## 3 Improvement of Envelope Analysis  
### Time-Frequency Domain Approach for Envelope Analysis  
In the previous section, the usefulness of envelope analysis was described in condition monitoring and fault detection related to the rotational speed of rotating machinery. In addition, the result of simulation showed that envelope analysis is no longer useful in cases contaminated by impulsive noises. In the present section, we propose a method that can utilize envelope analysis for defect detection even in the presence of intermittent impulsive noises. The method requires the transient signal analysis for the measured signals, unlike the analysis used in the previous section. In other words, a time-frequency analysis such as STFT or wavelet transform analysis should be used.  
Time-frequency envelope analysis can also be performed by extracting the feature value of the envelope spectrum for a short-enough time and observing the change over the entire time of the measured data. Figure 6a shows STFT with a 0.33-s-length time window (both time resolution and frequency resolution are considered), which was the case applied to the envelope of the faulty model without impulsive noises shown in Figure 3b. Figure 6b shows the magnitude at the rotational frequency over time in the STFT shown in Figure 6a. Since the short time window is applied, there is a slight delay with the time axis. It is shown that the normal model and the faulty model are clearly distinguished.  
Figure 5: Case where impulsive noises are contained four times at around 0.1, 0.4, 0.5, and 0.8 s. (**a**) Filtered normal signal and (**b**) envelope spectrum.  
Figure 6: Case without impulsive noise. (**a**) STFT of envelope of faulty signal model. (**b**) Magnitude at rotational frequency of 18 Hz over time with 0.33-s-length time window and 90% overlap.  
Figure 7a shows the STFT for the faulty envelope with impulsive noise mixed in one time shown in Figure 4. The impulsive noise at around 0.1 s shown in Figure 4b has a large value in a wide frequency range from 0 to 0.25 s, and the time range is longer than the duration of impulsive noise because of the time window and overlap. Figure 7b shows the magnitude at the rotation frequency over time in the STFT shown in Figure 7a. Comparing the normal and faulty models, it is difficult to set a threshold to differentiate the normal model from the faulty model because the magnitude is larger in the 0 s to 0.25 s section due to the presence of impulsive noise at around 0.1 s.  
Figure 8 shows a comparison of the peak value at the rotation frequency over time in the normal and faulty envelope spectrum when impulsive noises are mixed in four times, as shown in Figure 5. The area indicated by the red dotted square is the section affected by the impulsive noises. It can be seen that it is difficult to define a threshold for distinguishing between the normal model and faulty model in the entire time of measured signal. It is very contrasting with Figure 6b, which has no impulsive noise. Moreover, in practice, impulsive noises are not mixed in at the same time and the same magnitude for each measured signal. Therefore, we propose the method that can still detect defects through envelope analysis in the presence of impulsive noises as follows.  
Figure 8: Magnitude at rotational frequency of 18 Hz over time with 0.33-s-length time window and 90% overlap. Case where impulsive noises are contained four times at 0.1, 0.4, 0.5, and 0.8 s.  
Figure 7: Case where impulsive noise is contained one time at 0.1 seconds. (**a**) STFT of envelope of faulty signal model. (**b**) Magnitude at rotational frequency of 18 Hz over time with 0.33-s-length time window and 90% overlap.