## 3 Spectral Representation Using SGHWs
### Simulation Method of the Non-Stationary Stochastic Process Via SGHW
 \tag{30}\]  
in which \(b(\Theta_{l},\Omega_{d}/\Delta)\) are the amplitudes; \(\phi_{l,d}\) are still independent random phase angles and distributed uniformly over \([0,\ 2\pi]\); and \(\Theta_{l}\) and \(\Omega_{d}\) are independent random variables with PDFs \(p_{\Theta_{l}}(\omega)\) and \(p_{\Omega_{d}}(k)\), respectively. For application convenience, let the random component be subject to uniform distribution. This idea is also applied in SHF-II. Starting from the optimal choice of sampling frequency, the frequency components \(\Theta_{l}\) are randomly distributed in the subinterval \([\omega_{i}^{\ L},\omega_{i}^{\ L}]\) and \(\cup_{i=1}^{1}[\omega_{i}^{L},\omega_{i}^{U}]=[\omega_{L},\omega_{U}]\). The time translation components \(\Omega_{d}\) are randomly distributed in the subinterval \([k_{j}^{\ L},k_{j}^{\ L}]\) and \(\cup_{j=1}^{J}[k_{j}^{L},k_{j}^{U}]=[k_{L},k_{U}]\). In addition, \(\Omega_{d}\) should be integers. Here, the indicators \(L\) and \(U\)represent the lower and upper values, respectively. At the same time, different subintervals do not intersect with each other. The specific PDFs of \(\Theta_{l}\) and \(\Omega_{d}\) are expressed as follows:  
\[ \begin{split} p_{\Theta_{l}}(\omega)=\left\{\begin{array}{ll}\frac {1}{\omega_{l}^{U}-\omega_{l}^{U}}&\omega\in\left[\omega_{i}^{L},\omega_{i}^{ U}\right].\\ 0&\omega\
otin\left[\omega_{i}^{L},\omega_{i}^{U}\right].\\ \frac{1}{k_{j}^{U}-k_{j}^{U}}&k\in\left[k_{j}^{L},k_{j}^{U}\right].\\ 0&k\
otin\left[k_{j}^{L},k_{j}^{U}\right].\end{array}\right.\end{split}  \tag{31}\]  
In fact, the SHF method only randomizes the discretized frequency and is derived within the one-dimensional category, i.e., the PDF of the random frequencies is a one-dimensional PDF. However, the SGHW method randomizes two parameters. The proof is required in a two-dimensional interval consisting of angular frequencies and time translation factors. Therefore, the two-dimensional joint probability density function (JPDF) rather than the one-dimensional PDF should be applied in the SGHW method. The JPDF of \(\Theta_{l}\) and \(\Omega_{d}\) is expressed as follows:  
\[ \begin{split} p_{\Theta_{l}\Omega_{d}}(\omega,k)=\begin{cases} \frac{1}{(\omega_{l}^{U}-\omega_{l}^{U})\cdot(k_{j}^{U}-k_{j}^{U})}&\omega \in\left[\omega_{i}^{L},\omega_{i}^{U}\right],k\in\left[k_{j}^{L},k_{j}^{U} \right].\\ 0&\omega\
otin\left[\omega_{i}^{L},\omega_{i}^{U}\right],k\in\left[k_{j}^{ L},k_{j}^{U}\right].\end{cases}\end{split}  \tag{32}\]  
In Equation (30), the number of harmonic components is reduced to \(L\times D\). This is similar to Equation (23), which has been derived in [37]. Using the auto-correlation function of \(f_{(m,n),k}^{\text{RGHW}}(t)\), it can be easily proved that the JPDF \(p_{\Theta_{l}\Omega_{d}}(\omega,k)\) of \(\Theta_{l}\) and \(\Omega_{d}\) satisfies the following:  
\[ \begin{split}\left|b(\omega_{(m,n)^{\prime}}t_{(k)})\right|^{2} \cdot p_{\Theta_{l}\Omega_{d}}(\omega_{(m,n)},k)=2\Big{|}A(\omega_{(m,n)},t_{( k)})\Big{|}^{2}\cdot S_{ff}(\omega_{(m,n)})&=2S_{ff}(\omega_{(m,n)^{\prime}}t_{( k)})\end{split}  \tag{33}\]  
The amplitude is given by the following:  
\[b(\omega_{(m,n)},t_{(k)})=2\cdot A(\omega_{(m,n)},t_{(k)})\sqrt{\frac{S_{ff}( \omega_{(m,n)})}{p_{\Theta_{l}\Omega_{d}}\left(\omega_{(m,n)},k\right)}} \tag{34}\]  
When the above conditions are satisfied, the two-sided EPSD of \(f_{(m,n),k}^{\text{RGHW}}(t)\) is exactly \(S_{ff}(\omega_{(m,n)},t_{(k)})\) regardless of the number of components of the non-stationary random process. **Remark 1**. : _For the SGHW method, as shown in Equation (30), when the conditions in Equation (33) or (34) are satisfied, the frequencies, time translation factors, and phase angles are all random variables. These random variables are artificially defined as independent uniform distribution.