# On the Correlation of Symbols' Vibrational Behavior and Manufacturing Processes
## 2 Theoretical Background  
### Complex Frequency Domain Assurance Criterion  
The frequency response function (FRF) of a system \(h_{ij}(\omega)\) relates a single force \(f_{j}(\omega)\) excitation applied to point \(j\) to a response \(x_{i}(\omega)\) detected at point \(i\) by \(x_{i}(\omega)=h_{ij}(\omega)f_{j}(\omega)\). All FRFs contain information about the dynamic characteristics of a system, which in matrix notation can be expressed as \(H(\omega)=X(\omega)F^{-1}(\omega)\), where \(H(\omega)\) is the matrix containing all FRFs, \(X(\omega)\) is the matrix containing all responses, and \(F(\omega)\) the matrix containing all applied forces. To identify damage in a system, [PERSON] et al. [21, 22] proposed the complex frequency domain assurance criterion (CFDAC). The use of the CFDAC, a spectral domain indicator, offers advantages compared to modal, spatial, or temporal analysis. The CFDAC is calculated following a convenient condensing procedure that avoids information loss by applying the Fourier transform. It requires simplified post-processing and provides enhanced sensitivity to reinforcement or degradation [23]. This means that variations in the dynamic response of the structure, resulting from alterations, can be detected. The detection of such alterations becomes feasible by comparing two states.  
In the case of comparing a pristine state (denoted as \(p\)) with an altered state (denoted as \(a\)), after acquiring the response of the system at \(N\) points, the CFDAC is defined for each frequency pair \(\left(\omega_{f},\ \omega_{g}\right)\) as follows:  
\[\text{CFDAC}_{fg}=\frac{\left[\sum_{i=1}^{N}\sum_{j=1}^{N}h_{ij}^{(p)}\left( \omega_{f}\right)h_{ij}^{(a)}\left(\omega_{g}\right)\right]^{2}}{\left[\sum_{ i=1}^{N}\sum_{j=1}^{N}h_{ij}^{(p)}\left(\omega_{f}\right)h_{ij}^{*(p)}\left( \omega_{f}\right)\right]\left[\sum_{i=1}^{N}\sum_{j=1}^{N}h_{ij}^{(a)}\left( \omega_{g}\right)h_{ij}^{*(a)}\left(\omega_{g}\right)\right]}\in\mathbb{C} \tag{1}\]  
where \(h^{(p)}(\omega)\) is the set of FRFs for the pristine state, \(h^{(a)}(\omega)\) is the set of FRFs for the altered state, \(\omega_{f}\) and \(\omega_{g}\) denote the frequency (FFT line indices) of the pristine and altered states, respectively, \(i\) is the index for the location of the response point, \(j\) is the index for the excitation point, and \({}^{*}\) denotes the complex conjugate. For mathematical consistency, it must be ensured that all FRFs have the same amount of frequency lines (\(N_{f}=N_{g}\)) and resolution. Hence, \(\text{CFDAC}_{fg}\) is an \(N_{f}\times\ N_{f}\) array. The criterion is an indicator of the covariance between the two sets of FRFs \(h^{(p)}(\omega)\) and \(h^{(a)}(\omega)\), which are compared for each pair of spectral lines. The values of \(\text{CFDAC}_{fg}\) are confined to the \([-1,1]\), \([-j,j]\) region of the complex plane. If the same set of FRFs is used for both the pristine and altered states, then the \(\text{CFDAC}_{fg}\) array exhibits a perfect diagonal structure. The real part of the \(\text{CFDAC}_{fg}\) shows the correlation of the resonant peaks with their adjacent frequencies, whereas the imaginary part shows the correlation at the central frequencies of the lobes. Recently, the CFDAC was successfully extended to musical acoustics in terms of comparing results from simulations, measurements, and parts of musical instruments during additive manufacturing [23, 24, 25].