## 2 Related Works
### Simulation of Non-Stationary Process Using the Classic SRM  
\(\mathbf{S}(\omega,t)\) is the EPSD of a zero-mean non-stationary process with components \(x_{1}(t),x_{2}(t),\cdots,x_{n}(t)\), which can be expressed as [15]:  
\[\mathbf{S}(\omega,t)=\left[ \begin{array}{cccc}S_{11}(\omega,t)&S_{12}( \omega,t)&\cdots&S_{1n}(\omega,t)\\ S_{21}(\omega,t)&S_{22}(\omega,t)&\cdots&S_{2n}(\omega,t)\\ \vdots&\vdots&\ddots&\vdots\\ S_{n1}(\omega,t)&S_{n2}(\omega,t)&\cdots&S_{nn}(\omega,t)\\ \end{array} \right] \tag{3}\]  
Figure 3: Estimating and validating the evolutionary power spectral density (EPSD) of the measured turbulence. \[S_{ij}(\omega,t)=\sqrt{S_{ii}(\omega,t)S_{jj}(\omega,t)}\gamma_{ij}(\omega) \tag{4}\]  
where \(\omega\) represents the circular frequency; \(S_{ii}(\omega,t)=S_{ii}(-\omega,t)\) represents the auto-EPSD; \(S_{ij}(\omega,t)=S_{ij}^{*}(-\omega,t)\) represents the cross-EPSD (\"\({}^{*}\)\" denotes the complex conjugate); \(\gamma_{ij}(\omega)\) are complex coherence functions between \(x_{i}(t)\) and \(x_{j}(t)\). The coherence matrix \(\Gamma(\omega,t)\) can be represented by: [15]  
\[\mathbf{\Gamma}(\omega,t)=\left[ \begin{array}{cccc}1&\gamma_{12}(\omega,t)& \cdots&\gamma_{1n}(\omega,t)\\ \gamma_{21}(\omega,t)&1&\cdots&\gamma_{2n}(\omega,t)\\ \vdots&\vdots&\ddots&\vdots\\ \gamma_{n1}(\omega,t)&\gamma_{n2}(\omega,t)&\cdots&1\end{array} \right] \tag{5}\]  
The cross-correlation function can be determined by [15]:  
\[\mathbf{R}(t,t+\tau)=\left[ \begin{array}{cccc}R_{11}(t,t+\tau)&R_{12}(t,t+ \tau)&\cdots&R_{1n}(t,t+\tau)\\ R_{21}(t,t+\tau)&R_{22}(t,t+\tau)&\cdots&R_{2n}(t,t+\tau)\\ \vdots&\vdots&\ddots&\vdots\\ R_{n1}(t,t+\tau)&R_{n2}(t,t+\tau)&\cdots&R_{nn}(t,t+\tau)\end{array} \right] \tag{6}\]  
\[R_{ij}(t,t+\tau)=\int_{-\infty}^{\infty}\sqrt{S_{ii}(\omega,t)S_{jj}(\omega,t+ \tau)}\gamma_{ij}(\omega)e^{i\omega\tau}d\omega \tag{7}\]  
The cross-correlation matrix for non-stationary processes is determined by \(t\) and \(t+\tau\) (\(\tau\) is the time lag). To simulate the one-dimensional, \(n\) variate (1D-\(n\)V) non-stationary stochastic process \(\mathbf{x}(t)=\left[x_{1}(t),x_{2}(t),\cdots,x_{n}(t)\right]^{T}\), where \(T\) represents the transpose, the \(\mathbf{S}(\omega,t)\) should be decomposed at each time instant \(t\), into the following product [15]:  
\[\mathbf{S}(\omega,t)=\mathbf{H}(\omega,t)\mathbf{H}^{T*}(\omega,t) \tag{8}\]  
[PERSON]'s method can finish this decomposition, and \(\mathbf{H}(\omega,t)\) can be represented by:  
\[\mathbf{H}(\omega,t)=\left[ \begin{array}{cccc}H_{11}(\omega,t)&0&\cdots&0\\ H_{21}(\omega,t)&H_{22}(\omega,t)&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ H_{n1}(\omega,t)&H_{n2}(\omega,t)&\cdots&H_{nn}(\omega,t)\end{array} \right] \tag{9}\]  
where \(H_{ii}(\omega,t)\) are real and positive functions, while \(H_{ij}(\omega,t)\) are generally complex. The diagonal elements \(H_{ii}(\omega,t)\) satisfy the following relation:  
\[H_{ii}(\omega,t)=H_{ii}(-\omega,t) \tag{10}\]  
Furthermore, \(H_{ij}(\omega,t)\) can be written in polar form as:  
\[H_{ij}(\omega,t)=\left|H_{ij}(\omega,t)\right|e^{i\beta_{ij}(\omega,t)} \tag{11}\]  
\[\beta_{ij}(\omega,t)=\tan^{-1}\left(\frac{\text{Im}\left[H_{ij}(\omega,t) \right]}{\text{Re}\left[H_{ij}(\omega,t)\right]}\right) \tag{12}\]  
where Re and Im represent the real and imaginary parts of a complex number, respectively.