# A Scalable Spatial-Temporal Correlated Non-Stationary Channel Fading Generation Method
## 3 Scalable and Real-Time Hardware Generation Method  
### Overview of Hardware Generation Structure  
For MIMO systems, traditional hardware implementation methods for channel simulation consist of pre-storage and real-time methods. The pre-storage generation algorithm requires a large amount of storage, and it is challenging to support time-varying dynamic scenes. On the other hand, the real-time generation algorithm involves a large amount of computation, and it is difficult to support large-scale multi-channel scenarios. Therefore, a flexible and real-time hardware generation architecture is established using the matrix block approach shown in Figure 1. The correlation sub-matrices \(\mathbf{H}_{11}\), \(\mathbf{H}_{12}\), \(\mathbf{H}_{21}\), and \(\mathbf{H}_{22}\) are expressed as:  
\[\mathbf{H}_{(i+1)(j+1)}=\left[ \begin{array}{ccccc}h_{\big{(}\frac{P}{2}\cdot i +1\big{)}\big{(}\frac{Q}{2}\cdot j+1\big{)}}&h_{\big{(}\frac{P}{2}\cdot i+1 \big{)}\big{(}\frac{Q}{2}\cdot j+2\big{)}}&\cdots&h_{\big{(}\frac{P}{2}\cdot i +1\big{)}\big{(}\frac{Q}{2}\cdot j+\frac{Q}{2}\big{)}}\\ h_{\big{(}\frac{P}{2}\cdot i+2\big{)}\big{(}\frac{Q}{2}\cdot j+1\big{)}}&h_{ \big{(}\frac{P}{2}\cdot i+2\big{)}\big{(}\frac{Q}{2}\cdot j+2\big{)}}&\cdots&h _{\big{(}\frac{P}{2}\cdot i+2\big{)}\big{(}\frac{Q}{2}\cdot j+\frac{Q}{2} \big{)}}\\ \vdots&\vdots&\ddots&\vdots\\ h_{\big{(}\frac{P}{2}\cdot i+\frac{P}{2}\big{)}\big{(}\frac{Q}{2}\cdot i+1 \big{)}}&h_{\big{(}\frac{P}{2}\cdot i+\frac{P}{2}\big{)}\big{(}\frac{Q}{2} \cdot j+2\big{)}}&\cdots&h_{\big{(}\frac{P}{2}\cdot i+\frac{P}{2}\big{)}\big{(} \frac{Q}{2}\cdot j+\frac{Q}{2}\big{)}}\end{array} \right] \tag{9}\]  
where \(i,j\in\{0,1\},h_{qp}\) represents the sub-channel fading, and \(P\) and \(Q\) denote the number of transmitting antennas and the number of receiving antennas, respectively. It can be seen from the figure that \(\mathbf{X}_{1}\) and \(\mathbf{X}_{2}\) are channel transmission sub-matrices, and \(\mathbf{Y}_{1}\) and \(\mathbf{Y}_{2}\) are channel receiving sub-matrices, which can be expressed as  
\[\mathbf{X}_{1}=\left[ \begin{array}{ccccc}x_{1}&x_{2}&\cdots&x_{\frac{P}{2}} \end{array} \right]^{T},\mathbf{X}_{2}=\left[ \begin{array}{ccccc}x_{\big{(} \frac{P}{2}+1\big{)}}&x_{\big{(}\frac{P}{2}+2\big{)}}&\cdots&x_{P}\end{array}  \right]^{T} \tag{10}\]  
\[\mathbf{Y}_{1}=\left[ \begin{array}{ccccc}y_{1}&y_{2}&\cdots&y_{\frac{Q}{2}} \end{array} \right]^{T},\mathbf{Y}_{2}=\left[ \begin{array}{ccccc}y\big{(} \frac{Q}{2}+1\big{)}&y\big{(}\frac{Q}{2}+2\big{)}&\cdots&y_{Q}\end{array}  \right]^{T} \tag{11}\]  
In (10), \(x_{p}\) denotes the signal transmitted by the \(p\)-th antenna. In (11), \(y_{q}\) denotes the signal received by the \(q\)-th antenna, which can be expressed as  
\[y_{q}=y_{q,1}+y_{q,2}=(y_{q1}+y_{q2}+\cdots+y_{q\frac{P}{2}})+(y_{q}\big{(} \frac{p}{2}+1\big{)}+y_{q}\big{(}\frac{p}{2}+2\big{)}+\cdots+y_{qP}) \tag{12}\]  
where \(y_{q,1}\) and \(y_{q,2}\) denote the channel components output by the superposition of the two subsystems after the matrix block in the MIMO system, and \(y_{q}\) denotes the output of their superposition. In (12), \(y_{qp}=h_{qp}\cdot x_{p}(q=2,\ldots,Q,p=2,\ldots,P)\) and \(h_{qp}\) can be obtained by the generation method of spatial-temporal correlated channel fading. We divide the simulation unit into four subsystems with the same structure for parallel computing, which reduces system complexity, increases maintainability, and has better real-time performance. The above single-cluster correlated fading can be obtained by applying a series of nonlinear transformations to the Gaussian sequence. In practice, the channel fading in most scenariospresents non-stationary characteristics.