# Surface Roughness Prediction of Bearing Ring Precision Grinding Based on Feature Extraction
## 3 Grinding Surface Roughness Feature Extraction and Fusion  
### Time-Domain and Frequency-Domain Characteristics  
Time-domain features refer to the extraction of statistical characteristics of a signal in the time-domain. By observing the amplitude, time series, waveform, and other character istics of the signal, information such as its time-varying nature, stability, and peak values can be reflected. Table 1 presents the 11 time-domain statistical features extracted from vibration and acoustic emission signals along with their specific calculation formulas.  
The mean value, absolute mean value, root mean square value, and the amplitude and energy characteristics of the peak signal in the time-domain are given. The kurtosis index, margin index, skewness index, waveform index, and pulse index describe the distribution and characteristics of the signal in the time-domain, which can reveal the sharp impact and other characteristics existing in the signal.  
Frequency-domain features refer to the extraction of information about the frequency components of a signal by transforming the signal in the frequency-domain. During the grinding process of bearing rings, vibrations or energy releases at different frequencies may exist, and extracting frequency-domain features can help capture information about these frequencies. This section presents 13 statistical frequency-domain features extracted from vibration and acoustic emission signals, along with specific calculation formulas, as shown in Table 2. Among them, the mean of spectral amplitude samples reflects the amplitude and energy characteristics of the signal in the frequency-domain. Parameters such as frequency skewness, spectral amplitude skewness, frequency variance, spectral amplitude kurtosis, frequency kurtosis, spectral amplitude sample variance, coefficient of variation, and standardized spectral mean provide more specific descriptions of the signal's distribution characteristics in the frequency-domain. The mean frequency, root mean square frequency, main frequency band position variation, and frequency energy concentration reflect the main frequency peaks of the signal, representing changes in the position of the main frequency in the spectrum.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Time-Domain Features** & **Calculation Formulas** & \
\begin{tabular}{c} **Time-Domain** \\ **Features** \\ \end{tabular} &
\begin{tabular}{c} **Calculation Formulas** \\ \end{tabular} \\ \hline Mean Value & \(x_{1}=\frac{\sum_{n=1}^{N}x(n)}{N}\) & Kurtosis & \(x_{7}=\frac{\sum_{n=1}^{N}(x(n)-x_{1})^{4}}{(N-1)x_{2}^{4}}\) \\ Standard deviation & \(x_{2}=\sqrt{\frac{\sum_{n=1}^{N}(x(n)-x_{1})^{2}}{N-1}}\) & Peak value & \(x_{8}=\frac{x_{5}}{x_{4}}\) \\ Absolute mean value & \(x_{3}=\left(\frac{\sum_{n=1}^{N}\sqrt{\left|x(n)\right|}}{N}\right)^{2}\) & Kurtosis & \(x_{9}=\frac{x_{5}}{x_{3}}\) \\ Root Mean Square, RMS & \(x_{4}=\sqrt{\frac{\sum_{n=1}^{N}(x(n))^{2}}{N}}\) & Pulse indicator & \(x_{10}=\frac{x_{4}}{\frac{N}{N}\sum_{n=1}^{N}\left|x(n)\right|}\) \\ Peak value & \(x_{5}=max|x(n)|\) & Waveform factor & \(x_{11}=\frac{x_{4}}{N}\frac{\sum_{n=1}^{N}\left|x(n)\right|}{N}\) \\ Skewness & \(x_{6}=\frac{\sum_{n=1}^{N}(x(n)-P_{1})^{3}}{(N-1)P_{2}^{3}}\) & & \\ \hline \end{tabular} Note: (\(n\)) denotes the signal sequence, \(n=1\), \(2\), \(3\), \(\ldots\), \(N\); \(N\) denotes the number of data points.  
\end{table}
Table 1: Time-domain statistical features.