+1,z_{i}^{HR}\geq 0\\ -1,z_{i}^{HR}<0\end{array} \right. \tag{22}\]  
where \(z_{i}^{HR}\) denotes the remaining rotation angle, \(d_{i}^{HR}=\pm 1\) represents the rotation direction factor, and \(i\) denotes the number of current iterations. For the iterative process of the hyperbolic system, when the iteration number satisfies \(i=3k+1\), the iteration must be repeated. In CORDIC hyperbolic coordinate rotation mode, the final output value can be expressed as  
\[x_{n}^{HR} =A_{n}^{HR}(x_{1}^{HR}\cosh z_{1}^{HR}+y_{1}^{HR}\sinh z_{1}^{HR}) \tag{23}\] \[y_{n}^{HR} =A_{n}^{HR}(x_{1}^{HR}\sinh z_{1}^{HR}+y_{1}^{HR}\cosh z_{1}^{HR})\] \[z_{n}^{HR} =0\]  
where \(A_{n}^{HR}=\prod_{i=1}^{n-1}\sqrt{1-2^{-2i}}\) when \(n\rightarrow\infty\),\(A_{n}^{HR}\approx 0.82815936\); therefore, the modulo length compensation factor is \(K_{n}^{HR}\approx 1.20749706\). The sum of the rotation angles is \(\sum_{i=1}^{n}\tanh^{-1}2^{-i}=1.1181\), and the value \(z_{i}^{HR}\) range of the input angle is limited to \([-1.1181,+1.1181]\). Set the initial value \(x_{1}^{HR}=1,y_{1}^{HR}=0,z_{1}^{HR}=\theta\); then, Equation (23) can be further expressed as  
\[x_{n}^{HR} =\cosh\theta/K_{n}^{HR} \tag{24}\] \[y_{n}^{HR} =\sinh\theta/K_{n}^{HR}\] \[z_{n}^{HR} =0\]  
Figure 2: Hardware architecture of the CORDIC algorithm. However, in the nonlinear function transformation based on Gaussian sequence, taking lognormal fading as an example, the actual input angle will be outside the convergence region, so it is necessary to expand the convergence region.