## 2 Simulation Methods of Non-Stationary Stochastic Processes
### Spectral Representation Using the GHW Method
Later, [PERSON] and [PERSON] derived the method of expressing EPSD using continuous GHW transformation, and the relational expression is presented as follows:  
\[S\Big{(}\omega_{(m,n)},t_{(m,n)}\Big{)}=\frac{E\Big{[}\big{|}c_{m,n,k}\big{|} ^{2}\Big{]}}{2\pi(n-m)} \tag{11}\]  
A variety of approaches have been developed for obtaining the EPSD such as ARMA model [39], Hilbert-Huang transform [40], and S-transform [41]. The estimation of the EPSD using Equation (11) provides the possibility to simulate stochastic processes via GHW. It can be observed intuitively that Equation (10) is difficult to use directly to generate samples of random processes. The next step is to review a non-stationary stochastic process simulation formula. It can be obtained from Equation (9) that the time course \(f^{\text{GHW}}_{\infty}(t)\) reconstructed by the GHW function and coefficients is expressed as follows:  
\[f^{\text{GHW}}(t)=\sum_{m,n}\sum_{k}\bigg{\{}c_{(m,n),k}\cdot w_{(m,n),k} \bigg{(}t-\frac{k}{n-m}\bigg{)}+\widetilde{c}_{(m,n),k}\cdot w_{(m,n),k}^{*} \bigg{(}t-\frac{k}{n-m}\bigg{)}\bigg{\}} \tag{12}\]  
in which the superscript GHW means that the stochastic process is obtained by GHW and the range of values of the scale (\(m\),\(n\)) and translation \(k\) is infinite. The comparison of Equations (10) and (12) shows that the forms of the two equations are quite inconsistent, so a further derivation of Equation (12) is required. From the expression in Equations (7) and (9), it is obvious that both the GHW function and coefficients are complex; therefore, \(w_{(m,n),k}\) and \(c_{(m,n),k}\) can be written as follows:\[w_{(m,n),k} = \text{Re}(w_{(m,n),k})+\text{Im}(w_{(m,n),k}) \tag{13}\] \[c_{(m,n),k} = \text{Re}(c_{(m,n),k})+\text{Im}(c_{(m,n),k})\]  
in which Re (\(\cdot\)) and Im (\(\cdot\)) are used to represent the real and imaginary parts, respectively. Similar to the classical SRM as shown in Equation (6), in order to obtain a formula for simulating non-stationary stochastic processes by the superposition of trigonometric functions, the Euler formula needs to be applied and is expressed as follows:  
\[e^{ix}=\cos x+i\cdot\sin x \tag{14}\]  
Considering Equation (13), the Euler formula, and properties of trigonometric functions, the simulation formula for the one-dimensional non-stationary stochastic process via GHW is expressed as follows [30]:  
\[ \begin{array}{c}f^{\text{GHW}}(t)=\sum\limits_{m,n}\sum\limits_{k=-\infty}^{+ \infty}\sqrt{2\pi(n-m)\cdot S(\omega_{(m,n)},t_{(k,m,n)})}\cdot\frac{2\cdot\sin \left[\pi(n-m)\left(t-\frac{k}{n-m}\right)\right]}{\pi(n-m)\left(t-\frac{k}{n- m}\right)}\\ \cdot\cos\left[2\pi\left(\frac{m+n}{2}\right)\left(t-\frac{k}{n-m}\right)+ \phi_{(m,n),k}\right]\end{array}  \tag{15}\]  
Let \(b_{(m,n),k}\) be expressed as the amplitude and write it as follows:  
\[b(\omega_{(m,n)},t_{(k,m,n)})=\sqrt{2\pi(n-m)\cdot S(\omega_{(m,n)},t_{(k,m,n) })} \tag{16}\]  
In Equation (15), \(\phi_{(m,n),k}\) are identically independent random phases distributed uniformly over \([0,2\pi]\). In practical application, Equation (15) has to be truncated such that only a finite, i.e., \(k\), \(m\), \(n\), must be cut off. It can be obtained from Equation (11) that the estimated EPSD is the average EPSD on the rectangular element \([k/\Delta,(k+1)/\Delta)\times[m,n)\). Hence, Equation (15) can be rewritten as follows:  
\[ \begin{array}{c}f^{\text{GHW}}_{(m,n),k}(t)=\sqrt{2}\sum\limits_{m,n}\sum \limits_{k=k^{L}}^{k^{L}}b(\omega_{(m,n)},t_{(k,m,n)})\cdot\frac{\sqrt{2}\sin \left[\pi\Delta\left(t-\frac{k}{\Delta}\right)\right]}{\pi\Delta\left(t-\frac {k}{\Delta}\right)}\\ \cdot\cos\left[2\pi\left(\frac{m+n}{2}\right)\left(t-\frac{k}{\Delta}\right) +\phi_{(m,n),k}\right]\end{array}  \tag{17}\]  
which is deemed to be the approximation of \(f^{\text{GHW}}(t)\) in Equation (15). Equation (17) is also named the GHW method. When using the classical SRM, the number of components is determined by the frequency interval; a small frequency interval will cause more components.