# Chromaticity Recognition Technology of Colored Noise and Operational Modal Analysis
## 2 Theory
### Introduction to the OMA Method under Certain Colored Noise  
Reference [23] explains that, with known chromaticity, the modal parameters of the system can be identified. Here is a brief explanation:  
For a linear classical damped MDOF system with \(n\) degrees, the response of the system can be written as  
\[y(\omega)=\mathrm{H}(\omega)d(\omega) \tag{1}\]where \(y\) represents the response vector, H is the frequency response matrix, and \(d\) determines the excitation vector. Supposing that each excitation is uncorrelated with the other, the PSD of excitation can be written as:  
\[S_{dd}=\text{diag}\big{\{}\mu_{1}\omega^{\rho},\mu_{2}\omega^{\rho},\cdots,\mu_{ p}\omega^{\rho}\big{\}} \tag{2}\]  
where \(\mu_{i}(i=1,2,\cdots,p)\) determines the magnitude of excitation noise, \(\rho\) represents the color, and \(\text{diag}\{\}\) refers to constructing a diagonal matrix with corresponding elements. The response spectrum matrix is  
\[ \begin{split}&\mathbf{S_{yy}}(\omega)=\text{E}\big{(}\mathbf{y} \mathbf{y}^{*\text{T}}\big{)}=\text{E}\Big{(}\mathbf{Hd}\mathbf{d}^{*\text{T}} \mathbf{H}^{*\text{T}}\Big{)}=\mathbf{HS_{dd}}\mathbf{H}^{*\text{T}}\\ &=\sum\limits_{r=1}^{n}\left(\frac{\varphi_{r}\varphi_{r}^{\text {T}}}{K_{r}-M_{r}\omega^{2}+\text{j}\omega C_{r}}\right)\cdot\mathbf{S_{dd}} \cdot\sum\limits_{r=1}^{n}\left(\frac{\varphi_{r}\varphi_{r}^{\text{T}}}{K_{r} -M_{r}\omega^{2}-\text{j}\omega C_{r}}\right)^{\text{T}}\end{split}  \tag{3}\]  
where \(K_{r}\), \(M_{r}\), and \(C_{r}\) represent the \(r^{\text{th}}\) modal stiffness, modal mass, and modal damping, respectively. 'E', '*' and 'T' represent the mathematical expectation, the complex conjugate, and the transpose operation, respectively. \(\varphi_{r}\) is the \(r^{\text{th}}\) modal vector. When \(\omega\) is in the half bandwidth of the \(r^{\text{th}}\) mode resonance peak  
\[ \begin{split}\mathbf{S_{yy}}(\omega)&=\mathbf{ \Gamma\Lambda\Gamma^{\text{T}}}\\ &\approx\varphi_{r}\varphi_{r}^{\text{T}}\cdot\frac{\mathbf{S_{ dd}}}{(K_{r}-M_{r}\omega^{2})^{2}+(\omega C_{r})^{2}}\cdot\varphi_{r}\varphi_{r}^{ \text{T}}\end{split}  \tag{4}\]  
where \(\Gamma=\varphi_{r}e_{r}\varphi_{r}^{\text{T}},e_{r}\) is a constant. \(\Lambda=diag\{\sigma_{1},\sigma_{2}\cdots\sigma_{n}\}\), \((\sigma_{1}>\sigma_{2}>\cdots>\sigma_{n})\). If  
\[\mathbf{\overline{S}_{yy}}(\omega)=\mathbf{\Gamma^{\text{T}}S_{yy}}\Gamma \tag{5}\]  
we can have  
\[ \begin{split}&\mathbf{\overline{S}_{yy}}(1,1)=\text{max}(\mu_{1} \omega^{\rho},\mu_{2}\omega^{\rho},\cdots,\mu_{n}\omega^{\rho})\cdot\frac{1}{ \sigma_{r}^{2}M_{r}^{2}}\\ &\cdot\frac{1}{(\omega_{r}^{2}-\omega^{2})^{2}+(2\zeta_{r}\omega,\omega)^{2}}\\ &=\frac{\mu_{r}\omega^{\rho}}{\sigma_{r}^{2}M_{r}^{2}}\cdot\frac{1 }{(\omega_{r}^{2}-\omega^{2})^{2}+(2\zeta_{r}\omega\omega)^{2}}\end{split}  \tag{6}\]  
where \(\omega_{r}\), \(\zeta_{r}\), and \(\Gamma\) represent the \(r^{\text{th}}\) modal frequency, the \(r^{\text{th}}\) modal damping ratio, and the \(r^{\text{th}}\) mode shape, respectively. \(\mu_{x}=\text{max}(\mu_{1},\mu_{2},\cdots,\mu_{n})\). From Equation (6), we can obtain  
\[\frac{1}{\mathbf{\overline{S}_{yy}}(1,1)}=\frac{\varepsilon_{r}^{2}M_{r}^{2} }{\mu_{x}}\Big{[}\omega_{r}^{4}\omega^{-\rho-2\alpha}+\omega^{4-\rho-2\alpha} +(4\zeta_{r}^{2}-2)\omega_{r}{}^{2}\omega^{2-\rho-2\alpha}\Big{]} \tag{7}\]  
Let  
\[ \begin{split}& S(\chi_{r},\varepsilon_{r},\theta_{r})=\sum \limits_{i=1}^{N}\left[\frac{1}{\mathbf{\overline{S}_{yy}}(1,1)}-\frac{1}{ \mathbf{\overline{S}_{Y}}(\omega_{ri})}\right]^{2}\\ &=\sum\limits_{i=1}^{N}\left[\frac{\varepsilon_{r}^{2}M_{r}^{2} }{\mu_{x}}\big{(}\omega_{r}^{4}\omega_{ri}{}^{-\rho-2\alpha}+\omega_{ri}{}^{4 -\rho-2\alpha}+(4\zeta_{r}^{2}-2)\omega_{r}{}^{2}\omega_{ri}{}^{2-\rho-2 \alpha}\big{)}-\frac{1}{\mathbf{\overline{S}_{Y}}(\omega_{ri})}\right]^{2}\\ &=\sum\limits_{i=1}^{N}\left[\left(\chi_{r}\omega_{ri}{}^{-\rho-2 \alpha}+\varepsilon_{r}\omega_{ri}{}^{4-\rho-2\alpha}+\vartheta_{r}\omega_{ri} {}^{2-\rho-2\alpha}\right)-\frac{1}{\mathbf{\overline{S}_{Y}}(\omega_{ri})} \right]^{2}\end{split}