# Chromaticity Recognition Technology of Colored Noise and Operational Modal Analysis
## 2 Theory
### Introduction to the OMA Method under Certain Colored Noise
 \tag{8}\]  
where  
\[\chi_{r}=\frac{e_{r}^{2}M_{r}^{2}}{\mu_{x}}\omega_{r}^{4} \tag{9}\]  
\[\varepsilon_{r}=\frac{\varepsilon_{r}^{2}M_{r}^{2}}{\mu_{x}} \tag{10}\]\[\hat{\sigma}_{r}=\left(4\xi_{r}^{2}-2\right)\frac{e_{r}^{2}M_{r}^{2}}{\mu_{x}} \omega_{r}^{2} \tag{11}\]  
\(\omega_{ri}\) represents some frequency points near the \(r^{\text{th}}\) mode, and \(S_{Y}(\omega_{ri})\) is the spectrum near the \(r^{\text{th}}\) mode obtained from the experiment. If  
\[ \begin{cases}\frac{\partial S(\chi_{r},\varepsilon_{r},\vartheta_{r})}{ \partial\chi_{r}}=0\\ \frac{\partial S(\chi_{r},\varepsilon_{r},\vartheta_{r})}{\partial\varepsilon _{r}}=0\\ \frac{\partial S(\chi_{r},\varepsilon_{r},\vartheta_{r})}{\partial\vartheta_{r }}=0\end{cases}  \tag{12}\]  
we can have  
\[ \begin{cases}\frac{\partial S}{\partial\chi_{r}}=2\sum\limits_{i=1}^{N}\left[ \chi_{r}\omega_{ri}^{-\rho-2\alpha}+\varepsilon_{r}\omega_{ri}^{4-\rho-2\alpha }+\vartheta_{r}\omega_{ri}^{2-\rho-2\alpha}-\frac{1}{S_{Y}(\omega_{ri})} \right]\omega_{ri}^{-\rho-2\alpha}=0\\ \frac{\partial S}{\partial\varepsilon_{r}}=2\sum\limits_{i=1}^{N}\left[ \chi_{r}\omega_{ri}^{-\rho-2\alpha}+\varepsilon_{r}\omega_{ri}^{4-\rho-2 \alpha}+\vartheta_{r}\omega_{ri}^{2-\rho-2\alpha}-\frac{1}{S_{Y}(\omega_{ri})} \right]\omega_{ri}^{4-\rho-2\alpha}=0\\ \frac{\partial S}{\partial\theta_{r}}=2\sum\limits_{i=1}^{N}\left[ \chi_{r}\omega_{ri}^{-\rho-2\alpha}+\varepsilon_{r}\omega_{ri}^{4-\rho-2 \alpha}+\vartheta_{r}\omega_{ri}^{2-\rho-2\alpha}-\frac{1}{S_{Y}(\omega_{ri})} \right]\omega_{ri}^{2-\rho-2\alpha}=0\end{cases}  \tag{13}\]  
Or  
\[ \begin{cases}\sum\limits_{i=1}^{N}\omega_{ri}^{-2\rho-4\alpha}&\sum\limits_{i =1}^{N}\omega_{ri}^{4-2\rho-4\alpha}&\sum\limits_{i=1}^{N}\omega_{ri}^{2-2 \rho-4\alpha}\\ \sum\limits_{i=1}^{N}\omega_{ri}^{4-2\rho-4\alpha}&\sum\limits_{i=1}^{N} \omega_{ri}^{8-2\rho-4\alpha}&\sum\limits_{i=1}^{N}\omega_{ri}^{6-2\rho-4 \alpha}\\ \sum\limits_{i=1}^{N}\omega_{ri}^{2-2\rho-4\alpha}&\sum\limits_{i=1}^{N} \omega_{ri}^{6-2\rho-4\alpha}&\sum\limits_{i=1}^{N}\omega_{ri}^{4-2\rho-4 \alpha}\\ \sum\limits_{i=1}^{N}\frac{\omega_{ri}^{2-\rho-2\alpha}}{S_{Y}(\omega_{ri})} \end{cases}  \tag{14}\]  
In Equation (14), the left coefficient matrix is often reversible; \(a_{r}\), \(b_{r}\), and \(c_{r}\) can be written as  
\[ \begin{bmatrix}\chi_{r}\\ \varepsilon_{r}\\ \theta_{r}\end{bmatrix} = \begin{bmatrix}\sum\limits_{i=1}^{N}\omega_{ri}^{-2 \rho-4\alpha}&\sum\limits_{i=1}^{N}\omega_{ri}^{4-2\rho-4\alpha}&\sum\limits_{ i=1}^{N}\omega_{ri}^{2-2\rho-4\alpha}\\ \sum\limits_{i=1}^{N}\omega_{ri}^{4-2\rho-4\alpha}&\sum\limits_{i=1}^{N}\omega_ {ri}^{8-2\rho-4\alpha}&\sum\limits_{i=1}^{N}\omega_{ri}^{6-2\rho-4\alpha}\\ \sum\limits_{i=1}^{N}\omega_{ri}^{2-2\rho-4\alpha}&\sum\limits_{i=1}^{N}\omega_ {ri}^{6-2\rho-4\alpha}&\sum\limits_{i=1}^{N}\omega_{ri}^{4-2\rho-4\alpha}\\ \sum\limits_{i=1}^{N}\omega_{ri}^{2-2\rho-4\alpha}&\sum\limits_{i=1}^{N}\omega _{ri}^{6-2\rho-4\alpha}&\sum\limits_{i=1}^{N}\omega_{ri}^{4-2\rho-4\alpha}\\ \sum\limits_{i=1}^{N}\frac{\omega_{ri}^{2-\rho-2\alpha}}{S_{Y}(\omega_{ri})} \end{bmatrix}  \tag{15}\]  
From (14) and (15), the modal parameter can be obtained as  
\[\omega_{r}=\sqrt[4]{\frac{\chi_{r}}{\varepsilon_{r}}} \tag{16}\]  
\[\xi_{r}=\frac{1}{2}\sqrt{\frac{\theta_{r}}{\sqrt{\chi_{r}\varepsilon_{r}}}+2} \tag{17}\]  
\[\varphi_{r}=\frac{\Gamma(\ ;1)}{\max\left|\Gamma(\ ;1)\right|} \tag{18}\]  
For detailed simulation and experimentation of this method, please refer to reference [23]. In previous studies, the researchers could only estimate chromaticity \(\rho\) and incorporate it into the equation to perform modal analysis. However, in actual testing, the type of noise in the environment is usually unknown. Even if the type of noise is known, the actual noise acting on each mode is unknown due to interference or the effect of modal superposition method.