## 2 Review of Existing Mode-Pairing Criteria
### Extended Modal Assurance Criterion (MACX)  
As stated by [PERSON] et al. [22], the MAC criteria might lead to inconclusive results in the case of modes with a significant level of complexity, especially in the presence of close or repeated modes. According to [PERSON] et al. [20], these inconclusive results are due to the fact that the MAC provides different results depending on the combinations made with the pairs of complex and complex-conjugate of the two mode shapes being compared. In order to address those issues, [PERSON] et al. [20] proposed the Extended Modal Assurance Criterion (MACX), defined as:  
\[\text{MACX}_{ij}=\frac{\left(\left|\mathbf{\Phi}_{i}^{H}\mathbf{\[PERSON]}_{j}\right|+ \left|\mathbf{\[PERSON]}_{i}^{T}\mathbf{\[PERSON]}_{j}\right|\right)^{2}}{\left(\mathbf{\[PERSON]}_{i}^ {H}\mathbf{\[PERSON]}_{i}+\left|\mathbf{\[PERSON]}_{i}^{T}\mathbf{\[PERSON]}_{i}\right|\right)\left(\mathbf{ \[PERSON]}_{j}^{H}\mathbf{\[PERSON]}_{j}+\left|\mathbf{\[PERSON]}_{j}^{T}\mathbf{\[PERSON]}_{j}\right|\right)} \tag{2}\]  
in which \({}^{T}\) is the transpose operator. Compared to the MAC, the use of the MACX provides more consistent results in the case of complex mode shapes, but it is also influenced by the dimension of the modal ordinate vector and more sensible to variations in components of greater amplitude. Examples of its application can be found in references [28; 29]. Aiming to improve the MACX performance in situations where the modal ordinate vectors contain information from only a few points, [PERSON] et al. [20] proposed an enhancement of the criterion called Pole-Weighted MACX (MACXP). This criterion incorporates information regarding the natural frequencies and damping ratios of the structure by weighting the MACX with the poles of the dynamic system. The MACXP is defined as:  
\[\text{MACXP}_{ij}=\frac{\left(\frac{\left|\mathbf{\phi}_{i}^{H}\mathbf{\phi}_{j} \right|}{\left|\lambda_{i}^{*}+\lambda_{j}\right|}+\frac{\left|\mathbf{\phi}_{i}^ {T}\mathbf{\phi}_{j}\right|}{\left|\lambda_{i}+\lambda_{j  
\[\text{F}\text{D}\text{AC}\Big{(}\omega_{f},\omega_{g}\Big{)}=\frac{\left|\sum_{p=1 }^{N}\sum_{q=1}^{N}h_{pq}^{(x)}\Big{(}\omega_{f}\Big{)}h_{pq}^{*(a)}(\omega_{g} )\right|^{2}}{\left(\sum_{p=1}^{N}\sum_{q=1}^{N}h_{pq}^{(x)}\Big{(}\omega_{f} \Big{)}h_{pq}^{*(x)}\Big{(}\omega_{f}\Big{)}\right)\left(\sum_{p=1}^{N}\sum_{q =1}^{N}h_{pq}^{(a)}\big{(}\omega_{g}\big{)}h_{pq}^{*(a)}(\omega_{g})\right)} \tag{4}\]  
in which \(h_{pq}^{(x)}\Big{(}\omega_{f}\Big{)}\) and \(h_{pq}^{(a)}(\omega_{g})\) are the values of the FRFs corresponding to an excitation at the Degree Of Freedom (DOF) \(p\), measured at DOF \(q\) and at frequencies \(\omega_{f}\) and \(\omega_{g}\), respectively [30]. The FDAC parameter allows for the analysis of the correspondence between two FRFs for all frequencies within a selected range. Such an operation results in something similar to a MAC matrix, although it is much denser given the large number of frequency values which can be used compared to the restricted number of modes used to compute the MAC matrix [13]. Several variants of this criterion are also found in the literature, such as the Response Vector Assurance Criterion (RVAC) [31], in which only one column of the FRF matrix is used to compute the FDAC; the improved FDAC [15], which takes into account the lags between the FRFs and prevents the pairing of FRFs with a lag of \(180^{\circ}\); and the complex FDAC [30], which is calculated without the modulus and conjugated operator in the numerator to account for the real and imaginary parts of the criterion. The major drawbacks associated with the application of this mode-pairing criteria are related to the necessity of estimating experimental FRFs, which are not obtained when applying OMA techniques. Furthermore, the FDAC calculation implies a higher computational cost compared to the correlation criteria between modal order vectors, especially when there is a wide range and high resolution of frequencies.