## 4 Results and Discussion
### Fourier Transform Filter Implementation  
The filtering technique is applied to the reference signal using initially the Fourier transform. That is, the transform \(\mathcal{T}\) is the Fourier transform and the transform variable \(\chi\) is the frequency. The function \(M_{\phi,\tilde{\varepsilon}}(f)\) is calculated with a reference strain value, \(\tilde{\varepsilon}\), of about  
Figure 4: Unfiltered reference signal, \(h(0,t)\), obtained in the time-domain (**a**); its \(FT\) (**b**); and its \(STFT\) (**c**). Figure 5: Unfiltered focused signal, \(y(t)\). Figure 3: Experimental setup: (**A,B**) the transducers; (**C**) the fixing screws; (**D**) the threaded rod; (**E**) the hex nut; and (**F**) the mounting frame. 150 \(\mu\epsilon\) (where the unit \(\imath\epsilon\) is a short notation for a strain of \(\upmu\)m/m) as per Equation (10), and then used to compute the new reference, \(H_{\phi,\hat{z}}(f)\), according to Equation (11) and Figure 2. To illustrate the influence of the angle threshold, \(\phi\), Figure 6 shows the \(M_{\phi,\hat{z}}(f)\) function calculated with two values of \(\phi\), namely \(\phi=15^{\circ}\) and \(\phi=30^{\circ}\). One can see that they assume values zero or one, and, comparing their shapes, the \(M_{\phi,\hat{z}}(f)\) function with \(\phi=30^{\circ}\) (Figure 6b) is sparser around the unit because more frequency components are nullified than the one obtained with \(\phi=15^{\circ}\) (Figure 6a). This holds because there are fewer components that are \(\phi\)-sensitive to strain with a higher and more restrictive threshold. Figure 7 shows the respective reference signals obtained with these two \(M_{\phi,\hat{z}}(f)\) functions where amplitude is normalized with respect to the original signal, \(h(0,t)\). One can see that, with a higher value of \(\phi\), more frequency components are filtered out by \(M_{\phi,\hat{z}}(f)\) according to Figure 6. Consequently, the time-domain signal of the latter is less similar to the original reference signal \(h(0,t)\), shown in Figures 4a and 7b, for the sake of comparison. The focused signals after filtering, \(y_{\phi}(0,t)\), with \(\phi=15^{\circ}\) and \(\phi=30^{\circ}\), are shown in Figure 8b,c, respectively. Their amplitudes were normalized with respect to \(y(0,t)\) (Figure 8a). One can see that their shapes were changed due to the filtering procedure  
Figure 6: \(M_{\phi,\hat{z}}(f)\) with \(\phi=15^{\circ}\) (**a**) and \(\phi=30^{\circ}\) (**b**) over a narrower frequency band. Note that it assumes values zero or one, according to the phase difference between \(H(0,f)\) and \(H(\hat{z},f)\). Figure 7: \(FT\)–filtered reference signals in the frequency domain, \(H_{\phi,\hat{z}}(f)\) (**a,c,e**) and in time domain \(h_{\phi,\hat{z}}(f)\) (**b,d,f**), with \(\phi=0^{\circ}\) (**a,b**), \(\phi=15^{\circ}\) (**c,d**), and \(\phi=30^{\circ}\) (**e,f**). Amplitude is normalized with respect to \(h(0,t)\). when compared to the original focused signal, \(y(0,t)\). Furthermore, the focused signal obtained with \(\phi=30^{\circ}\) is clearly more altered than the one obtained with \(\phi=15^{\circ}\), due to the removal of more frequency components, as shown in Figure 7; those frequency components were the ones recognized by \(M_{\phi,2}(f)\) as low-sensitive with the corresponding threshold \(\phi\). One can also notice that the signal level away from the focus instant is more intense with \(\phi=30^{\circ}\) than with \(\phi=15^{\circ}\). That is, the former has a lower TREF. Figure 9 shows the behavior of peak amplitude decrease, in percent, as a function of strain for the original and filtered procedure for one realization of a loading experiment. In the former, the reference signal is \(h(0,t)\) (blue dots), whereas, in the latter, it is \(h_{\phi,2}(t)\) with \(\phi=30^{\circ}\) (red dots). As can be seen, the filtering procedure made the focus peak amplitude to decrease more as strain is imposed in the plate.