## 3 Experimental Methodology
### Feature Extraction Using Fast CWT Scalograms  
The CWT has several similarities to the Fourier transforms, such that it utilizes inner products in order to compute the similarity between the signal and an analysing function [53]. However, in the case of CWT, the analysing function is a wavelet, and the coefficients are the results of the comparison of the signal against shifted, scaled, and dilated versions of the wavelet, which are called constituent wavelets [53]. Compared with the STFT, wavelets provide better time-localization [30] and are more beneficial to non-stationary signals [53].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Training Data** & **Testing Data** \\ \hline Absence/Silence & 11,286 & 876 \\ Alarm & 2765 & 260 \\ Cat & 11,724 & 1080 \\ Dog & 6673 & 792 \\ Kitchen Activities & 12,291 & 1062 \\ Scream & 4308 & 376 \\ Shatter & 2877 & 370 \\ Shaver/toothbrush & 11,231 & 1077 \\ Slam & 1565 & 268 \\ Speech & 30,113 & 2374 \\ Water & 6796 & 829 \\
**TOTAL** & **101,629** & **9364** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of the Source Node Estimation Dataset.  
However, in order to reduce the computational requirements for deriving scalograms, this work proposes the use of the Fast Fourier Transform (FFT) algorithm for CWT coefficients computation [30]. Such that, if we define the mother wavelet (\(\Psi\)) to be [30], where \(t\) refers to continuous time:  
\[\psi_{ts}(u)=\frac{1}{\sqrt{t}}\psi\bigg{(}\frac{u-s}{t}\bigg{)} \tag{5}\]  
Then Equation (3), involving the _CWT_ coefficients, can be rewritten as follows [30], where \(y_{avg}\) refers to the average of the four-channels of the audio signal:  
\[CWT_{c}(s,t)=\int_{-\infty}^{\infty}y_{avg}(u)\psi_{t}^{*}(s-u)du \tag{6}\]  
This shows that _CWT_ coefficients can be expressed by the convolution of wavelets and signals. Thus, this can be written in the Fourier transform form domain, resulting in Equation (7) [30]:  
\[CWT_{c}(s,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}y_{avg}(\omega)\psi_{s,l}^{* }(\omega)d\omega \tag{7}\]  
where \(\psi_{s,l}^{*}(\omega)\) specifies the Fourier transform of the mother wavelet at scale \(t\):  
\[\psi_{s,l}^{*}(\omega)=\sqrt{t}\psi^{*}(t\omega)e^{i\omega s} \tag{8}\]  
Further, \(y_{avg}(\omega)\) then denotes the Fourier transform of the analysed signal \(y_{avg}(t)\):  
\[y_{avg}(\omega)=\int_{-\infty}^{\infty}y_{avg}(t)e^{i\omega t}dt \tag{9}\]  
Hence, the discrete versions of the convolutions can be represented as per Equation (10), where n is in discrete time domain:  
\[W(s)=\sum_{n=0}^{N-1}y_{avg}(n)\psi^{*}(s-n) \tag{10}\]  
From the sum in Equation (10), we can observe that CWT coefficients can be derived from the repetitive computation of the convolution of the signal, along with the wavelets, at every value of the scale per location [30]. This work follows this process in order to extract the DFT of the CWT coefficients at a faster rate compared to the traditional method.  
In summary, CWT coefficients are calculated through obtaining both the DFT of the signal, as per Equation (9), and the Morlet analysing function, as per Equation (8), via the FFT. The products of these are then derived and integrated, as per Equation (6), in order to extract the wavelet coefficients. Accordingly, the discrete version of the integration can be represented as a summation, which is observed in Equation (10).