## 3 Signal Analysis Method
### Fast Fourier Transform and MFCC Extraction  
The signals after monaural separation remained in the time domain. Figure 8 shows the time-domain plots of signals from seven categories. It can be observed that in the time domain, the signals of the different states show relatively subtle differences due to the similar interactions between gas and liquid phases inside the column. This similarity makes it difficult to achieve accurate recognition and classification of these different states based solely on time-domain features. It is evident that the features contained in the time domain for different categories cannot fully meet the requirements of classification tasks [35]. Therefore, it is necessary to transform the time-domain signals into frequency-domain signals through the steps of FFT (fast Fourier transform) and MFCC extraction.  
Figure 7: The MFCC spectrograms before (**a**) and after (**b**) Hampel filtering.  
Firstly, the sound signals after frame segmentation and windowing underwent FFT. The resulting spectrogram represents the time-frequency characteristics of the signal after overlapping the frames. The FFT is defined in Equation (3).  
\[G(k_{1})=\sum_{n=0}^{N_{\rm F}-1}g(n)\mathrm{e}^{-\mathrm{j}}\frac{2\pi nk_{1}}{N _{\rm F}},0\leq k_{1}\leq N_{\rm F}, \tag{3}\]  
where \(G(k_{1})\) represents the amplitude and phase information of the signal at frequency \(k_{1},g(n)\) denotes the time-domain information at time \(n\), and \(N_{\rm F}\) is the number of samples. Next, the energy spectrum \(E(k_{1})\) is calculated using Equation (4).  
\[E(k_{1})=\frac{1}{N_{\rm F}}|G(k_{1})|^{2}, \tag{4}\]  
The computed energy spectrum describes the characteristics of the audio signal in both the time and frequency domains. Using Mel filters, the frequencies in the frequency domain were converted into Mel frequencies to obtain the Mel spectrum. The calculation formula for the Mel filter is presented in Equation (5).  
\[Hm(k_{2})=\begin{cases}\frac{k_{2}-f(m-1)}{f(m)-f(m-1)}&,\quad f(m-1)\leq k_{2 }\leq f(m)\\ \frac{f(m+1)-k_{2}}{f(m+1)-f(m)}&,\quad f(m)\leq k_{2}\leq f(m+1)\\ \\ 0&,\quad\quad\quad\quad\quad\quad\text{others}\end{cases}, \tag{5}\]  
Figure 8: Time-domain spectrograms under different operating conditions: (**a**) normal; (**b**) potential fault 1; (**c**) flooding; (**d**) potential fault 2; (**e**) dry tray; (**f**) potential fault 3; (**g**) weeping.  
where \(N_{\text{Mel}}\) denotes the number of Mel filters in the Mel filter bank, and \(k_{2}\) represents the \(k_{2}\)-th point within an FFT. The variable mmm ranges from 1 to \(N\). The computation of the Mel spectrum from the energy spectrum of the passive acoustic signal using Mel filters is given by Equation (6).  
\[MelSpec(m)=\sum_{k_{2}=f(m-1)}^{f(m+1)}Hm(k_{2})\cdot E(k_{2}), \tag{6}\]  
where \(MelSpec(m)\) represents the Mel spectrum at Mel-frequency bin m, \(E(k_{2})\) denotes the energy spectrum at frequency bin \(k_{2}\), and \(Hm(k_{2})\) represents the response of the Mel filter m at frequency bin \(k_{2}\).  
After the aforementioned calculation, the logarithm of several outputs obtained for each frame was adopted, resulting in the logarithm Mel spectrum \(S_{m}\). Then, discrete cosine transform (DCT) was applied to the logarithm Mel spectrum to obtain the _Value_ of MFCCs. The computation process is given by Equation (7) as follows:  
\[Value=\sum_{m=1}^{M-1}S_{m}\cos\left[\frac{\pi n(m+1/2)}{M}\right],0\leq m\leq M, \tag{7}\]  
Finally, the obtained MFCCs were sorted in a time series, and after applying color mapping, the MFCC spectrogram of the passive acoustic signals was generated. In the spectrogram, the rows represent time, and the columns represent the values of the MFCCs.