## 2 Data Preprocessing for Feature Extraction  
This section describes the data preprocessing methods used to differentiate between leak and normal states based on 1D time-series acoustic signals obtained from several microphone sensors. As outlined in [34] and illustrated in Figure 1, 1D features representing RMS patterns and frequency patterns are extracted. At this stage, two signal processing techniques, designated as \((Path-\alpha,\ Path-\beta)\), are applied to extract the RMS pattern features. The detailed nomenclature and description of the data preprocessing methods are provided in Table 1.  
First, in \(Path-\alpha\), the input data \(x\) are sampled at intervals of \(T_{d}\) without any filtering, and the sampled data are represented as \(x_{n}=x(nT_{d})\). A moving average window is then applied to the sampled input data \(x=[x_{n}]\) to calculate their RMS level. Through this process, \(y_{\alpha}=\left[y_{n}\right]_{n=1,2,\ldots,T}\) is generated, where \(y_{n}\) is calculated as shown in Equation (1):  
\[y_{n}=\frac{1}{\mathcal{C}}\sqrt{\sum_{j=1}^{L_{n}-1}x_{n+j^{\prime}}^{2}}\ n=1,2,\ldots,T, \tag{1}\]  
where \(\mathcal{C}\) denotes a coefficient used to normalize the data and \(L_{n}\) represents the window size for the RMS level calculation of the received data. In \(Path-\beta\), the preprocessed signal \(y^{*}\) is first generated by passing the input data through a band-pass filter (BPF) block, as defined in Equation (2):  
\[y^{*}=BPF\{x\} \tag{2}\]  
Figure 1: Structure for RMS and frequency pattern extraction. where the \(BPF\) operates within the 20 kHz to 40 kHz frequency range, capturing ultrasonic leak signals primarily caused by turbulence at the leak site. This frequency range effectively reduces low-frequency mechanical noise, focusing on critical leak signal features to improve detection accuracy. Additional parameters, such as the amplitude weighting function and window size for RMS level calculation, are carefully selected to optimize signal processing. These parameters highlight key signal features, minimize irrelevant noise, and prepare the signal \(y^{*}\) for subsequent processing stages. The preprocessed signal \(y^{*}\) then passes through the \"amplitude weighting block\", where a weighting function \(W(f)\) is applied in the frequency domain. This function emphasizes desired frequency components, suppresses external noise, and amplifies signals with concentrated spectral presence. The processed signal \(y^{**}\) is calculated as shown in Equation (3):  
\[y^{**}=IFT\{W(f)\cdot FT\{y^{*}\}\}, \tag{3}\]  
where \(FT\{\cdot\}\) denotes the 1D Fourier transform (FT) and \(IFT\{\cdot\}\) represents the 1D inverse FT. The weighting function \(W(f)\) is defined as shown in Equation (4):  
\[W(f)=\left\{ \begin{array}{ll}k,&f\in[f_{L},f_{H}]\\ 0,&otherwise,\end{array} \right. \tag{4}\]  
where \(f_{L}\) and \(f_{H}\) are the beginning and ending frequencies for the windowing operation, and \(k\) represents the amplification ratio of the signal within this range. In the RMS block of \(Path-\beta\), the RMS level for \(y^{**}\) is calculated based on Equation (1), similar to \(Path-\alpha\), to generate \(y_{\beta}\). Finally, the \(y_{\alpha}\) and \(y_{\beta}\) feature vectors from both paths are combined to produce the final RMS pattern feature \(y_{rms}=\left[y_{\alpha},y_{\beta}\right]\). Frequency pattern features are then derived from the received time-series data by analyzing the frequency response magnitude of the leak signal. Similar to \(Path-\beta\) of the