## 2 Signal Processing  
### Short-Time Fourier Transform  
Wave propagation obtained in the time domain does not provide any information on the change in frequency content of the response due to the presence of a gap underneath a slab. It is useful to adapt spectral analysis with the analysis of the response in frequency domain. However, it is not easy to consider the time location of the changes in the spectrum when Fourier transform is used.  
Even though Fourier transform is a powerful signal processing technique, time information is not revealed properly in the frequency domain. Thus, Fourier transform is not good for nonstationary signals that vary over time. [PERSON] [17] suggested an alternative way of active Fourier transform that analyzes only a small section of a signal at a time. This technique is known as short-time Fourier transform (STFT) and contains both time and frequency information.  
For a slab with a gap, it may be supposed that wave reflections from the gap boundaries change both the time history and the spectrum of the response recorded at the slab surface. Therefore, time-frequency analysis of the surface response is superior to time domain or frequency domain only. Unlike Fourier analysis, STFT can be used to explain the change of spectrum in a time-frequency plane simultaneously.  
With the discrete time domain signal \(x\), the STFT can be obtained from the usual Fourier transform by multiplying the time signal \(x\) by an appropriate sliding time window \(w\)[18; 19; 20]. The location of the sliding window adds a time dimension and results in a time varying frequency analysis. The windowed signal wave propagation obtained in the time domain does not provide any information on the change in frequency content of the response due to the presence of a gap underneath a slab. It is useful to adapt spectral analysis with the analysis of the response in frequency domain. However, it is not easy to consider the time location of the changes in the spectrum when Fourier transform is used.  
The windowed signal \(y^{(k-th)}\) consisting of \(N_{w}\) points is computed for the \((k-th)\) window position, as follows [19]:  
\[y_{i}^{(k-th)}=w_{i}\cdot x_{i+kq} \tag{1}\]  
\[w_{k}=0.5-0.5\cdot\cos\left(2\pi\frac{k}{N_{w}}\right)\text{, }k=0\text{, } \cdots\text{, }N_{w}-1 \tag{2}\]  
where \(x\) is the time domain signal, \(w\) is the Hanning window, and \(N_{w}\) is the number of points in the Hanning window. By adapting this logical procedure that is applied for each of the windowed signals, the discrete Fourier transform (DFT) provides a \(k-th\) sub-matrix of the \(Y\) matrix:  
\[Y^{(k-th)}=DFT(y^{(k-th)}) \tag{3}\]  
Frequency characteristics are obtained at each time step. By repeating the procedure described, a \(Y\) matrix defines the STFT of the time signal \(x\) as follows:  
\[Y=STFT(x) \tag{4}\]As described by [PERSON] [20], the STFT algorithm can be briefly explained.  
It was pointed out that the time window width and the overlap between two successive time windows should be selected properly for computing the STFT since a wide window in the time domain improves the frequency resolution but impairs the time resolution. Therefore, the successful implementation of the STFT depends on balancing the trade-off between the frequency and time resolutions [19,20]. In this study, 1024 discrete time signal data points were obtained from FEA computed data and were short-time Fourier transformed with 128 window points at the center of each of the 16 time sections. A fixed single function called mother wavelet is scaled and translated into wavelets.