## 3 Time Domain Analysis of Characteristic Signal  
In the grinding process of single-crystal diamonds, the basic characteristics of the acoustic emission signal are determined by the microscopic contact characteristics between abrasive particles and the grinding surface. There are many forms of microscopic action between abrasive particles and the tool grinding surface, such as cutting, scratching, squeezing, or collision. In any form, each contact action can be regarded as an acoustic emission excitation limited by the manufacturing process. The size, shape, and distribution density of particles on the surface of the grinding wheel are random and intermittent; that is, the acoustic emission signal can be regarded as a pulse sequence, and the generation process of each acoustic emission pulse is independent and unrelated and can be regarded as an independent random event. Therefore, the excitation process of acoustic emission signal in the grinding process of diamond tools can be described by the Poisson process model in probability theory. According to the Poisson process model, the acoustic emission signal of the diamond tool in the grinding process can be regarded as a Poisson signal; the probability that k pulses of the acoustic emission signal are generated in unit time obeys the Poisson distribution. If \(X(\tau)\) is the number of pulses occurring at time \(\tau\), then the probability density function, mean value, variance, kurtosis, and skewness of \(k\) pulses occurring at time interval (0, \(\tau\)) are shown in Equations (6)-(10). \[p[X(\tau)=k]=\frac{\lambda k}{k!}e^{-\lambda} \tag{6}\]  
\[E(X)=\sum_{i=1}^{n}X_{i}/n=\lambda \tag{7}\]  
\[D(X)=E(X)\Big{[}(X-E(X))^{2}\Big{]}=\lambda \tag{8}\]  
\[\alpha=\frac{E\Big{\{}[X-E(X)]^{3}\Big{\}}}{E\Big{\{}[X-E(X)]^{2}\Big{\}}^{3 /2}}=\lambda^{-1/2} \tag{9}\]  
\[\beta=\frac{E\Big{\{}[X-E(X)]^{4}\Big{\}}}{E\Big{\{}[X-E(X)]^{2}\Big{\}}^{2} }=3+\lambda^{-1} \tag{10}\]  
\(p\) is the probability density, \(X\) is the sampling value of the signal, \(\lambda\) is the average pulse, \(E(X)\) is the mean value, \(D(X)\) is the variance, \(\alpha\) is the kurtosis, \(\beta\) is the skewness. For Poisson signals such as the acoustic emission signal in the diamond tool grinding process, both the mean value and variance are always \(\lambda\), it is not effective in distinguishing signal samples [11; 12; 13]. Therefore, low-order statistics cannot reflect the directional state information of the tool grinding process. The literature [6] shows that the signal energy is mainly concentrated in the low frequency (0-2.50 kHz) and the high frequency (5.00-10.00 kHz). The spectral peak in the low-frequency band is basically located near the rotation frequency of the spindle and the frequency doubling of each order, which belongs to the deterministic signal. The energy distribution in the high-frequency band has obvious randomness and belongs to the random signal. Therefore, it is particularly important to extract the sub-signal of the high-frequency band (5.00-10.00 KHz) to calculate mean value, variance, skewness, and kurtosis in the time domain. The 100 surface of a single-crystal diamond is taken as an example. It can be seen from the crystallographic theory that this surface has 4-order \begin{table}
\begin{tabular}{c c} \hline _Different Threshold Functions_ & _Signa-to-Noise Ratio_ \\ \hline _Hard threshold function_ & 15.073503 dB \\ _Soft threshold function_ & 20.661766 dB \\ _Improved threshold function_ & 21.815345 dB \\ \hline \end{tabular}
\end{table} 
Table 2: Signal-to-noise ratio of wavelet denoising with different thresholds. symmetry, which is 4 non-difficult grinding directions and 4 difficult grinding directions. The angle relationship of the non-difficult grinding directions and the difficult grinding directions in 100, 110, 111 typical crystal surfaces are shown in Figure 2.  
The non-difficult grinding directions and the difficult grinding directions are determined according to the grinding efficiency.