## 2 Background  
### Operational Modal Analysis  
In a linear N degree of freedom system, the relationship between the response and excitation of the power spectral density (PSD) matrices can be expressed as  
\[\mathbf{S}_{yy}(j\omega)=\mathbf{H}^{*}(j\omega)\mathbf{S}_{xx}(j\omega) \mathbf{H}^{\mathrm{T}}(j\omega), \tag{1}\]  
where \(\mathbf{S}_{yy}\) and \(\mathbf{S}_{xx}\) are the response and excitation PSD matrices, respectively. \({}^{**}\) is the conjugacy of the matrix, and \({}^{\mathrm{T}}\) denotes the transposition operation. Additionally, \(\mathbf{H}(j\omega)\) is the frequency response function (FRF) matrix:  
\[\mathbf{H}(j\omega)=\sum_{i=1}^{N}\left[\frac{\mathbf{R}_{r}}{|j\omega-s_{r} }+\frac{\mathbf{R}_{i}}{|j\omega-s_{i}^{*}|}\right], \tag{2}\]\[\mathbf{R}_{r}=\mathbf{\phi}_{r}\mathbf{\gamma}_{r}^{\rm T}, \tag{3}\]  
where \(s_{r}=-p_{r}+j\,q_{r}\) is the \(r\)th pole, \(p_{r}\,q_{r}\) is the damping factor, and \(q_{r}\) is the damped modal frequency. Additionally, \(\mathbf{R}_{r}\) is the residue, \(\mathbf{\phi}_{r}\) is the modal shape, and \(\mathbf{\gamma}_{r}\) is the modal participation vector. According to Equations (1) and (2):  
\[S_{yy}(j\omega)=\sum_{r=1}^{N}\Biggl{[}\frac{\mathbf{A}_{r}}{j\omega-s_{r}}+\frac {\mathbf{A}_{r}^{\rm H}}{-j\omega-s_{r}^{*}}+\frac{\mathbf{A}_{r}^{*}}{j\omega-s_{r}^ {*}}+\frac{\mathbf{A}_{r}^{\rm T}}{-j\omega-s_{r}}\Biggr{]}, \tag{4}\]  
\[\mathbf{A}_{r}=\sum_{b=1}^{N}\Biggl{[}\frac{\mathbf{R}_{b}^{*}S_{xx}\mathbf{R}_{r}^{\rm T }}{-s_{b}^{*}-s_{r}}+\frac{\mathbf{R}_{b}S_{xx}\mathbf{R}_{r}^{\rm T}}{-s_{b}-s_{r}} \Biggr{]}. \tag{5}\]  
In general, the damping of structure is small. Suppose that \(s_{r}=-p_{r}+j\,q_{r}\) is satisfied with \(p_{r}\approx q_{r},\mathbf{A}_{r}\approx d_{r}\mathbf{\phi}_{r}\mathbf{\phi}_{r}^{*},\mathbf{ \phi}_{r}^{\rm T}\), and \(d_{r}\) is a real number. \[S_{yy}(j\omega)\approx\sum_{r=1}^{N}\Biggl{[}\frac{d_{r}\mathbf{\phi}_{r}^{*}\bm {\phi}_{r}^{\rm T}}{j\omega-s_{r}}+\frac{d_{r}\mathbf{\phi}_{r}^{*}\mathbf{\phi}_{r}^{ \rm T}}{-j\omega-s_{r}^{*}}\Biggr{]}=\mathbf{\psi}^{*}\mathrm{diag}[2\mathrm{Re}( \frac{d_{r}}{j\omega-s_{r}})]\mathbf{\psi}^{\rm T},\;\omega\rightarrow\omega_{r}, \tag{6}\]  
\[S_{yy}^{\rm T}(j\omega)\approx\mathbf{\psi}\mathrm{diag}[2\mathrm{Re}(\frac{d_{r}} {j\omega-s_{r}})]\mathbf{\psi}^{\rm H},\;\omega\rightarrow\omega_{r}, \tag{7}\]  
The response PSD matrix can be decomposed by singular value decomposition (SVD) as  
\[S_{yy}^{\rm T}(j\omega)=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{\rm H}, \tag{8}\]  
\[\mathbf{\Lambda}=\mathrm{diag}(\lambda_{1},\lambda_{2},\cdots,\lambda_{N}), \tag{9}\]  
where the superscript 'H' denotes the conjugate transposition operation, \(\mathbf{U}\in\mathbf{C}^{\rm Q\times N}\) is a unitary matrix, which contains singular vectors, and it corresponds to \(\mathbf{\Psi}\) in Equation (7). \(\mathbf{\Lambda}\in\mathbf{R}^{\rm N\times N}\) is the diagonal matrix, which is composed of real singular values. Only the dominant modes are analyzed, and the \(r\)th mode of enhanced PSD can be expressed as  
\[\mathbf{P}_{sd}(j\omega)=\mathbf{U}_{r}^{\rm H}\mathbf{S}_{yy}^{\rm T}(j\omega)\mathbf{U}_{r}. \tag{10}\]  
On the near spectral line, the enhanced PSD function is made of the singular value of the response spectrum matrix corresponding to \(\mathbf{U}_{r}\). \[\mathbf{H}(j\omega)=\sum_{i=1}^{N}\left[\frac{\mathbf{R}_{r}}{|j\omega-s_{r} }+\frac{\mathbf{R}_{i}}{|j\omega-s_{i}^{*}|}\right], \tag{2}\]0  
Substitute \(s_{r}\) into Equation (11), so the enhanced PSD function can be written as:  
\[\mathbf{B}\left[ \begin{array}{c}\mathbf{P}_{sd}\\ \omega\mathbf{P}_{sd}\\ 1\end{array} \right]=-\omega^{2}\mathbf{P}_{sd}, \tag{12}\]  
\[\mathbf{B}=\left[ \begin{array}{cc}p_{r}^{2}+q_{r}^{2}&-2q_{r}&-2d_{r}p_{r}\end{array}  \right], \tag{13}\]  
Understandably, the response signal contains all the required modal information.