## 3 Spectral Representation Using SGHWs
### Simulation Method of the Non-Stationary Stochastic Process Via SGHW  
As shown in Equation (17), a stochastic process using the GHW method can be represented in the following form:  
\[f_{(m,n),k}^{\text{GHW}}(t)=\sqrt{2}{\sum_{m,n}}\sum_{k}f_{(m,n),k}(t) \tag{24}\]  
in which\(f_{(m,n),k}\) (\(t\)) is the harmonic component at scale \(m\), \(n\), \(k\) and can be written as follows:  
\[f_{(m,n),k}(t) =b(\omega_{(m,n)},t_{(k,m,n)})\cdot\frac{\sqrt{2}{\sin}[\pi\alpha (t-\frac{k}{\Delta})]}{\pi\alpha(t-\frac{k}{\Delta})} \tag{25}\] \[\cdot\cos\left[2\pi(\frac{m+n}{2})\left(t-\frac{k}{\Delta} \right)+\phi_{(m,n),k}\right]\]  
In Equation (25), to facilitate the calculation, the angular frequencies \(\omega\ _{(m,n)}\) are generally taken as the midpoint of the frequency band (\(m\),\(n\)). \(t_{(k,m,n)}\) is related to the parameter \(k\) and bandwidth \(\Delta\). At different scales (\(m_{i}\), \(n_{i}\)) and k\({}_{j}\), \(\omega_{(m_{i},n_{i})}\) and \(t_{(k_{j},m_{i},n_{i})}\) are expressed as follows:  
\[ \begin{array}{ll}\omega_{(m_{i},n_{i})}=\frac{m_{i}+n_{i}}{2}\cdot 2\pi= \omega_{i}&i=\ 1,2,\cdot\cdot\cdot,N,\\ t_{(k_{j},m_{i},n_{i})}=\frac{k_{j}}{n_{i}-m_{i}}=\frac{k_{j}}{\Delta}&j=1,\ 2,\cdot\cdot\cdot,M.\end{array}  \tag{26}\]  
Hence, the EPSD \(S(\omega_{(m,n)},t_{(k,m,n)})\) derived by Equation (11) can be rewritten as \(S(\pi\cdot(m+n),k/\Delta)\) and further extended to the form of the following matrix. \[S\bigg{(}\pi\cdot(m+n),\frac{k}{\Delta}\bigg{)}=S\bigg{(}\omega,\frac{k}{ \Delta}\bigg{)}= \begin{bmatrix}S_{(m_{1},n_{1}),k_{1}}&S_{(m_{1},n_{1}),k_{2}} &\cdots&S_{(m_{1},n_{1}),k_{M}}\\ S_{(m_{2},n_{2}),k_{1}}&S_{(m_{2},n_{2}),k_{2}}&\cdots&S_{(m_{2},n_{2}),k_{M}} \\ \vdots&\vdots&\ddots&\vdots\\ S_{(m_{N},n_{N}),k_{1}}&S_{(m_{N},n_{N}),k_{2}}&\cdots&S_{(m_{N},n_{N}),k_{M}} \end{bmatrix}  \tag{27}\]  
in which  
\[S_{(m_{i},n_{i}),k_{j}}=S\bigg{(}\pi\cdot(m_{i}+n_{i}),\frac{k_{j}}{\Delta} \bigg{)}=S\bigg{(}\omega_{i},\frac{k_{j}}{\Delta}\bigg{)}\ for\ \begin{array}{ll}i=1,2,\cdot\cdot\cdot,N,\\ j=1,\ 2,\cdot\cdot\cdot,M.\end{array}  \tag{28}\]  
Combining Equations (17), (26), and (28), Equation (17) can be rewritten as follows:  
\[ \begin{array}{ll}f^{\text{GHW}}_{(m,n),k}(t)=\sqrt{2}\sum\limits_{i=1}^{N} \sum\limits_{j=1}^{M}\sqrt{2\pi\Delta\cdot S\Big{(}\omega_{i},\frac{k_{j}}{ \Delta}\Big{)}}\cdot\frac{\sqrt{2}\sin\left[\pi\Delta\left(t-\frac{k_{j}}{ \Delta}\right)\right]}{\pi\Delta\left(t-\frac{k_{j}}{\Delta}\right)}\\ \cdot\cos\left[\omega_{i}\Big{(}t-\frac{k_{j}}{\Delta}\Big{)}+\phi_{i,j} \right]\end{array}  \tag{29}\]  
In the above formula, the number of harmonic components of the stochastic process is obviously \(N\times M\), while the random variables are \(2\times N\times M.\) Only when the number of terms of series truncation in Equation (17) and the number of random variables required for the simulation are large enough, the simulation effect can be very close to the actual situation. In the SHF method, besides the random phase angles, the frequencies are also taken as random variables. Different from the SHF method, because of the properties of HWs, in the SGHW method, both the wavelet levels and time translation factors should be considered when constructing random variables. Therefore, in the derivation of SGHW expression using the GHW method, to obtain higher simulation accuracy with fewer components, both angular frequencies \(\omega_{i}\) and the time shift factor \(k_{j}\) are taken as independent random variables. The random variables are denoted as \(\Theta_{l}\) and \(\Omega_{g}\), respectively. To this end, the SGHW representation of stochastic process is introduced as follows:  
\[ \begin{array}{ll}f^{\text{SGHW}}_{(m,n),k}(t)=\sqrt{2}\sum\limits_{l=1}^{L} \sum\limits_{d=1}^{D}b(\Theta_{l},\frac{\Omega_{d}}{\Delta})\cdot\frac{\sqrt{2 }\sin\left[\pi\Delta\left(t-\frac{\Omega_{d}}{\Delta}\right)\right]}{\pi\Delta \left(t-\frac{\Omega_{d}}{\Delta}\right)}\\ \cdot\cos\left[\Theta_{l}\left(t-\frac{\Omega_{d}}{\Delta}\right)+\phi_{l,d} \right]\end{array}