## 2 Blind Source Separation Techniques
### Extraction of Modal Parameters  
In general, blind source separation methods such as CP and GED can separate the mixed sources up to a scaling and permutation ambiguity. This means that the estimated sources are only determined up to a scaling factor and a permutation of the sources, and the exact sequence and amplitude of the modal responses and mode shapes cannot be recovered directly from the separated sources. However, this ambiguity can be resolved by rearranging the modes on the basis of their frequency values [22]. Specifically, the recovered modal response and its associated mode shape vector with the smallest frequency can be recognized as the first mode, and so on, until all modes have been assigned to their correct order. This is based on the assumption that the modal responses and mode shapes are ordered in frequency, which is often the case in practice.  
The modal assurance criterion (MAC) is a measure of the similarity between two mode shapes, which is commonly used to evaluate the correlation between the theoretical mode \(\boldsymbol{\varphi}_{i}\) and the estimated mode \(\boldsymbol{\widetilde{\varphi}}_{i}\) obtained from BSS methods such as CP and GED. The MAC is defined as  
\[MAC(\boldsymbol{\widetilde{\varphi}}_{i},\boldsymbol{\varphi}_{i})=\frac{ \left(\boldsymbol{\widetilde{\varphi}}_{i}^{\top}\cdot\boldsymbol{\varphi}_{i} \right)^{2}}{\left(\boldsymbol{\widetilde{\varphi}}_{i}^{\top}\cdot \boldsymbol{\widetilde{\varphi}}_{i}\right)\left(\boldsymbol{\varphi}_{i}^{ \top}\cdot\boldsymbol{\varphi}_{i}\right)}. \tag{31}\]  
The MAC ranges from 0 to 1, where a value of 1 indicates perfect correlation between the two mode shapes, and a value of 0 indicates no correlation. This study takes MAC values over 0.8 as a threshold criterion for identified mode shapes. In addition, it takes the error below 0.1 and 0.2 as the threshold criterion for identified frequencies and damping ratios, respectively. The natural frequency and damping can be obtained by measuring the time lapse and the logarithmic decrements between consecutive peaks of the trace of free-decay time history. Furthermore, accurately estimating the damping ratio in random vibration often requires additional preprocessing techniques. On the basis of RDT processing, the random response signal of a linear system is expected to exhibit similar characteristics to those observed in a free vibration response. In practical response histories, however, the peaks may not be captured exactly. For example, a peak may appear higher than its preceding peak because of nonlinearity, sampling, noise, and various other conditions, resulting in erroneous negative damping. The method presented in this study is based on the Hilbert transform. This method is more robust and considerably easier to apply than conventional methods.  
With the Hilbert transform H[\(\bullet\)], the analytic signal is defined as  
\[z_{i}(t)=\hat{q}_{i}(t)+j\text{H}[\hat{q}_{i}(t)]=A_{i}(t)e^{-\beta_{i}(t)}, \tag{32}\]  
where \(\boldsymbol{\hat{q}}(t)\) is the \(i\)th modal response \(q_{i}(t)\) processed by RDT method.  
The instantaneous amplitude is defined as  
\[A_{i}(t)=\sqrt{\hat{q}_{i}^{2}(t)+\text{H}^{2}[\hat{q}_{i}(t)]}. \tag{33}\]  
The phase function is defined as  
\[\theta_{i}(t)=\text{arctan}\bigg{(}\frac{\text{H}[\hat{q}_{i}(t)]}{\hat{q}_{i }(t)}\bigg{)}. \tag{34}\]  
The instantaneous frequency is simply expressed  
Thus, the damped angular frequencies \(\omega_{di}\) and \(\xi_{i}\)\(\omega_{i}\) can be obtained directly from the slopes of two straight lines of Equations (37) and (38). On the basis of Equation (39), t-he \(i\)th natural frequency \(\omega_{i}\) and the damping ratio \(\xi_{i}\) can be obtained.  
\[\omega_{di}=\omega_{i}\sqrt{1-\xi_{i}{}^{2}}. \tag{39}\]  
The output-only BSS modal identification algorithms (CP and GED) presented in this study are capable of performing blind modal identification, as they can extract modal information directly from the measured structural responses. Their implementations are summarized in the flowchart in Figure 3.