## 2 Methodology
### Formulation for Modal Parameter Identification  
The subspace-based method with instrumental variables, initially proposed by [PERSON] et al. [34], was restructured to collaborate with the SWPT. The application of the SWPT to Equations (4) and (6) results in  
\[\sum_{m=0}^{2^{k}-1}\sum_{l=0}\mathbf{Z}_{m,k,l+1}\mu_{m,k,l}(t)=\mathbf{A}\sum_ {m=0}^{2^{k}-1}\sum_{l=0}\mathbf{Z}_{m,k,l}\mu_{m,k,l}(t)+\mathbf{B}\sum_{m=0}^ {2^{k}-1}\sum_{l=0}\mathbf{F}_{m,k,l}\mu_{m,k,l}(t), \tag{11}\]  
\[\sum_{m=0}^{2^{k}-1}\sum_{l=0}\mathbf{Y}_{m,k,l}\mu_{m,k,l}(t)=\mathbf{E}\sum_ {m=0}^{2^{k}-1}\sum_{l=0}\mathbf{Z}_{m,k,l}\mu_{m,k,l}(t)+\mathbf{D}\sum_{m=0}^ {2^{k}-1}\sum_{l=0}\mathbf{F}_{m,k,l}\mu_{m,k,l}(t), \tag{12}\]  
where  
\[(\mathbf{Z}_{m,k,l},\mathbf{Y}_{m,k,l},\mathbf{F}_{m,k,l})=\int_{-\infty}^{ \infty}(\mathbf{z}(t),\mathbf{y}(t),\mathbf{f}(t))\mu_{m,k,l}(t)\ dt. \tag{13}\]  
It is noteworthy that in scenarios involving multiple support excitations, matrix \(\mathbf{D}\) should be nullified. Due to the orthonormality of the basis functions, Equations (11) and (12) can be further simplified to Equations (14) and (15), respectively. \[\mathbf{Z}_{m,k,l+1}=\mathbf{A}\mathbf{Z}_{m,k,l}+\mathbf{BF}_{m,k,l}. \tag{14}\]  
\[\mathbf{Y}_{m,k,l}=\mathbf{E}\mathbf{Z}_{m,k,l}+\mathbf{DF}_{m,k,l}. \tag{15}\]  
To conduct modal identification in the wavelet domain, all the measured responses undergo transformation into the wavelet domain. When utilizing the responses of a structure within the time interval \(t\in\left[\bar{l}\Delta t,\ \left(\bar{l}+2L\right)\Delta t\right]\) for modal parameter identification, \(\mathbf{Y}_{m,k,l}\) and \(\mathbf{F}_{m,k,l}\) with \(l\in\left[\bar{l},\ \bar{l}+2L\right]\) are employed to construct Equations (14) and (15). The parameters identified are then associated with a linear system equivalent to the actual system within the window \(t\in\left[\bar{l}\Delta t,\ \left(\bar{l}+2L\right)\Delta t\right]\) when dealing with a nonlinear or time-varying structural system. The following equation is obtained from Equation (14) and (15):  
\[\mathbf{Y}_{m,k,\bar{l}+s}=\mathbf{EA}^{s}\mathbf{Z}_{m,k,\bar{l}}+\mathbf{ DF}_{m,k,\bar{l}+s}+\sum_{i=1}^{s}\mathbf{EA}^{i-1}\mathbf{BF}_{m,k,\bar{l}+s-i-1}. \tag{16}\]  
Applying Equation (16) and following the methodology outlined by [PERSON] [52] for time-domain signal processing leads to the derivation of Equation (17). \[\mathbf{\hat{Y}}_{\bar{l}}=\mathbf{\Gamma}_{s}\mathbf{\hat{Z}}_{\bar{l}}+ \mathbf{H}_{s}\mathbf{\hat{F}}_{\bar{l}}, \tag{17}\]  
where \(\mathbf{\Gamma}_{s}\) is the extended observability matrix, and  
\[\mathbf{\Gamma}_{s}=\left[\mathbf{E}^{\mathrm{T}}\ \ \ \left(\mathbf{EA} \right)^{\mathrm{T}}\ \ \ \cdots\ \ \ \left(\mathbf{EA}^{s-1}\right)^{\mathrm{T}}\right]^{\mathrm{T}}, \tag{18}\]  
where superscript \(\mathrm{T}\) denotes the transpose operation of a matrix, \(\mathbf{H}_{s}\) is the Toeplitz matrix, and  
\[\mathbf{H}_{s}=\left[ \begin{array}{ccccc}\mathbf{D}&0&0&\cdots&0\\ \mathbf{EB}&\mathbf{D}&0&\cdots&0\\ \mathbf{E}\mathbf{AB}&\mathbf{EB}&\mathbf{D}&\cdots&0\\ \vdots&&&&\\ \mathbf{EA}^{s-1}\mathbf{B}&\mathbf{EA}^{s-2}\mathbf{B}&\cdots&&\mathbf{D} \end{array}