## 2 Pulse-Compression Thermography (PuCT)
This fact highlights another challenge in IRT inspection of CH items: these items are usually hand-made, multi-layer, complex structures, built-up with inhomogeneous materials and with an inherent variation of the local properties (color, material, curvature, etc.). To evaluate the presence of flaws, cracks, alterations, etc., one must be able to distinguish between true defects and the intrinsic variability of the structure. Hence, gathering as much information as possible is of utmost importance in CH inspection, as is avoiding any physical alteration of the item. In this framework, the use of coded modulated heating stimuli in combination with the PuC algorithm, i.e., PuCT, showed promise for achieving the desired SNR while inferring a high number of features from the impulse response analysis. The desired SNR is achieved by just increasing the time duration \(T\) of the coded heating stimuli, meaning that (i) low-power heat sources such as LED chips can be used, and (ii) the low temperature increment \(\Delta T\) can be maintained (the overall amount of energy is spread over a longer time with respect to PT). The desired information is retrieved by exploiting the PuC algorithm. Practically, PuC relies on the existence of a pair of signals \(\left\{s(t),\ \ \psi(t)\right\}\), such that their convolution (denoted with \"\(\star\)\") \(\overline{\delta}(t)\), well-approximates the \(\delta(t)\). See Equation (1):  
\[s(t)\star\psi(t)=\overline{\delta}(t)\approx A\delta(t), \tag{1}\]  
wherein \(s(t)\) is a coded signal having time duration \(T\) and bandwidth \(B\), \(\psi(t)\) is the so-called matched filter and \(A\) is a constant proportional to the energy of \(s(t)\). If \(s(t)\) is employed for modulating the heat source emission state, i.e., any on/off emission state, and \(y(t)=h(t)\star s(t)\) is the output signal captured by the IR camera, an estimate \(\overline{h}(t)\) of the impulse response \(h(t)\) is retrieved by convolving \(y(t)\) with \(\psi(t)\). This procedure is carried out for each point \(x\)-\(y\) of the captured thermograms, thus resulting in a collection of \(N_{x}\times N_{y}\) impulse responses \(\overline{h}\left(j_{x},j_{y},t\right)\), where \(N_{x}\) and \(N_{y}\) are the amount of collected pixels along the \(x\)-\(y\) directions respectively. The overall PuC process is shown in Equation (2) for a single pixel of the acquired thermograms, wherein the presence of Additive White Gaussian Noise (AWGN) \(e(t)\) is also considered:  
\[\overline{h}(t)=y(t)\star\psi(t)=h(t)\star\underbrace{s(t)\star\psi(t)}_{=A \overline{\delta}(t)}+\underbrace{e(t)\star\psi(t)}_{=\overline{e}(t)}=h(t) \star\overline{\delta}(t)+\overline{e}(t)\approx Ah(t)+\overline{e}(t). \tag{2}\]\(e(t)\) is here assumed as being uncorrelated to the \(\psi(t)\). If this condition holds, then the instantaneous power of the noise after PuC also increases by a factor \(A\), i.e., \(Var(\overline{e}(t))\sim A\ Var(e(t))\)[41]. This gives an insight into the SNR intrinsic gain mechanism of the PuC: while the estimated impulse response amplitude is multiplied by a factor \(A\), the equivalent noise amplitude is multiplied by a factor \(\sqrt{A}\), hence the SNR gain in the reconstructed impulse response is proportional to \(A\) itself, i.e., to the excitation energy. Therefore, an arbitrary SNR gain can be achieved by tuning \(B\) and \(T\). However, there are some drawbacks in employing PuC. \(B\) is limited by the SUT and the measurement system and \(T\) is limited by practical constraints, like limited measurement time, as well as physical ones, i.e., in IRT, the longer the measurement time is, the longer the unwanted effect of lateral diffusion is. Further, due to the limited values of \(T\) and \(B\), \"side-lobes\" are always present in \(\widetilde{\delta}(t)\), so that \(\overline{h}(t)\) could exhibit some mathematical noise [29; 30].