## 2 Pulse-Compression  
PuC was initially introduced in radar systems with the purpose of improving the inspection range and/or the range resolution of the system by using swept frequency \"chirps\" excitation and related modulated signals [31; 32]. The successful application in radar lead to the use of PuC in many other fields, such as ultrasonic inspections, sonar, and audio. [PERSON] and [PERSON] were among the first to implement PuC in ultrasonic NDT, and they demonstrated that the sensitivity of the inspection system can be increased without compromising the resolution [33]. [PERSON] and [PERSON] showed the high flexibility of chirp signals to cope with the measurement system [34], while [PERSON] and [PERSON] analyzed the performance of PuC for UT medical imaging [18].  
The basic principle of PuC lies in the convolution of a pair of signals \(s(t)\) and \(\Psi(t)\), such that \(\hat{\delta}(t)=[s*\Psi](t)\), where the symbol \(*\) denotes the convolution operator. This pair is used to estimate the impulse response of a linear time invariant (LTI) system, with the estimation quality being better as long as the \(\hat{\delta}(t)\) is close to a Dirac's delta function \(\delta(t)\).  
Under this assumption, if \(s(t)\) is the excitation signal and \(\mathcal{L}\{\cdot\}\) the operator describing the sample's response, the PuC output \(\hat{h}(t)\) estimated impulse response, is nothing but the convolution between the system's output \(y(t)=\mathcal{L}\{s(t)\}\) and the so-called matched filter \(\Psi(t)\) :  
\[\hat{h}(t)=[y*\Psi](t)=[[s*h]*\Psi](t)=[[s*\Psi]*h](t)=\big{[}\hat{\delta}*h \big{]}(t) \tag{1}\]  
It was found that the SNR of the estimated impulse response is maximized when the matched filter is the time reversed replica of the excitation, as established by matched filter theory [35]. This means that \(\Psi(t)=s(-t)\), and hence, \(\hat{\delta}(t)\) is the autocorrelation of \(s(t)\). At the same time, as mentioned above, the closer the \(\hat{\delta}(t)\) to the Dirac's delta, the better is the estimation accuracy. It can be argued, therefore, that suitable signals for PuC must have a \(\delta\)-like autocorrelation function. Chirp signals with proper bandwidth and phase-modulated signals, such as pseudo-noise binary sequences, are therefore suitable candidates for PuC [36]. Equation (1) is indeed valid in the general case and not only for frequency modulated signals.  
In practice, when using bandpass chirp signals, some sidelobes are always present in \(\hat{\delta}(t)\). To mitigate this effect, which could generate artifacts in the images, the matched filter amplitude is usually tapered by a proper window function \(w(t)\) that reduces the sidelobe level at the cost of a reduced range resolution, \(\Psi(t)=w(t)s(-t)\)[20; 21]. The results reported in this manuscript were obtained by using a linear chirp signal and reactance-Tukey window, as defined in [37].  
Chirp signals are more flexible in terms of frequency and time domains shaping compared to the binary coded sequences, and they are the best choice when using single-shot excitation, see [20] for details. In the experiments, the bandwidth of the chirp signal was optimized according to the resulting spectral characteristics of the measurement setup, consisting of the employed transducers and the SUT, to attain the best possible sensitivity and resolution. The duration of the excitation signal was the maximum possible for the given measurement condition, i.e., beam steering or full matrix capture.