## 3 Filtering Procedure for Synthesis of a New Reference Signal  
In this work, the aforementioned strain monitoring process is modified to increase the strain sensitivity of the signal \(y(\varepsilon,t)\) in Equation (3). This is accomplished by using a modified reference signal which is synthesized through a phase filter that relies on the prior knowledge of the system response at some non-zero strain state. The filtering technique is performed as follows. First, the impulse response signal under null strain, \(h(0,t)\), and under some non-zero strain, \(\tilde{\varepsilon}\), \(h(\tilde{\varepsilon},t)\), are acquired and addressed in the frequency domain, through one of the transforms described in Section 2.3, here generically represented as \(\mathcal{T}\{\cdot\}\):  
\[H(0,\chi)=\mathcal{T}\{h(0,t)\} \tag{7}\]  
and  
\[H(\tilde{\varepsilon},\chi)=\mathcal{T}\{h(\tilde{\varepsilon},t)\}, \tag{8}\]  
respectively, where \(\chi\) is the transform variable, i.e., either \(\chi=f\) for \(FT\) or \(\chi=(\tau,f)\) for \(STFT\). Then, these two signals are cross-correlated, which is performed by multiplying them in the transform domain:  
\[S_{\tilde{\varepsilon}}(\chi)=H^{*}(0,\chi)H(\tilde{\varepsilon},\chi), \tag{9}\]  
where the asterisk superscript means complex conjugate. Note that the phase of \(S_{\tilde{\varepsilon}}(\chi)\) represents the phase shift between the two impulse responses, \(h(0,t)\) and \(h(\tilde{\varepsilon},t)\), as a function of \(\chi\). Considering that the strain-induced speed and path changes mainly affect the phase of the received wave packets, and assuming that the major difference between \(h(0,t)\) and \(h(\tilde{\varepsilon},t)\) is caused by strain, it is interesting to analyze the phase of \(S_{\tilde{\varepsilon}}(\chi)\), which presents high absolute values for the spectral components that are more affected by strain, i.e., the more sensitive components. Based on this principle, one can define the \(\phi\)-sensitive coefficients as the ones whose phase of \(S_{\tilde{\varepsilon}}(\chi)\) is greater than \(\phi\). Therefore, a zero-one function, \(M_{\phi,\tilde{\varepsilon}}(\chi)\), is built, accordingly:  
\[M_{\phi,\tilde{\varepsilon}}(\chi)= \begin{cases}1,&\left|\angle S_{\tilde{ \varepsilon}}(\chi)\right|\geq\phi\\ 0,&\left|\angle S_{\tilde{\varepsilon}}(\chi)\right|<\phi\end{cases} . \tag{10}\]  
The actual filtering step consists of multiplying \(M_{\phi,\tilde{\varepsilon}}(\chi)\) by the spectrum of the original reference signal under null strain:  
\[H_{\phi,\tilde{\varepsilon}}(\chi)=M_{\phi,\tilde{\varepsilon}}(\chi)H(0, \chi). \tag{11}\]  
This process creates a new signal, \(H_{\phi,\tilde{\varepsilon}}(\chi)\), in which the coefficients that were deemed unfit, i.e., with low sensitivity, are eliminated. Lastly, \(H_{\phi,\tilde{\varepsilon}}(\chi)\) is transformed back to the time-domain, by the inverse of the chosen transform:  
\[h_{\phi,\tilde{\varepsilon}}(t)=\mathcal{T}^{-1}\{H_{\phi,\tilde{\varepsilon }}(\chi)\}. \tag{12}\]The filtering process is summarized in Figure 2. This new time-domain signal is used as reference for the original monitoring procedure, as explained in Section 2.1. More precisely, \(h_{\phi,\tilde{\varepsilon}}(t)\) should be used instead of \(h(0,t)\) in Equation (3), which becomes:  
\[y_{\phi}(\varepsilon,t)=h_{\phi,\tilde{\varepsilon}}(-t)*h(\varepsilon,t). \tag{13}\]  
It is worth highlighting that the present method considers that the main cause of phase-shifting is strain. However, temperature variation can also induce phase variations since it changes the wave speed and propagation path [41; 42]. In this paper, temperature is assumed constant. Alternatively, the present method can, in principle, be applied in scenarios where one intends to select the most temperature-sensitive components when strain remains constant.