## 5 Conclusions  
It was possible to identify the frequency components in the reference signal that are most affected by strain through their phase shift compared to a foreknown impulse response under non-zero strain. Then, a filtering procedure was proposed to create a new reference signal without the components that are less sensitive to strain, based on either the Fourier or short-time Fourier transforms. The former analyzes the phase of each frequency component considering the whole signal, whereas the latter considers the phase shift within time-windows. Both filtering techniques implemented in this study were able to increase the sensitivity to strain on the cross-correlated signal. However, an important tradeoff is the decrease of the energy concentration due to the use of the new reference signal.  
The _STFT_-based filter provides a lower decrease of the focusing capability and the sensitivity increased up to about fivefold considering all \(\phi\) range or twice within an interval that does not produce poorly focused signals. The better performance of the time-frequency implementation was due to the ability to distinguish wave packets that arrive at different time instants and share the same frequency, typically found at multimodal and reverberant media, such as the one adopted here. The proposed technique can use in principle any foreknown strain state, \(\tilde{\varepsilon}\). Here, two values were analyzed, namely low and high strain values. If \(\tilde{\varepsilon}\) is low, then, for the same threshold angle, the synthesized signal is more sensitive and can produce very low TREF values. Thus, whenever available, a higher strain state should be used since a higher \(\tilde{\varepsilon}\) makes the technique depend less abruptly on \(\phi\). It is  
Figure 14: Time–frequency representation of \(M_{\phi,\tilde{\varepsilon}}(\tau,f)\), with (**a**) \(\tilde{\varepsilon}=\varepsilon_{max}\) and \(\tilde{\varepsilon}=\varepsilon_{min}\) (**b**), at \(\phi=10^{\circ}\). Black and white spots represent values zero and one, respectively.  
worth highlighting that the need for a foreknown strain state is an intrinsic drawback of the proposed technique. However, the main objective here is to assess whether the sensitivity of the focused signal could be altered and how the proposed procedure is affected by the choice of the parameters and nature of the adopted transform.  
The results presented in this paper show that, in reverberant multimodal media, the phase of the signal retains much of the information on the medium strain and that a time-frequency representation is a more suitable analysis tool due to the nature of the signal, which is composed of several waves affected differently by strain and arriving over time with different frequency. The present technique could alternatively be used to analyze the influence of other phase-shifting factors, such as temperature, on the signal components, provided that there are no other relevant sources for phase variation.  
Conceptualization, L.M.M., A.C.K., and N.P.; methodology, L.M.M. and A.C.K.; software, L.M.; validation, L.M.M. and A.C.K.; investigation, L.M. and A.C.K.; resources, J.P.v.d.W.; writing--original draft preparation, L.M.M.; writing--review and editing, A.C.K., N.P., and J.P.v.d.W.; visualization, L.M.M. and A.C.K.; supervision, J.P.v.d.W.; and funding acquisition, J.P.v.d.W. All authors have read and agreed to the published version of the manuscript.  
This study was financed in part by the Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior-Brasil (CAPES)-Finance Code 001 and by the Brazilian National Council for Scientific and Technological Development, CNPq.  
The authors declare no conflict of interest.