## 2 Theoretical Background
### Strain Monitoring through Cross-Correlation or Time-Reversal Signals  
In a plate-like structure, the time-reversal technique is a process that compensates for the multimodal and dispersive behavior, allowing all the propagating modes to arrive synchronously at the reception point, producing a focused and recompressed signal [20]. This process is numerically equivalent to the cross-correlation of the system impulse response, as shown previously [27]. In this paper, the numerical approach is used. It can be summarized as follows.  
Consider a plate with two ultrasonic transducers positioned on its surface; one is used as a transmitter and the other as a receiver, where transmitter and receiver are separated by a predefined distance. First, an impulse-like signal is sent from the transmitter. This produces a response signal in the receiver, say \(h(t)\). This signal can be considered as the impulse response of the system that includes both transmitter and receiver acoustoelectric responses and all wave propagation related phenomena, such as dispersion, multimodal behavior, scattering and mode conversion. The signal \(h(t)\) obtained in this step is the reference signal for the remaining steps. In the next step, \(h(t)\) is convolved with its time-reversed version, \(h(-t)\), yielding  
\[y(t)=h(-t)*h(t)\quad, \tag{1}\]  
where the symbol \(*\) means convolution. Note that the signal \(y(t)\) is the cross-correlation of the system impulse response, \(h(t)\), with itself.  
The signal \(y(t)\) is numerically equivalent to the time-reversal focused signal, obtained by exciting the transmitter with \(h(-t)\), in which the dispersion and multimodal behavior are compensated [20; 27], making \(y(t)\) to be maximum at its focusing instant where it presents a high amplitude main peak. To quantify the focusing capability, time-reversal energy efficiency (TREF), introduced in [27], can be used. It is defined as the ratio of the energy in the central peak to the energy of the whole signal, as stated by Equation (2):  
\[\text{TREF}\{y(t)\}=\frac{\int_{t_{1}}^{t_{2}}y^{2}(t)dt}{\int_{-\infty}^{ \infty}y^{2}(t)dt}\quad. \tag{2}\]  
The central peak is defined as the signal between the first zero-crossing immediately before and after the main positive peak, \(t_{1}\) and \(t_{2}\) in Equation (2). The higher is the TREF, the more concentrated is the signal's energy in the central peak.  
When strain is imposed on the plate and the same reference signal remains, the focusing capability decreases with the strain level. This fact can be used as a way to monitor strain changes in the medium [27]. This can be understood by assuming that the transfer functions in the first and second steps are no longer the same due to the change in each mode's speed, according to the acoustoelastic theory [8; 9; 10], and propagating path. Considering that the first step is performed under no strain and the second step is performed under longitudinal strain, Equation (1) becomes the cross-correlation of both impulse responses:  
\[y(\varepsilon,t)=h(0,-t)*h(\varepsilon,t), \tag{3}\]  
where \(h(0,t)\) is the reference signal under zero-strain and \(h(\varepsilon,t)\) is the impulse response of the system when the plate is subjected to strain \(\varepsilon\). This process is summarized in Figure 1. Comparing Equations (1) and (3), one can conclude that strain changes the system from \(h(0,t)\) to \(h(\varepsilon,t)\), thus introducing a mismatch to the system response signal and consequently decreasing the main peak of \(y(\varepsilon,t)\), compared to \(y(t)\). Thus, the complex propagating behavior of the medium allows strain variation to be easily monitored by observing the changes on the main peak. The peak reduction is however subtle. In the next section, a procedure that provides a new reference signal, to be used instead of \(h(0,t)\), that increases the sensitivity to strain on \(y(\varepsilon,t)\) is presented.