## 2 Materials and Methods
### Other Techniques for Resonant Frequencies Identification
#### 2.4.2 Magnitude-Squared Coherence Method  
The magnitude-squared coherence (MSC) indicates the correlation of two signals at a certain frequency. The MSC is 1 when there is complete similarity in the spectral content at a given frequency and 0 when the signals are fully uncorrelated in that frequency band. One major advantage of the MSC is that a different amplitude in harmonics does not affect the MSC between the signals. In addition, the MSC is phase independent. This is important for analyzing superior modes of vibration, where two or more levels of the structure vibrate with the same harmonics but with different phases. However, external sources of vibration that were detected at the different measurement points may cause high MSCs at the frequencies of the origin of vibration. This happens for example when the structure is excited with some device (reference) or with different types of machinery working, i.e., air conditioning systems. In the case of the building under study, there was no source of noise and the structure was evicted so that ambient noise was the only source of excitation of the structure. The coherence spectrum is analogous to the conventional correlation coefficient and is defined as follows:  
\[C_{x,y}(f)=\frac{\left|P_{x,y}(f)\right|^{2}}{P_{x}(f)P_{y}(f)} \tag{4}\]  
Figure 15: The CPSDs (grey lines) of the combination between the selected Rbin and the rest of the Rbins and the mean CPSD (black). The frequencies 2.49 and 2.71 Hz are indicated with red and blue dashed lines, respectively. where \(P_{x}(f)\) and \(P_{y}(f)\) are the power spectral densities, and \(P_{xy}(f)\) is the cross-power spectral density between the signals \(x\) and \(y\). In other words, the coherence is a statistical indicator which specifies whether or not two signals are correlated. For cases in which the correlation degree between a selected signal \(s\) and a group of signals \(Z=\{z_{1},z_{2},\ldots,z_{n}\}\) is to be estimated, the Multiple Magnitude Coherence Method (MMSC) [33] can be used. The multiple coherence is defined as:  
\[ \begin{array}{l}M_{Zs}(f)=\frac{P_{Zs}^{*}(f)P_{Zs}^{*}(f)P_{Zs}(f)}{P_{s}(f) }\\ =\left[\begin{array}{ccc}P_{z_{1}s_{1}}^{*}(f)&\cdots&P_{z_{n}s_{1}}^{*}(f) \end{array}\right]\left[\begin{array}{ccc}P_{z_{1}z_{1}}&\cdots&P_{z_{1}z_{n} }\\ \vdots&\ddots&\vdots\\ P_{z_{n}z_{1}}&\cdots&P_{z_{n}z_{n}}\end{array}\right]\left[\begin{array}{ c}P_{z_{1}s_{1}}\\ \vdots\\ P_{z_{n}s_{1}}\end{array}\right]\end{array}  \tag{5}\]  
where \(M_{Zs}(f)\) is the multiple coherence function between the signal \(s\) and the array of signals \(Z\), \(P_{Zs}(f)\) is the \(n\)-dimensional vector of the cross-power spectral densities between the signal \(s\) and the series of signals \(Z\), \(P_{ZZ}^{-1}(f)\) is the inverse matrix of \(n\)\(x\)\(n\) dimension of the PSD and the CPSD of the series of signals \(Z\) and \(P_{s}(f)\) is the PSD of the signal \(s\). Figure 16 shows the MSC estimate for point 3 as an example. The gray lines represent the MSC of the different combinations between the selected channel and the other channels for the same direction--that is, between P3-L and P1-L, P2-L, P4-L and P5-L for the longitudinal direction (Figure 16a) and between P3-L and P1-L, P2-L, P4-L and P5-L for the transversal direction (Figure 16b). The mean spectra of the MSC is shown in black. It can be observed that, as in the CPSDs, there are predominant peaks in the frequencies corresponding to 2.49 and 2.79 Hz, highlighted with dashed red and blue lines, respectively. Figure 17 shows the MSC between the combinations of Rbin and the rest of the selected Rbins (grey lines), i.e., between Rbin41 and Rbin38, Rbin39, Rbin40, Rbin42 and Rbin43, and the mean MSC is shown in black. It can be observed that the highest MSC peaks are present in frequencies 2.49 and 2.71 Hz. The same frequencies are found in the peaks of both the PSD of Rbin41 and Rbin42 and in the mean cross-spectrum peaks. It can also be seen that the lower frequency peaks, between 1 and 2 Hz, are attenuated.