## 2 Methodology
### Stationary Wavelet Packet Transform  
It is widely recognized that the wavelet packet transform offers a superior time-frequency localization of signals compared to the discrete wavelet transform. The SWPT exhibits advantages over the wavelet packet transform in two key aspects. Firstly, the SWPTis a translation-invariant transformation, providing excellent computational efficiency for functions with time shifts. Additionally, the SWPT eliminates the need for down-samplers and up-samplers present in the wavelet packet transform [48].  
The SWPT decomposes a measured signal \(g(t)\) in space \(U_{0}^{(0)}\) into its subspaces as follows (Figure 1):  
\[\begin{split} U_{0}^{(0)}&=\ U_{1}^{(0)}\oplus U_{1 }^{(1)}=U_{2}^{(0)}\oplus U_{2}^{(1)}\oplus U_{2}^{(2)}\oplus U_{2}^{(3)}=\\ &\cdots=U_{k}^{(0)}\oplus U_{k}^{(1)}\oplus U_{k}^{(2)}\oplus \cdots\oplus U_{k}^{(2^{k}-2)}\oplus U_{k}^{(2^{k}-1)},\end{split} \tag{7}\]  
where \(\oplus\) represents the union of two subspaces; subscript \(k\) denotes the level of decomposition, and \(U_{k}^{(i)}\bot U_{k}^{(j)}\) for \(i\
eq j\). Space \(U_{k-1}^{(2^{k-1}-n)}\) is decomposed into \(U_{k}^{(2^{k}-2n+1)}\) and \(U_{k}^{(2^{k}-2n)}\) with \(n\leq k\). The subspace \(m\) at level \(k\), \(U_{k}^{(m)}\), is expanded through the set of orthonormal basis functions \(\left\{\mu_{m,k,l}(t),\,l\in Z\right\}\), where  
\[\mu_{m,k,l}(t)=\frac{1}{\sqrt{2^{k}}}\mu_{m}\left(\frac{t-l\Delta t}{2^{k}} \right), \tag{8}\]  
and \(\frac{1}{\Delta t}\) is the sampling rate for the signal under consideration. The basis functions are constructed by selecting a mother wavelet function and its corresponding scale function [49; 50]. The measured signal \(g(t)\) at the \(k^{\text{th}}\) level of decomposition, which corresponds to the subspaces \(U_{k}^{(m)}\) with \(m=0,1,2,\ldots,2^{k}-1\), is expressed as  
\[g(t)=\sum_{m=0}^{2^{k}-1}\sum_{l=0}\overline{a}(m,k,l)\mu_{m,k,l}(t), \tag{9}\]  
where \(\overline{a}(m,k,l)\) are the stationary wavelet packet coefficients in subspace \(U_{k}^{(m)}\). Using the property of translation invariance yields  
\[g(t-\tau\Delta t)=\sum_{m=0}^{2^{k}-1}\sum_{l=0}\overline{a}(m,k,l-\tau)\mu_ {m,k,l}(t). \tag{10}\]  
Figure 1: Decomposition of subspace.  
The frequency range of the signal decomposed into subspace \(U_{k}^{(m)}\) is dictated by the frequency range of the basis functions of \(U_{k}^{(m)}\). In this study, the Meyer mother wavelet \(\psi(t)\) and its corresponding scale function \(\phi(t)\)[51], as depicted in Figure 2a, with their Fourier spectra \(\left|\hat{\psi}(\omega)\right|\) and \(\left|\hat{\phi}(\omega)\right|\), shown in Figure 2b, are utilized to construct the basis functions of the subspaces. As illustrated in Figure 2b, \(\psi(t)\) and \(\phi(t)\) function as a band-pass and low-pass filter, respectively, with their frequency ranges partly overlapping.  
It is advantageous to choose a suitable subspace for modal parameter identification if distinct, main, and preserved frequency ranges are designated for each subspace at the same level of decomposition, ensuring non-overlapping frequency ranges between different subspaces. To facilitate this, the main preserved frequency ranges for \(\psi(t)\) and \(\phi(t)\) are defined as [0.5, 1] Hz and [0, 0.5] Hz, respectively (Figure 2b). The values of \(\left|\hat{\psi}(\omega)\right|\) and \(\left|\hat{\phi}(\omega)\right|\) exhibit larger values than \(50\sqrt{2}\%\) within these two frequency ranges, respectively. As a result, Figure 3 illustrates the main preserved frequency ranges in different subspaces applicable in this context.  
Figure 3: Main preserved frequency ranges in different subspaces.  
Figure 2: (**a**) Plots for [PERSON] mother wavelet \(\phi(t)\) and corresponding scale function \(\psi(t)\); (**b**) Fourier spectra of [PERSON] mother wavelet and scale function.