# Effects of Phase Shift Errors in Recurrence Plot for Rotating Machinery Fault Diagnosis
## 3 Results
### Smoothing Signal  
The smoothing process of vibration spectra is a source of phase shift. Fault diagnosis in rotating machinery transfers the vibration signal from machines to computers through the measuring instruments. Then the signal is processed, analyzed, stored, and transformed. Usually, measurements have two types of errors: systematic errors due to instruments and measurement techniques, and random errors which are commonly known as noise. One \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **For \(\mathbf{\phi}=\mathbf{0}\)** & **For \(\mathbf{\phi}=\mathbf{\pi}\)/8** & **Variation (\%)** \\ \hline \(RR\) & 0.1 & 0.1 & 0 \\ \hline \(DET\) & 0.9 & 0.9 & 0 \\ \hline \(ENTR\) & 3.6 & 3.6 & 0 \\ \hline \(LAM\) & 196.0 & 229.0 & 17 \\ \hline \(L\) & 1.5 & 1.5 & 0 \\ \hline \(L_{max}\) & 1.0 & 1.0 & 0 \\ \hline \(TT\) & 3.8 & 3.9 & 3 \\ \hline \(V_{max}\) & 28.0 & 27.0 & \(-4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: RQA quantification of the multiple asynchronous function with different phase shifts. Figure 8: Recurrence plot for the multiple asynchronous function of Equation (12): (**a**) without phase shift (\(\mathbf{\phi}=0\)); (**b**) with phase shift (\(\mathbf{\phi}=\pi/8\)). common technique to reduce the noise is the smoothing method; in general, it replaces each point of the signal with an average of \(m\) adjacent points. The most common smoothing algorithms are [42]: (a) rectangular or unweighted average smooth, that for an example with three points \((m=3)\) is \(sj=[y(j-1)+y(j)+y(j+1)]/3\) for \(j=2\) to \(n-1\), where \(s(j)\) is the \(jth\) point of the smoothed signal, \(y(j)\) is the \(jth\) point of the original signal, and \(n\) is the total number of points in the signal; (b) triangular smooth, which is the rectangular smooth with a weighted smoothing function. Another example of this method for five points \((m=5)\) is \(sj=[y(j-2)+2y(j-1)+3y(j)+2y(j+1)+y(j+2)]/3\) for \(j=3\) to \(n-2\). For better results, it is necessary to apply more than one pass of any smoothing method. However, low frequencies remain after the smoothing process and they affect signal precision. The following simulation analyzes the influence of smoothing on the shift phase of one single signal peak, although a real measurement consists of several peaks. We considered a Gaussian peak function with white noise, which is characterized by having the same power at all frequencies. Figure 9a represents a noisy Gaussian peak of 1000 signal points, with 500 points at the half-width of the peak, a peak width of 150 points and white noise. Figure 9b shows the Gaussian peak without noise. The noisy Gaussian peak from Figure 9a was smoothed with a triangular smooth of 50-point (W-50) and 30-point (W-30) width with three passes. Results of the smoothing process for both cases are presented in Figure 10a, which shows visible deviations of the base, position, height and width of the peak, and some remaining noise, mostly low-frequency. However, the focus of this work is only the change in the position which is related to the phase shift. Figure 10b illustrates the three normalized peaks to visualize the influence on the position of the top peak. Filtering is applied to eliminate the signal at the base and to visualize more clearly the variation in the position of the peaks after the smoothing with different smooth widths. Figure 9: The Gaussian peak for the analysis: (**a**) with white noise; (**b**) without noise. The RP was computed for the smoothed and normalized signals in Figure 10. The RP for the Gaussian peak without noise is shown in Figure 11a. Deviations after the smoothing process affected RP for W-50 in Figure 11b and W-30 in Figure 11c. The threshold distance considered in the three figures is \(\epsilon=0.10\). According to the results, there are significant differences in the RPs. More significant variations in position were observed for the smoothing with W-30.