## 3 Data Processing  
### Time-Frequency Filtering  
As explained in the previous section, a broadband excitation signal was used to make sure that potential LDRs were excited. The broadband nature of the excitation signal made higher harmonics extraction using the classical fast Fourier transform (FFT) impossible. This is graphically illustrated in Figure 3 for the CFRP plate with artificial delaminations. The component was excited using a burst chirp signal of duration 40 ms and frequency range 10 kHz to 60 kHz. For instance, at the time instance \(t=6\) ms, the instantaneous chirp excitation frequency was 15 kHz. This resulted in a linear response of the sample at \(f_{lin}=15\) kHz combined with the potential presence of nonlinear higher harmonic components: second harmonic at \(f_{HH2}=30\) kHz, third harmonic at\(f_{HH3}=45\) kHz, etc. (see Figure 3b,c, indicated in red). These higher harmonic components were of low amplitude and as such they were overshadowed by the linear part of the output response at 16 ms (i.e., \(f_{lin}=30\) kHz, indicated in green) and 28 ms (i.e., \(f_{lin}=45\) kHz) respectively. This is further indicated as 'overlap' on Figure 3c. Note again that reducing the frequency bandwidth of excitation was not an option as it would reduce the likelihood of exciting LDRs. As an alternative, a novel post-processing approach using short-time Fourier transform (STFT) and bandpass filtering in the time-frequency domain was proposed. The decomposition of the output response into its linear and nonlinear components using bandpass filtering in the time-frequency domain is graphically illustrated in Figure 4.  
First, the measured velocity response \(V_{Z}(x,y,t)\) is transformed from the time domain to the time-frequency domain using STFT:  
\[ \begin{split}&\widetilde{V}_{Z}(x,y,t(l),f(k))=\frac{1}{M}\sum \limits_{m=1}^{M}V_{Z}^{l}(x,y,t(m))\;e^{-2\pi i\frac{\text{$\text{$\text{$ \text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$ \text{$\text{$}}}}}${}}${}}$}}}}}}}\\ &\text{with }V_{Z}^{l}(x,y,t(m))=V_{Z}(x,y,t(m+lH))\;w(m)\end{split}  \tag{1}\]where \(L\) is the number of time-divisions and each time-division is represented by index \(l=1,2\ldots L\). \(M\) is the length of each time-division expressed in number of samples, while \(m=1,2\ldots M\) is the local time index within the time-division \(l\). \(H\) is the hop size or distance between successive time-divisions, expressed in number of samples. The array \(w(m)\) is a window which is multiplied with each time-division \(l\) to avoid spectral leakage. The output after STFT is \(\widetilde{V}_{Z}(x,y,t(l),f(k))\) in which \(k=1,2\ldots K\) is the frequency index and \(K=\frac{M}{2}\) (Nyquist). The '\(-\)' indicate that the signal is expressed in the time-frequency domain. The STFT is performed in Matlab using the implementation according to reference [37], with hop size \(H=66\), window length \(M=512\) and number of time divisions \(L=1+\frac{N-M}{H}=372\) (with \(N=25\ 000\), total number of time samples). A Hanning window is employed for \(w(m)\). Figure 4: Extraction of the second higher harmonic component \(HH2\) for the test specimen with artificial delaminations: (**a**) Average short-time Fourier transform of the measured out-of-plane velocity response, (**b**) Bandpass filter around the \(HH2\) curve, (**c**) Extracted \(HH2\) component in time-frequency domain. Figure 3: (**a**) Burst chirp excitation signal in time domain, (**b**,**c**) velocity response in time and frequency domain (using fast Fourier transformation (FFT)), respectively. The absolute value of \(\overline{V}_{Z}(x,y,t,f)\), averaged over all scan points is shown in Figure 4a for the coupon with artificial delaminations. The linear response of the component is visible as a line of increased amplitude from \(f_{start}\) = 10 kHz to \(f_{end}\) = 60 kHz. Next to this linear response, multiple lines are found corresponding to the higher harmonic components. The second and third higher harmonic components are marked \(HH_{2}\) and \(HH_{3}\).