# Chromaticity Recognition Technology of Colored Noise and Operational Modal Analysis
## 2 Theory
### Single Degree of Freedom System  
Assuming the PSD of \(G_{xx}\) excited in the frequency domain is not flat, they can be expressed as:  
\[G_{xx}=\mu\omega^{\rho} \tag{19}\]  
where \(\mu\) represents the level of noise, and \(\rho\) determines the color (\(-2\) represents brown noise, \(-1\) represents pink noise, \(1\) represents blue noise, and \(2\) represents purple noise). \(\omega\) is the frequency of excitation.  
For linear classical damping single degree of freedom systems, the acceleration response PSD of the system under colored noise can be expressed as  
\[\begin{array}{ll}G_{yy}(\omega)&=\mu\omega^{\gamma}{\left|H\right|}^{2}=\frac {\mu}{m^{2}}\frac{\omega^{\gamma\gamma}}{\left(\frac{\omega_{1}^{2}-\omega^{2} }{2}\right)^{2}+\left(2\zeta\omega_{n}\omega\right)^{2}}\\ &=\frac{\mu}{m^{2}}\left[\omega_{1}^{2}+\omega^{4}+(4\zeta^{2}-2)\omega_{n} \omega^{2}\right]\omega^{-(\gamma+\rho)}\end{array} \tag{20}\]  
In Equation (20), \(m\) represents the mass of a single degree of freedom system, \(\omega_{n}\) represents the natural frequency, and \(\zeta\) represents the damping ratio of the system. \(G(\omega)\) represents the PSD of the acceleration response. \(\alpha\) is related to the type of sensor that we use (for example, for strain \(\alpha=0\), for acceleration \(\alpha=2\), for velocity \(\alpha=1\), for displacement \(\alpha=0\)).  
Let  
\[\gamma=\alpha+\rho \tag{21}\]  
Equation (20) can be written as  
\[\frac{1}{G_{yy}(\omega)}=\frac{m^{2}}{\mu}\left[\omega_{n}^{4}+\omega^{4}+(4 \zeta^{2}-2)\omega_{n}{}^{2}\omega^{2}\right]\omega^{-\gamma} \tag{22}\]  
Let  
\[Y=\frac{1}{G_{yy}(\omega)} \tag{23}\]  
\[a={\omega_{n}}^{2} \tag{24}\]  
\[b=\left(4\zeta^{2}-2\right){\omega_{n}}^{2} \tag{25}\]  
\[c=\frac{\mu}{m^{2}} \tag{26}\]  
Equation (22) can be written as  
\[cY\omega^{\gamma}=\omega^{4}+b\omega^{2}+a^{2} \tag{27}\]  
If \(\omega_{i}\sim\omega_{\tau}\) represent the frequency points near the natural frequency \(\omega_{n}\), respectively, and \(Y_{i}\sim Y_{\tau}\) represent the corresponding response PSD, where (\(1\leq i<j<k<\tau\leq N\)), we can have  
\[cY_{i}{\omega_{i}}^{\gamma}-\omega_{i}^{4}-b\omega_{i}^{2}=cY_{i}{\omega_{j}} ^{\gamma}-{\omega_{j}}^{4}-b{\omega_{j}}^{2}=a \tag{28}\]  
\[cY_{i}{\omega_{i}}^{\gamma}-{\omega_{i}^{4}}-b{\omega_{i}^{2}}=cY_{i}{\omega_ {k}}^{\gamma}-{\omega_{k}}^{4}-b{\omega_{k}}^{2}=a \tag{29}\]  
\[cY_{i}{\omega_{i}}^{\gamma}-{\omega_{i}^{4}}-b{\omega_{i}^{2}}=cY_{\tau}{ \omega_{\tau}}^{\gamma}-{\omega_{\tau}}^{4}-b{\omega_{\tau}  
Let  
\[p_{ijk} =\frac{Y_{i}}{\big{(}\omega_{j}{}^{2}-\omega_{i}{}^{2}\big{)}(\omega_{ k}{}^{2}-\omega_{i}{}^{2})} \tag{32}\] \[p_{jik} =\frac{Y_{j}}{\big{(}\omega_{i}{}^{2}-\omega{}^{2}\big{)}\big{(} \omega_{k}{}^{2}-\omega{}^{2}\big{)}}\] (33) \[p_{kij} =\frac{Y_{k}}{(\omega_{i}{}^{2}-\omega_{k}{}^{2})\big{(}\omega_{ j}{}^{2}-\omega_{k}{}^{2}\big{)}}\] (34) \[p_{j\kappa\tau} =\frac{Y_{j}}{\big{(}\omega_{k}{}^{2}-\omega_{j}{}^{2}\big{)} \big{(}\omega_{\tau}{}^{2}-\omega_{j}{}^{2}\big{)}}\] (35) \[p_{kj\tau} =\frac{Y_{k}}{\big{(}\omega_{j}{}^{2}-\omega_{k}{}^{2}\big{)}( \omega_{\tau}{}^{2}-\omega_{k}{}^{2})}\] (36) \[p_{\tau jk} =\frac{Y_{\tau}}{\big{(}\omega_{j}{}^{2}-\omega_{\tau}{}^{2} \big{)}(\omega_{k}{}^{2}-\omega_{\tau}{}^{2})} \tag{37}\]  
Eliminating \(b\), Equation (31) can be rewritten as  
\[p_{ijk}\omega_{i}{}^{\gamma}+p_{jik}\omega_{j}^{\gamma}+p_{kij}\omega_{k}^{ \gamma}=p_{j\kappa\tau}\omega_{j}^{\gamma}+p_{kj\tau}\omega_{k}^{\gamma}+p_{ \tau jk}\omega_{\tau}{}^{\gamma}=\frac{1}{c} \tag{38}\]  
i.e.,  
\[p_{ijk}\omega_{i}{}^{\gamma}+\Big{(}p_{kij}-p_{kj\tau}\Big{)}\omega_{k}^{ \gamma}=p_{\tau jk}\omega_{\tau}{}^{\gamma}+\Big{(}p_{jk\tau}-p_{jik}\Big{)} \omega_{j}^{\gamma} \tag{39}\]  
Defining Equation (39) as the Chromaticity Recognition Function (CRF), it can be seen that the CRF is related to the four frequency points \(\omega_{i}\sim\omega_{\tau}\), as well as their corresponding PSD \(Y_{i}\sim Y_{\tau}\).