## 3 Audio Characterization and Recognition Tests
### Data Analysis
#### 3.1.3 Frequency Band Energy Share and Harmonic Component
The frequency domain spectrum of PD has three cases, including a 100 Hz fundamental frequency, 50 Hz fundamental frequency, and no fundamental frequency, which correspond to a time domain waveform with a 10 ms or 20 ms period of acoustic pulse group, or no periodic acoustic pulse.This comparison reveals that PDs with 100 Hz fundamental frequency and mechanical vibration are not distinguishable in terms of harmonic components. Neither the PD without fundamental frequency nor the background noise had harmonic components and they were indistinguishable, and only the PD with 50 Hz fundamental frequency was distinguishable. Figure 7: Background noise frequency domain spectrum. As shown in Table 3, five features \(d_{1}\)-\(d_{5}\) in the time domain were selected, as also shown in Table 5. \(d_{1}\) calculates the number of times the signal passes the zero value in each frame, \(d_{2}\) and \(d_{3}\) describe the direction of waveform skew and the sharpness of waveform kurtosis, \(d_{4}\) describes the extremity of the waveform, and \(d_{5}\) measures the correlation of the waveform within each frame. In Table 5, \(y(n)\) is the time domain data of each frame, \(\mu\) represents the mean value of \(y(n)\), and the formula \(\mu=E[y(n)]\), \(\sigma\) is the mean squared deviation, \(\sigma=\sqrt{E(y(n)^{2})-\mu^{2}}\). \(d_{5}\) first divides \(y(n)\) into 10 equal segments and uses Equation (7) to calculate the correlation coefficient among the 10 segments. According to Table 4, five frequency domain features \(v_{1}\)-\(v_{5}\) were selected, as shown in Table 6. Among them, \(v_{1}\) determines whether the average frequency energy is concentrated in 5 kHz-16 kHz, \(v_{2}\) and \(v_{3}\) determine whether the frequency energy is concentrated in the 50 Hz or 100 Hz component in 0-1 kHz, and \(v_{4}\) and \(v_{5}\) evaluate the 100 Hz harmonic component in 0-1 kHz. \(Y(f)\) in Table 6 is the amplitude of the audio signal at frequency \(f\). The prerequisite for calculating \(v_{4}\) is to determine whether the maximum amplitude in 0-1 kHz is at 100 Hz, and if it is, then it is calculated, otherwise it is recorded as 0. \(max(Y(100f))\) represents the maximum amplitude of the 100 Hz harmonic in 0-1 kHz. For the recognition of different audio types, the correct selection of features and computation is a very important issue. Too few and too many features do not signify a high recognition rate, but in fact only features that characterize significant differences should be selected. Harmonic components are effective for identifying different types of mechanical vibration, but PDs also have similar harmonic components. At the same time, there is a \begin{table}
\begin{tabular}{c c c} \hline \hline
**Feature** & **Feature Name** & **Formula** \\
**Number** & **Feature Name** & **Formula** \\ \hline
1 &
\begin{tabular}{c} Average High-frequency \\ share \\ \end{tabular} & \(v_{1}=\sum_{f=1}^{10}(\sum_{f=5000}^{16,000}Y(f)/\sum_{0}^{16,000}Y(f))\times \frac{1}{10}\times 100\%\) \\
2 & 100 Hz component share & \(v_{2}=\sum_{f=1}^{10}Y(100f)/\sum_{0}^{1000}Y(f)\times 100\%\) \\
3 &
\begin{tabular}{c} 50 Hz even/odd \\ harmonic sum \\ \end{tabular} & \(v_{3}=\frac{\sum_{f=2}^{20}Y(50f)}{\sum_{f=1}^{19}Y(50f)},f=\begin{cases}2,4, \cdots,20\\ 1,3,\cdots,19\end{cases}\) \\
4 &
\begin{tabular}{c} Harmonic maximum \\ value/fundamental \\ frequency value \\ \end{tabular} & \(v_{4}=\frac{max(Y(100f))}{Y(100)}\times 100\%,f=2,3,\cdots,10\) \\
5 &
\begin{tabular}{c} Fundamental frequency \\ value/harmonic sum \\ \end{tabular} & \(v_{5}=Y(100)/\sum_{f=2}^{10}Y(100f),f=2,3,\cdots,10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Selected frequency domain features and formula.